From 5befc5260a7df7e15a8e4a841e9037f275b3ae13 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Mon, 5 Feb 2024 18:00:35 -0500 Subject: [PATCH] update --- doc/pub/week4/html/week4-bs.html | 598 +++-- doc/pub/week4/html/week4-reveal.html | 585 +++-- doc/pub/week4/html/week4-solarized.html | 587 +++-- doc/pub/week4/html/week4.html | 587 +++-- doc/pub/week4/ipynb/ipynb-week4-src.tar.gz | Bin 997116 -> 997116 bytes doc/pub/week4/ipynb/week4.ipynb | 2435 ++++++++++++-------- doc/pub/week4/pdf/week4.pdf | Bin 1171503 -> 1137599 bytes doc/src/week3/programs/#simplenn.py# | 65 + doc/src/week3/programs/.#simplenn.py | 1 + doc/src/week4/week4.do.txt | 22 +- 10 files changed, 2697 insertions(+), 2183 deletions(-) create mode 100644 doc/src/week3/programs/#simplenn.py# create mode 120000 doc/src/week3/programs/.#simplenn.py diff --git a/doc/pub/week4/html/week4-bs.html b/doc/pub/week4/html/week4-bs.html index d036474c..bd949981 100644 --- a/doc/pub/week4/html/week4-bs.html +++ b/doc/pub/week4/html/week4-bs.html @@ -62,6 +62,14 @@ 2, None, 'optimizing-the-parameters'), + ('Implementing the simple perceptron model', + 2, + None, + 'implementing-the-simple-perceptron-model'), + ('Exercise 1: Extensions to the above code', + 2, + None, + 'exercise-1-extensions-to-the-above-code'), ('Adding a hidden layer', 2, None, 'adding-a-hidden-layer'), ('Layout of a simple neural network with one hidden layer', 2, @@ -70,10 +78,11 @@ ('The derivatives', 2, None, 'the-derivatives'), ('Important observations', 2, None, 'important-observations'), ('The training', 2, None, 'the-training'), - ('Code examples for the simple models', + ('Code example', 2, None, 'code-example'), + ('Exercise 2: Including more data', 2, None, - 'code-examples-for-the-simple-models'), + 'exercise-2-including-more-data'), ('Simple neural network and the back propagation equations', 2, None, @@ -253,25 +262,6 @@ 2, None, 'a-more-efficient-way-of-coding-the-above-convolution'), - ('Convolution Examples: Principle of Superposition and Periodic ' - 'Forces (Fourier Transforms)', - 2, - None, - 'convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms'), - ('Principle of Superposition', - 2, - None, - 'principle-of-superposition'), - ('Simple Code Example', 2, None, 'simple-code-example'), - ('Wrapping up Fourier transforms', - 2, - None, - 'wrapping-up-fourier-transforms'), - ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), - ('Final words on Fourier Transforms', - 2, - None, - 'final-words-on-fourier-transforms'), ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'), ('Cross-Correlation', 2, None, 'cross-correlation'), ('More on Dimensionalities', 2, None, 'more-on-dimensionalities'), @@ -321,12 +311,15 @@
  • First network example, simple percepetron with one input
  • Layout of a simple neural network with no hidden layer
  • Optimizing the parameters
    • +
  • Implementing the simple perceptron model
    • +
  • Exercise 1: Extensions to the above code
  • Adding a hidden layer
  • Layout of a simple neural network with one hidden layer
  • The derivatives
  • Important observations
  • The training
    • -
  • Code examples for the simple models
    • +
  • Code example
    • +
  • Exercise 2: Including more data
  • Simple neural network and the back propagation equations
  • Layout of a simple neural network with two input nodes, one hidden layer and one output node
  • The ouput layer
    • @@ -395,12 +388,6 @@
  • Convolution Examples: Polynomial multiplication
  • Efficient Polynomial Multiplication
  • A more efficient way of coding the above Convolution
    • -
  • Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)
    • -
  • Principle of Superposition
    • -
  • Simple Code Example
    • -
  • Wrapping up Fourier transforms
    • -
  • Finding the Coefficients
    • -
  • Final words on Fourier Transforms
  • Two-dimensional Objects
  • Cross-Correlation
  • More on Dimensionalities
    • @@ -433,7 +420,7 @@

    February 5-9: Advanced machine learning and data analysis for the physical s
    -

    Feb 5, 2024

    +

    February 6, 2024


    @@ -567,10 +554,118 @@

    Optimizing the parameters

    +

    Implementing the simple perceptron model

    + +

    In the example code here we implement the above equations (with explict +expressions for the derivatives) with just one input variable \( x \) and +one output variable. The target value \( y=2x+1 \) is a simple linear +function in \( x \). Since this is a regression problem, we define the cost function to be proportional to the least squares error +

    +$$ +C(y,w_1,b_1)=\frac{1}{2}(a_1-y)^2, +$$ + +

    with \( a_1 \) the output from the network.

    + + + +
    +
    +
    +
    +
    + 
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+
+def feed_forward(x):
+    # weighted sum of inputs to the output layer
+    z_1 = x*output_weights + output_bias
+    # Output from output node (one node only)
+    # Here the output is equal to the input
+    a_1 = z_1
+    return a_1
+
+def backpropagation(x, y):
+    a_1 = feed_forward(x)
+    # derivative of cost function
+    derivative_cost = a_1 - y
+    # the variable delta in the equations, note that output a_1 = z_1, its derivatives wrt z_o is thus 1
+    delta_1 = derivative_cost
+    # gradients for the output layer
+    output_weights_gradient = delta_1*x
+    output_bias_gradient = delta_1
+    # The cost function is 0.5*(a_1-y)^2. This gives a measure of the error for each iteration
+    return output_weights_gradient, output_bias_gradient
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = 4.0
+# Target values
+y = 2*x+1.0
+
+# Defining the neural network
+n_inputs = 1
+n_outputs = 1
+# Initialize the network
+# weights and bias in the output layer
+output_weights = np.random.randn()
+output_bias = np.random.randn()
+
+# implementing a simple gradient descent approach with fixed learning rate
+eta = 0.01
+for i in range(40):
+    # calculate gradients from back propagation
+    derivative_w1, derivative_b1 = backpropagation(x, y)
+    # update weights and biases
+    output_weights -= eta * derivative_w1
+    output_bias -= eta * derivative_b1
+# our final prediction after training
+ytilde = output_weights*x+output_bias
+print(0.5*((ytilde-y)**2))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    Running this code gives us an acceptable results after some 40-50 iterations. Note that the results depend on the value of the learning rate.

    + + + + +

    Exercise 1: Extensions to the above code

    + +

    Feel free to add more input nodes and weights to the above +code. Furthermore, try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +

    + +

    You could also try to change the function \( f(x)=y \) from a linear polynomial in \( x \) to a higher-order polynomial. +Comment your results. +

    + +

    Hint: Increasing the number of input variables and input nodes requires a rewrite of the input data in terms of a matrix. You need to figure out the correct dimensionalities.

    + + +

    Adding a hidden layer

    -

    We change our simple model to (see graph) +

    We change our simple model to (see graph below) a network with just one hidden layer but with scalar variables only.

    @@ -661,7 +756,117 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.

    -

    Code examples for the simple models

    +

    Code example

    + +

    The code here implements the above model with one hidden layer and +scalar variables for the same function we studied in the previous +example. The code is however set up so that we can add multiple +inputs \( x \) and target values \( y \). Note also that we have the +possibility of defining a feature matrix \( \boldsymbol{X} \) with more than just +one column for the input values. This will turn useful in our next example. We have also defined matrices and vectors for all of our operations although it is not necessary here. +

    + + + +
    +
    +
    +
    +
    + 
    import numpy as np
+# We use the Sigmoid function as activation function
+def sigmoid(z):
+    return 1.0/(1.0+np.exp(-z))
+
+def forwardpropagation(x):
+    # weighted sum of inputs to the hidden layer
+    z_1 = np.matmul(x, w_1) + b_1
+    # activation in the hidden layer
+    a_1 = sigmoid(z_1)
+    # weighted sum of inputs to the output layer
+    z_2 = np.matmul(a_1, w_2) + b_2
+    a_2 = z_2
+    return a_1, a_2
+
+def backpropagation(x, y):
+    a_1, a_2 = forwardpropagation(x)
+    # parameter delta for the output layer, note that a_2=z_2 and its derivative wrt z_2 is just 1
+    delta_2 = a_2 - y
+    print(0.5*((a_2-y)**2))
+    # delta for  the hidden layer
+    delta_1 = np.matmul(delta_2, w_2.T) * a_1 * (1 - a_1)
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_1.T, delta_2)
+    output_bias_gradient = np.sum(delta_2, axis=0)
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(x.T, delta_1)
+    hidden_bias_gradient = np.sum(delta_1, axis=0)
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = np.array([4.0],dtype=np.float64)
+# Target values
+y = 2*x+1.0 
+
+# Defining the neural network, only scalars here
+n_inputs = x.shape
+n_features = 1
+n_hidden_neurons = 1
+n_outputs = 1
+
+# Initialize the network
+# weights and bias in the hidden layer
+w_1 = np.random.randn(n_features, n_hidden_neurons)
+b_1 = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+w_2 = np.random.randn(n_hidden_neurons, n_outputs)
+b_2 = np.zeros(n_outputs) + 0.01
+
+eta = 0.1
+for i in range(50):
+    # calculate gradients
+    derivW2, derivB2, derivW1, derivB1 = backpropagation(x, y)
+    # update weights and biases
+    w_2 -= eta * derivW2
+    b_2 -= eta * derivB2
+    w_1 -= eta * derivW1
+    b_1 -= eta * derivB1
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    We see that after some few iterations (the results do depend on the learning rate however), we get an error which is rather small.

    + + + + +

    Exercise 2: Including more data

    + +

    Try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +Discuss your results as functions of the amount of training data and various learning rates. +

    + +

    Challenge: Try to change the activation functions and replace the hard-coded analytical expressions with automatic derivation via either autograd or JAX.

    + +

    Simple neural network and the back propagation equations

    @@ -866,6 +1071,41 @@

    Gradient expressions

    Program example

    +

    We extend our simple code to a function which depends on two variable \( x_0 \) and \( x_1 \), that is

    +$$ +y=f(x_0,x_1)=x_0^2+3x_0x_1+x_1^2+5. +$$ + +

    We feed our network with \( n=100 \) entries \( x_0 \) and \( x_1 \). We have thus two features represented by these variable and an input matrix/design matrix \( \boldsymbol{X}\in \mathbf{R}^{n\times 2} \)

    +$$ +\boldsymbol{X}=\begin{bmatrix} x_{00} & x_{01} \\ x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \\ \dots & \dots \\ \dots & \dots \\ x_{n-20} & x_{n-21} \\ x_{n-10} & x_{n-11} \end{bmatrix}. +$$ + + + + +
    +
    +
    +
    +
    + 
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    Getting serious, the back propagation equations for a neural network

    @@ -3503,14 +3743,21 @@

    Efficient Polynomial

    $$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} +\begin{align} +\delta_0=&\alpha_0\beta_0 +\label{_auto4}\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0 +\label{_auto5}\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0 +\label{_auto6}\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3 +\label{_auto7}\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3 +\label{_auto8}\\ +\delta_5=&\alpha_2\beta_3. +\label{_auto9}\\ +\label{_auto10} +\end{align} $$

    We note that \( \alpha_i=0 \) except for \( i\in \left\{0,1,2\right\} \) and \( \beta_i=0 \) except for \( i\in\left\{0,1,2,3\right\} \).

    @@ -3565,279 +3812,6 @@

    A m

    Does the number of floating point operations change here when we use the commutative property?

    - -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    - -

    For problems with so-called harmonic oscillations, given by for example the following differential equation

    -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ - -

    where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.

    - -

    If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like -

    - -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\label{_auto4} -\end{equation} -$$ - - - -

    Principle of Superposition

    - -

    This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. -

    - -

    Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) -

    - -$$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} -$$ - -

    One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. -

    - - -

    Simple Code Example

    - -

    The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \).

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -

    For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, -

    - -$$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\label{_auto5} -\end{equation} -$$ - - - -

    Wrapping up Fourier transforms

    - -

    We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), -

    - -$$ -\begin{equation} -\label{eq:fourierdef1} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). -\end{equation} -$$ - -

    The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, -

    - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ - - - -

    Finding the Coefficients

    - -

    Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). -

    - -

    The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by -

    - -$$ -\begin{eqnarray} -\label{eq:fourierdef2} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -

    To check the consistency of these expressions and to verify -Eq. \eqref{eq:fourierdef2}, one can insert the expansion of \( F(t) \) in -Eq. \eqref{eq:fourierdef1} into the expression for the coefficients in -Eq. \eqref{eq:fourierdef2} and see whether -

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -

    Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives -

    - -$$ -\begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\label{_auto6} -\end{equation} -$$ - -

    and

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ -\nonumber -&=&f_n~\checkmark. -\end{eqnarray} -$$ - -

    The same method can be used to check for the consistency of \( g_n \).

    - - -

    Final words on Fourier Transforms

    - -

    The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. -

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-T =0.2
-# Max value of square signal                                                                             
-Fmax= 2.0
-# Width of signal   
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
-    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -

    Two-dimensional Objects

    diff --git a/doc/pub/week4/html/week4-reveal.html b/doc/pub/week4/html/week4-reveal.html index 20d46049..1aa642b6 100644 --- a/doc/pub/week4/html/week4-reveal.html +++ b/doc/pub/week4/html/week4-reveal.html @@ -184,7 +184,7 @@

    February 5-9: Advanced machine learning and data
    -

    Feb 5, 2024

    +

    February 6, 2024


    @@ -335,10 +335,122 @@

    Optimizing the parameters

     
    +

    +

    Implementing the simple perceptron model

    + +

    In the example code here we implement the above equations (with explict +expressions for the derivatives) with just one input variable \( x \) and +one output variable. The target value \( y=2x+1 \) is a simple linear +function in \( x \). Since this is a regression problem, we define the cost function to be proportional to the least squares error +

    +

     
    +$$ +C(y,w_1,b_1)=\frac{1}{2}(a_1-y)^2, +$$ +

     
    + +

    with \( a_1 \) the output from the network.

    + + + +
    +
    +
    +
    +
    + 
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+
+def feed_forward(x):
+    # weighted sum of inputs to the output layer
+    z_1 = x*output_weights + output_bias
+    # Output from output node (one node only)
+    # Here the output is equal to the input
+    a_1 = z_1
+    return a_1
+
+def backpropagation(x, y):
+    a_1 = feed_forward(x)
+    # derivative of cost function
+    derivative_cost = a_1 - y
+    # the variable delta in the equations, note that output a_1 = z_1, its derivatives wrt z_o is thus 1
+    delta_1 = derivative_cost
+    # gradients for the output layer
+    output_weights_gradient = delta_1*x
+    output_bias_gradient = delta_1
+    # The cost function is 0.5*(a_1-y)^2. This gives a measure of the error for each iteration
+    return output_weights_gradient, output_bias_gradient
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = 4.0
+# Target values
+y = 2*x+1.0
+
+# Defining the neural network
+n_inputs = 1
+n_outputs = 1
+# Initialize the network
+# weights and bias in the output layer
+output_weights = np.random.randn()
+output_bias = np.random.randn()
+
+# implementing a simple gradient descent approach with fixed learning rate
+eta = 0.01
+for i in range(40):
+    # calculate gradients from back propagation
+    derivative_w1, derivative_b1 = backpropagation(x, y)
+    # update weights and biases
+    output_weights -= eta * derivative_w1
+    output_bias -= eta * derivative_b1
+# our final prediction after training
+ytilde = output_weights*x+output_bias
+print(0.5*((ytilde-y)**2))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    Running this code gives us an acceptable results after some 40-50 iterations. Note that the results depend on the value of the learning rate.

    +
    + +
    + + +

    Exercise 1: Extensions to the above code

    + +

    Feel free to add more input nodes and weights to the above +code. Furthermore, try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +

    + +

    You could also try to change the function \( f(x)=y \) from a linear polynomial in \( x \) to a higher-order polynomial. +Comment your results. +

    + +

    Hint: Increasing the number of input variables and input nodes requires a rewrite of the input data in terms of a matrix. You need to figure out the correct dimensionalities.

    + + +
    +

    Adding a hidden layer

    -

    We change our simple model to (see graph) +

    We change our simple model to (see graph below) a network with just one hidden layer but with scalar variables only.

    @@ -450,7 +562,118 @@

    The training

    -

    Code examples for the simple models

    +

    Code example

    + +

    The code here implements the above model with one hidden layer and +scalar variables for the same function we studied in the previous +example. The code is however set up so that we can add multiple +inputs \( x \) and target values \( y \). Note also that we have the +possibility of defining a feature matrix \( \boldsymbol{X} \) with more than just +one column for the input values. This will turn useful in our next example. We have also defined matrices and vectors for all of our operations although it is not necessary here. +

    + + + +
    +
    +
    +
    +
    + 
    import numpy as np
+# We use the Sigmoid function as activation function
+def sigmoid(z):
+    return 1.0/(1.0+np.exp(-z))
+
+def forwardpropagation(x):
+    # weighted sum of inputs to the hidden layer
+    z_1 = np.matmul(x, w_1) + b_1
+    # activation in the hidden layer
+    a_1 = sigmoid(z_1)
+    # weighted sum of inputs to the output layer
+    z_2 = np.matmul(a_1, w_2) + b_2
+    a_2 = z_2
+    return a_1, a_2
+
+def backpropagation(x, y):
+    a_1, a_2 = forwardpropagation(x)
+    # parameter delta for the output layer, note that a_2=z_2 and its derivative wrt z_2 is just 1
+    delta_2 = a_2 - y
+    print(0.5*((a_2-y)**2))
+    # delta for  the hidden layer
+    delta_1 = np.matmul(delta_2, w_2.T) * a_1 * (1 - a_1)
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_1.T, delta_2)
+    output_bias_gradient = np.sum(delta_2, axis=0)
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(x.T, delta_1)
+    hidden_bias_gradient = np.sum(delta_1, axis=0)
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = np.array([4.0],dtype=np.float64)
+# Target values
+y = 2*x+1.0 
+
+# Defining the neural network, only scalars here
+n_inputs = x.shape
+n_features = 1
+n_hidden_neurons = 1
+n_outputs = 1
+
+# Initialize the network
+# weights and bias in the hidden layer
+w_1 = np.random.randn(n_features, n_hidden_neurons)
+b_1 = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+w_2 = np.random.randn(n_hidden_neurons, n_outputs)
+b_2 = np.zeros(n_outputs) + 0.01
+
+eta = 0.1
+for i in range(50):
+    # calculate gradients
+    derivW2, derivB2, derivW1, derivB1 = backpropagation(x, y)
+    # update weights and biases
+    w_2 -= eta * derivW2
+    b_2 -= eta * derivB2
+    w_1 -= eta * derivW1
+    b_1 -= eta * derivB1
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    We see that after some few iterations (the results do depend on the learning rate however), we get an error which is rather small.

    +
    + +
    + + +

    Exercise 2: Including more data

    + +

    Try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +Discuss your results as functions of the amount of training data and various learning rates. +

    + +

    Challenge: Try to change the activation functions and replace the hard-coded analytical expressions with automatic derivation via either autograd or JAX.

    + +
    @@ -710,6 +933,44 @@

    Gradient expressions

    Program example

    + +

    We extend our simple code to a function which depends on two variable \( x_0 \) and \( x_1 \), that is

    +

     
    +$$ +y=f(x_0,x_1)=x_0^2+3x_0x_1+x_1^2+5. +$$ +

     
    + +

    We feed our network with \( n=100 \) entries \( x_0 \) and \( x_1 \). We have thus two features represented by these variable and an input matrix/design matrix \( \boldsymbol{X}\in \mathbf{R}^{n\times 2} \)

    +

     
    +$$ +\boldsymbol{X}=\begin{bmatrix} x_{00} & x_{01} \\ x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \\ \dots & \dots \\ \dots & \dots \\ x_{n-20} & x_{n-21} \\ x_{n-10} & x_{n-11} \end{bmatrix}. +$$ +

     
    + + + + +

    +
    +
    +
    +
    + 
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    @@ -3452,14 +3713,21 @@

    Efficient Polynomial Multiplication

     
    $$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} +\begin{align} +\delta_0=&\alpha_0\beta_0 +\tag{4}\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0 +\tag{5}\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0 +\tag{6}\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3 +\tag{7}\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3 +\tag{8}\\ +\delta_5=&\alpha_2\beta_3. +\tag{9}\\ +\tag{10} +\end{align} $$

     
    @@ -3525,301 +3793,6 @@

    A more efficient w

    Does the number of floating point operations change here when we use the commutative property?

    -
    -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    - -

    For problems with so-called harmonic oscillations, given by for example the following differential equation

    -

     
    -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ -

     
    - -

    where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.

    - -

    If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like -

    - -

     
    -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\tag{4} -\end{equation} -$$ -

     
    -

    - -
    -

    Principle of Superposition

    - -

    This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. -

    - -

    Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) -

    - -

     
    -$$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} -$$ -

     
    - -

    One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. -

    -
    - -
    -

    Simple Code Example

    - -

    The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \).

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -

    For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, -

    - -

     
    -$$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\tag{5} -\end{equation} -$$ -

     
    -

    - -
    -

    Wrapping up Fourier transforms

    - -

    We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), -

    - -

     
    -$$ -\begin{equation} -\tag{6} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). -\end{equation} -$$ -

     
    - -

    The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, -

    - -

     
    -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ -

     
    -

    - -
    -

    Finding the Coefficients

    - -

    Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). -

    - -

    The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by -

    - -

     
    -$$ -\begin{eqnarray} -\tag{7} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ -

     
    - -

    To check the consistency of these expressions and to verify -Eq. (7), one can insert the expansion of \( F(t) \) in -Eq. (6) into the expression for the coefficients in -Eq. (7) and see whether -

    - -

     
    -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ -

     
    - -

    Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives -

    - -

     
    -$$ -\begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\tag{8} -\end{equation} -$$ -

     
    - -

    and

    - -

     
    -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ -\nonumber -&=&f_n~\checkmark. -\end{eqnarray} -$$ -

     
    - -

    The same method can be used to check for the consistency of \( g_n \).

    -
    - -
    -

    Final words on Fourier Transforms

    - -

    The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. -

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-T =0.2
-# Max value of square signal                                                                             
-Fmax= 2.0
-# Width of signal   
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
-    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -

    Two-dimensional Objects

    diff --git a/doc/pub/week4/html/week4-solarized.html b/doc/pub/week4/html/week4-solarized.html index 988681dd..c6955d17 100644 --- a/doc/pub/week4/html/week4-solarized.html +++ b/doc/pub/week4/html/week4-solarized.html @@ -89,6 +89,14 @@ 2, None, 'optimizing-the-parameters'), + ('Implementing the simple perceptron model', + 2, + None, + 'implementing-the-simple-perceptron-model'), + ('Exercise 1: Extensions to the above code', + 2, + None, + 'exercise-1-extensions-to-the-above-code'), ('Adding a hidden layer', 2, None, 'adding-a-hidden-layer'), ('Layout of a simple neural network with one hidden layer', 2, @@ -97,10 +105,11 @@ ('The derivatives', 2, None, 'the-derivatives'), ('Important observations', 2, None, 'important-observations'), ('The training', 2, None, 'the-training'), - ('Code examples for the simple models', + ('Code example', 2, None, 'code-example'), + ('Exercise 2: Including more data', 2, None, - 'code-examples-for-the-simple-models'), + 'exercise-2-including-more-data'), ('Simple neural network and the back propagation equations', 2, None, @@ -280,25 +289,6 @@ 2, None, 'a-more-efficient-way-of-coding-the-above-convolution'), - ('Convolution Examples: Principle of Superposition and Periodic ' - 'Forces (Fourier Transforms)', - 2, - None, - 'convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms'), - ('Principle of Superposition', - 2, - None, - 'principle-of-superposition'), - ('Simple Code Example', 2, None, 'simple-code-example'), - ('Wrapping up Fourier transforms', - 2, - None, - 'wrapping-up-fourier-transforms'), - ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), - ('Final words on Fourier Transforms', - 2, - None, - 'final-words-on-fourier-transforms'), ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'), ('Cross-Correlation', 2, None, 'cross-correlation'), ('More on Dimensionalities', 2, None, 'more-on-dimensionalities'), @@ -343,7 +333,7 @@

    February 5-9: Advanced machine learning and data analysis for the physical s
    -

    Feb 5, 2024

    +

    February 6, 2024


    @@ -471,10 +461,118 @@

    Optimizing the parameters

    $$ +









    +

    Implementing the simple perceptron model

    + +

    In the example code here we implement the above equations (with explict +expressions for the derivatives) with just one input variable \( x \) and +one output variable. The target value \( y=2x+1 \) is a simple linear +function in \( x \). Since this is a regression problem, we define the cost function to be proportional to the least squares error +

    +$$ +C(y,w_1,b_1)=\frac{1}{2}(a_1-y)^2, +$$ + +

    with \( a_1 \) the output from the network.

    + + + +
    +
    +
    +
    +
    + 
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+
+def feed_forward(x):
+    # weighted sum of inputs to the output layer
+    z_1 = x*output_weights + output_bias
+    # Output from output node (one node only)
+    # Here the output is equal to the input
+    a_1 = z_1
+    return a_1
+
+def backpropagation(x, y):
+    a_1 = feed_forward(x)
+    # derivative of cost function
+    derivative_cost = a_1 - y
+    # the variable delta in the equations, note that output a_1 = z_1, its derivatives wrt z_o is thus 1
+    delta_1 = derivative_cost
+    # gradients for the output layer
+    output_weights_gradient = delta_1*x
+    output_bias_gradient = delta_1
+    # The cost function is 0.5*(a_1-y)^2. This gives a measure of the error for each iteration
+    return output_weights_gradient, output_bias_gradient
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = 4.0
+# Target values
+y = 2*x+1.0
+
+# Defining the neural network
+n_inputs = 1
+n_outputs = 1
+# Initialize the network
+# weights and bias in the output layer
+output_weights = np.random.randn()
+output_bias = np.random.randn()
+
+# implementing a simple gradient descent approach with fixed learning rate
+eta = 0.01
+for i in range(40):
+    # calculate gradients from back propagation
+    derivative_w1, derivative_b1 = backpropagation(x, y)
+    # update weights and biases
+    output_weights -= eta * derivative_w1
+    output_bias -= eta * derivative_b1
+# our final prediction after training
+ytilde = output_weights*x+output_bias
+print(0.5*((ytilde-y)**2))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    Running this code gives us an acceptable results after some 40-50 iterations. Note that the results depend on the value of the learning rate.

    + +









    + + +

    Exercise 1: Extensions to the above code

    + +

    Feel free to add more input nodes and weights to the above +code. Furthermore, try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +

    + +

    You could also try to change the function \( f(x)=y \) from a linear polynomial in \( x \) to a higher-order polynomial. +Comment your results. +

    + +

    Hint: Increasing the number of input variables and input nodes requires a rewrite of the input data in terms of a matrix. You need to figure out the correct dimensionalities.

    + + +









    Adding a hidden layer

    -

    We change our simple model to (see graph) +

    We change our simple model to (see graph below) a network with just one hidden layer but with scalar variables only.

    @@ -564,7 +662,117 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.











    -

    Code examples for the simple models

    +

    Code example

    + +

    The code here implements the above model with one hidden layer and +scalar variables for the same function we studied in the previous +example. The code is however set up so that we can add multiple +inputs \( x \) and target values \( y \). Note also that we have the +possibility of defining a feature matrix \( \boldsymbol{X} \) with more than just +one column for the input values. This will turn useful in our next example. We have also defined matrices and vectors for all of our operations although it is not necessary here. +

    + + + +
    +
    +
    +
    +
    + 
    import numpy as np
+# We use the Sigmoid function as activation function
+def sigmoid(z):
+    return 1.0/(1.0+np.exp(-z))
+
+def forwardpropagation(x):
+    # weighted sum of inputs to the hidden layer
+    z_1 = np.matmul(x, w_1) + b_1
+    # activation in the hidden layer
+    a_1 = sigmoid(z_1)
+    # weighted sum of inputs to the output layer
+    z_2 = np.matmul(a_1, w_2) + b_2
+    a_2 = z_2
+    return a_1, a_2
+
+def backpropagation(x, y):
+    a_1, a_2 = forwardpropagation(x)
+    # parameter delta for the output layer, note that a_2=z_2 and its derivative wrt z_2 is just 1
+    delta_2 = a_2 - y
+    print(0.5*((a_2-y)**2))
+    # delta for  the hidden layer
+    delta_1 = np.matmul(delta_2, w_2.T) * a_1 * (1 - a_1)
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_1.T, delta_2)
+    output_bias_gradient = np.sum(delta_2, axis=0)
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(x.T, delta_1)
+    hidden_bias_gradient = np.sum(delta_1, axis=0)
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = np.array([4.0],dtype=np.float64)
+# Target values
+y = 2*x+1.0 
+
+# Defining the neural network, only scalars here
+n_inputs = x.shape
+n_features = 1
+n_hidden_neurons = 1
+n_outputs = 1
+
+# Initialize the network
+# weights and bias in the hidden layer
+w_1 = np.random.randn(n_features, n_hidden_neurons)
+b_1 = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+w_2 = np.random.randn(n_hidden_neurons, n_outputs)
+b_2 = np.zeros(n_outputs) + 0.01
+
+eta = 0.1
+for i in range(50):
+    # calculate gradients
+    derivW2, derivB2, derivW1, derivB1 = backpropagation(x, y)
+    # update weights and biases
+    w_2 -= eta * derivW2
+    b_2 -= eta * derivB2
+    w_1 -= eta * derivW1
+    b_1 -= eta * derivB1
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    We see that after some few iterations (the results do depend on the learning rate however), we get an error which is rather small.

    + +









    + + +

    Exercise 2: Including more data

    + +

    Try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +Discuss your results as functions of the amount of training data and various learning rates. +

    + +

    Challenge: Try to change the activation functions and replace the hard-coded analytical expressions with automatic derivation via either autograd or JAX.

    + +









    Simple neural network and the back propagation equations

    @@ -769,6 +977,41 @@

    Gradient expressions











    Program example

    +

    We extend our simple code to a function which depends on two variable \( x_0 \) and \( x_1 \), that is

    +$$ +y=f(x_0,x_1)=x_0^2+3x_0x_1+x_1^2+5. +$$ + +

    We feed our network with \( n=100 \) entries \( x_0 \) and \( x_1 \). We have thus two features represented by these variable and an input matrix/design matrix \( \boldsymbol{X}\in \mathbf{R}^{n\times 2} \)

    +$$ +\boldsymbol{X}=\begin{bmatrix} x_{00} & x_{01} \\ x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \\ \dots & \dots \\ \dots & \dots \\ x_{n-20} & x_{n-21} \\ x_{n-10} & x_{n-11} \end{bmatrix}. +$$ + + + + +
    +
    +
    +
    +
    + 
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +









    Getting serious, the back propagation equations for a neural network

    @@ -3406,14 +3649,21 @@

    Efficient Polynomial Multiplication

    $$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} +\begin{align} +\delta_0=&\alpha_0\beta_0 +\label{_auto4}\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0 +\label{_auto5}\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0 +\label{_auto6}\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3 +\label{_auto7}\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3 +\label{_auto8}\\ +\delta_5=&\alpha_2\beta_3. +\label{_auto9}\\ +\label{_auto10} +\end{align} $$

    We note that \( \alpha_i=0 \) except for \( i\in \left\{0,1,2\right\} \) and \( \beta_i=0 \) except for \( i\in\left\{0,1,2,3\right\} \).

    @@ -3468,279 +3718,6 @@

    A more efficient w

    Does the number of floating point operations change here when we use the commutative property?

    -









    -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    - -

    For problems with so-called harmonic oscillations, given by for example the following differential equation

    -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ - -

    where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.

    - -

    If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like -

    - -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\label{_auto4} -\end{equation} -$$ - - -









    -

    Principle of Superposition

    - -

    This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. -

    - -

    Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) -

    - -$$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} -$$ - -

    One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. -

    - -









    -

    Simple Code Example

    - -

    The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \).

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -

    For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, -

    - -$$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\label{_auto5} -\end{equation} -$$ - - -









    -

    Wrapping up Fourier transforms

    - -

    We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), -

    - -$$ -\begin{equation} -\label{eq:fourierdef1} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). -\end{equation} -$$ - -

    The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, -

    - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ - - -









    -

    Finding the Coefficients

    - -

    Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). -

    - -

    The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by -

    - -$$ -\begin{eqnarray} -\label{eq:fourierdef2} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -

    To check the consistency of these expressions and to verify -Eq. \eqref{eq:fourierdef2}, one can insert the expansion of \( F(t) \) in -Eq. \eqref{eq:fourierdef1} into the expression for the coefficients in -Eq. \eqref{eq:fourierdef2} and see whether -

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -

    Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives -

    - -$$ -\begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\label{_auto6} -\end{equation} -$$ - -

    and

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ -\nonumber -&=&f_n~\checkmark. -\end{eqnarray} -$$ - -

    The same method can be used to check for the consistency of \( g_n \).

    - -









    -

    Final words on Fourier Transforms

    - -

    The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. -

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-T =0.2
-# Max value of square signal                                                                             
-Fmax= 2.0
-# Width of signal   
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
-    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -









    Two-dimensional Objects

    diff --git a/doc/pub/week4/html/week4.html b/doc/pub/week4/html/week4.html index 8fe6142f..50772586 100644 --- a/doc/pub/week4/html/week4.html +++ b/doc/pub/week4/html/week4.html @@ -166,6 +166,14 @@ 2, None, 'optimizing-the-parameters'), + ('Implementing the simple perceptron model', + 2, + None, + 'implementing-the-simple-perceptron-model'), + ('Exercise 1: Extensions to the above code', + 2, + None, + 'exercise-1-extensions-to-the-above-code'), ('Adding a hidden layer', 2, None, 'adding-a-hidden-layer'), ('Layout of a simple neural network with one hidden layer', 2, @@ -174,10 +182,11 @@ ('The derivatives', 2, None, 'the-derivatives'), ('Important observations', 2, None, 'important-observations'), ('The training', 2, None, 'the-training'), - ('Code examples for the simple models', + ('Code example', 2, None, 'code-example'), + ('Exercise 2: Including more data', 2, None, - 'code-examples-for-the-simple-models'), + 'exercise-2-including-more-data'), ('Simple neural network and the back propagation equations', 2, None, @@ -357,25 +366,6 @@ 2, None, 'a-more-efficient-way-of-coding-the-above-convolution'), - ('Convolution Examples: Principle of Superposition and Periodic ' - 'Forces (Fourier Transforms)', - 2, - None, - 'convolution-examples-principle-of-superposition-and-periodic-forces-fourier-transforms'), - ('Principle of Superposition', - 2, - None, - 'principle-of-superposition'), - ('Simple Code Example', 2, None, 'simple-code-example'), - ('Wrapping up Fourier transforms', - 2, - None, - 'wrapping-up-fourier-transforms'), - ('Finding the Coefficients', 2, None, 'finding-the-coefficients'), - ('Final words on Fourier Transforms', - 2, - None, - 'final-words-on-fourier-transforms'), ('Two-dimensional Objects', 2, None, 'two-dimensional-objects'), ('Cross-Correlation', 2, None, 'cross-correlation'), ('More on Dimensionalities', 2, None, 'more-on-dimensionalities'), @@ -420,7 +410,7 @@

    February 5-9: Advanced machine learning and data analysis for the physical s
    -

    Feb 5, 2024

    +

    February 6, 2024


    @@ -548,10 +538,118 @@

    Optimizing the parameters

    $$ +









    +

    Implementing the simple perceptron model

    + +

    In the example code here we implement the above equations (with explict +expressions for the derivatives) with just one input variable \( x \) and +one output variable. The target value \( y=2x+1 \) is a simple linear +function in \( x \). Since this is a regression problem, we define the cost function to be proportional to the least squares error +

    +$$ +C(y,w_1,b_1)=\frac{1}{2}(a_1-y)^2, +$$ + +

    with \( a_1 \) the output from the network.

    + + + +
    +
    +
    +
    +
    + 
    # import necessary packages
+import numpy as np
+import matplotlib.pyplot as plt
+
+def feed_forward(x):
+    # weighted sum of inputs to the output layer
+    z_1 = x*output_weights + output_bias
+    # Output from output node (one node only)
+    # Here the output is equal to the input
+    a_1 = z_1
+    return a_1
+
+def backpropagation(x, y):
+    a_1 = feed_forward(x)
+    # derivative of cost function
+    derivative_cost = a_1 - y
+    # the variable delta in the equations, note that output a_1 = z_1, its derivatives wrt z_o is thus 1
+    delta_1 = derivative_cost
+    # gradients for the output layer
+    output_weights_gradient = delta_1*x
+    output_bias_gradient = delta_1
+    # The cost function is 0.5*(a_1-y)^2. This gives a measure of the error for each iteration
+    return output_weights_gradient, output_bias_gradient
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = 4.0
+# Target values
+y = 2*x+1.0
+
+# Defining the neural network
+n_inputs = 1
+n_outputs = 1
+# Initialize the network
+# weights and bias in the output layer
+output_weights = np.random.randn()
+output_bias = np.random.randn()
+
+# implementing a simple gradient descent approach with fixed learning rate
+eta = 0.01
+for i in range(40):
+    # calculate gradients from back propagation
+    derivative_w1, derivative_b1 = backpropagation(x, y)
+    # update weights and biases
+    output_weights -= eta * derivative_w1
+    output_bias -= eta * derivative_b1
+# our final prediction after training
+ytilde = output_weights*x+output_bias
+print(0.5*((ytilde-y)**2))
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    Running this code gives us an acceptable results after some 40-50 iterations. Note that the results depend on the value of the learning rate.

    + +









    + + +

    Exercise 1: Extensions to the above code

    + +

    Feel free to add more input nodes and weights to the above +code. Furthermore, try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +

    + +

    You could also try to change the function \( f(x)=y \) from a linear polynomial in \( x \) to a higher-order polynomial. +Comment your results. +

    + +

    Hint: Increasing the number of input variables and input nodes requires a rewrite of the input data in terms of a matrix. You need to figure out the correct dimensionalities.

    + + +









    Adding a hidden layer

    -

    We change our simple model to (see graph) +

    We change our simple model to (see graph below) a network with just one hidden layer but with scalar variables only.

    @@ -641,7 +739,117 @@

    The training

    For the first hidden layer \( a_{i-1}=a_0=x \) for this simple model.











    -

    Code examples for the simple models

    +

    Code example

    + +

    The code here implements the above model with one hidden layer and +scalar variables for the same function we studied in the previous +example. The code is however set up so that we can add multiple +inputs \( x \) and target values \( y \). Note also that we have the +possibility of defining a feature matrix \( \boldsymbol{X} \) with more than just +one column for the input values. This will turn useful in our next example. We have also defined matrices and vectors for all of our operations although it is not necessary here. +

    + + + +
    +
    +
    +
    +
    + 
    import numpy as np
+# We use the Sigmoid function as activation function
+def sigmoid(z):
+    return 1.0/(1.0+np.exp(-z))
+
+def forwardpropagation(x):
+    # weighted sum of inputs to the hidden layer
+    z_1 = np.matmul(x, w_1) + b_1
+    # activation in the hidden layer
+    a_1 = sigmoid(z_1)
+    # weighted sum of inputs to the output layer
+    z_2 = np.matmul(a_1, w_2) + b_2
+    a_2 = z_2
+    return a_1, a_2
+
+def backpropagation(x, y):
+    a_1, a_2 = forwardpropagation(x)
+    # parameter delta for the output layer, note that a_2=z_2 and its derivative wrt z_2 is just 1
+    delta_2 = a_2 - y
+    print(0.5*((a_2-y)**2))
+    # delta for  the hidden layer
+    delta_1 = np.matmul(delta_2, w_2.T) * a_1 * (1 - a_1)
+    # gradients for the output layer
+    output_weights_gradient = np.matmul(a_1.T, delta_2)
+    output_bias_gradient = np.sum(delta_2, axis=0)
+    # gradient for the hidden layer
+    hidden_weights_gradient = np.matmul(x.T, delta_1)
+    hidden_bias_gradient = np.sum(delta_1, axis=0)
+    return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient
+
+
+# ensure the same random numbers appear every time
+np.random.seed(0)
+# Input variable
+x = np.array([4.0],dtype=np.float64)
+# Target values
+y = 2*x+1.0 
+
+# Defining the neural network, only scalars here
+n_inputs = x.shape
+n_features = 1
+n_hidden_neurons = 1
+n_outputs = 1
+
+# Initialize the network
+# weights and bias in the hidden layer
+w_1 = np.random.randn(n_features, n_hidden_neurons)
+b_1 = np.zeros(n_hidden_neurons) + 0.01
+
+# weights and bias in the output layer
+w_2 = np.random.randn(n_hidden_neurons, n_outputs)
+b_2 = np.zeros(n_outputs) + 0.01
+
+eta = 0.1
+for i in range(50):
+    # calculate gradients
+    derivW2, derivB2, derivW1, derivB1 = backpropagation(x, y)
+    # update weights and biases
+    w_2 -= eta * derivW2
+    b_2 -= eta * derivB2
+    w_1 -= eta * derivW1
+    b_1 -= eta * derivB1
+
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +

    We see that after some few iterations (the results do depend on the learning rate however), we get an error which is rather small.

    + +









    + + +

    Exercise 2: Including more data

    + +

    Try to increase the amount of input and +target/output data. Try also to perform calculations for more values +of the learning rates. Feel free to add either hyperparameters with an +\( l_1 \) norm or an \( l_2 \) norm and discuss your results. +Discuss your results as functions of the amount of training data and various learning rates. +

    + +

    Challenge: Try to change the activation functions and replace the hard-coded analytical expressions with automatic derivation via either autograd or JAX.

    + +









    Simple neural network and the back propagation equations

    @@ -846,6 +1054,41 @@

    Gradient expressions











    Program example

    +

    We extend our simple code to a function which depends on two variable \( x_0 \) and \( x_1 \), that is

    +$$ +y=f(x_0,x_1)=x_0^2+3x_0x_1+x_1^2+5. +$$ + +

    We feed our network with \( n=100 \) entries \( x_0 \) and \( x_1 \). We have thus two features represented by these variable and an input matrix/design matrix \( \boldsymbol{X}\in \mathbf{R}^{n\times 2} \)

    +$$ +\boldsymbol{X}=\begin{bmatrix} x_{00} & x_{01} \\ x_{00} & x_{01} \\ x_{10} & x_{11} \\ x_{20} & x_{21} \\ \dots & \dots \\ \dots & \dots \\ x_{n-20} & x_{n-21} \\ x_{n-10} & x_{n-11} \end{bmatrix}. +$$ + + + + +
    +
    +
    +
    +
    + 
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    +
    + +









    Getting serious, the back propagation equations for a neural network

    @@ -3483,14 +3726,21 @@

    Efficient Polynomial Multiplication

    $$ -\begin{split} -\delta_0=&\alpha_0\beta_0\\ -\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0\\ -\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0\\ -\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3\\ -\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3\\ -\delta_5=&\alpha_2\beta_3.\\ -\end{split} +\begin{align} +\delta_0=&\alpha_0\beta_0 +\label{_auto4}\\ +\delta_1=&\alpha_1\beta_0+\alpha_1\beta_0 +\label{_auto5}\\ +\delta_2=&\alpha_0\beta_2+\alpha_1\beta_1+\alpha_2\beta_0 +\label{_auto6}\\ +\delta_3=&\alpha_1\beta_2+\alpha_2\beta_1+\alpha_0\beta_3 +\label{_auto7}\\ +\delta_4=&\alpha_2\beta_2+\alpha_1\beta_3 +\label{_auto8}\\ +\delta_5=&\alpha_2\beta_3. +\label{_auto9}\\ +\label{_auto10} +\end{align} $$

    We note that \( \alpha_i=0 \) except for \( i\in \left\{0,1,2\right\} \) and \( \beta_i=0 \) except for \( i\in\left\{0,1,2,3\right\} \).

    @@ -3545,279 +3795,6 @@

    A more efficient w

    Does the number of floating point operations change here when we use the commutative property?

    -









    -

    Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)

    - -

    For problems with so-called harmonic oscillations, given by for example the following differential equation

    -$$ -m\frac{d^2x}{dt^2}+\eta\frac{dx}{dt}+x(t)=F(t), -$$ - -

    where \( F(t) \) is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.

    - -

    If one has several driving forces, \( F(t)=\sum_n F_n(t) \), one can find -the particular solution to each \( F_n \), \( x_{pn}(t) \), and the particular -solution for the entire driving force is then given by a series like -

    - -$$ -\begin{equation} -x_p(t)=\sum_nx_{pn}(t). -\label{_auto4} -\end{equation} -$$ - - -









    -

    Principle of Superposition

    - -

    This is known as the principle of superposition. It only applies when -the homogenous equation is linear. If there were an anharmonic term -such as \( x^3 \) in the homogenous equation, then when one summed various -solutions, \( x=(\sum_n x_n)^2 \), one would get cross -terms. Superposition is especially useful when \( F(t) \) can be written -as a sum of sinusoidal terms, because the solutions for each -sinusoidal (sine or cosine) term is analytic. -

    - -

    Driving forces are often periodic, even when they are not -sinusoidal. Periodicity implies that for some time \( \tau \) -

    - -$$ -\begin{eqnarray} -F(t+\tau)=F(t). -\end{eqnarray} -$$ - -

    One example of a non-sinusoidal periodic force is a square wave. Many -components in electric circuits are non-linear, e.g. diodes, which -makes many wave forms non-sinusoidal even when the circuits are being -driven by purely sinusoidal sources. -

    - -









    -

    Simple Code Example

    - -

    The code here shows a typical example of such a square wave generated using the functionality included in the scipy Python package. We have used a period of \( \tau=0.2 \).

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t)
-plt.plot(t, SqrSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
    - -

    For the sinusoidal example the -period is \( \tau=2\pi/\omega \). However, higher harmonics can also -satisfy the periodicity requirement. In general, any force that -satisfies the periodicity requirement can be expressed as a sum over -harmonics, -

    - -$$ -\begin{equation} -F(t)=\frac{f_0}{2}+\sum_{n>0} f_n\cos(2n\pi t/\tau)+g_n\sin(2n\pi t/\tau). -\label{_auto5} -\end{equation} -$$ - - -









    -

    Wrapping up Fourier transforms

    - -

    We can write down the answer for -\( x_{pn}(t) \), by substituting \( f_n/m \) or \( g_n/m \) for \( F_0/m \). By -writing each factor \( 2n\pi t/\tau \) as \( n\omega t \), with \( \omega\equiv -2\pi/\tau \), -

    - -$$ -\begin{equation} -\label{eq:fourierdef1} -F(t)=\frac{f_0}{2}+\sum_{n>0}f_n\cos(n\omega t)+g_n\sin(n\omega t). -\end{equation} -$$ - -

    The solutions for \( x(t) \) then come from replacing \( \omega \) with -\( n\omega \) for each term in the particular solution, -

    - -$$ -\begin{eqnarray} -x_p(t)&=&\frac{f_0}{2k}+\sum_{n>0} \alpha_n\cos(n\omega t-\delta_n)+\beta_n\sin(n\omega t-\delta_n),\\ -\nonumber -\alpha_n&=&\frac{f_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\beta_n&=&\frac{g_n/m}{\sqrt{((n\omega)^2-\omega_0^2)+4\beta^2n^2\omega^2}},\\ -\nonumber -\delta_n&=&\tan^{-1}\left(\frac{2\beta n\omega}{\omega_0^2-n^2\omega^2}\right). -\end{eqnarray} -$$ - - -









    -

    Finding the Coefficients

    - -

    Because the forces have been applied for a long time, any non-zero -damping eliminates the homogenous parts of the solution, so one need -only consider the particular solution for each \( n \). -

    - -

    The problem is considered solved if one can find expressions for the -coefficients \( f_n \) and \( g_n \), even though the solutions are expressed -as an infinite sum. The coefficients can be extracted from the -function \( F(t) \) by -

    - -$$ -\begin{eqnarray} -\label{eq:fourierdef2} -f_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\cos(2n\pi t/\tau),\\ -\nonumber -g_n&=&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~F(t)\sin(2n\pi t/\tau). -\end{eqnarray} -$$ - -

    To check the consistency of these expressions and to verify -Eq. \eqref{eq:fourierdef2}, one can insert the expansion of \( F(t) \) in -Eq. \eqref{eq:fourierdef1} into the expression for the coefficients in -Eq. \eqref{eq:fourierdef2} and see whether -

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~\left\{ -\frac{f_0}{2}+\sum_{m>0}f_m\cos(m\omega t)+g_m\sin(m\omega t) -\right\}\cos(n\omega t). -\end{eqnarray} -$$ - -

    Immediately, one can throw away all the terms with \( g_m \) because they -convolute an even and an odd function. The term with \( f_0/2 \) -disappears because \( \cos(n\omega t) \) is equally positive and negative -over the interval and will integrate to zero. For all the terms -\( f_m\cos(m\omega t) \) appearing in the sum, one can use angle addition -formulas to see that \( \cos(m\omega t)\cos(n\omega -t)=(1/2)(\cos[(m+n)\omega t]+\cos[(m-n)\omega t] \). This will integrate -to zero unless \( m=n \). In that case the \( m=n \) term gives -

    - -$$ -\begin{equation} -\int_{-\tau/2}^{\tau/2}dt~\cos^2(m\omega t)=\frac{\tau}{2}, -\label{_auto6} -\end{equation} -$$ - -

    and

    - -$$ -\begin{eqnarray} -f_n&=?&\frac{2}{\tau}\int_{-\tau/2}^{\tau/2} dt~f_n/2\\ -\nonumber -&=&f_n~\checkmark. -\end{eqnarray} -$$ - -

    The same method can be used to check for the consistency of \( g_n \).

    - -









    -

    Final words on Fourier Transforms

    - -

    The code here uses the Fourier series applied to a -square wave signal. The code here -visualizes the various approximations given by Fourier series compared -with a square wave with period \( T=0.2 \) (dimensionless time), width \( 0.1 \) and max value of the force \( F=2 \). We -see that when we increase the number of components in the Fourier -series, the Fourier series approximation gets closer and closer to the -square wave signal. -

    - - - -
    -
    -
    -
    -
    - 
    import numpy as np
-import math
-from scipy import signal
-import matplotlib.pyplot as plt
-
-# number of points                                                                                       
-n = 500
-# start and final times                                                                                  
-t0 = 0.0
-tn = 1.0
-# Period                                                                                                 
-T =0.2
-# Max value of square signal                                                                             
-Fmax= 2.0
-# Width of signal   
-Width = 0.1
-t = np.linspace(t0, tn, n, endpoint=False)
-SqrSignal = np.zeros(n)
-FourierSeriesSignal = np.zeros(n)
-SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)
-a0 = Fmax*Width/T
-FourierSeriesSignal = a0
-Factor = 2.0*Fmax/np.pi
-for i in range(1,500):
-    FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)
-plt.plot(t, SqrSignal)
-plt.plot(t, FourierSeriesSignal)
-plt.ylim(-0.5, 2.5)
-plt.show()
-
    -
    -
    -
    -
    -
    -
    -
    -
    -
    -
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    -
    -
    - -









    Two-dimensional Objects

    diff --git a/doc/pub/week4/ipynb/ipynb-week4-src.tar.gz b/doc/pub/week4/ipynb/ipynb-week4-src.tar.gz index 5467f5f129bb37da8b3fb634383b062293068f46..47710ca1a394ca5ea197ae279c10368dc6c1362a 100644 GIT binary patch delta 63 zcmWN_DFJ{$002S$J^l!^KwThWfLKKy0ubmV4gn+QP4P{Ux$a=Fq#|Lxj5f}gf diff --git a/doc/pub/week4/ipynb/week4.ipynb b/doc/pub/week4/ipynb/week4.ipynb index 85bf3458..67c9708e 100644 --- a/doc/pub/week4/ipynb/week4.ipynb +++ b/doc/pub/week4/ipynb/week4.ipynb @@ -2,8 +2,10 @@ "cells": [ { "cell_type": "markdown", - "id": "879fc731", - "metadata": {}, + "id": "2a49cdef", + "metadata": { + "editable": true + }, "source": [ "\n", @@ -12,21 +14,23 @@ }, { "cell_type": "markdown", - "id": "91036413", - "metadata": {}, + "id": "bf32b8a0", + "metadata": { + "editable": true + }, "source": [ "# February 5-9: Advanced machine learning and data analysis for the physical sciences\n", "**Morten Hjorth-Jensen**, Department of Physics and Center for Computing in Science Education, University of Oslo, Norway and Department of Physics and Astronomy and Facility for Rare Isotope Beams, Michigan State University, East Lansing, Michigan, USA\n", "\n", - "Date: **Feb 5, 2024**\n", - "\n", - "Copyright 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license" + "Date: **February 6, 2024**" ] }, { "cell_type": "markdown", - "id": "dd7d32bf", - "metadata": {}, + "id": "1b029486", + "metadata": { + "editable": true + }, "source": [ "## Overview of fourth week, February 5-9\n", "\n", @@ -39,8 +43,10 @@ }, { "cell_type": "markdown", - "id": "28516142", - "metadata": {}, + "id": "58975d34", + "metadata": { + "editable": true + }, "source": [ "## Mathematics of deep learning\n", "\n", @@ -53,8 +59,10 @@ }, { "cell_type": "markdown", - "id": "de74a183", - "metadata": {}, + "id": "cce1ea60", + "metadata": { + "editable": true + }, "source": [ "## Reminder on books with hands-on material and codes\n", "* [Sebastian Rashcka et al, Machine learning with Sickit-Learn and PyTorch](https://sebastianraschka.com/blog/2022/ml-pytorch-book.html)\n", @@ -69,8 +77,10 @@ }, { "cell_type": "markdown", - "id": "cf8c3660", - "metadata": {}, + "id": "f8b35434", + "metadata": { + "editable": true + }, "source": [ "## Reading recommendations\n", "\n", @@ -83,8 +93,10 @@ }, { "cell_type": "markdown", - "id": "92b2be74", - "metadata": {}, + "id": "2efd0b20", + "metadata": { + "editable": true + }, "source": [ "## Videos on CNNs\n", "\n", @@ -98,8 +110,10 @@ }, { "cell_type": "markdown", - "id": "971612a9", - "metadata": {}, + "id": "f0c3d4d5", + "metadata": { + "editable": true + }, "source": [ "## First network example, simple percepetron with one input\n", "\n", @@ -111,8 +125,10 @@ }, { "cell_type": "markdown", - "id": "845a012f", - "metadata": {}, + "id": "faf3f729", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z_1 = w_1x+b_1,\n", @@ -121,8 +137,10 @@ }, { "cell_type": "markdown", - "id": "aa374473", - "metadata": {}, + "id": "98927ae9", + "metadata": { + "editable": true + }, "source": [ "where $w_1$ is the weight and $b_1$ is the bias. These are the\n", "parameters we want to optimize. The output is $a_1=\\sigma(z_1)$ (see\n", @@ -133,8 +151,10 @@ }, { "cell_type": "markdown", - "id": "0e6ff7df", - "metadata": {}, + "id": "617faccb", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(x;w_1,b_1)=\\frac{1}{2}(a_1-y)^2.\n", @@ -143,8 +163,10 @@ }, { "cell_type": "markdown", - "id": "40414282", - "metadata": {}, + "id": "a2099bb8", + "metadata": { + "editable": true + }, "source": [ "## Layout of a simple neural network with no hidden layer\n", "\n", @@ -157,8 +179,10 @@ }, { "cell_type": "markdown", - "id": "95c5cea2", - "metadata": {}, + "id": "38079fa5", + "metadata": { + "editable": true + }, "source": [ "## Optimizing the parameters\n", "\n", @@ -171,8 +195,10 @@ }, { "cell_type": "markdown", - "id": "ba92330f", - "metadata": {}, + "id": "7d05fd35", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_1} \\hspace{0.1cm}\\mathrm{and}\\hspace{0.1cm}\\frac{\\partial C}{\\partial b_1}.\n", @@ -181,16 +207,20 @@ }, { "cell_type": "markdown", - "id": "a3b97c07", - "metadata": {}, + "id": "5cd537c9", + "metadata": { + "editable": true + }, "source": [ "Using the chain rule we find" ] }, { "cell_type": "markdown", - "id": "5bd81aba", - "metadata": {}, + "id": "3031de7c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_1}=\\frac{\\partial C}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial w_1}=(a_1-y)\\sigma_1'x,\n", @@ -199,16 +229,20 @@ }, { "cell_type": "markdown", - "id": "63dfefdb", - "metadata": {}, + "id": "80132233", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "90845a74", - "metadata": {}, + "id": "1d6acd30", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial b_1}=\\frac{\\partial C}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial b_1}=(a_1-y)\\sigma_1',\n", @@ -217,16 +251,20 @@ }, { "cell_type": "markdown", - "id": "d6892c66", - "metadata": {}, + "id": "ddf3b2ab", + "metadata": { + "editable": true + }, "source": [ "which we later will just define as" ] }, { "cell_type": "markdown", - "id": "e7206a8e", - "metadata": {}, + "id": "31af8685", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}=\\delta_1.\n", @@ -235,12 +273,146 @@ }, { "cell_type": "markdown", - "id": "0804ce98", - "metadata": {}, + "id": "618670a4", + "metadata": { + "editable": true + }, + "source": [ + "## Implementing the simple perceptron model\n", + "\n", + "In the example code here we implement the above equations (with explict\n", + "expressions for the derivatives) with just one input variable $x$ and\n", + "one output variable. The target value $y=2x+1$ is a simple linear\n", + "function in $x$. Since this is a regression problem, we define the cost function to be proportional to the least squares error" + ] + }, + { + "cell_type": "markdown", + "id": "76bdf642", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "C(y,w_1,b_1)=\\frac{1}{2}(a_1-y)^2,\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "58adb380", + "metadata": { + "editable": true + }, + "source": [ + "with $a_1$ the output from the network." + ] + }, + { + "cell_type": "code", + "execution_count": 1, + "id": "d741b7b1", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "%matplotlib inline\n", + "\n", + "# import necessary packages\n", + "import numpy as np\n", + "import matplotlib.pyplot as plt\n", + "\n", + "def feed_forward(x):\n", + " # weighted sum of inputs to the output layer\n", + " z_1 = x*output_weights + output_bias\n", + " # Output from output node (one node only)\n", + " # Here the output is equal to the input\n", + " a_1 = z_1\n", + " return a_1\n", + "\n", + "def backpropagation(x, y):\n", + " a_1 = feed_forward(x)\n", + " # derivative of cost function\n", + " derivative_cost = a_1 - y\n", + " # the variable delta in the equations, note that output a_1 = z_1, its derivatives wrt z_o is thus 1\n", + " delta_1 = derivative_cost\n", + " # gradients for the output layer\n", + " output_weights_gradient = delta_1*x\n", + " output_bias_gradient = delta_1\n", + " # The cost function is 0.5*(a_1-y)^2. This gives a measure of the error for each iteration\n", + " return output_weights_gradient, output_bias_gradient\n", + "\n", + "# ensure the same random numbers appear every time\n", + "np.random.seed(0)\n", + "# Input variable\n", + "x = 4.0\n", + "# Target values\n", + "y = 2*x+1.0\n", + "\n", + "# Defining the neural network\n", + "n_inputs = 1\n", + "n_outputs = 1\n", + "# Initialize the network\n", + "# weights and bias in the output layer\n", + "output_weights = np.random.randn()\n", + "output_bias = np.random.randn()\n", + "\n", + "# implementing a simple gradient descent approach with fixed learning rate\n", + "eta = 0.01\n", + "for i in range(40):\n", + " # calculate gradients from back propagation\n", + " derivative_w1, derivative_b1 = backpropagation(x, y)\n", + " # update weights and biases\n", + " output_weights -= eta * derivative_w1\n", + " output_bias -= eta * derivative_b1\n", + "# our final prediction after training\n", + "ytilde = output_weights*x+output_bias\n", + "print(0.5*((ytilde-y)**2))" + ] + }, + { + "cell_type": "markdown", + "id": "ca04d604", + "metadata": { + "editable": true + }, + "source": [ + "Running this code gives us an acceptable results after some 40-50 iterations. Note that the results depend on the value of the learning rate." + ] + }, + { + "cell_type": "markdown", + "id": "fa4a5e9b", + "metadata": { + "editable": true + }, + "source": [ + "## Exercise 1: Extensions to the above code\n", + "\n", + "Feel free to add more input nodes and weights to the above\n", + "code. Furthermore, try to increase the amount of input and\n", + "target/output data. Try also to perform calculations for more values\n", + "of the learning rates. Feel free to add either hyperparameters with an\n", + "$l_1$ norm or an $l_2$ norm and discuss your results.\n", + "\n", + "You could also try to change the function $f(x)=y$ from a linear polynomial in $x$ to a higher-order polynomial.\n", + "Comment your results.\n", + "\n", + "**Hint**: Increasing the number of input variables and input nodes requires a rewrite of the input data in terms of a matrix. You need to figure out the correct dimensionalities." + ] + }, + { + "cell_type": "markdown", + "id": "f11ee5f4", + "metadata": { + "editable": true + }, "source": [ "## Adding a hidden layer\n", "\n", - "We change our simple model to (see graph)\n", + "We change our simple model to (see graph below)\n", "a network with just one hidden layer but with scalar variables only.\n", "\n", "Our output variable changes to $a_2$ and $a_1$ is now the output from the hidden node and $a_0=x$.\n", @@ -249,8 +421,10 @@ }, { "cell_type": "markdown", - "id": "771f280d", - "metadata": {}, + "id": "6a49c65a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z_1 = w_1a_0+b_1 \\hspace{0.1cm} \\wedge a_1 = \\sigma_1(z_1),\n", @@ -259,8 +433,10 @@ }, { "cell_type": "markdown", - "id": "d83d827a", - "metadata": {}, + "id": "6efbbffd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z_2 = w_2a_1+b_2 \\hspace{0.1cm} \\wedge a_2 = \\sigma_2(z_2),\n", @@ -269,16 +445,20 @@ }, { "cell_type": "markdown", - "id": "c682942f", - "metadata": {}, + "id": "ffc70d6c", + "metadata": { + "editable": true + }, "source": [ "and the cost function" ] }, { "cell_type": "markdown", - "id": "1db78ad4", - "metadata": {}, + "id": "d594824b", + "metadata": { + "editable": true + }, "source": [ "$$\n", "C(x;\\boldsymbol{\\Theta})=\\frac{1}{2}(a_2-y)^2,\n", @@ -287,16 +467,20 @@ }, { "cell_type": "markdown", - "id": "e9b46133", - "metadata": {}, + "id": "e040d4ea", + "metadata": { + "editable": true + }, "source": [ "with $\\boldsymbol{\\Theta}=[w_1,w_2,b_1,b_2]$." ] }, { "cell_type": "markdown", - "id": "915f9948", - "metadata": {}, + "id": "ca0122ea", + "metadata": { + "editable": true + }, "source": [ "## Layout of a simple neural network with one hidden layer\n", "\n", @@ -309,8 +493,10 @@ }, { "cell_type": "markdown", - "id": "9c84f296", - "metadata": {}, + "id": "8f5593f6", + "metadata": { + "editable": true + }, "source": [ "## The derivatives\n", "\n", @@ -319,8 +505,10 @@ }, { "cell_type": "markdown", - "id": "a60ed43a", - "metadata": {}, + "id": "039af9c8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_2}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial w_2}=(a_2-y)\\sigma_2'a_1=\\delta_2a_1,\n", @@ -329,8 +517,10 @@ }, { "cell_type": "markdown", - "id": "9091ec3a", - "metadata": {}, + "id": "972a14a8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial b_2}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial b_2}=(a_2-y)\\sigma_2'=\\delta_2,\n", @@ -339,8 +529,10 @@ }, { "cell_type": "markdown", - "id": "a12c7bb0", - "metadata": {}, + "id": "4d7a8fb4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_1}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial w_1}=(a_2-y)\\sigma_2'a_1\\sigma_1'a_0,\n", @@ -349,8 +541,10 @@ }, { "cell_type": "markdown", - "id": "87aa212e", - "metadata": {}, + "id": "76fc0395", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial b_1}=\\frac{\\partial C}{\\partial a_2}\\frac{\\partial a_2}{\\partial z_2}\\frac{\\partial z_2}{\\partial a_1}\\frac{\\partial a_1}{\\partial z_1}\\frac{\\partial z_1}{\\partial b_1}=(a_2-y)\\sigma_2'\\sigma_1'=\\delta_1.\n", @@ -359,16 +553,20 @@ }, { "cell_type": "markdown", - "id": "2650ec97", - "metadata": {}, + "id": "760178f3", + "metadata": { + "editable": true + }, "source": [ "Can you generalize this to more than one hidden layer?" ] }, { "cell_type": "markdown", - "id": "8de864f4", - "metadata": {}, + "id": "34997a9c", + "metadata": { + "editable": true + }, "source": [ "## Important observations\n", "\n", @@ -381,8 +579,10 @@ }, { "cell_type": "markdown", - "id": "cdd2825d", - "metadata": {}, + "id": "1dd2e4d4", + "metadata": { + "editable": true + }, "source": [ "## The training\n", "\n", @@ -391,8 +591,10 @@ }, { "cell_type": "markdown", - "id": "0bad5df8", - "metadata": {}, + "id": "392b83e6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{i}\\leftarrow w_{i}- \\eta \\delta_i a_{i-1},\n", @@ -401,16 +603,20 @@ }, { "cell_type": "markdown", - "id": "1b83d170", - "metadata": {}, + "id": "aee6cfcb", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "2b3550e1", - "metadata": {}, + "id": "51dbce46", + "metadata": { + "editable": true + }, "source": [ "$$\n", "b_i \\leftarrow b_i-\\eta \\delta_i,\n", @@ -419,8 +625,10 @@ }, { "cell_type": "markdown", - "id": "f52ebd8b", - "metadata": {}, + "id": "c8074da9", + "metadata": { + "editable": true + }, "source": [ "with $\\eta$ is the learning rate.\n", "\n", @@ -431,16 +639,129 @@ }, { "cell_type": "markdown", - "id": "8563e051", - "metadata": {}, + "id": "9434e7f9", + "metadata": { + "editable": true + }, + "source": [ + "## Code example\n", + "\n", + "The code here implements the above model with one hidden layer and\n", + "scalar variables for the same function we studied in the previous\n", + "example. The code is however set up so that we can add multiple\n", + "inputs $x$ and target values $y$. Note also that we have the\n", + "possibility of defining a feature matrix $\\boldsymbol{X}$ with more than just\n", + "one column for the input values. This will turn useful in our next example. We have also defined matrices and vectors for all of our operations although it is not necessary here." + ] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "f7c5fc08", + "metadata": { + "collapsed": false, + "editable": true + }, + "outputs": [], + "source": [ + "import numpy as np\n", + "# We use the Sigmoid function as activation function\n", + "def sigmoid(z):\n", + " return 1.0/(1.0+np.exp(-z))\n", + "\n", + "def forwardpropagation(x):\n", + " # weighted sum of inputs to the hidden layer\n", + " z_1 = np.matmul(x, w_1) + b_1\n", + " # activation in the hidden layer\n", + " a_1 = sigmoid(z_1)\n", + " # weighted sum of inputs to the output layer\n", + " z_2 = np.matmul(a_1, w_2) + b_2\n", + " a_2 = z_2\n", + " return a_1, a_2\n", + "\n", + "def backpropagation(x, y):\n", + " a_1, a_2 = forwardpropagation(x)\n", + " # parameter delta for the output layer, note that a_2=z_2 and its derivative wrt z_2 is just 1\n", + " delta_2 = a_2 - y\n", + " print(0.5*((a_2-y)**2))\n", + " # delta for the hidden layer\n", + " delta_1 = np.matmul(delta_2, w_2.T) * a_1 * (1 - a_1)\n", + " # gradients for the output layer\n", + " output_weights_gradient = np.matmul(a_1.T, delta_2)\n", + " output_bias_gradient = np.sum(delta_2, axis=0)\n", + " # gradient for the hidden layer\n", + " hidden_weights_gradient = np.matmul(x.T, delta_1)\n", + " hidden_bias_gradient = np.sum(delta_1, axis=0)\n", + " return output_weights_gradient, output_bias_gradient, hidden_weights_gradient, hidden_bias_gradient\n", + "\n", + "\n", + "# ensure the same random numbers appear every time\n", + "np.random.seed(0)\n", + "# Input variable\n", + "x = np.array([4.0],dtype=np.float64)\n", + "# Target values\n", + "y = 2*x+1.0 \n", + "\n", + "# Defining the neural network, only scalars here\n", + "n_inputs = x.shape\n", + "n_features = 1\n", + "n_hidden_neurons = 1\n", + "n_outputs = 1\n", + "\n", + "# Initialize the network\n", + "# weights and bias in the hidden layer\n", + "w_1 = np.random.randn(n_features, n_hidden_neurons)\n", + "b_1 = np.zeros(n_hidden_neurons) + 0.01\n", + "\n", + "# weights and bias in the output layer\n", + "w_2 = np.random.randn(n_hidden_neurons, n_outputs)\n", + "b_2 = np.zeros(n_outputs) + 0.01\n", + "\n", + "eta = 0.1\n", + "for i in range(50):\n", + " # calculate gradients\n", + " derivW2, derivB2, derivW1, derivB1 = backpropagation(x, y)\n", + " # update weights and biases\n", + " w_2 -= eta * derivW2\n", + " b_2 -= eta * derivB2\n", + " w_1 -= eta * derivW1\n", + " b_1 -= eta * derivB1" + ] + }, + { + "cell_type": "markdown", + "id": "c2f5319e", + "metadata": { + "editable": true + }, + "source": [ + "We see that after some few iterations (the results do depend on the learning rate however), we get an error which is rather small." + ] + }, + { + "cell_type": "markdown", + "id": "1565ed87", + "metadata": { + "editable": true + }, "source": [ - "## Code examples for the simple models" + "## Exercise 2: Including more data\n", + "\n", + "Try to increase the amount of input and\n", + "target/output data. Try also to perform calculations for more values\n", + "of the learning rates. Feel free to add either hyperparameters with an\n", + "$l_1$ norm or an $l_2$ norm and discuss your results.\n", + "Discuss your results as functions of the amount of training data and various learning rates.\n", + "\n", + "**Challenge:** Try to change the activation functions and replace the hard-coded analytical expressions with automatic derivation via either **autograd** or **JAX**." ] }, { "cell_type": "markdown", - "id": "f4e4d7f0", - "metadata": {}, + "id": "0910f8d7", + "metadata": { + "editable": true + }, "source": [ "## Simple neural network and the back propagation equations\n", "\n", @@ -454,8 +775,10 @@ }, { "cell_type": "markdown", - "id": "c8462694", - "metadata": {}, + "id": "05c99305", + "metadata": { + "editable": true + }, "source": [ "$$\n", "x_0 = a_0^{(0)} \\wedge x_1 = a_1^{(0)}.\n", @@ -464,16 +787,20 @@ }, { "cell_type": "markdown", - "id": "ba24c083", - "metadata": {}, + "id": "3b95bed8", + "metadata": { + "editable": true + }, "source": [ "The hidden layer (layer $(1)$) has nodes which yield the outputs $a_0^{(1)}$ and $a_1^{(1)}$) with weight $\\boldsymbol{w}$ and bias $\\boldsymbol{b}$ parameters" ] }, { "cell_type": "markdown", - "id": "0fefc397", - "metadata": {}, + "id": "20f590bf", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{ij}^{(1)}=\\left\\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)}\\right\\} \\wedge b^{(1)}=\\left\\{b_0^{(1)},b_1^{(1)}\\right\\}.\n", @@ -482,8 +809,10 @@ }, { "cell_type": "markdown", - "id": "5bf20787", - "metadata": {}, + "id": "2945b726", + "metadata": { + "editable": true + }, "source": [ "## Layout of a simple neural network with two input nodes, one hidden layer and one output node\n", "\n", @@ -496,8 +825,10 @@ }, { "cell_type": "markdown", - "id": "25dd346b", - "metadata": {}, + "id": "b5474112", + "metadata": { + "editable": true + }, "source": [ "## The ouput layer\n", "\n", @@ -506,8 +837,10 @@ }, { "cell_type": "markdown", - "id": "df5073dc", - "metadata": {}, + "id": "f3569208", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{i}^{(2)}=\\left\\{w_{0}^{(2)},w_{1}^{(2)}\\right\\} \\wedge b^{(2)}.\n", @@ -516,8 +849,10 @@ }, { "cell_type": "markdown", - "id": "be0d19fd", - "metadata": {}, + "id": "d9f7e7b1", + "metadata": { + "editable": true + }, "source": [ "Our output is $\\tilde{y}=a^{(2)}$ and we define a generic cost function $C(a^{(2)},y;\\boldsymbol{\\Theta})$ where $y$ is the target value (a scalar here).\n", "The parameters we need to optimize are given by" @@ -525,8 +860,10 @@ }, { "cell_type": "markdown", - "id": "23785a0f", - "metadata": {}, + "id": "8eb82456", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\Theta}=\\left\\{w_{00}^{(1)},w_{01}^{(1)},w_{10}^{(1)},w_{11}^{(1)},w_{0}^{(2)},w_{1}^{(2)},b_0^{(1)},b_1^{(1)},b^{(2)}\\right\\}.\n", @@ -535,8 +872,10 @@ }, { "cell_type": "markdown", - "id": "dd7bc4ab", - "metadata": {}, + "id": "6095d861", + "metadata": { + "editable": true + }, "source": [ "## Compact expressions\n", "\n", @@ -546,8 +885,10 @@ }, { "cell_type": "markdown", - "id": "6adf73c2", - "metadata": {}, + "id": "d497a76a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{bmatrix}z_0^{(1)} \\\\ z_1^{(1)} \\end{bmatrix}=\\begin{bmatrix}w_{00}^{(1)} & w_{01}^{(1)}\\\\ w_{10}^{(1)} &w_{11}^{(1)} \\end{bmatrix}\\begin{bmatrix}a_0^{(0)} \\\\ a_1^{(0)} \\end{bmatrix}+\\begin{bmatrix}b_0^{(1)} \\\\ b_1^{(1)} \\end{bmatrix},\n", @@ -556,16 +897,20 @@ }, { "cell_type": "markdown", - "id": "47e9be47", - "metadata": {}, + "id": "7255fd68", + "metadata": { + "editable": true + }, "source": [ "with outputs" ] }, { "cell_type": "markdown", - "id": "4b22099a", - "metadata": {}, + "id": "fa1b10dc", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\begin{bmatrix}a_0^{(1)} \\\\ a_1^{(1)} \\end{bmatrix}=\\begin{bmatrix}\\sigma^{(1)}(z_0^{(1)}) \\\\ \\sigma^{(1)}(z_1^{(1)}) \\end{bmatrix}.\n", @@ -574,8 +919,10 @@ }, { "cell_type": "markdown", - "id": "55126ac7", - "metadata": {}, + "id": "5fce6ea2", + "metadata": { + "editable": true + }, "source": [ "## Output layer\n", "\n", @@ -584,8 +931,10 @@ }, { "cell_type": "markdown", - "id": "e5ca0b40", - "metadata": {}, + "id": "e0030d69", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z^{(2)} = w_{0}^{(2)}a_0^{(1)} +w_{1}^{(2)}a_1^{(1)}+b^{(2)},\n", @@ -594,16 +943,20 @@ }, { "cell_type": "markdown", - "id": "0b040522", - "metadata": {}, + "id": "fa3adace", + "metadata": { + "editable": true + }, "source": [ "resulting in the output" ] }, { "cell_type": "markdown", - "id": "bdca6660", - "metadata": {}, + "id": "80a081d6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "a^{(2)}=\\sigma^{(2)}(z^{(2)}).\n", @@ -612,8 +965,10 @@ }, { "cell_type": "markdown", - "id": "d9c83455", - "metadata": {}, + "id": "ae4a540b", + "metadata": { + "editable": true + }, "source": [ "## Explicit derivatives\n", "\n", @@ -626,8 +981,10 @@ }, { "cell_type": "markdown", - "id": "f9e08474", - "metadata": {}, + "id": "1d27632c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_{i}^{(2)}}=\\frac{\\partial C}{\\partial a^{(2)}}\\frac{\\partial a^{(2)}}{\\partial z^{(2)}}\\frac{\\partial z^{(2)}}{\\partial w_{i}^{(2)}}=\\delta^{(2)}a_i^{(1)},\n", @@ -636,16 +993,20 @@ }, { "cell_type": "markdown", - "id": "9093e24c", - "metadata": {}, + "id": "fae4f891", + "metadata": { + "editable": true + }, "source": [ "with" ] }, { "cell_type": "markdown", - "id": "f33ab690", - "metadata": {}, + "id": "7bf95fa1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta^{(2)}=\\frac{\\partial C}{\\partial a^{(2)}}\\frac{\\partial a^{(2)}}{\\partial z^{(2)}}\n", @@ -654,16 +1015,20 @@ }, { "cell_type": "markdown", - "id": "3ec55ccc", - "metadata": {}, + "id": "88a9152e", + "metadata": { + "editable": true + }, "source": [ "and finally" ] }, { "cell_type": "markdown", - "id": "a6d2bf02", - "metadata": {}, + "id": "9d93608d", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial b^{(2)}}=\\frac{\\partial C}{\\partial a^{(2)}}\\frac{\\partial a^{(2)}}{\\partial z^{(2)}}\\frac{\\partial z^{(2)}}{\\partial b^{(2)}}=\\delta^{(2)}.\n", @@ -672,8 +1037,10 @@ }, { "cell_type": "markdown", - "id": "af5094f3", - "metadata": {}, + "id": "0769c631", + "metadata": { + "editable": true + }, "source": [ "## Derivatives of the hidden layer\n", "\n", @@ -682,8 +1049,10 @@ }, { "cell_type": "markdown", - "id": "51832846", - "metadata": {}, + "id": "4ae22460", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_{00}^{(1)}}=\\frac{\\partial C}{\\partial a^{(2)}}\\frac{\\partial a^{(2)}}{\\partial z^{(2)}}\n", @@ -693,16 +1062,20 @@ }, { "cell_type": "markdown", - "id": "9508fd87", - "metadata": {}, + "id": "45035d54", + "metadata": { + "editable": true + }, "source": [ "which, noting that" ] }, { "cell_type": "markdown", - "id": "ea53ac8b", - "metadata": {}, + "id": "76285cd7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z^{(2)} =w_0^{(2)}a_0^{(1)}+w_1^{(2)}a_1^{(1)}+b^{(2)},\n", @@ -711,16 +1084,20 @@ }, { "cell_type": "markdown", - "id": "32d6bb31", - "metadata": {}, + "id": "58dbd4ae", + "metadata": { + "editable": true + }, "source": [ "allows us to rewrite" ] }, { "cell_type": "markdown", - "id": "e4f49c29", - "metadata": {}, + "id": "c77ab0c7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial z^{(2)}}{\\partial z_0^{(1)}}\\frac{\\partial z_0^{(1)}}{\\partial w_{00}^{(1)}}=w_0^{(2)}\\frac{\\partial a_0^{(1)}}{\\partial z_0^{(1)}}a_0^{(1)}.\n", @@ -729,8 +1106,10 @@ }, { "cell_type": "markdown", - "id": "f7af2ca2", - "metadata": {}, + "id": "6552759a", + "metadata": { + "editable": true + }, "source": [ "## Final expression\n", "Defining" @@ -738,8 +1117,10 @@ }, { "cell_type": "markdown", - "id": "12fe9da5", - "metadata": {}, + "id": "24b3d604", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_0^{(1)}=w_0^{(2)}\\frac{\\partial a_0^{(1)}}{\\partial z_0^{(1)}}\\delta^{(2)},\n", @@ -748,16 +1129,20 @@ }, { "cell_type": "markdown", - "id": "c54c2412", - "metadata": {}, + "id": "2700cbc9", + "metadata": { + "editable": true + }, "source": [ "we have" ] }, { "cell_type": "markdown", - "id": "5fe91d9b", - "metadata": {}, + "id": "956bbd58", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_{00}^{(1)}}=\\delta_0^{(1)}a_0^{(1)}.\n", @@ -766,16 +1151,20 @@ }, { "cell_type": "markdown", - "id": "f042a6f9", - "metadata": {}, + "id": "cc177db7", + "metadata": { + "editable": true + }, "source": [ "Similarly, we obtain" ] }, { "cell_type": "markdown", - "id": "5a95ebab", - "metadata": {}, + "id": "41cadbed", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_{01}^{(1)}}=\\delta_0^{(1)}a_1^{(1)}.\n", @@ -784,8 +1173,10 @@ }, { "cell_type": "markdown", - "id": "60c88229", - "metadata": {}, + "id": "874c9b15", + "metadata": { + "editable": true + }, "source": [ "## Completing the list\n", "\n", @@ -794,8 +1185,10 @@ }, { "cell_type": "markdown", - "id": "038d6c37", - "metadata": {}, + "id": "1afae5de", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_{10}^{(1)}}=\\delta_1^{(1)}a_0^{(1)},\n", @@ -804,16 +1197,20 @@ }, { "cell_type": "markdown", - "id": "9fdc15f7", - "metadata": {}, + "id": "1f091fbc", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "b1bece6d", - "metadata": {}, + "id": "f135f4a8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial w_{11}^{(1)}}=\\delta_1^{(1)}a_1^{(1)},\n", @@ -822,16 +1219,20 @@ }, { "cell_type": "markdown", - "id": "2fa78f33", - "metadata": {}, + "id": "4c0ef834", + "metadata": { + "editable": true + }, "source": [ "where we have defined" ] }, { "cell_type": "markdown", - "id": "f73efc8f", - "metadata": {}, + "id": "6697b4a7", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_1^{(1)}=w_1^{(2)}\\frac{\\partial a_1^{(1)}}{\\partial z_1^{(1)}}\\delta^{(2)}.\n", @@ -840,8 +1241,10 @@ }, { "cell_type": "markdown", - "id": "95c412c9", - "metadata": {}, + "id": "fa5c9e17", + "metadata": { + "editable": true + }, "source": [ "## Final expressions for the biases of the hidden layer\n", "\n", @@ -850,8 +1253,10 @@ }, { "cell_type": "markdown", - "id": "3b0108a4", - "metadata": {}, + "id": "2eabfcf8", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial b_{0}^{(1)}}=\\delta_0^{(1)},\n", @@ -860,16 +1265,20 @@ }, { "cell_type": "markdown", - "id": "c1351c93", - "metadata": {}, + "id": "beda6195", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "c34447a1", - "metadata": {}, + "id": "f83f0501", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial C}{\\partial b_{1}^{(1)}}=\\delta_1^{(1)}.\n", @@ -878,16 +1287,20 @@ }, { "cell_type": "markdown", - "id": "771908e8", - "metadata": {}, + "id": "c2a33331", + "metadata": { + "editable": true + }, "source": [ "As we will see below, these expressions can be generalized in a more compact form." ] }, { "cell_type": "markdown", - "id": "22dd2c66", - "metadata": {}, + "id": "3f41bed5", + "metadata": { + "editable": true + }, "source": [ "## Gradient expressions\n", "\n", @@ -897,8 +1310,10 @@ }, { "cell_type": "markdown", - "id": "0dbf02f8", - "metadata": {}, + "id": "bcfea242", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{i}^{(2)}\\leftarrow w_{i}^{(2)}- \\eta \\delta^{(2)} a_{i}^{(1)},\n", @@ -907,16 +1322,20 @@ }, { "cell_type": "markdown", - "id": "b1b953ed", - "metadata": {}, + "id": "1825c107", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "25fd9b43", - "metadata": {}, + "id": "dca9e0ff", + "metadata": { + "editable": true + }, "source": [ "$$\n", "b^{(2)} \\leftarrow b^{(2)}-\\eta \\delta^{(2)},\n", @@ -925,16 +1344,20 @@ }, { "cell_type": "markdown", - "id": "66064302", - "metadata": {}, + "id": "634f15c5", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "43ff1b4f", - "metadata": {}, + "id": "312b2244", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{ij}^{(1)}\\leftarrow w_{ij}^{(1)}- \\eta \\delta_{i}^{(1)} a_{j}^{(0)},\n", @@ -943,16 +1366,20 @@ }, { "cell_type": "markdown", - "id": "e77e08eb", - "metadata": {}, + "id": "8f6d7e62", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "4281f07f", - "metadata": {}, + "id": "19a9f01e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "b_{i}^{(1)} \\leftarrow b_{i}^{(1)}-\\eta \\delta_{i}^{(1)},\n", @@ -961,24 +1388,66 @@ }, { "cell_type": "markdown", - "id": "b48954bd", - "metadata": {}, + "id": "5b42983c", + "metadata": { + "editable": true + }, "source": [ "where $\\eta$ is the learning rate." ] }, { "cell_type": "markdown", - "id": "fe64d304", - "metadata": {}, + "id": "b2d3eb15", + "metadata": { + "editable": true + }, + "source": [ + "## Program example\n", + "\n", + "We extend our simple code to a function which depends on two variable $x_0$ and $x_1$, that is" + ] + }, + { + "cell_type": "markdown", + "id": "c06c23a7", + "metadata": { + "editable": true + }, + "source": [ + "$$\n", + "y=f(x_0,x_1)=x_0^2+3x_0x_1+x_1^2+5.\n", + "$$" + ] + }, + { + "cell_type": "markdown", + "id": "2e90169a", + "metadata": { + "editable": true + }, + "source": [ + "We feed our network with $n=100$ entries $x_0$ and $x_1$. We have thus two features represented by these variable and an input matrix/design matrix $\\boldsymbol{X}\\in \\mathbf{R}^{n\\times 2}$" + ] + }, + { + "cell_type": "markdown", + "id": "10ed4cdd", + "metadata": { + "editable": true + }, "source": [ - "## Program example" + "$$\n", + "\\boldsymbol{X}=\\begin{bmatrix} x_{00} & x_{01} \\\\ x_{00} & x_{01} \\\\ x_{10} & x_{11} \\\\ x_{20} & x_{21} \\\\ \\dots & \\dots \\\\ \\dots & \\dots \\\\ x_{n-20} & x_{n-21} \\\\ x_{n-10} & x_{n-11} \\end{bmatrix}.\n", + "$$" ] }, { "cell_type": "markdown", - "id": "1fa99bcc", - "metadata": {}, + "id": "3295a93e", + "metadata": { + "editable": true + }, "source": [ "## Getting serious, the back propagation equations for a neural network\n", "\n", @@ -989,8 +1458,10 @@ }, { "cell_type": "markdown", - "id": "0605e5da", - "metadata": {}, + "id": "43569962", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial{\\cal C}((\\boldsymbol{\\Theta}^L)}{\\partial w_{jk}^L} = \\left(a_j^L - y_j\\right)a_j^L(1-a_j^L)a_k^{L-1},\n", @@ -999,16 +1470,20 @@ }, { "cell_type": "markdown", - "id": "b56d6604", - "metadata": {}, + "id": "c8295a1b", + "metadata": { + "editable": true + }, "source": [ "Defining" ] }, { "cell_type": "markdown", - "id": "3af335b5", - "metadata": {}, + "id": "d42484a4", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^L = a_j^L(1-a_j^L)\\left(a_j^L - y_j\\right) = \\sigma'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)},\n", @@ -1017,16 +1492,20 @@ }, { "cell_type": "markdown", - "id": "0d7e3bbc", - "metadata": {}, + "id": "f4e63102", + "metadata": { + "editable": true + }, "source": [ "and using the Hadamard product of two vectors we can write this as" ] }, { "cell_type": "markdown", - "id": "885cbf88", - "metadata": {}, + "id": "36c73de9", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\boldsymbol{\\delta}^L = \\sigma'(\\hat{z}^L)\\circ\\frac{\\partial {\\cal C}}{\\partial (\\boldsymbol{a}^L)}.\n", @@ -1035,8 +1514,10 @@ }, { "cell_type": "markdown", - "id": "19f2afb0", - "metadata": {}, + "id": "29dc45ed", + "metadata": { + "editable": true + }, "source": [ "## Analyzing the last results\n", "\n", @@ -1051,8 +1532,10 @@ }, { "cell_type": "markdown", - "id": "1ca6c734", - "metadata": {}, + "id": "9dabfc48", + "metadata": { + "editable": true + }, "source": [ "## More considerations\n", "\n", @@ -1067,8 +1550,10 @@ }, { "cell_type": "markdown", - "id": "1aaaf7d0", - "metadata": {}, + "id": "7a6a91d2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}\n", @@ -1077,16 +1562,20 @@ }, { "cell_type": "markdown", - "id": "ce7bac62", - "metadata": {}, + "id": "6b8c64bc", + "metadata": { + "editable": true + }, "source": [ "With the definition of $\\delta_j^L$ we have a more compact definition of the derivative of the cost function in terms of the weights, namely" ] }, { "cell_type": "markdown", - "id": "fbf219b5", - "metadata": {}, + "id": "17312d4f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\frac{\\partial{\\cal C}}{\\partial w_{jk}^L} = \\delta_j^La_k^{L-1}.\n", @@ -1095,8 +1584,10 @@ }, { "cell_type": "markdown", - "id": "c6895875", - "metadata": {}, + "id": "015c30c4", + "metadata": { + "editable": true + }, "source": [ "## Derivatives in terms of $z_j^L$\n", "\n", @@ -1105,8 +1596,10 @@ }, { "cell_type": "markdown", - "id": "0ca5f7d4", - "metadata": {}, + "id": "8e0dfdef", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^L =\\frac{\\partial {\\cal C}}{\\partial z_j^L}= \\frac{\\partial {\\cal C}}{\\partial a_j^L}\\frac{\\partial a_j^L}{\\partial z_j^L},\n", @@ -1115,16 +1608,20 @@ }, { "cell_type": "markdown", - "id": "a47e0eae", - "metadata": {}, + "id": "26284ee3", + "metadata": { + "editable": true + }, "source": [ "which can also be interpreted as the partial derivative of the cost function with respect to the biases $b_j^L$, namely" ] }, { "cell_type": "markdown", - "id": "e45b4232", - "metadata": {}, + "id": "96a7cadd", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^L = \\frac{\\partial {\\cal C}}{\\partial b_j^L}\\frac{\\partial b_j^L}{\\partial z_j^L}=\\frac{\\partial {\\cal C}}{\\partial b_j^L},\n", @@ -1133,16 +1630,20 @@ }, { "cell_type": "markdown", - "id": "8998e08b", - "metadata": {}, + "id": "9d37c79f", + "metadata": { + "editable": true + }, "source": [ "That is, the error $\\delta_j^L$ is exactly equal to the rate of change of the cost function as a function of the bias." ] }, { "cell_type": "markdown", - "id": "ed4fbf34", - "metadata": {}, + "id": "261a9ed2", + "metadata": { + "editable": true + }, "source": [ "## Bringing it together\n", "\n", @@ -1151,8 +1652,10 @@ }, { "cell_type": "markdown", - "id": "0614eae1", - "metadata": {}, + "id": "52dd9ed1", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1167,16 +1670,20 @@ }, { "cell_type": "markdown", - "id": "52ca9277", - "metadata": {}, + "id": "4f82051f", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "02bd2548", - "metadata": {}, + "id": "15113121", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1191,16 +1698,20 @@ }, { "cell_type": "markdown", - "id": "97544d9f", - "metadata": {}, + "id": "913b2332", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "8fb882f7", - "metadata": {}, + "id": "39ac9261", + "metadata": { + "editable": true + }, "source": [ "\n", "
    \n", @@ -1215,8 +1726,10 @@ }, { "cell_type": "markdown", - "id": "2b262287", - "metadata": {}, + "id": "6ce19052", + "metadata": { + "editable": true + }, "source": [ "## Final back propagating equation\n", "\n", @@ -1225,8 +1738,10 @@ }, { "cell_type": "markdown", - "id": "3313d858", - "metadata": {}, + "id": "7ebd8b54", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^l =\\frac{\\partial {\\cal C}}{\\partial z_j^l}.\n", @@ -1235,16 +1750,20 @@ }, { "cell_type": "markdown", - "id": "09edf05a", - "metadata": {}, + "id": "f91e51cf", + "metadata": { + "editable": true + }, "source": [ "We want to express this in terms of the equations for layer $l+1$." ] }, { "cell_type": "markdown", - "id": "e4d61a11", - "metadata": {}, + "id": "b5009124", + "metadata": { + "editable": true + }, "source": [ "## Using the chain rule and summing over all $k$ entries\n", "\n", @@ -1253,8 +1772,10 @@ }, { "cell_type": "markdown", - "id": "481e2936", - "metadata": {}, + "id": "9dfc6485", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^l =\\sum_k \\frac{\\partial {\\cal C}}{\\partial z_k^{l+1}}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}}=\\sum_k \\delta_k^{l+1}\\frac{\\partial z_k^{l+1}}{\\partial z_j^{l}},\n", @@ -1263,16 +1784,20 @@ }, { "cell_type": "markdown", - "id": "9f802864", - "metadata": {}, + "id": "17e57a98", + "metadata": { + "editable": true + }, "source": [ "and recalling that" ] }, { "cell_type": "markdown", - "id": "b9e80e32", - "metadata": {}, + "id": "c49a1372", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z_j^{l+1} = \\sum_{i=1}^{M_{l}}w_{ij}^{l+1}a_i^{l}+b_j^{l+1},\n", @@ -1281,16 +1806,20 @@ }, { "cell_type": "markdown", - "id": "35303801", - "metadata": {}, + "id": "ca4a90a3", + "metadata": { + "editable": true + }, "source": [ "with $M_l$ being the number of nodes in layer $l$, we obtain" ] }, { "cell_type": "markdown", - "id": "dcf208eb", - "metadata": {}, + "id": "426d58c2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^l =\\sum_k \\delta_k^{l+1}w_{kj}^{l+1}\\sigma'(z_j^l),\n", @@ -1299,8 +1828,10 @@ }, { "cell_type": "markdown", - "id": "194f0157", - "metadata": {}, + "id": "55bcbfd7", + "metadata": { + "editable": true + }, "source": [ "This is our final equation.\n", "\n", @@ -1309,8 +1840,10 @@ }, { "cell_type": "markdown", - "id": "e991a9eb", - "metadata": {}, + "id": "271b0412", + "metadata": { + "editable": true + }, "source": [ "## Setting up the back propagation algorithm\n", "\n", @@ -1330,8 +1863,10 @@ }, { "cell_type": "markdown", - "id": "0f74addd", - "metadata": {}, + "id": "e517e9af", + "metadata": { + "editable": true + }, "source": [ "## Setting up the back propagation algorithm, part 2\n", "\n", @@ -1340,8 +1875,10 @@ }, { "cell_type": "markdown", - "id": "4f2df2d9", - "metadata": {}, + "id": "559e7234", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^L = \\sigma'(z_j^L)\\frac{\\partial {\\cal C}}{\\partial (a_j^L)}.\n", @@ -1350,16 +1887,20 @@ }, { "cell_type": "markdown", - "id": "bb7d67d4", - "metadata": {}, + "id": "e630fa01", + "metadata": { + "editable": true + }, "source": [ "Then we compute the back propagate error for each $l=L-1,L-2,\\dots,1$ as" ] }, { "cell_type": "markdown", - "id": "2a073c51", - "metadata": {}, + "id": "4e41356a", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}\\sigma'(z_j^l).\n", @@ -1368,8 +1909,10 @@ }, { "cell_type": "markdown", - "id": "1bcf4181", - "metadata": {}, + "id": "ceb0ec78", + "metadata": { + "editable": true + }, "source": [ "## Setting up the Back propagation algorithm, part 3\n", "\n", @@ -1380,8 +1923,10 @@ }, { "cell_type": "markdown", - "id": "0159cb38", - "metadata": {}, + "id": "e5ce7168", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", @@ -1390,8 +1935,10 @@ }, { "cell_type": "markdown", - "id": "77fcc68f", - "metadata": {}, + "id": "08ff2a57", + "metadata": { + "editable": true + }, "source": [ "$$\n", "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", @@ -1400,16 +1947,20 @@ }, { "cell_type": "markdown", - "id": "df2afa77", - "metadata": {}, + "id": "cdb6a0c4", + "metadata": { + "editable": true + }, "source": [ "with $\\eta$ being the learning rate." ] }, { "cell_type": "markdown", - "id": "3f52677b", - "metadata": {}, + "id": "8498cde5", + "metadata": { + "editable": true + }, "source": [ "## Updating the gradients\n", "\n", @@ -1418,8 +1969,10 @@ }, { "cell_type": "markdown", - "id": "083f7900", - "metadata": {}, + "id": "98b6255f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\delta_j^l = \\sum_k \\delta_k^{l+1}w_{kj}^{l+1}sigma'(z_j^l),\n", @@ -1428,16 +1981,20 @@ }, { "cell_type": "markdown", - "id": "8fcaded7", - "metadata": {}, + "id": "441e1569", + "metadata": { + "editable": true + }, "source": [ "we update the weights and the biases using gradient descent for each $l=L-1,L-2,\\dots,1$ and update the weights and biases according to the rules" ] }, { "cell_type": "markdown", - "id": "cada9606", - "metadata": {}, + "id": "9a5f267f", + "metadata": { + "editable": true + }, "source": [ "$$\n", "w_{jk}^l\\leftarrow = w_{jk}^l- \\eta \\delta_j^la_k^{l-1},\n", @@ -1446,8 +2003,10 @@ }, { "cell_type": "markdown", - "id": "2d24f67a", - "metadata": {}, + "id": "2cb53cf6", + "metadata": { + "editable": true + }, "source": [ "$$\n", "b_j^l \\leftarrow b_j^l-\\eta \\frac{\\partial {\\cal C}}{\\partial b_j^l}=b_j^l-\\eta \\delta_j^l,\n", @@ -1456,8 +2015,10 @@ }, { "cell_type": "markdown", - "id": "68eaedda", - "metadata": {}, + "id": "1bcc664c", + "metadata": { + "editable": true + }, "source": [ "## Fine-tuning neural network hyperparameters\n", "\n", @@ -1483,8 +2044,10 @@ }, { "cell_type": "markdown", - "id": "f8bd9fde", - "metadata": {}, + "id": "2015500e", + "metadata": { + "editable": true + }, "source": [ "## Hidden layers\n", "\n", @@ -1507,8 +2070,10 @@ }, { "cell_type": "markdown", - "id": "bf1dd910", - "metadata": {}, + "id": "56293dd9", + "metadata": { + "editable": true + }, "source": [ "## Which activation function should I use?\n", "\n", @@ -1535,8 +2100,10 @@ }, { "cell_type": "markdown", - "id": "dd5da40b", - "metadata": {}, + "id": "6e83dbe7", + "metadata": { + "editable": true + }, "source": [ "## Is the Logistic activation function (Sigmoid) our choice?\n", "\n", @@ -1565,8 +2132,10 @@ }, { "cell_type": "markdown", - "id": "bfbe8008", - "metadata": {}, + "id": "5110093d", + "metadata": { + "editable": true + }, "source": [ "## The derivative of the Logistic funtion\n", "\n", @@ -1591,8 +2160,10 @@ }, { "cell_type": "markdown", - "id": "179c592c", - "metadata": {}, + "id": "784d1093", + "metadata": { + "editable": true + }, "source": [ "## Insights from the paper by Glorot and Bengio\n", "\n", @@ -1609,8 +2180,10 @@ }, { "cell_type": "markdown", - "id": "8a546ef2", - "metadata": {}, + "id": "61ce8415", + "metadata": { + "editable": true + }, "source": [ "## The RELU function family\n", "\n", @@ -1632,8 +2205,10 @@ }, { "cell_type": "markdown", - "id": "73eb718c", - "metadata": {}, + "id": "b40a6a04", + "metadata": { + "editable": true + }, "source": [ "$$\n", "ELU(z) = \\left\\{\\begin{array}{cc} \\alpha\\left( \\exp{(z)}-1\\right) & z < 0,\\\\ z & z \\ge 0.\\end{array}\\right.\n", @@ -1642,8 +2217,10 @@ }, { "cell_type": "markdown", - "id": "11bf93b4", - "metadata": {}, + "id": "52fc7adb", + "metadata": { + "editable": true + }, "source": [ "## Which activation function should we use?\n", "\n", @@ -1662,8 +2239,10 @@ }, { "cell_type": "markdown", - "id": "0ad1c73a", - "metadata": {}, + "id": "1e321810", + "metadata": { + "editable": true + }, "source": [ "## More on activation functions, output layers\n", "\n", @@ -1682,8 +2261,10 @@ }, { "cell_type": "markdown", - "id": "d8079475", - "metadata": {}, + "id": "835eb96f", + "metadata": { + "editable": true + }, "source": [ "## Batch Normalization\n", "\n", @@ -1706,8 +2287,10 @@ }, { "cell_type": "markdown", - "id": "42c1aad4", - "metadata": {}, + "id": "b3ae6a56", + "metadata": { + "editable": true + }, "source": [ "## Dropout\n", "\n", @@ -1724,8 +2307,10 @@ }, { "cell_type": "markdown", - "id": "604a7d24", - "metadata": {}, + "id": "2c1d78bc", + "metadata": { + "editable": true + }, "source": [ "## Gradient Clipping\n", "\n", @@ -1742,8 +2327,10 @@ }, { "cell_type": "markdown", - "id": "c4faef07", - "metadata": {}, + "id": "d7cb1360", + "metadata": { + "editable": true + }, "source": [ "## A top-down perspective on Neural networks\n", "\n", @@ -1766,8 +2353,10 @@ }, { "cell_type": "markdown", - "id": "b1a0db4b", - "metadata": {}, + "id": "8b520d2c", + "metadata": { + "editable": true + }, "source": [ "## More top-down perspectives\n", "\n", @@ -1793,8 +2382,10 @@ }, { "cell_type": "markdown", - "id": "1ebd27f8", - "metadata": {}, + "id": "1c337b64", + "metadata": { + "editable": true + }, "source": [ "## Limitations of supervised learning with deep networks\n", "\n", @@ -1809,8 +2400,10 @@ }, { "cell_type": "markdown", - "id": "b6057083", - "metadata": {}, + "id": "83f52f29", + "metadata": { + "editable": true + }, "source": [ "## Limitations of NNs\n", "\n", @@ -1823,8 +2416,10 @@ }, { "cell_type": "markdown", - "id": "e928856e", - "metadata": {}, + "id": "4c8b5b07", + "metadata": { + "editable": true + }, "source": [ "## Homogeneous data\n", "\n", @@ -1833,8 +2428,10 @@ }, { "cell_type": "markdown", - "id": "0e33d3f8", - "metadata": {}, + "id": "32b86419", + "metadata": { + "editable": true + }, "source": [ "## More limitations\n", "\n", @@ -1845,8 +2442,10 @@ }, { "cell_type": "markdown", - "id": "7a78a423", - "metadata": {}, + "id": "5650569f", + "metadata": { + "editable": true + }, "source": [ "## Building a neural network code\n", "\n", @@ -1863,8 +2462,10 @@ }, { "cell_type": "markdown", - "id": "55f9130f", - "metadata": {}, + "id": "527bda49", + "metadata": { + "editable": true + }, "source": [ "## Learning rate methods\n", "\n", @@ -1881,9 +2482,12 @@ }, { "cell_type": "code", - "execution_count": 1, - "id": "344bffe7", - "metadata": {}, + "execution_count": 3, + "id": "1d822247", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "import autograd.numpy as np\n", @@ -2020,8 +2624,10 @@ }, { "cell_type": "markdown", - "id": "a98a137d", - "metadata": {}, + "id": "15941f5a", + "metadata": { + "editable": true + }, "source": [ "## Usage of the above learning rate schedulers\n", "\n", @@ -2033,9 +2639,12 @@ }, { "cell_type": "code", - "execution_count": 2, - "id": "b27f99be", - "metadata": {}, + "execution_count": 4, + "id": "981c4c76", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "momentum_scheduler = Momentum(eta=1e-3, momentum=0.9)\n", @@ -2044,8 +2653,10 @@ }, { "cell_type": "markdown", - "id": "9fb52260", - "metadata": {}, + "id": "f0d0a9d7", + "metadata": { + "editable": true + }, "source": [ "Here is a small example for how a segment of code using schedulers\n", "could look. Switching out the schedulers is simple." @@ -2053,9 +2664,12 @@ }, { "cell_type": "code", - "execution_count": 3, - "id": "421e2f19", - "metadata": {}, + "execution_count": 5, + "id": "6e2b4226", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "weights = np.ones((3,3))\n", @@ -2073,8 +2687,10 @@ }, { "cell_type": "markdown", - "id": "bc58c677", - "metadata": {}, + "id": "cb1b8fdf", + "metadata": { + "editable": true + }, "source": [ "## Cost functions\n", "\n", @@ -2086,9 +2702,12 @@ }, { "cell_type": "code", - "execution_count": 4, - "id": "c4d0edd0", - "metadata": {}, + "execution_count": 6, + "id": "07bf8776", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "import autograd.numpy as np\n", @@ -2122,8 +2741,10 @@ }, { "cell_type": "markdown", - "id": "a72c5bb9", - "metadata": {}, + "id": "c8354e37", + "metadata": { + "editable": true + }, "source": [ "Below we give a short example of how these cost function may be used\n", "to obtain results if you wish to test them out on your own using\n", @@ -2132,9 +2753,12 @@ }, { "cell_type": "code", - "execution_count": 5, - "id": "3f8dac2d", - "metadata": {}, + "execution_count": 7, + "id": "7fb2cdab", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "from autograd import grad\n", @@ -2151,8 +2775,10 @@ }, { "cell_type": "markdown", - "id": "85fee0ff", - "metadata": {}, + "id": "8f5e76a9", + "metadata": { + "editable": true + }, "source": [ "## Activation functions\n", "\n", @@ -2164,9 +2790,12 @@ }, { "cell_type": "code", - "execution_count": 6, - "id": "70cdd1cc", - "metadata": {}, + "execution_count": 8, + "id": "9ff25295", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "import autograd.numpy as np\n", @@ -2220,8 +2849,10 @@ }, { "cell_type": "markdown", - "id": "0284d572", - "metadata": {}, + "id": "015ea5b6", + "metadata": { + "editable": true + }, "source": [ "Below follows a short demonstration of how to use an activation\n", "function. The derivative of the activation function will be important\n", @@ -2232,9 +2863,12 @@ }, { "cell_type": "code", - "execution_count": 7, - "id": "bad46ab5", - "metadata": {}, + "execution_count": 9, + "id": "563ea0ed", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "z = np.array([[4, 5, 6]]).T\n", @@ -2251,8 +2885,10 @@ }, { "cell_type": "markdown", - "id": "6dc4dce5", - "metadata": {}, + "id": "8b5705ae", + "metadata": { + "editable": true + }, "source": [ "## The Neural Network\n", "\n", @@ -2272,9 +2908,12 @@ }, { "cell_type": "code", - "execution_count": 8, - "id": "e940dc92", - "metadata": {}, + "execution_count": 10, + "id": "065ddb72", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "import math\n", @@ -2742,8 +3381,10 @@ }, { "cell_type": "markdown", - "id": "d40a92a4", - "metadata": {}, + "id": "f1914481", + "metadata": { + "editable": true + }, "source": [ "Before we make a model, we will quickly generate a dataset we can use\n", "for our linear regression problem as shown below" @@ -2751,9 +3392,12 @@ }, { "cell_type": "code", - "execution_count": 9, - "id": "0a3b0d90", - "metadata": {}, + "execution_count": 11, + "id": "227bc44c", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "import autograd.numpy as np\n", @@ -2793,8 +3437,10 @@ }, { "cell_type": "markdown", - "id": "6d2a942d", - "metadata": {}, + "id": "91cd4c2d", + "metadata": { + "editable": true + }, "source": [ "Now that we have our dataset ready for the regression, we can create\n", "our regressor. Note that with the seed parameter, we can make sure our\n", @@ -2806,9 +3452,12 @@ }, { "cell_type": "code", - "execution_count": 10, - "id": "989f33d1", - "metadata": {}, + "execution_count": 12, + "id": "ff2faaa1", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "input_nodes = X_train.shape[1]\n", @@ -2819,17 +3468,22 @@ }, { "cell_type": "markdown", - "id": "0e7b10af", - "metadata": {}, + "id": "f9ad7a83", + "metadata": { + "editable": true + }, "source": [ "We then fit our model with our training data using the scheduler of our choice." ] }, { "cell_type": "code", - "execution_count": 11, - "id": "f1b71ba7", - "metadata": {}, + "execution_count": 13, + "id": "4d4fee47", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", @@ -2840,8 +3494,10 @@ }, { "cell_type": "markdown", - "id": "17faa298", - "metadata": {}, + "id": "b011d02a", + "metadata": { + "editable": true + }, "source": [ "Due to the progress bar we can see the MSE (train$\\_$error) throughout\n", "the FFNN's training. Note that the fit() function has some optional\n", @@ -2853,9 +3509,12 @@ }, { "cell_type": "code", - "execution_count": 12, - "id": "0c6d600d", - "metadata": {}, + "execution_count": 14, + "id": "9bef44ae", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "linear_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", @@ -2865,8 +3524,10 @@ }, { "cell_type": "markdown", - "id": "b0b68b5d", - "metadata": {}, + "id": "5f4ef1fe", + "metadata": { + "editable": true + }, "source": [ "We see that given more epochs to train on, the regressor reaches a lower MSE.\n", "\n", @@ -2877,9 +3538,12 @@ }, { "cell_type": "code", - "execution_count": 13, - "id": "5394a6db", - "metadata": {}, + "execution_count": 15, + "id": "5b26c63c", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "from sklearn.datasets import load_breast_cancer\n", @@ -2900,9 +3564,12 @@ }, { "cell_type": "code", - "execution_count": 14, - "id": "dc7df60b", - "metadata": {}, + "execution_count": 16, + "id": "f6704bdf", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "input_nodes = X_train.shape[1]\n", @@ -2913,17 +3580,22 @@ }, { "cell_type": "markdown", - "id": "89a578f2", - "metadata": {}, + "id": "aab5b5e6", + "metadata": { + "editable": true + }, "source": [ "We will now make use of our validation data by passing it into our fit function as a keyword argument" ] }, { "cell_type": "code", - "execution_count": 15, - "id": "2bd2d521", - "metadata": {}, + "execution_count": 17, + "id": "cb8f133a", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "logistic_regression.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", @@ -2934,17 +3606,22 @@ }, { "cell_type": "markdown", - "id": "6e97fc8e", - "metadata": {}, + "id": "d48cbb60", + "metadata": { + "editable": true + }, "source": [ "Finally, we will create a neural network with 2 hidden layers with activation functions." ] }, { "cell_type": "code", - "execution_count": 16, - "id": "349a39dc", - "metadata": {}, + "execution_count": 18, + "id": "c24f5054", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "input_nodes = X_train.shape[1]\n", @@ -2959,9 +3636,12 @@ }, { "cell_type": "code", - "execution_count": 17, - "id": "74a388a0", - "metadata": {}, + "execution_count": 19, + "id": "4754b999", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "neural_network.reset_weights() # reset weights such that previous runs or reruns don't affect the weights\n", @@ -2972,8 +3652,10 @@ }, { "cell_type": "markdown", - "id": "df19c400", - "metadata": {}, + "id": "69c218e2", + "metadata": { + "editable": true + }, "source": [ "## Multiclass classification\n", "\n", @@ -2984,9 +3666,12 @@ }, { "cell_type": "code", - "execution_count": 18, - "id": "3a633f3b", - "metadata": {}, + "execution_count": 20, + "id": "d7240200", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "from sklearn.datasets import load_digits\n", @@ -3019,8 +3704,10 @@ }, { "cell_type": "markdown", - "id": "bb31ccbd", - "metadata": {}, + "id": "36c7cb2f", + "metadata": { + "editable": true + }, "source": [ "## Testing the XOR gate and other gates\n", "\n", @@ -3029,9 +3716,12 @@ }, { "cell_type": "code", - "execution_count": 19, - "id": "2dfeafac", - "metadata": {}, + "execution_count": 21, + "id": "38e8056e", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "X = np.array([ [0, 0], [0, 1], [1, 0],[1, 1]],dtype=np.float64)\n", @@ -3050,16 +3740,20 @@ }, { "cell_type": "markdown", - "id": "208f9f0d", - "metadata": {}, + "id": "d4f915f2", + "metadata": { + "editable": true + }, "source": [ "Not bad, but the results depend strongly on the learning reate. Try different learning rates." ] }, { "cell_type": "markdown", - "id": "7897099c", - "metadata": {}, + "id": "054b1fe6", + "metadata": { + "editable": true + }, "source": [ "## Building neural networks in Tensorflow and Keras\n", "\n", @@ -3074,8 +3768,10 @@ }, { "cell_type": "markdown", - "id": "cc3a3331", - "metadata": {}, + "id": "4dde2edc", + "metadata": { + "editable": true + }, "source": [ "## Tensorflow\n", "\n", @@ -3106,9 +3802,12 @@ }, { "cell_type": "code", - "execution_count": 20, - "id": "29aa9481", - "metadata": {}, + "execution_count": 22, + "id": "b9e26ea8", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "pip3 install tensorflow" @@ -3116,8 +3815,10 @@ }, { "cell_type": "markdown", - "id": "e01cd26f", - "metadata": {}, + "id": "8c0efa60", + "metadata": { + "editable": true + }, "source": [ "and/or if you use **anaconda**, just write (or install from the graphical user interface)\n", "(current release of CPU-only TensorFlow)" @@ -3125,9 +3826,12 @@ }, { "cell_type": "code", - "execution_count": 21, - "id": "1e404598", - "metadata": {}, + "execution_count": 23, + "id": "9608ce22", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "conda create -n tf tensorflow\n", @@ -3136,17 +3840,22 @@ }, { "cell_type": "markdown", - "id": "c3a2a398", - "metadata": {}, + "id": "8bf24057", + "metadata": { + "editable": true + }, "source": [ "To install the current release of GPU TensorFlow" ] }, { "cell_type": "code", - "execution_count": 22, - "id": "e26fa0f8", - "metadata": {}, + "execution_count": 24, + "id": "e7ccc57b", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "conda create -n tf-gpu tensorflow-gpu\n", @@ -3155,8 +3864,10 @@ }, { "cell_type": "markdown", - "id": "511fca8e", - "metadata": {}, + "id": "a795908c", + "metadata": { + "editable": true + }, "source": [ "## Using Keras\n", "\n", @@ -3167,9 +3878,12 @@ }, { "cell_type": "code", - "execution_count": 23, - "id": "49d62f89", - "metadata": {}, + "execution_count": 25, + "id": "345ed534", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "conda install keras" @@ -3177,8 +3891,10 @@ }, { "cell_type": "markdown", - "id": "34cd8c66", - "metadata": {}, + "id": "d1595854", + "metadata": { + "editable": true + }, "source": [ "You can look up the [instructions here](https://keras.io/) for more information.\n", "\n", @@ -3187,8 +3903,10 @@ }, { "cell_type": "markdown", - "id": "d3ea346a", - "metadata": {}, + "id": "947e5ac6", + "metadata": { + "editable": true + }, "source": [ "## Collect and pre-process data\n", "\n", @@ -3197,13 +3915,14 @@ }, { "cell_type": "code", - "execution_count": 24, - "id": "7c500038", - "metadata": {}, + "execution_count": 26, + "id": "f37d4651", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ - "%matplotlib inline\n", - "\n", "# import necessary packages\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", @@ -3251,9 +3970,12 @@ }, { "cell_type": "code", - "execution_count": 25, - "id": "d6b15ed5", - "metadata": {}, + "execution_count": 27, + "id": "4053a683", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "from tensorflow.keras.layers import Input\n", @@ -3277,9 +3999,12 @@ }, { "cell_type": "code", - "execution_count": 26, - "id": "b9feced1", - "metadata": {}, + "execution_count": 28, + "id": "a0dc2eca", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "\n", @@ -3304,9 +4029,12 @@ }, { "cell_type": "code", - "execution_count": 27, - "id": "76eab831", - "metadata": {}, + "execution_count": 29, + "id": "39dca61d", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "DNN_keras = np.zeros((len(eta_vals), len(lmbd_vals)), dtype=object)\n", @@ -3328,9 +4056,12 @@ }, { "cell_type": "code", - "execution_count": 28, - "id": "82d63436", - "metadata": {}, + "execution_count": 30, + "id": "77d454a5", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "# optional\n", @@ -3368,17 +4099,22 @@ }, { "cell_type": "markdown", - "id": "21d299c0", - "metadata": {}, + "id": "0ab65251", + "metadata": { + "editable": true + }, "source": [ "## And using PyTorch (more discussions to follow)" ] }, { "cell_type": "code", - "execution_count": 29, - "id": "b9b3c94f", - "metadata": {}, + "execution_count": 31, + "id": "982a5609", + "metadata": { + "collapsed": false, + "editable": true + }, "outputs": [], "source": [ "# Simple neural-network (NN) code using pytorch\n", @@ -3444,8 +4180,10 @@ }, { "cell_type": "markdown", - "id": "dbe12a08", - "metadata": {}, + "id": "35f5793b", + "metadata": { + "editable": true + }, "source": [ "## Convolutional Neural Networks (recognizing images)\n", "\n", @@ -3469,8 +4207,10 @@ }, { "cell_type": "markdown", - "id": "dc550024", - "metadata": {}, + "id": "844d0a67", + "metadata": { + "editable": true + }, "source": [ "## What is the Difference\n", "\n", @@ -3489,8 +4229,10 @@ }, { "cell_type": "markdown", - "id": "f67bb593", - "metadata": {}, + "id": "9b42ad1c", + "metadata": { + "editable": true + }, "source": [ "## Neural Networks vs CNNs\n", "\n", @@ -3505,8 +4247,10 @@ }, { "cell_type": "markdown", - "id": "6b0f70b7", - "metadata": {}, + "id": "8cf42912", + "metadata": { + "editable": true + }, "source": [ "## Why CNNS for images, sound files, medical images from CT scans etc?\n", "\n", @@ -3533,8 +4277,10 @@ }, { "cell_type": "markdown", - "id": "f4aa9714", - "metadata": {}, + "id": "ce4ed3b6", + "metadata": { + "editable": true + }, "source": [ "## Regular NNs don’t scale well to full images\n", "\n", @@ -3561,8 +4307,10 @@ }, { "cell_type": "markdown", - "id": "418e433b", - "metadata": {}, + "id": "492861c2", + "metadata": { + "editable": true + }, "source": [ "## 3D volumes of neurons\n", "\n", @@ -3599,8 +4347,10 @@ }, { "cell_type": "markdown", - "id": "87f09d30", - "metadata": {}, + "id": "9c1aa339", + "metadata": { + "editable": true + }, "source": [ "## Layers used to build CNNs\n", "\n", @@ -3626,8 +4376,10 @@ }, { "cell_type": "markdown", - "id": "adfb2e74", - "metadata": {}, + "id": "20ca361b", + "metadata": { + "editable": true + }, "source": [ "## Transforming images\n", "\n", @@ -3646,8 +4398,10 @@ }, { "cell_type": "markdown", - "id": "30a38c2c", - "metadata": {}, + "id": "b6c90c10", + "metadata": { + "editable": true + }, "source": [ "## CNNs in brief\n", "\n", @@ -3673,8 +4427,10 @@ }, { "cell_type": "markdown", - "id": "a3e0cc0a", - "metadata": {}, + "id": "638ee230", + "metadata": { + "editable": true + }, "source": [ "## Key Idea\n", "\n", @@ -3688,8 +4444,10 @@ }, { "cell_type": "markdown", - "id": "ed9660a2", - "metadata": {}, + "id": "01df261a", + "metadata": { + "editable": true + }, "source": [ "## Mathematics of CNNs\n", "\n", @@ -3706,8 +4464,10 @@ }, { "cell_type": "markdown", - "id": "677ffdf8", - "metadata": {}, + "id": "1c4dcc67", + "metadata": { + "editable": true + }, "source": [ "$$\n", "y(t) = \\int x(a) w(t-a) da,\n", @@ -3716,8 +4476,10 @@ }, { "cell_type": "markdown", - "id": "b79f846e", - "metadata": {}, + "id": "8d1d3cc7", + "metadata": { + "editable": true + }, "source": [ "where $x(a)$ represents a so-called input and $w(t-a)$ is normally called the weight function or kernel.\n", "\n", @@ -3726,8 +4488,10 @@ }, { "cell_type": "markdown", - "id": "0b63b45b", - "metadata": {}, + "id": "ad261c08", + "metadata": { + "editable": true + }, "source": [ "$$\n", "y(t) = \\left(x * w\\right)(t).\n", @@ -3736,16 +4500,20 @@ }, { "cell_type": "markdown", - "id": "980f6dcf", - "metadata": {}, + "id": "32156ca3", + "metadata": { + "editable": true + }, "source": [ "The discretized version reads" ] }, { "cell_type": "markdown", - "id": "490010c9", - "metadata": {}, + "id": "3b74661e", + "metadata": { + "editable": true + }, "source": [ "$$\n", "y(t) = \\sum_{a=-\\infty}^{a=\\infty}x(a)w(t-a).\n", @@ -3754,8 +4522,10 @@ }, { "cell_type": "markdown", - "id": "14d5aebb", - "metadata": {}, + "id": "29e0f225", + "metadata": { + "editable": true + }, "source": [ "Computing the inverse of the above convolution operations is known as deconvolution.\n", "\n", @@ -3764,8 +4534,10 @@ }, { "cell_type": "markdown", - "id": "4aaa7918", - "metadata": {}, + "id": "b201b426", + "metadata": { + "editable": true + }, "source": [ "## Convolution Examples: Polynomial multiplication\n", "\n", @@ -3777,8 +4549,10 @@ }, { "cell_type": "markdown", - "id": "64dd0e7a", - "metadata": {}, + "id": "8a34abf2", + "metadata": { + "editable": true + }, "source": [ "$$\n", "p(t) = \\alpha_0+\\alpha_1 t+\\alpha_2 t^2,\n", @@ -3787,16 +4561,20 @@ }, { "cell_type": "markdown", - "id": "a1b27c4f", - "metadata": {}, + "id": "e1b2c1dc", + "metadata": { + "editable": true + }, "source": [ "and" ] }, { "cell_type": "markdown", - "id": "3cff8ff8", - "metadata": {}, + "id": "e29cff91", + "metadata": { + "editable": true + }, "source": [ "$$\n", "s(t) = \\beta_0+\\beta_1 t+\\beta_2 t^2+\\beta_3 t^3.\n", @@ -3805,16 +4583,20 @@ }, { "cell_type": "markdown", - "id": "dfa19b7e", - "metadata": {}, + "id": "d712a03f", + "metadata": { + "editable": true + }, "source": [ "The polynomial multiplication gives us a new polynomial of degree $5$" ] }, { "cell_type": "markdown", - "id": "77633eee", - "metadata": {}, + "id": "3ab32e1c", + "metadata": { + "editable": true + }, "source": [ "$$\n", "z(t) = \\delta_0+\\delta_1 t+\\delta_2 t^2+\\delta_3 t^3+\\delta_4 t^4+\\delta_5 t^5.\n", @@ -3823,8 +4605,10 @@ }, { "cell_type": "markdown", - "id": "c6ecec98", - "metadata": {}, + "id": "72b6c7b7", + "metadata": { + "editable": true + }, "source": [ "## Efficient Polynomial Multiplication\n", "\n", @@ -3834,521 +4618,266 @@ }, { "cell_type": "markdown", - "id": "d44c1c9c", - "metadata": {}, - "source": [ - "$$\n", - "\\begin{split}\n", - "\\delta_0=&\\alpha_0\\beta_0\\\\\n", - "\\delta_1=&\\alpha_1\\beta_0+\\alpha_1\\beta_0\\\\\n", - "\\delta_2=&\\alpha_0\\beta_2+\\alpha_1\\beta_1+\\alpha_2\\beta_0\\\\\n", - "\\delta_3=&\\alpha_1\\beta_2+\\alpha_2\\beta_1+\\alpha_0\\beta_3\\\\\n", - "\\delta_4=&\\alpha_2\\beta_2+\\alpha_1\\beta_3\\\\\n", - "\\delta_5=&\\alpha_2\\beta_3.\\\\\n", - "\\end{split}\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "4f18e71f", - "metadata": {}, + "id": "7ea239c4", + "metadata": { + "editable": true + }, "source": [ - "We note that $\\alpha_i=0$ except for $i\\in \\left\\{0,1,2\\right\\}$ and $\\beta_i=0$ except for $i\\in\\left\\{0,1,2,3\\right\\}$.\n", + "\n", + "
    \n", "\n", - "We can then rewrite the coefficients $\\delta_j$ using a discrete convolution as" - ] - }, - { - "cell_type": "markdown", - "id": "5bb487d5", - "metadata": {}, - "source": [ "$$\n", - "\\delta_j = \\sum_{i=-\\infty}^{i=\\infty}\\alpha_i\\beta_{j-i}=(\\alpha * \\beta)_j,\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "9084870f", - "metadata": {}, - "source": [ - "or as a double sum with restriction $l=i+j$" - ] - }, - { - "cell_type": "markdown", - "id": "4dffe3d4", - "metadata": {}, - "source": [ - "$$\n", - "\\delta_l = \\sum_{ij}\\alpha_i\\beta_{j}.\n", + "\\begin{equation}\n", + "\\delta_0=\\alpha_0\\beta_0\n", + "\\label{_auto4} \\tag{4}\n", + "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "b63e74e3", - "metadata": {}, - "source": [ - "Do you see a potential drawback with these equations?" - ] - }, - { - "cell_type": "markdown", - "id": "5776bddd", - "metadata": {}, + "id": "510ca485", + "metadata": { + "editable": true + }, "source": [ - "## A more efficient way of coding the above Convolution\n", + "\n", + "
    \n", "\n", - "Since we only have a finite number of $\\alpha$ and $\\beta$ values\n", - "which are non-zero, we can rewrite the above convolution expressions\n", - "as a matrix-vector multiplication" - ] - }, - { - "cell_type": "markdown", - "id": "3347c411", - "metadata": {}, - "source": [ "$$\n", - "\\boldsymbol{\\delta}=\\begin{bmatrix}\\alpha_0 & 0 & 0 & 0 \\\\\n", - " \\alpha_1 & \\alpha_0 & 0 & 0 \\\\\n", - "\t\t\t \\alpha_2 & \\alpha_1 & \\alpha_0 & 0 \\\\\n", - "\t\t\t 0 & \\alpha_2 & \\alpha_1 & \\alpha_0 \\\\\n", - "\t\t\t 0 & 0 & \\alpha_2 & \\alpha_1 \\\\\n", - "\t\t\t 0 & 0 & 0 & \\alpha_2\n", - "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\\\ \\beta_2 \\\\ \\beta_3\\end{bmatrix}.\n", - "$$" - ] - }, - { - "cell_type": "markdown", - "id": "d6a74f32", - "metadata": {}, - "source": [ - "The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding $\\beta$ and a vector holding $\\alpha$.\n", - "In this case we have" - ] - }, - { - "cell_type": "markdown", - "id": "b51509fe", - "metadata": {}, - "source": [ - "$$\n", - "\\boldsymbol{\\delta}=\\begin{bmatrix}\\beta_0 & 0 & 0 \\\\\n", - " \\beta_1 & \\beta_0 & 0 \\\\\n", - "\t\t\t \\beta_2 & \\beta_1 & \\beta_0 \\\\\n", - "\t\t\t \\beta_3 & \\beta_2 & \\beta_1 \\\\\n", - "\t\t\t 0 & \\beta_3 & \\beta_2 \\\\\n", - "\t\t\t 0 & 0 & \\beta_3\n", - "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\alpha_0 \\\\ \\alpha_1 \\\\ \\alpha_2\\end{bmatrix}.\n", + "\\begin{equation} \n", + "\\delta_1=\\alpha_1\\beta_0+\\alpha_1\\beta_0\n", + "\\label{_auto5} \\tag{5}\n", + "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "dd13bf97", - "metadata": {}, - "source": [ - "Note that the use of these matrices is for mathematical purposes only and not implementation purposes.\n", - "When implementing the above equation we do not encode (and allocate memory) the matrices explicitely.\n", - "We rather code the convolutions in the minimal memory footprint that they require.\n", - "\n", - "Does the number of floating point operations change here when we use the commutative property?" - ] - }, - { - "cell_type": "markdown", - "id": "c4975e6e", - "metadata": {}, + "id": "9ad65976", + "metadata": { + "editable": true + }, "source": [ - "## Convolution Examples: Principle of Superposition and Periodic Forces (Fourier Transforms)\n", + "\n", + "
    \n", "\n", - "For problems with so-called harmonic oscillations, given by for example the following differential equation" - ] - }, - { - "cell_type": "markdown", - "id": "9b2be9a8", - "metadata": {}, - "source": [ "$$\n", - "m\\frac{d^2x}{dt^2}+\\eta\\frac{dx}{dt}+x(t)=F(t),\n", + "\\begin{equation} \n", + "\\delta_2=\\alpha_0\\beta_2+\\alpha_1\\beta_1+\\alpha_2\\beta_0\n", + "\\label{_auto6} \\tag{6}\n", + "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "63120ef0", - "metadata": {}, - "source": [ - "where $F(t)$ is an applied external force acting on the system (often called a driving force), one can use the theory of Fourier transformations to find the solutions of this type of equations.\n", - "\n", - "If one has several driving forces, $F(t)=\\sum_n F_n(t)$, one can find\n", - "the particular solution to each $F_n$, $x_{pn}(t)$, and the particular\n", - "solution for the entire driving force is then given by a series like" - ] - }, - { - "cell_type": "markdown", - "id": "4879e2f3", - "metadata": {}, + "id": "eb7e7521", + "metadata": { + "editable": true + }, "source": [ "\n", - "
    \n", + "
    \n", "\n", "$$\n", - "\\begin{equation}\n", - "x_p(t)=\\sum_nx_{pn}(t).\n", - "\\label{_auto4} \\tag{4}\n", + "\\begin{equation} \n", + "\\delta_3=\\alpha_1\\beta_2+\\alpha_2\\beta_1+\\alpha_0\\beta_3\n", + "\\label{_auto7} \\tag{7}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "22f818a5", - "metadata": {}, + "id": "a5a48723", + "metadata": { + "editable": true + }, "source": [ - "## Principle of Superposition\n", - "\n", - "This is known as the principle of superposition. It only applies when\n", - "the homogenous equation is linear. If there were an anharmonic term\n", - "such as $x^3$ in the homogenous equation, then when one summed various\n", - "solutions, $x=(\\sum_n x_n)^2$, one would get cross\n", - "terms. Superposition is especially useful when $F(t)$ can be written\n", - "as a sum of sinusoidal terms, because the solutions for each\n", - "sinusoidal (sine or cosine) term is analytic. \n", + "\n", + "
    \n", "\n", - "Driving forces are often periodic, even when they are not\n", - "sinusoidal. Periodicity implies that for some time $\\tau$" - ] - }, - { - "cell_type": "markdown", - "id": "fa845422", - "metadata": {}, - "source": [ "$$\n", - "\\begin{eqnarray}\n", - "F(t+\\tau)=F(t). \n", - "\\end{eqnarray}\n", + "\\begin{equation} \n", + "\\delta_4=\\alpha_2\\beta_2+\\alpha_1\\beta_3\n", + "\\label{_auto8} \\tag{8}\n", + "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "97ec2c12", - "metadata": {}, - "source": [ - "One example of a non-sinusoidal periodic force is a square wave. Many\n", - "components in electric circuits are non-linear, e.g. diodes, which\n", - "makes many wave forms non-sinusoidal even when the circuits are being\n", - "driven by purely sinusoidal sources." - ] - }, - { - "cell_type": "markdown", - "id": "23e2f51a", - "metadata": {}, - "source": [ - "## Simple Code Example\n", - "\n", - "The code here shows a typical example of such a square wave generated using the functionality included in the **scipy** Python package. We have used a period of $\\tau=0.2$." - ] - }, - { - "cell_type": "code", - "execution_count": 30, - "id": "1dfe357f", - "metadata": {}, - "outputs": [], - "source": [ - "import numpy as np\n", - "import math\n", - "from scipy import signal\n", - "import matplotlib.pyplot as plt\n", - "\n", - "# number of points \n", - "n = 500\n", - "# start and final times \n", - "t0 = 0.0\n", - "tn = 1.0\n", - "# Period \n", - "t = np.linspace(t0, tn, n, endpoint=False)\n", - "SqrSignal = np.zeros(n)\n", - "SqrSignal = 1.0+signal.square(2*np.pi*5*t)\n", - "plt.plot(t, SqrSignal)\n", - "plt.ylim(-0.5, 2.5)\n", - "plt.show()" - ] - }, - { - "cell_type": "markdown", - "id": "d5c69ef7", - "metadata": {}, - "source": [ - "For the sinusoidal example the\n", - "period is $\\tau=2\\pi/\\omega$. However, higher harmonics can also\n", - "satisfy the periodicity requirement. In general, any force that\n", - "satisfies the periodicity requirement can be expressed as a sum over\n", - "harmonics," - ] - }, - { - "cell_type": "markdown", - "id": "5e90f9bd", - "metadata": {}, + "id": "11df6292", + "metadata": { + "editable": true + }, "source": [ "\n", - "
    \n", + "
    \n", "\n", "$$\n", - "\\begin{equation}\n", - "F(t)=\\frac{f_0}{2}+\\sum_{n>0} f_n\\cos(2n\\pi t/\\tau)+g_n\\sin(2n\\pi t/\\tau).\n", - "\\label{_auto5} \\tag{5}\n", + "\\begin{equation} \n", + "\\delta_5=\\alpha_2\\beta_3.\n", + "\\label{_auto9} \\tag{9}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "6f80b787", - "metadata": {}, - "source": [ - "## Wrapping up Fourier transforms\n", - "\n", - "We can write down the answer for\n", - "$x_{pn}(t)$, by substituting $f_n/m$ or $g_n/m$ for $F_0/m$. By\n", - "writing each factor $2n\\pi t/\\tau$ as $n\\omega t$, with $\\omega\\equiv\n", - "2\\pi/\\tau$," - ] - }, - { - "cell_type": "markdown", - "id": "93a1bdd1", - "metadata": {}, + "id": "ffb31cb8", + "metadata": { + "editable": true + }, "source": [ "\n", - "
    \n", + "
    \n", "\n", "$$\n", - "\\begin{equation}\n", - "\\label{eq:fourierdef1} \\tag{6}\n", - "F(t)=\\frac{f_0}{2}+\\sum_{n>0}f_n\\cos(n\\omega t)+g_n\\sin(n\\omega t).\n", + "\\begin{equation} \n", + "\\label{_auto10} \\tag{10}\n", "\\end{equation}\n", "$$" ] }, { "cell_type": "markdown", - "id": "ec2e37bb", - "metadata": {}, + "id": "ecd7d1fc", + "metadata": { + "editable": true + }, "source": [ - "The solutions for $x(t)$ then come from replacing $\\omega$ with\n", - "$n\\omega$ for each term in the particular solution," + "We note that $\\alpha_i=0$ except for $i\\in \\left\\{0,1,2\\right\\}$ and $\\beta_i=0$ except for $i\\in\\left\\{0,1,2,3\\right\\}$.\n", + "\n", + "We can then rewrite the coefficients $\\delta_j$ using a discrete convolution as" ] }, { "cell_type": "markdown", - "id": "09716945", - "metadata": {}, + "id": "09a70a12", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\begin{eqnarray}\n", - "x_p(t)&=&\\frac{f_0}{2k}+\\sum_{n>0} \\alpha_n\\cos(n\\omega t-\\delta_n)+\\beta_n\\sin(n\\omega t-\\delta_n),\\\\\n", - "\\nonumber\n", - "\\alpha_n&=&\\frac{f_n/m}{\\sqrt{((n\\omega)^2-\\omega_0^2)+4\\beta^2n^2\\omega^2}},\\\\\n", - "\\nonumber\n", - "\\beta_n&=&\\frac{g_n/m}{\\sqrt{((n\\omega)^2-\\omega_0^2)+4\\beta^2n^2\\omega^2}},\\\\\n", - "\\nonumber\n", - "\\delta_n&=&\\tan^{-1}\\left(\\frac{2\\beta n\\omega}{\\omega_0^2-n^2\\omega^2}\\right).\n", - "\\end{eqnarray}\n", + "\\delta_j = \\sum_{i=-\\infty}^{i=\\infty}\\alpha_i\\beta_{j-i}=(\\alpha * \\beta)_j,\n", "$$" ] }, { "cell_type": "markdown", - "id": "66d30b11", - "metadata": {}, + "id": "b1b8db79", + "metadata": { + "editable": true + }, "source": [ - "## Finding the Coefficients\n", - "\n", - "Because the forces have been applied for a long time, any non-zero\n", - "damping eliminates the homogenous parts of the solution, so one need\n", - "only consider the particular solution for each $n$.\n", - "\n", - "The problem is considered solved if one can find expressions for the\n", - "coefficients $f_n$ and $g_n$, even though the solutions are expressed\n", - "as an infinite sum. The coefficients can be extracted from the\n", - "function $F(t)$ by" + "or as a double sum with restriction $l=i+j$" ] }, { "cell_type": "markdown", - "id": "99c86e27", - "metadata": {}, + "id": "c62e68b4", + "metadata": { + "editable": true + }, "source": [ - "\n", - "
    \n", - "\n", "$$\n", - "\\begin{eqnarray}\n", - "\\label{eq:fourierdef2} \\tag{7}\n", - "f_n&=&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~F(t)\\cos(2n\\pi t/\\tau),\\\\\n", - "\\nonumber\n", - "g_n&=&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~F(t)\\sin(2n\\pi t/\\tau).\n", - "\\end{eqnarray}\n", + "\\delta_l = \\sum_{ij}\\alpha_i\\beta_{j}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "b5563921", - "metadata": {}, - "source": [ - "To check the consistency of these expressions and to verify\n", - "Eq. ([7](#eq:fourierdef2)), one can insert the expansion of $F(t)$ in\n", - "Eq. ([6](#eq:fourierdef1)) into the expression for the coefficients in\n", - "Eq. ([7](#eq:fourierdef2)) and see whether" - ] - }, - { - "cell_type": "markdown", - "id": "fe12a873", - "metadata": {}, + "id": "61d1db60", + "metadata": { + "editable": true + }, "source": [ - "$$\n", - "\\begin{eqnarray}\n", - "f_n&=?&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~\\left\\{\n", - "\\frac{f_0}{2}+\\sum_{m>0}f_m\\cos(m\\omega t)+g_m\\sin(m\\omega t)\n", - "\\right\\}\\cos(n\\omega t).\n", - "\\end{eqnarray}\n", - "$$" + "Do you see a potential drawback with these equations?" ] }, { "cell_type": "markdown", - "id": "71723078", - "metadata": {}, + "id": "a9b512a8", + "metadata": { + "editable": true + }, "source": [ - "Immediately, one can throw away all the terms with $g_m$ because they\n", - "convolute an even and an odd function. The term with $f_0/2$\n", - "disappears because $\\cos(n\\omega t)$ is equally positive and negative\n", - "over the interval and will integrate to zero. For all the terms\n", - "$f_m\\cos(m\\omega t)$ appearing in the sum, one can use angle addition\n", - "formulas to see that $\\cos(m\\omega t)\\cos(n\\omega\n", - "t)=(1/2)(\\cos[(m+n)\\omega t]+\\cos[(m-n)\\omega t]$. This will integrate\n", - "to zero unless $m=n$. In that case the $m=n$ term gives" + "## A more efficient way of coding the above Convolution\n", + "\n", + "Since we only have a finite number of $\\alpha$ and $\\beta$ values\n", + "which are non-zero, we can rewrite the above convolution expressions\n", + "as a matrix-vector multiplication" ] }, { "cell_type": "markdown", - "id": "6db2c3aa", - "metadata": {}, + "id": "ebbaa615", + "metadata": { + "editable": true + }, "source": [ - "\n", - "
    \n", - "\n", "$$\n", - "\\begin{equation}\n", - "\\int_{-\\tau/2}^{\\tau/2}dt~\\cos^2(m\\omega t)=\\frac{\\tau}{2},\n", - "\\label{_auto6} \\tag{8}\n", - "\\end{equation}\n", + "\\boldsymbol{\\delta}=\\begin{bmatrix}\\alpha_0 & 0 & 0 & 0 \\\\\n", + " \\alpha_1 & \\alpha_0 & 0 & 0 \\\\\n", + "\t\t\t \\alpha_2 & \\alpha_1 & \\alpha_0 & 0 \\\\\n", + "\t\t\t 0 & \\alpha_2 & \\alpha_1 & \\alpha_0 \\\\\n", + "\t\t\t 0 & 0 & \\alpha_2 & \\alpha_1 \\\\\n", + "\t\t\t 0 & 0 & 0 & \\alpha_2\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\beta_0 \\\\ \\beta_1 \\\\ \\beta_2 \\\\ \\beta_3\\end{bmatrix}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "3bad1b07", - "metadata": {}, + "id": "ae36d14d", + "metadata": { + "editable": true + }, "source": [ - "and" + "The process is commutative and we can easily see that we can rewrite the multiplication in terms of a matrix holding $\\beta$ and a vector holding $\\alpha$.\n", + "In this case we have" ] }, { "cell_type": "markdown", - "id": "cd7fe4ed", - "metadata": {}, + "id": "41daf5bc", + "metadata": { + "editable": true + }, "source": [ "$$\n", - "\\begin{eqnarray}\n", - "f_n&=?&\\frac{2}{\\tau}\\int_{-\\tau/2}^{\\tau/2} dt~f_n/2\\\\\n", - "\\nonumber\n", - "&=&f_n~\\checkmark.\n", - "\\end{eqnarray}\n", + "\\boldsymbol{\\delta}=\\begin{bmatrix}\\beta_0 & 0 & 0 \\\\\n", + " \\beta_1 & \\beta_0 & 0 \\\\\n", + "\t\t\t \\beta_2 & \\beta_1 & \\beta_0 \\\\\n", + "\t\t\t \\beta_3 & \\beta_2 & \\beta_1 \\\\\n", + "\t\t\t 0 & \\beta_3 & \\beta_2 \\\\\n", + "\t\t\t 0 & 0 & \\beta_3\n", + "\t\t\t \\end{bmatrix}\\begin{bmatrix} \\alpha_0 \\\\ \\alpha_1 \\\\ \\alpha_2\\end{bmatrix}.\n", "$$" ] }, { "cell_type": "markdown", - "id": "bf3342a0", - "metadata": {}, - "source": [ - "The same method can be used to check for the consistency of $g_n$." - ] - }, - { - "cell_type": "markdown", - "id": "fa05f48c", - "metadata": {}, - "source": [ - "## Final words on Fourier Transforms\n", - "\n", - "The code here uses the Fourier series applied to a \n", - "square wave signal. The code here\n", - "visualizes the various approximations given by Fourier series compared\n", - "with a square wave with period $T=0.2$ (dimensionless time), width $0.1$ and max value of the force $F=2$. We\n", - "see that when we increase the number of components in the Fourier\n", - "series, the Fourier series approximation gets closer and closer to the\n", - "square wave signal." - ] - }, - { - "cell_type": "code", - "execution_count": 31, - "id": "fc921d9d", - "metadata": {}, - "outputs": [], + "id": "f468150e", + "metadata": { + "editable": true + }, "source": [ - "import numpy as np\n", - "import math\n", - "from scipy import signal\n", - "import matplotlib.pyplot as plt\n", + "Note that the use of these matrices is for mathematical purposes only and not implementation purposes.\n", + "When implementing the above equation we do not encode (and allocate memory) the matrices explicitely.\n", + "We rather code the convolutions in the minimal memory footprint that they require.\n", "\n", - "# number of points \n", - "n = 500\n", - "# start and final times \n", - "t0 = 0.0\n", - "tn = 1.0\n", - "# Period \n", - "T =0.2\n", - "# Max value of square signal \n", - "Fmax= 2.0\n", - "# Width of signal \n", - "Width = 0.1\n", - "t = np.linspace(t0, tn, n, endpoint=False)\n", - "SqrSignal = np.zeros(n)\n", - "FourierSeriesSignal = np.zeros(n)\n", - "SqrSignal = 1.0+signal.square(2*np.pi*5*t+np.pi*Width/T)\n", - "a0 = Fmax*Width/T\n", - "FourierSeriesSignal = a0\n", - "Factor = 2.0*Fmax/np.pi\n", - "for i in range(1,500):\n", - " FourierSeriesSignal += Factor/(i)*np.sin(np.pi*i*Width/T)*np.cos(i*t*2*np.pi/T)\n", - "plt.plot(t, SqrSignal)\n", - "plt.plot(t, FourierSeriesSignal)\n", - "plt.ylim(-0.5, 2.5)\n", - "plt.show()" + "Does the number of floating point operations change here when we use the commutative property?" ] }, { "cell_type": "markdown", - "id": "41918e8c", - "metadata": {}, + "id": "eea7671c", + "metadata": { + "editable": true + }, "source": [ "## Two-dimensional Objects\n", "\n", @@ -4359,8 +4888,10 @@ }, { "cell_type": "markdown", - "id": "3e9dbfaa", - "metadata": {}, + "id": "8efebdfe", + "metadata": { + "editable": true + }, "source": [ "$$\n", "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(m,n)K(i-m,j-n).\n", @@ -4369,16 +4900,20 @@ }, { "cell_type": "markdown", - "id": "9cc5bc9a", - "metadata": {}, + "id": "e3b3c8cb", + "metadata": { + "editable": true + }, "source": [ "Convolution is a commutatitave process, which means we can rewrite this equation as" ] }, { "cell_type": "markdown", - "id": "c7563b24", - "metadata": {}, + "id": "18d84ac3", + "metadata": { + "editable": true + }, "source": [ "$$\n", "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i-m,j-n)K(m,n).\n", @@ -4387,16 +4922,20 @@ }, { "cell_type": "markdown", - "id": "d05c84ea", - "metadata": {}, + "id": "e28b62a8", + "metadata": { + "editable": true + }, "source": [ "Normally the latter is more straightforward to implement in a machine elarning library since there is less variation in the range of values of $m$ and $n$." ] }, { "cell_type": "markdown", - "id": "b037b494", - "metadata": {}, + "id": "a9034bb8", + "metadata": { + "editable": true + }, "source": [ "## Cross-Correlation\n", "\n", @@ -4405,8 +4944,10 @@ }, { "cell_type": "markdown", - "id": "628a0441", - "metadata": {}, + "id": "d3d234f1", + "metadata": { + "editable": true + }, "source": [ "$$\n", "S_(i,j)=(I * K)(i,j) = \\sum_m\\sum_n I(i+m,j-+)K(m,n).\n", @@ -4415,8 +4956,10 @@ }, { "cell_type": "markdown", - "id": "fb12b21f", - "metadata": {}, + "id": "c32b1e5f", + "metadata": { + "editable": true + }, "source": [ "## More on Dimensionalities\n", "\n", @@ -4440,8 +4983,10 @@ }, { "cell_type": "markdown", - "id": "b524f11c", - "metadata": {}, + "id": "63926684", + "metadata": { + "editable": true + }, "source": [ "$$\n", "\\mathrm{NumberParameters}=10^{10}+10^4+10^4+1 \\approx 10^{10},\n", @@ -4450,16 +4995,20 @@ }, { "cell_type": "markdown", - "id": "0dfba800", - "metadata": {}, + "id": "656d1fb9", + "metadata": { + "editable": true + }, "source": [ "that is ten billion parameters to determine." ] }, { "cell_type": "markdown", - "id": "f9070d27", - "metadata": {}, + "id": "65320e73", + "metadata": { + "editable": true + }, "source": [ "## Further Dimensionality Remarks\n", "\n", @@ -4481,25 +5030,7 @@ ] } ], - "metadata": { - "kernelspec": { - "display_name": "Python 3 (ipykernel)", - "language": "python", - "name": "python3" - }, - "language_info": { - "codemirror_mode": { - "name": "ipython", - "version": 3 - }, - "file_extension": ".py", - "mimetype": "text/x-python", - "name": "python", - "nbconvert_exporter": "python", - "pygments_lexer": "ipython3", - "version": "3.9.10" - } - }, + "metadata": {}, "nbformat": 4, "nbformat_minor": 5 } diff --git a/doc/pub/week4/pdf/week4.pdf b/doc/pub/week4/pdf/week4.pdf index 7aa90eb3c6650edf67aa2a576b6caf0f31d7dae7..03e153a96fd8a4a880719679ebe50556b2cee945 100644 GIT binary patch delta 458227 zcmZs?V~}P+w=~+e-P1OwZQHhOW7>Gywr$(CF>Twn-F@dh=X^Kr7x(Ol9Z^-kDt6^w zYh|vi+CK5#R#7rC1rf2|jC8CpWV4IIt1wJVgbalCMpiI9JTUY!rgr9kEC|_|QuF{V1r(Qg%t?Oo=*UNh~?EV)}aEUvA^@;q5^`5O9E` 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