01-machine-learning.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Machine learning and artificial neural networks (ANNs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-warning\">\n",
    "The code you will see in this notebook is julia. A recent high-level and fast programming language. But do not worry since the basic syntax, is similar to Python.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Machine learning** (ML) is the study of computer **algorithms** that **improve automatically** through experience. It is seen as a subset of **artificial intelligence**. Machine learning algorithms build a **mathematical model based on sample data.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Artificial neural networks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "<img src=\"https://res.cloudinary.com/dyd911kmh/image/upload/f_auto,q_auto:best/v1547672259/2_i1cdwq.png\" width=550px/>\n",
    "\n",
    "A simple example: **One input** and **no activation** function. You actually know this as **linear regression**."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = -0.3;\n",
    "b = 7.5;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f (generic function with 1 method)"
      ]
     },
     "execution_count": 145,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x) = a*x+b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "6.9"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(2.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b> Did you know the body mass of animals animals is correlated to its heart rate? </b>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "names = [\"mouse\", \"rat\", \"rabbit\", \"dog\", \"human\"]\n",
    "mass = [25.0, 200.0, 2000.0, 30000.0, 70000.0]\n",
    "pulse = [670.0, 420.0, 205.0, 85.0, 72.0];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "Y₁ = log.(pulse);\n",
    "X₁ = log.(mass);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "scatter(X₁, Y₁,series_annotations = text.(names,:bottom),\n",
    "    label=\"\", xlabel=\"log mass (g)\", ylabel=\"log pulse rate (BPM)\", \n",
    "    ylims=(4,7), xlims=(2,12)) \n",
    "\n",
    "plot!(X₁, f.(X₁), color=:grey, lw=2, ls=:dash, label=\"f(x) = $(a)*x+$(b)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    <b>Neural networks are a non-linear combination of linear models.</b>\n",
    "    The values of the linear models pass through a non-linear activation function generating the input values for the next layer.\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 171,
   "metadata": {},
   "outputs": [],
   "source": [
    "a₁, a₂ = -2, 10\n",
    "b₁, b₂, b₃ = -1, -3, 3;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [],
   "source": [
    "f₁(x) = tanh(f(x))\n",
    "f₂(x) = (f₁(x) + a₁*f₁(x+b₁) + f₁(x+b₂))\n",
    "f₃(x) = tanh(a₂*f₂(x)+a₁*f₂(x+b₃));"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 176,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 176,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X = 20:0.1:40\n",
    "plot(X, f₁.(X), \n",
    "    label=\"f₁(x) = tanh($(a)*x+$(b))\",xlabel = \"X\", ylabel=\"f(X)\",\n",
    "    lw = 2, legend=:bottomright)\n",
    "plot!(X, f₂.(X), label = \"f₂(x) =f₁(x) + a₁*f₁(x+b₁) + f₁(x+b₂)\", lw = 2)\n",
    "plot!(X, f₃.(X), label = \"f₃(x) = tanh(a₂*f₂(x)+a₁*f₂(x+b₃))\", lw = 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The final equation for f<sub>3</sub>(x) is,\n",
    "\n",
    "$$f_3(x)= \\mathrm{tanh}\\left(a_2\\,(\\mathrm{tanh}(-0.3\\,x + 7.5) + a₁\\,\\mathrm{tanh}(-0.3\\,x + 7.5) + \\mathrm{tanh}(-0.3\\,x + 7.5))) + a_1\\,(\\mathrm{tanh}(-0.3\\,x + 7.5) + a₁\\,\\mathrm{tanh}(-0.3\\,x + 7.5) + \\mathrm{tanh}(-0.3\\,x + 7.5))\\right)$$\n",
    "\n",
    "and is actually a simple, 3-layer neural-network where the non-linear function (activation function) is a hyperbolic tangent."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-info\">\n",
    "    It is clear that the more layers you add the more non-linear and funky the functions become. <b>Training</b> a neural network is just finding the parameter a<sub>i</sub> and b<sub>i</sub> that best fit a certain dataset, using optimisation routines. This is discussed in the <a href=\"./02-NODEs.ipynb\">next notebook </a>.\n",
    "</div>"
   ]
  }
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