notebooks/python-6.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Arrays of numbers & Data Fitting\n",
    "In this notebook we look at how to create / read in an array of numbers, fit the data to a polynomial or any arbitrary function."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ndarray from numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For any function you want to know more, place a question mark after the name and shift-enter to see a pop-up showing help about that function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.arange?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Change the argument to the function arange from float (3.0) to int (3) and see the output array and its datatype."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "numpy.ndarray"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a = np.arange(3.0)\n",
    "type(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "dtype('float64')"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.dtype"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 2.])"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can inspect the datatypes available in numpy."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "finfo(resolution=0.001, min=-6.55040e+04, max=6.55040e+04, dtype=float16)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.finfo(np.float16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "finfo(resolution=1e-15, min=-1.7976931348623157e+308, max=1.7976931348623157e+308, dtype=float64)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.finfo(np.float64)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "iinfo(min=-32768, max=32767, dtype=int16)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.iinfo(np.int16)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "iinfo(min=-9223372036854775808, max=9223372036854775807, dtype=int64)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.iinfo(np.int64)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can instruct which datatype to be used to create the array."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "a1 = np.arange(10, dtype='f')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0., 1., 2., 3., 4., 5., 6., 7., 8., 9.], dtype=float32)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "a2 = np.arange(10, dtype='i')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0, 1, 2, 3, 4, 5, 6, 7, 8, 9], dtype=int32)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Shape of array"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(10,)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a1.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "a2 = a1.reshape(5,2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 1.],\n",
       "       [2., 3.],\n",
       "       [4., 5.],\n",
       "       [6., 7.],\n",
       "       [8., 9.]], dtype=float32)"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(5, 2)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 1. 2. 3. 4. 5. 6. 7. 8. 9.]\n"
     ]
    }
   ],
   "source": [
    "a3 = a2.ravel()\n",
    "print(a3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[0. 1. 2. 3. 4. 5. 6. 7. 8. 9.]\n"
     ]
    }
   ],
   "source": [
    "a4 = a2.flatten()\n",
    "print(a4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0., 2., 4., 6., 8.],\n",
       "       [1., 3., 5., 7., 9.]], dtype=float32)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a2.transpose()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reading array from a local file\n",
    "\n",
    "Check the working directory of the notebook to ensure the filename is relative to the path."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'/ehome/gphani/Programs/JupyterNotebooks/Python'"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%pwd"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We assume that there is a file \"expdata.csv\" in the above path with two columns of data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "d1 = np.genfromtxt('expdata.csv', delimiter=',', usecols=(0,1), skip_header=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ 1.  ,  1.  ],\n",
       "       [ 2.  ,  3.89],\n",
       "       [ 3.  ,  9.05],\n",
       "       [ 4.  , 15.87],\n",
       "       [ 5.  , 25.08]])"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f0274848630>]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(d1[:,0], d1[:,1], 'ro-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Fitting the data to a polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "poly = np.polyfit(d1[:,0], d1[:,1], 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 6.014, -7.064])"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10.978"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.polyval(poly, 3.0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "yf = np.polyval(poly, d1[:,0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([-1.05 ,  4.964, 10.978, 16.992, 23.006])"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "yf"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x7f027480cc18>]"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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8VPQlQj8KpUHpBpa9eNkyMwxRP2YykYuLdQsVKSaBLkQ6FxkdycjQkUzYMoGcWXMyqcUkPGp5kCWTBb/+V69Cjx6m/7x0abhwwXwQeleuXGaav3AKSfahK6XmKaXClVL77nvMWyl1Rim1K+Gfm23LFEIkl9aaJfuXUH5aecZsGkOHlzpwyPMQvWr3sizMAwKgcmX46Sfw9oYjR8w489KlTWu9dGmzAFfHjjb/XoRlLGmhfw/4Agseenyi1nq81SsSQqTawYsH6RnYk9///Z2Xi73M4naLqedSz7IX37gB/fqZ8K5UCVauhOrVzXMdO0qAO7EkA11rHaqUKmP7UoQQqRVxJ4IRoSOY+OdEcmfNzdRWU+nu2t2yFjmYzSg+/thMGBo0yLTMs2dP8mXCOaSmD91TKfUhsB3op7W++riDlFLugDuAi3x4IoRNaK35af9P9Fvbj7MRZ/m46seMaTaGormLWnaCyEgT4L6+ULas2VmoTh3bFi2sLqXj0GcAzwNVgXPAhMQO1FrP1lq7aq1dixQpksK3E0IkZn/4fpouaEqHpR0olrsYm7tsZl7reZaH+ebNULWqCfNevWDXLgnzNCpFLXSt9YW7t5VSc4BVVqtICGGRiDsR+GzwYfLWyeTNlpfpbtNxr+FO5kyZLTtBVBQMGwbjx5sPONevlxmfaVyKAl0pVVxrfS7hbltg35OOF0JYj9aaH/f9SP+1/Tl38xyfVPuE0U1HUyR3Mv4C3rEDPvwQDhww0/nHj4e8eW1XtLCLJANdKfUj0AgorJQ6DXgBjZRSVQENnAC62bBGIUSCfeH78PT3ZMPJDbg+48qv7/1KrRK1LD9BdLQZNz5qFDz9tBma2LKl7QoWdmXJKJcOj3l4rg1qEUIk4nrUdbxDvJn611Ty5cjHrDdm0bVaV8u7VwD27oXOnWHnTrOz0OTJstdnOiMzRYVwYlprftjzAwOCBhAeGY57DXdGNRlFoVyFLD9JbCyMGwdeXibAly+XdcvTKQl0IZzU7vO78QzwZGPYRmqVqMWq91fh+oxr8k5y6JBplW/daraImz4dZLRZuiWBLoSTuRZ1jWHrhzFt2zQK5CjAnDfn0KVaFzKpZIwyjo83mzV/8QXkzAk//gjvviv7fKZzEuhCOIl4Hc/C3QsZ+PtALkZepLtrd0Y2GUnBnAWTd6Ljx81szw0b4I03zHorxS1YHlekeRLoQjiBXed34eHvweZTm6ldsjYBHQOoXrx68k6itQnvfv0gc2aYNw8++kha5RmIBLoQDnT19lWGrh/KjO0zKJSzEPP+bx6dq3ZOXvcKmLVXunaFtWuhWTOYO1fWKc+AJNCFcIB4Hc/3u75n0O+DuHz7Mp+5fsbwxsMpkDOZwwi1hoULoWdPs1nz9OnQvbu0yjMoCXQh7GzH2R14+Huw9cxW6pWqh6+bL1Wfrpr8E50/D926wYoVUL8+fP89PP+81esVaYcEuhB2cuX2FYYED2HWjlkUyV2E+W3m06lKJ1RKWtNLlsCnn8LNmzBhgllUK3MyJhmJdEkCXQgbi9fxzP17Ll8Ef8HVqKv0qNUDn8Y+5M+RP/knu3wZPDzMLkI1a8L8+VChgvWLFmmSBLoQNrTtzDY8/D3YdnYb9V3qM81tGlWKVUnZyVauhP/9D65cgZEj4fPPIYv8Cot75KdBCBu4fOsyg4MHM+fvORTLU4yFbRfS8aWOKeteuX4devc2feRVqsCaNfDyy1avWaR9EuhCWFFcfBzf/v0tg9cN5nrUdXrX7o13I2+eyv5Uyk4YFARdusC5c/DllzB0KGTLZt2iRbohgS6ElWw9vRUPfw92nNtBw9IN8XXzpXLRyik72c2bMHAgzJgB5cubXYVqJWOZXJEhSaALkUoXIy/yRfAXzN05l+J5irPorUW8V/m9lHWvAISGmqn7x4+bWZ8jRpj1WIRIggS6ECkUFx/HrB2zGLJuCDejb9K/Tn+GNRxG3uwp3Pnn9m0YMgQmTYJnnzVrsTRoYN2iRbomgS5ECmw5tQUPfw92nt9J4zKN8XXzpWKRiik/4datZpnbQ4fgs89g7FjIk8d6BYsMIZkLRgiRsYVHhvPxbx9Td15dwiPDWdxuMcEfBqc8zO/cMa3yunXh1i3zIei0aRLmIkWkhS6EBWLjY5mxbQZD1w8lMiaSgXUHMrThUPJkS0Xw7tplWuV79pg+84kTIV8+6xUtMhwJdCGSsClsEx7+Huy+sJtmzzVjaquplC9cPuUnjI2FMWPAxwcKFzYTht54w3oFiwxLAl2IRJy/eZ7Pf/+cBbsXUOqpUixpv4R2FdqlfPQKwIEDplW+fTt06ABTp0KhZOwPKsQTSKAL8ZDY+Fh8//LFK8SL2zG3+aL+FwxpMITc2XKn/KRxcWb0ypAhpn/855+hfXvrFS0E8qGoEAB4h3gDEHoylOqzqtNnTR/qlKzDvs/2Mbrp6NSF+dGj0LAh9O8PrVrB/v0S5sImpIUuBOCzwYejV47it9cPl3wuLHtnGW3Kt0ld90p8vJnpOXAgZM0KCxbABx/I5hPCZiTQRYYWExfD1L+mArDkwBKGNBjC4AaDyZU1V+pOHBZm1mAJDoYWLeDbb6FkSStULETiJNBFhvXRrx8xf/f8/+5Hx0Uz6o9RZMmUBe9G3ik7qdbw3XdmdcS7mzZ/8om0yoVdSKCLDOfMjTP0D+rP4n2LKZO/DJNaTKLNT23QXjp1Jz571qxX7u9v+sy/+85M4RfCTiTQRYYRHRfN5D8nMzx0ODFxMQx7dRiD6g8iZ9ZULnylNfz4I3h6QlQUTJ5sbmeSMQfCviTQRYYQ/G8wngGe/HPpH94o9waTWkzi+YL3NlT2auiVshNfvGj29ly6FGrXNlvClStnpaqFSJ4kmxBKqXlKqXCl1L77HiuolApSSh1J+FrAtmUKkTKnb5zmnSXv0GxhM6LjolnZYSUrO6x8IMyBlPWZL18OlSqZmZ5jxsDGjRLmwqEs+Zvwe6DlQ48NAoK11mWB4IT7QjiN6Lhoxm4cS3nf8qw8vBKfRj7s/2w/b5SzwhT7q1ehUyd46y0zcmXHDrO/Z+bMqT+3EKmQZJeL1jpUKVXmoYdbA40Sbs8HQoDPrViXECkWdCyIHgE9OHT5EK1fbM3EFhN5toCVPpwMCDCjVsLDwcvLzPzMmtU65xYilVLah15Ma30OQGt9TilVNLEDlVLugDuAi4tLCt9OiKSFXQ+j75q+LD24lBcKvoD/+/60KtvKOie/ccPsHvTtt6abZcUKqFHDOucWwkps/qGo1no2MBvA1dU1lePChHjUndg7TNgygVF/jEJrzcjGI+lXtx85suSwzhusX2+Wtz11ysz69PGBHFY6txBWlNJAv6CUKp7QOi8OhFuzKCEsFXg0kJ4BPTly5QhvVXiLb5p/Q+n8pa1z8lu3YNAgsyJi2bLmQ886daxzbiFsIKUDZVcAnRNudwZ+s045QljmxLUTtP2pLa38TJdKYMdAlr6z1HphvnkzVK1qwrxnT7MZhYS5cHJJttCVUj9iPgAtrJQ6DXgBY4CflVJdgTBAlo4TdhEVG8W4TeMYvXE0mVQmvmr6FX1q9yF7luxWeoMo82Hn+PFQqhSsWweNG1vn3ELYmCWjXDok8lRTK9cixBOtPryaXoG9OHb1GO0rtmdC8wmUylcqdSf18zMjVcLCoFgxM7vz7hT+CRMgb17rFC+EHchMUeH0jl89Tu81vVlxaAUvFnqRoE5BNHuuWepP7OcH7u6mrxzg/HmziNaAAfD116k/vxB2JotNCKd1O+Y2PiE+VJxekeB/gxnbbCx7Pt1jnTAHGDz4XpjfpbXZTUiINEha6MIprTy0kl6BvTh+7TjvVnqX8c3HU/IpK60nHhcHixaZbpbHSexxIZycBLpwKseuHKNXYC9WH1lNhcIVCP4wmCbPNrHOyePj4ZdfwNsbDh40MzxjYh49TibAiTRKulyEU7gVc4th64dRaXolNpzcwPjXxrO7+27rhLnW8NtvUK0avPuu+eDzl19g3jzI9dDORLlywahRqX9PIRxAWujCobTW/HboN3oH9ubk9ZO8/9L7jHttHM/kfcYaJ4fAQBg2DLZvN5OD/PxMqN9dSEupe6NcXFxMmHfsmPr3FsIBJNCFwxy5fISegT0JPBpI5aKVCekcQsMyDa1z8nXrYOhQM0GoTBmze9AHH0CWh37kO3aUABfphgS6sLvI6EhG/zGa8VvGkz1zdia2mIhHTQ+yZrbCqoUbN5ogDwkxS9vOnGnWYcmWLfXnFsLJSaALu9Fas/yf5fRZ04ew62F8UOUDvm72NcXzFk/9ybdtM0G+Zo2ZIDR5shljLotoiQxEAl3YxaFLh+gZ2JO1x9ZSpVgVfmj7Aw1KN0j9iXfvNn3kK1ZAoUIwbhx89tmjH3YKkQFIoAubuhl9k5GhI/lmyzfkzJqTKS2n8GnNT8mSKZU/egcOmDVXfvkF8ueHkSPNIloyVV9kYBLowia01vxy4Bf6ru3L6Run6fxyZ8Y2G0uxPMVSd+IjR8x65IsWQZ48ppulb18T6kJkcBLowuoOXjxIj4AeBB8PpurTVfnp7Z+oW6pu6k564gSMGAHz50P27GajiQEDTDeLEAKQQBdWFHEnghGhI5j450TyZMuDbytfurt2J3OmVGyefPq0GRs+d66ZENSjh9l0olgqW/pCpEMS6CLVtNb8tP8n+q3tx9mIs3Sp2oWvmn1F0dyJbjWbtPPnYcwYM+wwPt5szDxkCJQoYb3ChUhnJNBFquwP30+PgB6sP7Ge6sWrs/SdpdQuWTvlJ7x0yYxUmToVoqPho4/gyy/N5CAhxBNJoIsUuXHnBj4hPkz5awp5s+Vlxusz+F/1/6W8e+XaNbOhxKRJEBlpZm96ecELL1i3cCHSMQl0kSxaaxbtXcSAoAGcv3meT6p/wuimoymcq3DKThgRYSYBTZhgQr19e7MaYsWKVq1biIxAAl1YbO+FvXgGeBJ6MhTXZ1z59b1fqVWiVspOFhkJ06aZnYEuX4bWrc1wxJdftm7RQmQgEugiSdejruMd4s3Uv6aSL0c+Zr0xi67VuqaseyUqCmbNgq++ggsXoGVLGD4cata0fuFCZDAS6CJRWmt+2PMDA4IGEB4ZTrca3RjZZCSFcqVg7Hd0tFl/fORIOHMGGjeGpUuhXj3rFy5EBiWBLh5r9/ndeAZ4sjFsI6+UeIVV76/C9RnX5J8oNhYWLDCTgk6cMAG+cKEJdCGEVUmgiwdci7rGsPXDmLZtGgVzFuTbN7/l42ofk0klc3OruDhYvNj0ix85Aq6uMGMGtGhhNpUQQlidBLoAIF7Hs2D3Aj7//XMu3bpE9xrdGdFkBAVzFkzmieJh2TIz5PDAAahSxWz/9uabEuRC2JgEumDnuZ14+Huw5fQW6pSsQ2DHQKoVr5a8k2gNK1eapWx374YKFeDnn6FdOzNlXwhhcxLoGdjV21f5ct2XzNwxk0I5C/Fd6+/48OUPk9e9ojWsXWtWPdy2zUwEWrgQOnS4t2+nEMIuJNAzoHgdz/e7vufz3z/nyu0reNT0YHjj4eTPkcwlaENCzLT8TZugdGmzgNaHHz66b6cQwi7kNy+D2XF2Bx7+Hmw9s5X6LvXxbeXLy08nczLP5s2mRb5uHTzzDEyfDl27yr6dQjhYqgJdKXUCiADigFitdQrGtQl7uHL7CkOChzBrxyyK5i7KgjYL+KDKB6jkfFC5fbvpIw8IgKJFYeJE6NYNcua0XeFCCItZ49OqxlrrqhLmzsU7xBsw3Stzdsyh3NRyzPl7Dr1e6cUhz0N0ermT5WG+Zw+0aWNmc27dCmPHwr//Qu/eEuZCOBHpckmnfDb48HrZ1/Hw92Db2W00cGnANLdpvFTsJctPcvCgWSjr558hXz4zRb9XL3jqKZvVLYRIudQGugbWKqU0MEtrPdsKNYlUunTrEgCvfPsKxfIU44e2P/D+S+9b3iI/etSEt58f5MplNpbo1w8KFLBh1UKI1EptoNfTWp9VShUFgpRS/2itQ+8/QCnlDrgDuLi4pPLtxJMMWz+MEaEj/ruv0Zy/eZ4jV45YFuYnT5op+t9/bz7g7NfP7N1ZOIVL4woh7CpVga61PpvwNVwptRxuu9dyAAAN50lEQVSoBYQ+dMxsYDaAq6urTs37icRtPb0V/yP+ADQq04iQEyFoLwsv95kzMHo0zJljZnN6eMAXX8DTT9uwYiGEtaX4Q1GlVG6lVN67t4HmwD5rFSYsczHyIp+s+ITac2tz7uY5fmz3I+s+XGfZi8PDoW9feP55mD0bunQx3S2TJ0uYC5EGpaaFXgxYnvCnfBZgkdY60CpViSTFxccxc/tMvlz/JTejbzKg7gCGvjqUvNnzAuDV0CvxF1++fG/fzqgo6NzZjCt/9lk7VS+EsIUUB7rW+l9AtpdxgM2nNuPh78Gu87to+mxTpraaSoUiFR44xruR96MvvHbNjB2fOBFu3jTT8728oFw5+xQuhLApGbaYhoRHhvP575/z/a7vKZG3BD+//TNvV3w76Q88b96EKVNMq/zaNbNglo8PVKpkn8KFEHYhgZ4GxMbHMmPbDIauH8qtmFsMqjeIIa8OIU+2PI8e7OdnhhmGhUHJklC/PgQFwaVLZglbHx+olsyVFIUQaYIEupPbGLYRD38P9lzYQ/PnmzOl5RReLPzi4w/28wN3d7h1y9w/dQp+/BEqV4ZVq+CVV+xXuBDC7iTQndT5m+cZGDSQhXsWUuqpUix9Zylty7d9fPeK1vDPP+DpeS/M7xcRIWEuRAYgge5kYuNj8f3LF68QL6JioxhcfzCDGwwmd7bcDx4YGQnr14O/v1ks68SJxE8aFmbTmoUQzkEC3YmEngzFw9+DfeH7aPlCS6a0nELZQmXNk1qbvTnvBviGDXDnDuTODU2bwqBBZpbnmTOPnlhm6AqRIUigO4GzEWcZEDSARXsXUTpfaZa/u5zWL7ZGRUWZ8L4b4seOmReUL29mc7ZqBQ0aQPbs5vE8eR7sQwezFsuoUfb/poQQdieB7kAxcTFM2ToF7w3exMTFMPTVoQx65h1yBYVA39dNl0pUlFmitkkTM6uzVavEJwB17Gi+3h3l4uJiwvzu40KIdE1pbb/lVVxdXfX27dvt9n7OLORECB7+Hhy4eAC3/LWYfLICL6zeAocPmwPKlgU3NxPgDRtCjhyOLVgI4TBKqR2W7DkhLXQ7O3PjDP2Xf8riEyt59k4uVqzKxht7/0Ll2AONGpmRKq1amc2WhRAiGSTQ7eHOHaJD1zN53WiGZ95ELPF4b4SBZ4uSs8UbMKaVCfNcuRxdqRAiDZNAt5WwMPNBZkAAwYfX4Nkkin+KwJtXCjOpdDeem9XJrKGSnD09hRDiCSTQrSU6GjZtujcqZf9+Tj0F/drmYsm7UTyX7WlWve7L61XaObpSIUQ6JYGeGmfO/NcKJyjIzMjMmpXohvX5pmNpRsStJ17FM7z+cAbUG0COLPLBphDCdiTQkyMmBrZsudcK37PHPF6ypFmKtlUr1j4PPTZ8zuHLh2lTvg0TW0ykTP4yDi1bCJExSKAn5dw5CAw0AR4UBNevQ5YsZhXDsWPN0MJKlQi7cYo+a/qwbNkyXij4AgEdA2j5QktHVy+EyEAk0B8WGwtbt95rhe/caR4vXhzeftsEeNOmkC8fAHdi7zD+j9GM+sPMxhzVZBT96vQje5bsjvoOhBAZlAQ6mL0177bC166Fq1chc2aoW9dsnuzmBlWqPDIiJeBIAD0De3L0ylHaVWjHNy2+wSWfrJsihHCMjBnocXGwbdu9Vvjd2avFikHr1ibAX3sN8ud/7MtPXDtB78De/HboN8oVKseaD9bQ/PnmdvwGhBDiURkn0C9dgjVrTICvWWM2Ss6UCWrXNqsUurlB1armsURExUYxbtM4Rm8cTSaVia+afkWf2n2ke0UI4RTSb6DHx8OOHfdWKvzrL7MEbZEiJrzd3KB5cyhY0KLTrT68ml6BvTh29RjtK7ZnQvMJlMpXysbfhBBCWC59BfqVK6YP3N/f9IlfvGj6vWvVAm9vs0ZKjRpPbIU/7N+r/9I7sDcrD6+kfOHyBHUKotlzzWz3PQghRAo5f6Dfv+nxw8vBxsfDrl33WuF//mkeK1QIWrS41wovUiTZb3s75jZjN41lzMYxZMmUha+bfU2v2r3Iljmblb9BIYSwDucO9Ic3PT55Ev73PzOsMCLChPiFC+Y5V1cT/G5uULOmGaWSAlprVh5eSe/A3hy/dpz3Kr/H+NfGU+KpElb6poQQwjacO9CHDHl00+Pbt2HqVChQwLS+3dxMa7xYsVS/3dErR+kV2Av/I/5ULFKRdR+uo/GzjVN9XiGEsAfnDvTENjdWyowdz2Kd8m/F3OKrP77i681fkz1zdiY0n0CPWj3ImjmrVc4vhBD24NyB7uJiulke97gVwlxrzW+HfqN3YG9OXj9Jx5c6Mu61cRTPWzzV5xZCCHuzfLiHI4wa9eimD1ba9PjI5SO4LXKj7U9tyZs9LyGdQ/jhrR8kzIUQaZZzt9BtsOlxZHQko/8Yzfgt48mRJQcTW0zEo6aHdK8IIdK8VAW6UqolMBnIDHyrtR5jlaru17GjVXat11qz7OAy+qzpw6kbp+hUpRNfv/Y1T+d52gpFCiGE46U40JVSmYFpwGvAaWCbUmqF1vqAtYqzlkOXDtEjoAdB/wZRpVgVFrVbRH2X+o4uSwghrCo1LfRawFGt9b8ASqnFQGvAaQL9ZvRNRoaO5Jst35Aray6mtJzCpzU/JUsm5+5pEkKIlEhNspUATt13/zTwSurKsQ6tNUsOLKHf2n6cvnGaj6p+xJimYyiWJ/Vj1YUQwlmlJtAft129fuQgpdwBdwAXF9uvFX7w4kF6BPQg+HgwVZ+uyk9v/0TdUnVt/r5CCOFoqQn008D9yw2WBM4+fJDWejYwG8DV1fWRwLeWiDsRDN8wnElbJ5EnWx6muU2jW41uZM6UsiUAhBAirUlNoG8DyiqlngXOAO8B71ulqmTQWrN432L6B/XnbMRZulTtwphmYyiSO/kLcgkhRFqW4kDXWscqpTyBNZhhi/O01vutVpkF9ofvxzPAk5ATIVQvXp2l7yyldsna9ixBCCGcRqqGe2it/QF/K9VisRt3buAT4sPkrZN5KvtTzHh9Bv+r/j/pXhFCZGjOPfX/Pt4h3mit8dvjx4u+LzLxz4l0qdaFwz0O0921u4S5ECLDSzMDsn02+LD+xHpCT4ZS85marHhvBTVL1HR0WUII4TTSRKCPDB0JmD7z2W/Mpmv1rmRSaeaPCyGEsAunDnTvEG98Nvj8d//y7cu4r3LnTMQZvBt5O64wIYRwQkprmw0Nf4Srq6vevn17il6rfBTay361CiGEs1BK7dBauyZ1nPRbCCFEOpFmAt2roZejSxBCCKeWZgJd+syFEOLJ0kygCyGEeDIJdCGESCck0IUQIp2QQBdCiHRCAl0IIdIJu04sUkpdBE6m8OWFgUtWLMdapK7kkbqSR+pKHmetC1JXW2mtdZKbPNg10FNDKbXdkplS9iZ1JY/UlTxSV/I4a11gn9qky0UIIdIJCXQhhEgn0lKgz3Z0AYmQupJH6koeqSt5nLUusENtaaYPXQghxJOlpRa6EEKIJ3CqQFdKzVNKhSul9iXyvFJKTVFKHVVK7VFKVXeSuhoppa4rpXYl/Btmp7pKKaXWK6UOKqX2K6V6PeYYu18zC+uy+zVTSuVQSv2llNqdUJfPY45xxPWypC6H/IwlvHdmpdROpdSqxzznkN9JC+py1O/kCaXU3oT3fGTzB5tfL6210/wDXgWqA/sSed4NCAAUUBvY6iR1NQJWOeB6FQeqJ9zOCxwGKjr6mllYl92vWcI1yJNwOyuwFajtBNfLkroc8jOW8N59gUWPe39H/U5aUJejfidPAIWf8LxNr5dTtdC11qHAlScc0hpYoI0/gfxKqeJOUJdDaK3Paa3/TrgdARwESjx0mN2vmYV12V3CNbiZcDdrwr+HP0RyxPWypC6HUEqVBF4Hvk3kEIf8TlpQl7Oy6fVyqkC3QAng1H33T+MEQZGgTsKfzAFKqUr2fnOlVBmgGqZ1dz+HXrMn1AUOuGYJf6bvAsKBIK21U1wvC+oCx/yMTQIGAvGJPO+on6+k6gLHXC8NrFVK7VBKuT/meZter7QW6OoxjzlDS+ZvzNTcl4GpwK/2fHOlVB5gKdBba33j4acf8xK7XLMk6nLINdNax2mtqwIlgVpKqcoPHeKQ62VBXXa/XkqpN4BwrfWOJx32mMdser0srMtRv5P1tNbVgVaAh1Lq1Yeet+n1SmuBfhoodd/9ksBZB9XyH631jbt/Mmut/YGsSqnC9nhvpVRWTGj6aa2XPeYQh1yzpOpy5DVLeM9rQAjQ8qGnHPozllhdDrpe9YD/U0qdABYDTZRSPzx0jCOuV5J1OernS2t9NuFrOLAcqPXQITa9Xmkt0FcAHyZ8UlwbuK61PufoopRSTyulVMLtWpjretkO76uAucBBrfU3iRxm92tmSV2OuGZKqSJKqfwJt3MCzYB/HjrMEdcryboccb201l9orUtqrcsA7wHrtNYfPHSY3a+XJXU56Ocrt1Iq793bQHPg4ZFxNr1eWax1ImtQSv2I+XS6sFLqNOCF+YAIrfVMwB/zKfFR4BbwsZPU9TbwqVIqFrgNvKcTPtK2sXpAJ2BvQv8rwGDA5b7aHHHNLKnLEdesODBfKZUZ8wv+s9Z6lVKq+311OeJ6WVKXo37GHuEE18uSuhxxvYoByxP+H8kCLNJaB9rzeslMUSGESCfSWpeLEEKIREigCyFEOiGBLoQQ6YQEuhBCpBMS6EIIkU5IoAshRDohgS6EEOmEBLoQQqQT/w+KxZfvSD/PhQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(d1[:,0], d1[:,1], 'ro-')\n",
    "plt.plot(d1[:,0], yf, 'g+-')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Fitting data to any function using scipy\n",
    "\n",
    "We create a data that is a bit noisy but otherwise sinusoidal in appearance. We then use the optimize sub-package under scipy to fit the user defined function to the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_data = np.linspace(-5, 5, num=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math as math"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here is a reminder that you cannot use functions in the math module to pass an array in. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "only size-1 arrays can be converted to Python scalars",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-36-249a7a6b9f59>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0my_data\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmath\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msin\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx_data\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m: only size-1 arrays can be converted to Python scalars"
     ]
    }
   ],
   "source": [
    "y_data = math.sin(x_data)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You can do that using the same functions in numpy module. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_data = 2.9*np.sin(1.5*x_data) + np.random.normal(size=50)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_data, y_data)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy import optimize"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test_func(x,a,b):\n",
    "    return a*np.sin(b*x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "params, params_covariance = optimize.curve_fit(test_func, x_data, y_data, p0=[2, 2])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[3.05931973 1.45754553]\n"
     ]
    }
   ],
   "source": [
    "print(params)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.scatter(x_data, y_data, c='r')\n",
    "plt.plot(x_data,test_func(x_data,params[0], params[1]), label='fitted a*sin(b)')\n",
    "plt.legend(loc='best')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "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": "3.7.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}