ML-Kurs-SS2023/notebooks/02_fit_numpyFit.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fit 3rd order Polynomial to graph data using numpy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "plt.rcParams[\"font.size\"] = 20\n",
    "import numpy as np"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10], dtype='d')\n",
    "dx =  np.array([0.1,0.1,0.5,0.1,0.5,0.1,0.5,0.1,0.5,0.1], dtype='d')\n",
    "y = np.array([1.1 ,2.3 ,2.7 ,3.2 ,3.1 ,2.4 ,1.7 ,1.5 ,1.5  ,1.7 ], dtype='d')\n",
    "dy = np.array([0.15,0.22,0.29,0.39,0.31,0.21,0.13,0.15,0.19,0.13], dtype='d')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "create numpy array with weights as 1/error "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "ones = np.ones(10, dtype='d')\n",
    "weight = ones/dy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "various fit options"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 3 \n",
    "#model = np.polyfit(x, y, n, 0 , 0, weight, cov='unscaled' )\n",
    "#model, fitCov = np.polyfit(x, y, n , 0 , 0, weight, cov='unscaled' )\n",
    "model, fitCov = np.polyfit(x, y, n , 0 , 0, weight, cov='unscaled' )\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "monitor printout of coefficient polynomial vector (model)  and covariance matrix (fitCov)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[ 0.02590585 -0.48405665  2.52045767 -0.98073638]\n",
      "[[ 1.19801683e-05 -2.07833174e-04  9.95675425e-04 -9.97263377e-04]\n",
      " [-2.07833174e-04  3.65835615e-03 -1.78482005e-02  1.83096634e-02]\n",
      " [ 9.95675425e-04 -1.78482005e-02  8.93452262e-02 -9.59887600e-02]\n",
      " [-9.97263377e-04  1.83096634e-02 -9.59887600e-02  1.19001618e-01]]\n"
     ]
    }
   ],
   "source": [
    "print (model)\n",
    "print (fitCov)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "prepare errors for plotting, see\n",
    "https://stackoverflow.com/questions/28505008/numpy-polyfit-how-to-get-1-sigma-uncertainty-around-the-estimated-curve/28528966\n",
    "write polynomial as np.dot(yy, model)  with yy=[x**n, x*n-1, ..., 1] and  x can be a single value or \n",
    "or a vector and  model as coefficient vector. Since this a linear equation, with the covariance matrix\n",
    "fitCov of model, the covariance matrix of the values is np.dot(yy, np.dot(model, yy.T))."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Do the interpolation for plotting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "x_plot = np.linspace(0.1, 10.5, 500)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Matrix with rows 1, x, x**2, ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "YY = np.vstack([x_plot**(n-i) for i in range(n+1)]).T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "matrix multiplication calculates the polynomial values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "y_plot = np.dot(YY, model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Covariance_y = YY*Covariance*YY.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "fitCov_y_plot = np.dot(YY, np.dot(fitCov, YY.T))\n",
    "Sigma_y_plot = np.sqrt(np.diag(fitCov_y_plot))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "model contains parameters in order highest power first! Define fit function "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "def fitpol3(model):\n",
    "    return model[3] + model[2]*x + model[1]*x**2 + model[0]*x**3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "construct polynomial from coefficients in model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "predict = np.poly1d(model)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "r_2 coefficient of the determination of the regression score function\n",
    "input are the y data values and the prediction from the fit with x data values \n",
    "this is borrowed from scikit-learn"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.9486643210376879\n"
     ]
    }
   ],
   "source": [
    "from sklearn.metrics import r2_score\n",
    "p = r2_score(y, predict(x))\n",
    "print (p)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "plot data"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()\n",
    "plt.errorbar(x, y, dy , dx, fmt=\"o\")\n",
    "plt.plot(x, fitpol3(model) )\n",
    "plt.title(\"numpy Fit Test\")\n",
    "plt.xlim(-0.1, 10.1)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "plot data with covariance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fg, ax = plt.subplots(1, 1)\n",
    "ax.set_title(\"numpy Fit Test with $\\pm1\\sigma$ interval\")\n",
    "ax.fill_between(x_plot, y_plot+Sigma_y_plot, y_plot-Sigma_y_plot, alpha=.25)\n",
    "ax.plot(x_plot, y_plot,'-')\n",
    "ax.errorbar(x, y,dy , dx, fmt='o')\n",
    "ax.axis('tight')\n",
    "\n",
    "fg.canvas.draw()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.8.16"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}