07.04. Нелинейные наименьшие квадраты.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Нелинейные наименьшие квадраты"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from scipy import linalg\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math\n",
    "from scipy.optimize import minimize, rosen, rosen_der, rosen_hess, curve_fit, OptimizeResult"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Логистическая функция:\n",
    "$$\n",
    "y =   \\frac{a-d}{1+   (x/c)^b  }  + d\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "def logistic4(x, a, b, c, d):\n",
    "    return ((a-d)/(1.0+(b**(-x/c)))) + d"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "nobs = 24\n",
    "xdata = np.linspace(-3.5, 3.5, nobs)\n",
    "ptrue = [1, 3, 1, 0]\n",
    "ydata = logistic4(xdata, *ptrue) + 0.2*np.random.random(nobs)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "popt, pcov = curve_fit(logistic4, xdata, ydata)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Param\tTrue\tEstim (+/- 1 SD)\n",
      "a\t 1.00\t 0.11 (+/- 0.05)\n",
      "b\t 3.00\t 1.24 (+/-540420.22)\n",
      "c\t 1.00\t-0.20 (+/-412958.24)\n",
      "d\t 0.00\t 1.15 (+/- 0.05)\n"
     ]
    }
   ],
   "source": [
    "perr = yerr=np.sqrt(np.diag(pcov))\n",
    "print('Param\\tTrue\\tEstim (+/- 1 SD)')\n",
    "for p, pt, po, pe  in zip('abcd', ptrue, popt, perr):\n",
    "    print('%s\\t%5.2f\\t%5.2f (+/-%5.2f)' % (p, pt, po, pe))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-4, 4, 100)\n",
    "y = logistic4(x, *popt)\n",
    "plt.plot(x, y)\n",
    "plt.plot(xdata, ydata, 'o')\n",
    "pass"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "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": "3.8.3"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}