algorithms/approx.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Аппроксимация линейной функцией с помощью МНК"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Теория"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Любая линейная функция может быть записана уравнением:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$y=a \\cdot x+b$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Аппроксимация заключается в отыскании коэффициентов $a$ и $b$ уравнения таких, чтобы все экспериментальные точки лежали наиболее близко к аппроксимирующей прямой."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "С этой целью чаще всего используется метод наименьших квадратов (МНК), суть которого заключается в следующем: сумма квадратов отклонений значения точки от аппроксимирующей точки принимает минимальное значение:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$F(a,b)= \\sum_{i=1}^{n} (y_i-(a \\cdot x_i+b))^2 \\rightarrow min$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Решение поставленной задачи сводится к нахождению экстремума указанной функции двух переменных. С этой целью находим частные производные функции функции по коэффициентам $a$ и $b$ и приравниваем их к нулю:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\left\\{\\begin{matrix} \\frac{\\partial F(a,b)}{\\partial a} = -2 \\sum_{i=1}^{n} (y_i-(a \\cdot x_i+b)) \\cdot x_i = 0 \\\\ \\frac{\\partial F(a,b)}{\\partial b} = -2 \\sum_{i=1}^{n} (y_i-(a \\cdot x_i+b)) = 0 \\end{matrix}\\right.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Решаем полученную систему уравнений и определяем значения коэффициентов:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\\left\\{\\begin{matrix}\n",
    "a = \\frac{n \\cdot \\sum_{i=1}^{n} x_i \\cdot y_i - \\sum_{i=1}^{n} x_i \\cdot \\sum_{i=1}^{n} y_i}{n \\cdot \\sum_{i=1}^{n} x_i^2 - (\\sum_{i=1}^{n} x_i)^2  }\n",
    "\\\\ b = \\frac{\\sum_{i=1}^{n} y_i - a \\cdot \\sum_{i=1}^{n} x_i}{n}\n",
    "\\end{matrix}\\right.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Реализация и тест"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def get_approx(x, y):\n",
    "    '''\n",
    "    Принимает на вход два массива точек (x, y) и возвращает\n",
    "    коэффициенты a и b, которые соответствуют прямой, наилучшим\n",
    "    образом апроксимирующей данные точки.\n",
    "    '''\n",
    "    sumx, sumy, sumx2, sumxy = 0, 0, 0, 0\n",
    "    n = len(x)\n",
    "    \n",
    "    for (curr_x, curr_y) in zip(x, y):\n",
    "        sumx += curr_x\n",
    "        sumy += curr_y\n",
    "        sumx2 += curr_x ** 2\n",
    "        sumxy += curr_x * curr_y\n",
    "        \n",
    "    a = (n * sumxy - (sumx * sumy)) / (n * sumx2 - sumx ** 2)\n",
    "    b = (sumy - a * sumx) / n\n",
    "    \n",
    "    return (a, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def get_points(f):\n",
    "    '''\n",
    "    Принимает на вход линейную функцию f и генерирует в соответствии\n",
    "    с ней точки + случайный шум.\n",
    "    '''\n",
    "    x = np.arange(0, 10.5, 0.5)\n",
    "    y = f(x) + np.random.random(len(x)) * 30\n",
    "    return (x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def get_approx_line(x, y):\n",
    "    '''\n",
    "    Возвращает точки прямой, наилучшим образом апроксимирующей\n",
    "    исходные точки.\n",
    "    '''\n",
    "    a, b = get_approx(x, y)\n",
    "    f = lambda x: a * x + b\n",
    "    x = np.arange(0, 11, 1)\n",
    "    y = f(x)\n",
    "    return(x, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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4JqX01Kjtm4H9K1GPNC7btsHnPgdXXAGHHJI1D4cdlndVkpS7sjcNEfFasmsW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\nJEklsWmQJEklsWmQJEklsWmQJEkl+T9aDGJKbd7CoAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x105e72ba8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x, y = get_points(lambda x: 8 * x + 3)\n",
    "plt.scatter(x, y)\n",
    "\n",
    "x_approx, y_approx = get_approx_line(x, y)\n",
    "plot = plt.plot(x_approx, y_approx, c='red')"
   ]
  }
 ],
 "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": "3.5.2"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}