07.01. Аппроксимация методом наименьших квадратов.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "%matplotlib inline\n",
    "from scipy import linalg"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Аппроксимация данных методом наименьших квадратов\n",
    "\n",
    "Задача *аппроксимации* похожа на задачу интерполяции. Имеются некоторые точки и мы хотим найти функцию, график которой проходит через эти точки. Главное отличие от интерполяции в том, что график не обязательно точно проходит через эти точки, а может немного отклоняться от них, однако в целом функция правильно описывает тренд в данных.\n",
    "\n",
    "Пусть, например, у нас есть такие точки:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "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.array([0, 1, 2, 3, 4])\n",
    "y = np.array([1, 2, 2, 3, 4])\n",
    "plt.plot(x, y, 'o', color='tab:orange')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Точки лежат вдоль некоторой прямой. Например, такой\n",
    "(пока вы можете не обращать внимание на то, как мы ее построили):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = len(x)\n",
    "X = np.stack((np.ones(m), x)).T\n",
    "beta, residuals, rank, sv = linalg.lstsq(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999993, 0.7000000000000002)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta0, beta1 = beta\n",
    "beta0, beta1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bCmwCfunueRE+fzQwGiArK6uus4oEwq4DZTwwq4BXF21Qn6cEStSBbmYtgb8Cd7r73hq7lwBd3H2/mQ0FpgE9aj6Gu08BpgBkZ2f7cU8tkgBVVc6rizbwwFsF7C+tYHS4z7Ol+jwlIKL6m2hmGYTC/M/u/nrN/dUD3t3fNLPHzaydu++I3agiiZO7sYQx03L5bMMeBnZrw73q85QAiuYqFwOeAfLd/aGjrOkAbHV3N7OBQCNgZ0wnFUmAkkPl/OGdFbz08TratGjKw98+j+vO76TTKxJI0bxDvxi4BfjczD4Lb/stkAXg7k8CNwA/MrMK4BAw0t11SkWS1pE+z/tm5bPrQBm3XNiFn191tvo8JdCiucplAXDMtyPuPhmYHKuhRBJpxZZ9jM3J5ZM1uzi/c2uev019npIc9N0ckbD9hyt49N2VPPf3tbRs1oT7v3kuN2arz1OShwJd0t6RPs97Z+azZW9pqM9zSE/aqM9TkowCXdJa0fb9jJ+ex/xVoT7Px2/uT3/1eUqSUqBLWjpUVskf3y9kyrwimmY04p7hvbn5QvV5SnJToEvaeXf5Vu6ekUfxbvV5SmpRoEva2LAr1Of5bn6oz/OV0Rdyofo8JYUo0CXlHa6oZMoHRUx+v5DGjYzfDu3JbRd3U5+npBwFuqS0eSu3M356Hmt2HOCaczsy5lr1eUrqUqBLStpccoh7Z+bzxueb6dauBS9+fyCXnKU+T0ltCnRJKeWVVTz39zU88m6oz/MXV57F6EvV5ynpQYEuKWNh0U7G5uSycut+rjjnVMYP660+T0krCnRJeurzFAlRoEvSqqxyXvp4Hf/99gr1eYqgQJcktWT9bsZOyyVvk/o8RY5QoEtS2X2gjAfeKuCVT0N9nn/8Tn+Gnqs+TxFQoEuSqKpypi7awP1vFbBPfZ4iEenVIIGXu7GEsTm5/GP9HgZ2bcPE6/pwdgf1eYrUpECXwCo5VM5D76zgTx+vo02LE3joxvP4Rj/1eYocjQJdAsfdmfbZRn73RgG7Dhzm5gu78Av1eYrUSoEugbJy6z7GTstl4ZpdnNe5Nc/fdoH6PEWipECXQDhwuIJH56zi2QVraNmsCfd981y+rT5PkTpRoEtCuTuzcrcwceZyNpeU8u3szvz6avV5ihwPBbokzJodBxiXk8v8VTvo1fFkJn+nPwO6qM9T5Hgp0CXuSstDfZ7/80ERTZs04u5hvbj5wi40UeGESL3UGuhm1hl4EegAVAFT3P3RGmsMeBQYChwERrn7ktiPK3W2bCrMmQAlxdAqEwaPg743Jmyc6n2e3+jXibuG9uTUk5olbJ4vCdjxEqmLaN6hVwC/cPclZnYSsNjMZrv78mprrgZ6hD8GAU+Ef5VEWjYVZtwB5YdC90s2hO5D3EMq1Oe5nHfzt9Lj1Jb85V8v5KIzA9bnGaDjJXI8ag10d98MbA7f3mdm+UAnoHqgjwBedHcHPjaz1mbWMfy5kihzJvxfOB1Rfii0PU4Bdbiikqfmhfo8G5lx19U9+f5XA9rnGYDjJVIfdTqHbmZdgX7Awhq7OgEbqt0vDm/7QqCb2WhgNEBWVlbdJpW6Kymu2/YYm79qO+Nz8ijacYCh53ZgzDW9OL11gPs8E3y8ROor6kA3s5bAX4E73X1vzd0RPsW/tMF9CjAFIDs7+0v7JcZaZYZOG0Ta3oC2lJQy8Y3lvLFsM13bnsgL3x/IpcnQ55mg4yUSK1F93WtmGYTC/M/u/nqEJcVA52r3M4FN9R9P6mXwOMio8Y44o3loewMor6ziqXlFDP7DXN5dvpWfX3kWb915SXKEOcT9eInEWjRXuRjwDJDv7g8dZdl04Cdm9gqhb4aW6Px5ABw57xuHqzYWFu1kXE4eK7bu4/Kep3L3sN5ktU2yPs84Hi+RhmCh72MeY4HZV4H5wOeELlsE+C2QBeDuT4ZDfzIwhNBli7e5+6JjPW52drYvWnTMJZIEtu87zH1v5vP6PzbSqXVzxg/rxZW9TtP/iCjSQMxssbtnR9oXzVUuC4h8jrz6Ggf+/fjGk2RUWeX8eeE6Hnx7BaXllfz7187kJ1/roT5PkQTST4pKnf1j/W7G5uSSu3EvX+3ejntG9OZM9XmKJJwCXaK2+0AZv397Ba98up5TT2rKYzf149q+HXV6RSQgFOhSq6oq538Xb+D+WQXsLa3g9ou7ceeVZ6nPUyRg9IqUY8rbVMLYabksWb+HC7qewsTr+tCzw8mJHktEIlCgS0R7S8t56J2VvPjRWtq0OIE/fOs8vtlffZ4iQaZAly9wd3I+28S9b+Sz88BhblGfp0jSUKDLP63auo+xObl8XBTq83xu1AWcm6k+T5FkoUAXDhyuYNKcVTyzYA0tmjbh/33jXEZeoD5PkWSjQE9j7s5buVuYoD5PkZSgQE9Ta3YcYPz0POat3M45HU9m8nf6MaBLm0SPJSL1oEBPM6XllTz+fiFPflDECU0aMX5YL25Rn6dISlCgp5E5+aE+zw27DjHi/NP5r6HncOrJAerzFJF6UaCngeLdoT7P2cu30v3Ulrz8r4P4ypntEj2WiMSYAj2FHa6o5On5a3jsvVUYxm+u7sn3L+7GCU10ekUkFSnQU9SCVTsYl5NL0Y4DDOndgbHDetEpyH2eIlJvCvQUs6WklHvfWM7MZZvp0vZEnr/tAi47+9REjyUicaBATxHllVW88OFaHp69kvIq52dXnMW/XXoGzTJUOCGSLhToKeCTNbsYOy03ufs8RaTeFOhJbMf+w9z3ZgF/XVJMp9bNmXLLAPV5iqQxBXoSqqxyXg73eR4qr+THl53JTy7vzokn6I9TJJ0pAZLMZxv2MHZaLp9vLOHi7m25Z3gfup+qPk8RUaAnjT0Hy3jgrVCfZ/uW6vMUkS9ToAdcVZXz2uJi7n+rgJJD5Xz/4m7ceUUPTmqmwgkR+SIFeoDV7POcMKIP53RUn6eIRFZroJvZs8C1wDZ37xNh/2VADrAmvOl1d58QyyHTTfU+z1NOPIH//tZ5XK8+TxGpRTTv0J8HJgMvHmPNfHe/NiYTpbEjfZ6/ezOfHfsPc/OgLvzyqrNpdaJOr4hI7WoNdHefZ2ZdG36U9PaFPs/MVjxzazZ9M1sneiwRSSKxOod+kZktBTYBv3T3vEiLzGw0MBogKysrRk+d3A4crmDSe6t4Zn6oz/N33+jDyAuyaKw+TxGpo1gE+hKgi7vvN7OhwDSgR6SF7j4FmAKQnZ3tMXjupHWkz3PizOVsKinlxuxMfj2kJ21bNk30aCKSpOod6O6+t9rtN83scTNr5+476vvYqWptuM/zg5Xb6dnhJB5Tn6eIxEC9A93MOgBb3d3NbCDQCNhZ78lSUGl5JY/PXc2TH6zmhMaNGHdtL753kfo8RSQ2orls8S/AZUA7MysGxgMZAO7+JHAD8CMzqwAOASPdPa1Pp0TyXsFW7p6+nPW7DqrPU0QaRDRXudxUy/7JhC5rlAjU5yki8aKfFG0gZRVVPDW/6J99nr8e0pPbv6o+TxFpOAr0BvD3wh2MzcmlaLv6PEUkfhToMbR1bykTZ6rPU0QSQ4EeAxWVVTz/4VoeeXcVZZVV3HlFD3546Znq8xSRuFKg19Ona0N9ngVb9vG1s9tz9/DedGnbItFjiUgaUqAfp5p9nv9zywCuUp+niCSQAr2OKquclz9Zz4NvFXCovJIfXXYmP1Wfp4gEgFKoDpZu2MPYnFyWFZfwlTPbMmGE+jxFJDgU6FHYc7CM37+9gr98EurznHRTP4apz1NEAkaBfgzq8xSRZKJAP4rlm/YyNieXxet2k93lFCZepz5PEQk2BXoN+0rLeWj2Sl74MNTn+eANfbm+fyaNVDghIgGnQA9zd6Yv3cS9b4T6PL87KIv/vKqn+jxFJGko0IHCbfsYOy2Pj4p20jezFU9/L5vzOqvPU0SSS1oH+sGyCibNKeTp+UW0aNqEe6/rw00D1ecpIskpLQPd3Xk7bwsTZoT6PL81IJPfXK0+TxFJbmkX6DX7PCfd1I/srurzFJHklzaBXlpeyRNzV/OE+jxFJEWlRaC/X7CN8dPzWL/rIMPPO53/uuYcTlOfp4ikmJQO9OLdB5kwYznvLN/Kme1b8PIPBvGV7urzFJHUlJKBXlZRxdMLipg0R32eIpI+Ui7QPwz3ea7efoCv9z6NccN6q89TRNJCygT61r2l3PtGPjOWbqJL2xN57rYL+Jr6PEUkjSR9oFdUVvHCR+t4ePZK9XmKSFqrNdDN7FngWmCbu/eJsN+AR4GhwEFglLsvifWgACybCnMmQEkxtMqkqO/P+fHn3SnYso/Lzm7PPerzFJE0Fs13CZ8Hhhxj/9VAj/DHaOCJ+o8VwbKpMOMOKNkAOJRsoMO8XzNo3xyevHkAz426QGEuImmt1kB393nArmMsGQG86CEfA63NrGOsBvynOROg/NAXNp1oZYxv8RpD+nRQe5CIpL1YXMfXCdhQ7X5xeNuXmNloM1tkZou2b99et2cpKY64udHejXV7HBGRFBWLQI/01tgjLXT3Ke6e7e7Z7du3r9uztMqs23YRkTQTi0AvBjpXu58JbIrB437R4HGQUeN68ozmoe0iIhKTQJ8OfM9CLgRK3H1zDB73i/reCMMmQavOgIV+HTYptF1ERKK6bPEvwGVAOzMrBsYDGQDu/iTwJqFLFgsJXbZ4W0MNS98bFeAiIkdRa6C7+0217Hfg32M2kYiIHBf9b1UiIilCgS4ikiIU6CIiKUKBLiKSIiz0Pc0EPLHZdmDdcX56O2BHDMeJlaDOBcGdTXPVjeaqm1Scq4u7R/zJzIQFen2Y2SJ3z070HDUFdS4I7myaq240V92k21w65SIikiIU6CIiKSJZA31Kogc4iqDOBcGdTXPVjeaqm7SaKynPoYuIyJcl6zt0ERGpQYEuIpIiAh3oZjbEzFaYWaGZ/SbC/qZm9mp4/0Iz6xqQuUaZ2XYz+yz88YM4zfWsmW0zs9yj7DczmxSee5mZ9Q/IXJeZWUm149Xg/8m9mXU2s/fNLN/M8szsPyKsifvxinKuuB+v8PM2M7NPzGxpeLZ7IqyJ+2syyrkS9ZpsbGb/MLOZEfbF/li5eyA/gMbAauAM4ARgKdCrxpofA0+Gb48EXg3IXKOAyQk4ZpcA/YHco+wfCswi1DJ1IbAwIHNdBsyM87HqCPQP3z4JWBnhzzHuxyvKueJ+vMLPa0DL8O0MYCFwYY01iXhNRjNXol6TPwdejvTn1RDHKsjv0AcChe5e5O5lwCuECqmrGwG8EL79GjDYGr4tOpq5EsKDUuhd97nizt03u/uS8O19QD5f7sKN+/GKcq6ECB+H/eG7GeGPmldVxP01GeVccWdmmcA1wNNHWRLzYxXkQI+mfPqfa9y9AigB2gZgLoDrw1+mv2ZmnSPsT4SoC70T4KLwl8yzzKx3PJ84/KVuP0Lv7KpL6PE6xlyQoOMVPoXwGbANmO3uRz1mcXxNRjMXxP81+QjwK6DqKPtjfqyCHOjRlE9HXVAdQ9E85wygq7v3Bd7l//4VTrREHK9oLCH0/1OcBzwGTIvXE5tZS+CvwJ3uvrfm7gifEpfjVctcCTte7l7p7ucT6g4eaGZ9aixJyDGLYq64vibN7Fpgm7svPtayCNvqdayCHOjRlE//c42ZNQFa0fBf2tc6l7vvdPfD4btPAQMaeKZoxafQu47cfe+RL5nd/U0gw8zaNfTzmlkGodD8s7u/HmFJQo5XbXMl6njVmGEPMBcYUmNXIl6Ttc6VgNfkxcBwM1tL6LTs5Wb2Uo01MT9WQQ70T4EeZtbNzE4g9E2D6TXWTAduDd++AXjPw99hSORcNc6zDid0HjQI4lPoXUdm1uHIuUMzG0jo7+XOBn5OA54B8t39oaMsi/vximauRByv8HO1N7PW4dvNgSuAghrL4v6ajGaueL8m3f0ud890966EMuI9d7+5xrKYH6taO0UTxd0rzOwnwNuErix51t3zzGwCsMjdpxP6i/8nMysk9C/byIDMdYeZDQcqwnONaui5IGCF3nWb6wbgR2ZWARwCRsbhH+aLgVuAz8PnXgF+C2RVmysRxyuauRJxvCB0Bc4LZtaY0D8iU919ZqJfk1HOlZDXZE0Nfaz0oxG+aLkAAAAzSURBVP8iIikiyKdcRESkDhToIiIpQoEuIpIiFOgiIilCgS4ikiIU6CIiKUKBLiKSIv4/AOVmICcuGxYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, X@beta)\n",
    "plt.plot(x, y, 'o')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1. Простая линейная регрессия\n",
    "\n",
    "Один из самых распространенных методов нахождения такой аппроксимирующей прямой - *метод наименьших квадратов* (МНК).\n",
    "Итак, пусть имеются точки\n",
    "$$\n",
    "(x_1, y_1),\\, (x_2, y_2),\\, \\dots,\\, (x_m, y_m).\n",
    "$$\n",
    "Ищем линейную функцию (в математической статистике ее называют *простой линейной регрессией*)\n",
    "$$\n",
    "y = \\beta_0 + \\beta_1 x,\n",
    "$$\n",
    "такую, что\n",
    "$$\n",
    "y_i \\approx \\beta_0 + \\beta_1 x_i\n",
    "\\qquad\n",
    "(i=1,2,\\dots, m).\n",
    "$$\n",
    "А именно, ищем $\\beta_0$, $\\beta_1$, на которых достигается минимум суммы квадратов невязок (*остаточной суммы квадратов*)\n",
    "$$\n",
    "R(\\beta_0,\\, \\beta_1) = \\sum_{i=1}^m \\bigl(y_i - \\beta_0 - \\beta_1 x_i\\bigr)^2 \\to \\min_{\\beta_0,\\, \\beta_1} \\tag{*}\n",
    "$$\n",
    "$R(\\beta_0,\\, \\beta_1)$ - это сумма квадратов отклонений предсказанных значений $\\beta_0 + \\beta_1 x_i$ от истинных $y_i$. \n",
    "\n",
    "Чтобы найти миниимум $(*)$ возьмум частные производные по каждому неизвестному параметру $\\beta_0$ и $\\beta_1$ и приравняем их нулю:\n",
    "$$\n",
    "\\frac{\\partial R}{\\partial \\beta_0} = -2\\sum_{i=1}^m \\bigl(y_i - \\beta_0 - \\beta_1 x_i\\bigr) = 0,\n",
    "\\qquad\n",
    "\\frac{\\partial R}{\\partial \\beta_0} = -2\\sum_{i=1}^m \\bigl(y_i - \\beta_0 - \\beta_1 x_i\\bigr) x_i = 0.\n",
    "$$\n",
    "После очевидных преобразований получаем\n",
    "$$\n",
    "\\left\\{\n",
    "\\begin{array}{l}\n",
    "        \\beta_0 m \\hskip{.8cm}   + \\beta_1 \\sum\\limits_{i=1}^m   x_i  \\hskip{.44cm} = \\sum\\limits_{i=1}^m y_i, \\\\\n",
    " \\beta_0 \\sum\\limits_{i=1}^m x_i + \\beta_1 \\sum\\limits_{i=1}^m x_i^2 = \\sum\\limits_{i=1}^m x_iy_i. \\\\\n",
    "\\end{array} \n",
    "\\right.  \\tag{**}\n",
    "$$\n",
    "Получили систему линейных уравнений относительно неизвестных $\\beta_0$, $\\beta_1$.\n",
    "Она называется *системой нормальных уравнений*.\n",
    "Можно доказать, что решение этой системы доставляет минимум функции ($*$) (для этого надо проверить вторые производные, но мы не будем этим сейчас заниматься).\n",
    "\n",
    "**Пример.** Пусть имеются точки $(x_i, y_i)$ $(i=1,2,\\dots,m)$, $m=5$,\n",
    "$$\n",
    "\\begin{array}{|c|c|c|c|}\n",
    "\\hline\n",
    "x_i & 0 & 1 & 2 & 3 & 4 \\\\\n",
    "\\hline                  \n",
    "y_i & 1 & 2 & 2 & 3 & 4 \\\\\n",
    "\\hline\n",
    "\\end{array}\n",
    "$$\n",
    "Находим $\\sum\\limits_{i=1}^5 x_i = 10$, $\\sum\\limits_{i=1}^5 x_i^2 = 30$, $\\sum\\limits_{i=1}^5 y_i = 12$, \n",
    "$\\sum\\limits_{i=1}^5 x_iy_i = 31$. Система ($**$) приобретает вид\n",
    "$$\n",
    "\\left\\{\n",
    "\\begin{array}{r}\n",
    "       5 \\beta_0 + 10 \\beta_1 = 12, \\\\\n",
    "      10 \\beta_0 + 30 \\beta_1 = 31.\n",
    "\\end{array}      \n",
    "\\right.      \n",
    "$$\n",
    "Находим решение этой системы $\\beta_0 = 1$, $\\beta=0.7$.\n",
    "Таким образом, уравнение искомой аппроксимирующей прямой - $y=1+0.7x$.\n",
    "Именно она изображена на рисунке выше.\n",
    "\n",
    "**Замечание.**\n",
    "Если ввести матрицы\n",
    "$$\n",
    "X = \n",
    "\\left(\n",
    "\\begin{array}{cc}\n",
    "1 & x_1 \\\\\n",
    "1 & x_2 \\\\\n",
    "\\vdots & \\vdots \\\\\n",
    "1 & x_m \\\\\n",
    "\\end{array}\n",
    "\\right),\n",
    "\\qquad\n",
    "y =\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "y_1 \\\\\n",
    "y_2 \\\\\n",
    "\\vdots \\\\\n",
    "y_m \\\\\n",
    "\\end{array}\n",
    "\\right),\n",
    "\\qquad\n",
    "\\beta =\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "\\beta_0 \\\\\n",
    "\\beta_1 \\\\\n",
    "\\end{array}\n",
    "\\right),\n",
    "$$\n",
    "тогда систему нормальных уравнений ($**$) можно записать как\n",
    "$$\n",
    "(X^{\\rm T} X)\\, \\beta = X^{\\rm T} y. \\tag{***}\n",
    "$$\n",
    "\n",
    "В частности, для рассмотренного примера\n",
    "$$\n",
    "X^{\\rm T} X =\n",
    "\\left(\n",
    "\\begin{array}{ccccc}\n",
    "  1 & 1 & 1 & 1 & 1 \\\\\n",
    "  0 & 1 & 2 & 3 & 4\n",
    "\\end{array}\n",
    "\\right)\n",
    "\\cdot\n",
    "\\left(\n",
    "\\begin{array}{cc}\n",
    "  1 & 0 \\\\\n",
    "  1 & 1 \\\\\n",
    "  1 & 2 \\\\\n",
    "  1 & 3 \\\\\n",
    "  1 & 4 \\\\\n",
    "\\end{array}\n",
    "\\right)\n",
    "=\n",
    "\\left(\n",
    "\\begin{array}{cc}\n",
    "  5 & 10 \\\\\n",
    " 10 & 30 \\\\\n",
    "\\end{array}\n",
    "\\right),\n",
    "\\qquad\n",
    "X^{\\rm T} \\beta =\n",
    "\\left(\n",
    "\\begin{array}{ccccc}\n",
    "  1 & 1 & 1 & 1 & 1 \\\\\n",
    "  0 & 1 & 2 & 3 & 4\n",
    "\\end{array}\n",
    "\\right)\n",
    "\\cdot\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "  1 \\\\\n",
    "  2 \\\\\n",
    "  2 \\\\\n",
    "  3 \\\\\n",
    "  4 \\\\\n",
    "\\end{array}\n",
    "\\right)\n",
    "=\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "  12 \\\\\n",
    "  31 \\\\\n",
    "\\end{array}\n",
    "\\right).\n",
    "$$\n",
    "\n",
    "Давайте все это опробуем в Питоне. Напрямую будем вычислять матрицы $X^{\\rm T} X$, $X^{\\rm T} y$.\n",
    "А для решения системы ($***$) воспользуемся функцией `linalg.solve`. \n",
    "На практике мы рекомендуем пользоваться функцией `linalg.lstsq`, описанной ниже."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.0000000000000002, 0.7)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "x = np.array([0, 1, 2, 3, 4])\n",
    "y = np.array([1, 2, 2, 3, 4])\n",
    "\n",
    "n = len(x)\n",
    "X = np.stack((np.ones(n), x)).T\n",
    "beta = np.linalg.solve(X.T@X, X.T@y)\n",
    "beta0, beta1 = beta\n",
    "beta0, beta1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "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": [
    "plt.plot(x, y, 'o')\n",
    "plt.plot(x, X@beta)\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2. МНК в общей постановке\n",
    "\n",
    "В более общей постановке задачи аппроксимации предполагают, что данные \n",
    "$$\n",
    "(x_1, y_1),\\, (x_2, y_2),\\, \\dots,\\, (x_m, y_m)\n",
    "$$\n",
    "удовлетворяют соотношению\n",
    "$$\n",
    "y_i = f(x_i) + \\mbox{ошибка} \\qquad (i=1,2,\\dots,m),\n",
    "$$\n",
    "где $f(x)$ - некоторая функция, называемая *модельной функцией* или *моделью*.\n",
    "Иными словами,\n",
    "$$\n",
    "y_i \\approx f(x_i) \\qquad (i=1,2,\\dots,m). \\tag{+}\n",
    "$$\n",
    "\n",
    "Часто модель ищут в виде \n",
    "$$\n",
    "f(x) = \\beta_1 f_1(x) + \\beta_2 f_2(x) + \\dots + \\beta_n f_n(x), \\tag{++}\n",
    "$$\n",
    "где $f_1(x),f_2(x),\\dots,f_n(x)$ - некоторые заданные функции.\n",
    "Например, при аппроксимации данных линейной функцией (как было выше) $f_1(x) = 1$, $f_2(x)=x$. \n",
    "В более общем случае рассматривают аппроксимацию полиномами (многочленами), тогда\n",
    "$f_1(x) = 1,\\, f_2(x)=x\\, \\dots,\\, f_n(x)=x^{n-1}$.\n",
    "\n",
    "Есть разные подходы, как понимать приближенное равенство в ($+$).\n",
    "Наиболее популярный подход - *метод наименьших квадратов*, когда ищут значения параметров $\\beta_1, \\dots, \\beta_n$,\n",
    "на которых достигается минимум\n",
    "$$\n",
    " \\sum_{i=1}^m \\left(\\beta_1 f_1(x_i) + \\dots + \\beta_n f_n(x_i) - y_i \\strut\\right)^2 \\to \\min_{\\beta_1, \\dots, \\beta_n}. \\tag{+++}\n",
    "$$\n",
    "При этом\n",
    "$$\n",
    "{\\rm MSE} = \\frac{1}{m}\\sum_{i=1}^m \\left(\\beta_1 f_1(x_i) + \\dots + \\beta_n f_n(x_i) - y_i \\strut\\right)^2\n",
    "$$\n",
    "называют *средним квадратом ошибки* (*mean squared error*).\n",
    "\n",
    "Так как параметры $\\beta_1, \\dots, \\beta_n$ в ($++$) входят в модель линейно, то метод называется *линейным* методом наименьших квадратов (при этом функции $f_1(x),f_2(x),\\dots,f_n(x)$ не обязаны быть линейными).\n",
    "\n",
    "Заметим, что $x$ не обязан быть скаляром, а может быть вектором.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 3. МНК на матричном языке\n",
    "\n",
    "Очень удобно задачу наименьших квадратов исследовать, используя матричные обозначения.\n",
    "Введем матрицы \n",
    "$$\n",
    "X = \n",
    "\\left(\n",
    "\\begin{array}{cc}\n",
    "f_1(x_1) & f_2(x_1) & \\dots & f_n(x_1) \\\\\n",
    "f_1(x_2) & f_2(x_2) & \\dots & f_n(x_2) \\\\\n",
    "\\dots \\\\\n",
    "f_1(x_m) & f_2(x_m) & \\dots & f_n(x_m) \\\\\n",
    "\\end{array}\n",
    "\\right),\n",
    "\\qquad\n",
    "y =\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "y_1 \\\\\n",
    "y_2 \\\\\n",
    "\\vdots \\\\\n",
    "y_m \\\\\n",
    "\\end{array}\n",
    "\\right),\n",
    "\\qquad\n",
    "\\beta =\n",
    "\\left(\n",
    "\\begin{array}{c}\n",
    "\\beta_1 \\\\\n",
    "\\beta_2 \\\\\n",
    "\\vdots \\\\\n",
    "\\beta_n\n",
    "\\end{array}\n",
    "\\right).\n",
    "$$\n",
    "\n",
    "Обратите внимание, что все элементы в $X$ и $y$ известны, а коэффициенты в $\\beta$ пока нет.\n",
    "\n",
    "Формулы $(+)$ тогда можно записать как\n",
    "$$\n",
    "X\\beta \\approx y\n",
    "$$\n",
    "а задачу минимизации $(+++)$ - следующим образом:\n",
    "$$\n",
    "\\|X\\beta - y\\|_2^2 \\to \\min_{\\beta}.\n",
    "$$\n",
    "Заметим, что минимум нормы $\\|X\\beta - y\\|_2$ совпадает с минимумом ее квадрата $\\|X\\beta - y\\|_2^2$ (так как квадрат - функция монотонная для неотрицательного аргумента), но с $\\|X\\beta - y\\|_2^2$ работать проще, так как здесь отсутствуют радикалы.\n",
    "\n",
    "Обратите внимание, что равенство $X\\beta \\approx y$ означает, что мы желаем приближенно представить столбец $y$ в виде линейной комбинации столбцов матрицы $X$. Требование $\\|X\\beta - y\\|_2 \\to \\min_{\\beta}$ специфицирует, *что конкретно* означает здесь это *приближение* - минимизацию евклидовой нормы невязки $X\\beta - y$.\n",
    "\n",
    "Если бы систему $X\\beta = y$ удавалось решить точно, то норма невязки равнялась бы нулю.\n",
    "В этом случае график модельной зависимости проходил бы точно через заданные точки.\n",
    "Однако, как правило, при аппроксимации данных получаемая с.л.у. $X\\beta = y$ переопределена, т.е. не имеет решений.\n",
    "Вектор $\\beta$, на котором минимизируется евклидова норма  $\\|X\\beta - y\\|_2$ называется *псевдорешением* системы $X\\beta = y$.\n",
    "Итак, решение линейной задачи наименьших квадратов есть псевдорешение системы $X\\beta = y$.\n",
    "\n",
    "Для нахождения решения задачи наименьших квадратов можно также найти все частные производные по каждому неизвестному параметру $\\beta_1, \\beta_2, \\dots, \\beta_n$ и приравнять их к нулю, как мы это делали для частного случая. Оказывается, мы придем к *системе нормальных уравнений* того же вида ($**$)\n",
    "$$\n",
    "(X^{\\rm T} X)\\, \\beta = X^{\\rm T} y. \\tag{**}\n",
    "$$\n",
    "Псевдорешение системы $X\\beta = y$ есть решение системы нормальных уравнений.\n",
    "\n",
    "Итак, решение линейной задачи наименьших квадратов сводится к решение системы нормальных линейных уравнений.\n",
    "\n",
    "**Замечание 1.** При решении больших задач обычно явно не вычилсяют матрицу $X^{\\rm T} X$ (она может оказаться плохо обусловленной), а используют более продвинутые методы решения линейной задачи наименьших квадратов, например, использующие \n",
    "$QR$-разложение, сингулярное разложение и др.\n",
    "\n",
    "**Замечание 2.** Вместо минимизации евклидовой нормы (т.е. вместо минимизации суммы квадратов) мы могли бы минимизировать другую норму, например, чебышевскую (минимизировать максимальную ошибку) или манхеттеновскую (минимизировать сумму модулей ошибок). В некоторых практических приложениях такие постановки задачи аппроксимации более естественны. Однако евклидову норму минимизировать проще. Все сводится к решению системы нормальных уравнений. Для минимизации других норм используют более сложные методы (методы оптимизации).\n",
    "\n",
    "**Замечание 3.** Часто встречаются задачи аппроксимации, в которых неизвестные параметры входят нелинейным образом. Например, по данным мы ищем зависимость вида\n",
    "$$\n",
    "f(x) = \\beta_1 e^{\\lambda_1 x} + \\beta_2 e^{\\lambda_2 x}.\n",
    "$$\n",
    "Если $\\lambda_1$, $\\lambda_2$ известны, а требуется найти $\\beta_1$, $\\beta_2$, то это *линейная задача* наименьших квадратов.\n",
    "Если же требуется найти $\\lambda_1$, $\\lambda_2$, то это *нелинейная задача* наименьших квадратов. Для решения нелинейных задач наименьших квадратов используют более сложные методы (методы оптимизации).\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " ## 4. Функция `linalg.lstsq()`\n",
    " Линейную задачу наименьших квадратов решает функция `linalg.lstsq(X, y)`.\n",
    " Вернемся к нашему модельному примеру."
   ]
  },
  {
   "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.array([0, 1, 2, 3, 4])\n",
    "y = np.array([1, 2, 2, 3, 4])\n",
    "plt.plot(x, y, 'o', color='tab:orange')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Формируем матрицу `X` и обращаемся к функции `linalg.lstsq(X, y)`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = len(x)\n",
    "X = np.stack((np.ones(m), x)).T\n",
    "beta, residuals, rank, sv = linalg.lstsq(X, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`lstsq()` возвращает коэффициенты `beta`, сумму квадратов `residuals`, ранг `rank` матрицы `X` и ее сингулярные числа `sv`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.9999999999999993, 0.7000000000000002)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta0, beta1 = beta\n",
    "beta0, beta1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bCmwCfunueRE+fzQwGiArK6uus4oEwq4DZTwwq4BXF21Qn6cEStSBbmYtgb8Cd7r73hq7lwBd3H2/mQ0FpgE9aj6Gu08BpgBkZ2f7cU8tkgBVVc6rizbwwFsF7C+tYHS4z7Ol+jwlIKL6m2hmGYTC/M/u/nrN/dUD3t3fNLPHzaydu++I3agiiZO7sYQx03L5bMMeBnZrw73q85QAiuYqFwOeAfLd/aGjrOkAbHV3N7OBQCNgZ0wnFUmAkkPl/OGdFbz08TratGjKw98+j+vO76TTKxJI0bxDvxi4BfjczD4Lb/stkAXg7k8CNwA/MrMK4BAw0t11SkWS1pE+z/tm5bPrQBm3XNiFn191tvo8JdCiucplAXDMtyPuPhmYHKuhRBJpxZZ9jM3J5ZM1uzi/c2uev019npIc9N0ckbD9hyt49N2VPPf3tbRs1oT7v3kuN2arz1OShwJd0t6RPs97Z+azZW9pqM9zSE/aqM9TkowCXdJa0fb9jJ+ex/xVoT7Px2/uT3/1eUqSUqBLWjpUVskf3y9kyrwimmY04p7hvbn5QvV5SnJToEvaeXf5Vu6ekUfxbvV5SmpRoEva2LAr1Of5bn6oz/OV0Rdyofo8JYUo0CXlHa6oZMoHRUx+v5DGjYzfDu3JbRd3U5+npBwFuqS0eSu3M356Hmt2HOCaczsy5lr1eUrqUqBLStpccoh7Z+bzxueb6dauBS9+fyCXnKU+T0ltCnRJKeWVVTz39zU88m6oz/MXV57F6EvV5ynpQYEuKWNh0U7G5uSycut+rjjnVMYP660+T0krCnRJeurzFAlRoEvSqqxyXvp4Hf/99gr1eYqgQJcktWT9bsZOyyVvk/o8RY5QoEtS2X2gjAfeKuCVT0N9nn/8Tn+Gnqs+TxFQoEuSqKpypi7awP1vFbBPfZ4iEenVIIGXu7GEsTm5/GP9HgZ2bcPE6/pwdgf1eYrUpECXwCo5VM5D76zgTx+vo02LE3joxvP4Rj/1eYocjQJdAsfdmfbZRn73RgG7Dhzm5gu78Av1eYrUSoEugbJy6z7GTstl4ZpdnNe5Nc/fdoH6PEWipECXQDhwuIJH56zi2QVraNmsCfd981y+rT5PkTpRoEtCuTuzcrcwceZyNpeU8u3szvz6avV5ihwPBbokzJodBxiXk8v8VTvo1fFkJn+nPwO6qM9T5Hgp0CXuSstDfZ7/80ERTZs04u5hvbj5wi40UeGESL3UGuhm1hl4EegAVAFT3P3RGmsMeBQYChwERrn7ktiPK3W2bCrMmQAlxdAqEwaPg743Jmyc6n2e3+jXibuG9uTUk5olbJ4vCdjxEqmLaN6hVwC/cPclZnYSsNjMZrv78mprrgZ6hD8GAU+Ef5VEWjYVZtwB5YdC90s2hO5D3EMq1Oe5nHfzt9Lj1Jb85V8v5KIzA9bnGaDjJXI8ag10d98MbA7f3mdm+UAnoHqgjwBedHcHPjaz1mbWMfy5kihzJvxfOB1Rfii0PU4Bdbiikqfmhfo8G5lx19U9+f5XA9rnGYDjJVIfdTqHbmZdgX7Awhq7OgEbqt0vDm/7QqCb2WhgNEBWVlbdJpW6Kymu2/YYm79qO+Nz8ijacYCh53ZgzDW9OL11gPs8E3y8ROor6kA3s5bAX4E73X1vzd0RPsW/tMF9CjAFIDs7+0v7JcZaZYZOG0Ta3oC2lJQy8Y3lvLFsM13bnsgL3x/IpcnQ55mg4yUSK1F93WtmGYTC/M/u/nqEJcVA52r3M4FN9R9P6mXwOMio8Y44o3loewMor6ziqXlFDP7DXN5dvpWfX3kWb915SXKEOcT9eInEWjRXuRjwDJDv7g8dZdl04Cdm9gqhb4aW6Px5ABw57xuHqzYWFu1kXE4eK7bu4/Kep3L3sN5ktU2yPs84Hi+RhmCh72MeY4HZV4H5wOeELlsE+C2QBeDuT4ZDfzIwhNBli7e5+6JjPW52drYvWnTMJZIEtu87zH1v5vP6PzbSqXVzxg/rxZW9TtP/iCjSQMxssbtnR9oXzVUuC4h8jrz6Ggf+/fjGk2RUWeX8eeE6Hnx7BaXllfz7187kJ1/roT5PkQTST4pKnf1j/W7G5uSSu3EvX+3ejntG9OZM9XmKJJwCXaK2+0AZv397Ba98up5TT2rKYzf149q+HXV6RSQgFOhSq6oq538Xb+D+WQXsLa3g9ou7ceeVZ6nPUyRg9IqUY8rbVMLYabksWb+HC7qewsTr+tCzw8mJHktEIlCgS0R7S8t56J2VvPjRWtq0OIE/fOs8vtlffZ4iQaZAly9wd3I+28S9b+Sz88BhblGfp0jSUKDLP63auo+xObl8XBTq83xu1AWcm6k+T5FkoUAXDhyuYNKcVTyzYA0tmjbh/33jXEZeoD5PkWSjQE9j7s5buVuYoD5PkZSgQE9Ta3YcYPz0POat3M45HU9m8nf6MaBLm0SPJSL1oEBPM6XllTz+fiFPflDECU0aMX5YL25Rn6dISlCgp5E5+aE+zw27DjHi/NP5r6HncOrJAerzFJF6UaCngeLdoT7P2cu30v3Ulrz8r4P4ypntEj2WiMSYAj2FHa6o5On5a3jsvVUYxm+u7sn3L+7GCU10ekUkFSnQU9SCVTsYl5NL0Y4DDOndgbHDetEpyH2eIlJvCvQUs6WklHvfWM7MZZvp0vZEnr/tAi47+9REjyUicaBATxHllVW88OFaHp69kvIq52dXnMW/XXoGzTJUOCGSLhToKeCTNbsYOy03ufs8RaTeFOhJbMf+w9z3ZgF/XVJMp9bNmXLLAPV5iqQxBXoSqqxyXg73eR4qr+THl53JTy7vzokn6I9TJJ0pAZLMZxv2MHZaLp9vLOHi7m25Z3gfup+qPk8RUaAnjT0Hy3jgrVCfZ/uW6vMUkS9ToAdcVZXz2uJi7n+rgJJD5Xz/4m7ceUUPTmqmwgkR+SIFeoDV7POcMKIP53RUn6eIRFZroJvZs8C1wDZ37xNh/2VADrAmvOl1d58QyyHTTfU+z1NOPIH//tZ5XK8+TxGpRTTv0J8HJgMvHmPNfHe/NiYTpbEjfZ6/ezOfHfsPc/OgLvzyqrNpdaJOr4hI7WoNdHefZ2ZdG36U9PaFPs/MVjxzazZ9M1sneiwRSSKxOod+kZktBTYBv3T3vEiLzGw0MBogKysrRk+d3A4crmDSe6t4Zn6oz/N33+jDyAuyaKw+TxGpo1gE+hKgi7vvN7OhwDSgR6SF7j4FmAKQnZ3tMXjupHWkz3PizOVsKinlxuxMfj2kJ21bNk30aCKSpOod6O6+t9rtN83scTNr5+476vvYqWptuM/zg5Xb6dnhJB5Tn6eIxEC9A93MOgBb3d3NbCDQCNhZ78lSUGl5JY/PXc2TH6zmhMaNGHdtL753kfo8RSQ2orls8S/AZUA7MysGxgMZAO7+JHAD8CMzqwAOASPdPa1Pp0TyXsFW7p6+nPW7DqrPU0QaRDRXudxUy/7JhC5rlAjU5yki8aKfFG0gZRVVPDW/6J99nr8e0pPbv6o+TxFpOAr0BvD3wh2MzcmlaLv6PEUkfhToMbR1bykTZ6rPU0QSQ4EeAxWVVTz/4VoeeXcVZZVV3HlFD3546Znq8xSRuFKg19Ona0N9ngVb9vG1s9tz9/DedGnbItFjiUgaUqAfp5p9nv9zywCuUp+niCSQAr2OKquclz9Zz4NvFXCovJIfXXYmP1Wfp4gEgFKoDpZu2MPYnFyWFZfwlTPbMmGE+jxFJDgU6FHYc7CM37+9gr98EurznHRTP4apz1NEAkaBfgzq8xSRZKJAP4rlm/YyNieXxet2k93lFCZepz5PEQk2BXoN+0rLeWj2Sl74MNTn+eANfbm+fyaNVDghIgGnQA9zd6Yv3cS9b4T6PL87KIv/vKqn+jxFJGko0IHCbfsYOy2Pj4p20jezFU9/L5vzOqvPU0SSS1oH+sGyCibNKeTp+UW0aNqEe6/rw00D1ecpIskpLQPd3Xk7bwsTZoT6PL81IJPfXK0+TxFJbmkX6DX7PCfd1I/srurzFJHklzaBXlpeyRNzV/OE+jxFJEWlRaC/X7CN8dPzWL/rIMPPO53/uuYcTlOfp4ikmJQO9OLdB5kwYznvLN/Kme1b8PIPBvGV7urzFJHUlJKBXlZRxdMLipg0R32eIpI+Ui7QPwz3ea7efoCv9z6NccN6q89TRNJCygT61r2l3PtGPjOWbqJL2xN57rYL+Jr6PEUkjSR9oFdUVvHCR+t4ePZK9XmKSFqrNdDN7FngWmCbu/eJsN+AR4GhwEFglLsvifWgACybCnMmQEkxtMqkqO/P+fHn3SnYso/Lzm7PPerzFJE0Fs13CZ8Hhhxj/9VAj/DHaOCJ+o8VwbKpMOMOKNkAOJRsoMO8XzNo3xyevHkAz426QGEuImmt1kB393nArmMsGQG86CEfA63NrGOsBvynOROg/NAXNp1oZYxv8RpD+nRQe5CIpL1YXMfXCdhQ7X5xeNuXmNloM1tkZou2b99et2cpKY64udHejXV7HBGRFBWLQI/01tgjLXT3Ke6e7e7Z7du3r9uztMqs23YRkTQTi0AvBjpXu58JbIrB437R4HGQUeN68ozmoe0iIhKTQJ8OfM9CLgRK3H1zDB73i/reCMMmQavOgIV+HTYptF1ERKK6bPEvwGVAOzMrBsYDGQDu/iTwJqFLFgsJXbZ4W0MNS98bFeAiIkdRa6C7+0217Hfg32M2kYiIHBf9b1UiIilCgS4ikiIU6CIiKUKBLiKSIiz0Pc0EPLHZdmDdcX56O2BHDMeJlaDOBcGdTXPVjeaqm1Scq4u7R/zJzIQFen2Y2SJ3z070HDUFdS4I7myaq240V92k21w65SIikiIU6CIiKSJZA31Kogc4iqDOBcGdTXPVjeaqm7SaKynPoYuIyJcl6zt0ERGpQYEuIpIiAh3oZjbEzFaYWaGZ/SbC/qZm9mp4/0Iz6xqQuUaZ2XYz+yz88YM4zfWsmW0zs9yj7DczmxSee5mZ9Q/IXJeZWUm149Xg/8m9mXU2s/fNLN/M8szsPyKsifvxinKuuB+v8PM2M7NPzGxpeLZ7IqyJ+2syyrkS9ZpsbGb/MLOZEfbF/li5eyA/gMbAauAM4ARgKdCrxpofA0+Gb48EXg3IXKOAyQk4ZpcA/YHco+wfCswi1DJ1IbAwIHNdBsyM87HqCPQP3z4JWBnhzzHuxyvKueJ+vMLPa0DL8O0MYCFwYY01iXhNRjNXol6TPwdejvTn1RDHKsjv0AcChe5e5O5lwCuECqmrGwG8EL79GjDYGr4tOpq5EsKDUuhd97nizt03u/uS8O19QD5f7sKN+/GKcq6ECB+H/eG7GeGPmldVxP01GeVccWdmmcA1wNNHWRLzYxXkQI+mfPqfa9y9AigB2gZgLoDrw1+mv2ZmnSPsT4SoC70T4KLwl8yzzKx3PJ84/KVuP0Lv7KpL6PE6xlyQoOMVPoXwGbANmO3uRz1mcXxNRjMXxP81+QjwK6DqKPtjfqyCHOjRlE9HXVAdQ9E85wygq7v3Bd7l//4VTrREHK9oLCH0/1OcBzwGTIvXE5tZS+CvwJ3uvrfm7gifEpfjVctcCTte7l7p7ucT6g4eaGZ9aixJyDGLYq64vibN7Fpgm7svPtayCNvqdayCHOjRlE//c42ZNQFa0fBf2tc6l7vvdPfD4btPAQMaeKZoxafQu47cfe+RL5nd/U0gw8zaNfTzmlkGodD8s7u/HmFJQo5XbXMl6njVmGEPMBcYUmNXIl6Ttc6VgNfkxcBwM1tL6LTs5Wb2Uo01MT9WQQ70T4EeZtbNzE4g9E2D6TXWTAduDd++AXjPw99hSORcNc6zDid0HjQI4lPoXUdm1uHIuUMzG0jo7+XOBn5OA54B8t39oaMsi/vximauRByv8HO1N7PW4dvNgSuAghrL4v6ajGaueL8m3f0ud890966EMuI9d7+5xrKYH6taO0UTxd0rzOwnwNuErix51t3zzGwCsMjdpxP6i/8nMysk9C/byIDMdYeZDQcqwnONaui5IGCF3nWb6wbgR2ZWARwCRsbhH+aLgVuAz8PnXgF+C2RVmysRxyuauRJxvCB0Bc4LZtaY0D8iU919ZqJfk1HOlZDXZE0Nfaz0oxG+aLkAAAAzSURBVP8iIikiyKdcRESkDhToIiIpQoEuIpIiFOgiIilCgS4ikiIU6CIiKUKBLiKSIv4/AOVmICcuGxYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.plot(x, X@beta)\n",
    "plt.plot(x, y, 'o')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Сумма квадратов отклонений равна"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.29999999999999954"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "residuals"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Проверим это:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.2999999999999998"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linalg.norm(X@beta - y)**2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 5. Функция `polyfit()`\n",
    "\n",
    "Для аппроксимации данных полиномами можно воспользоваться функцией `polyfit(x,y,n)`. Здесь `n` - степень искомого многочлена. Коэффициенты многочлена возвращаются в массиве в порядке убывания степеней."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.7, 1. ])"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.polyfit(x, y, 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 6. Еще один простой пример\n",
    "\n",
    "Давайте аппроксимируем данные, сгенерированные вручную.\n",
    "Точки будут лежать вдоль параболы.\n",
    "Запишем сами функцию и добавим к координате $y$ небольшую ошибку (распределенную по нормальному закону."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "def my_quadratic_function(x):\n",
    "    return x**2 - 0.6*x + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Парабола:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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STkKsrmgJJH4dQzfGTDLGZBhjMgoKCvy5aRFpoLJKF7fNzOBAcTl/vyWdDi3inY4kJ/BroVtrp1lr06216SkpOiMuEiyqXW7um7Wa9blF/G3CQAZ1bO50JDkJXYcuIqdkrWXKvCy+3JTPk+P7MKpPa6cjSR102aKInNLLi3bw9rI93HVhV24elup0HDmFeo/QjTGzgBFAsjEmF3gciAaw1r5ijGkNZABNAbcx5ldAmrX2iM9Si4hffLQml+c+28L4s9oy+bKeTseRetRb6NbaCfUsPwC091oiEQkI324/yOT31zG0Swueu7Y/Ebo8MeBpyEVEfmTD3mLufGsVnZMb8+pN6TSKinQ6knhAhS4i/2FnQQm3TF9BYlw0M28bTGKcrjUPFip0EfnBvqIybnp9BQBv3T6YNolxDieS06HLFkUEgMJjldz0+nKKy6qYPWkoXVKaOB1JTpOO0EWEkopqJr6xgtzDZbx2Szp92+nhzsFIR+giYa68ysWkNzPI2neEV288m6FdkpyOJGdIR+giYaza5eb+2Wv4bsch/nxtfy5Ja+V0JGkAFbpImHK5LQ++l8mCrDweuyKNqwfpdpJgp0IXCUNut+XhD9bxz7X7mDy6J7ed19npSOIFKnSRMGOt5Xcfb+D9Vbn86pLu/GJEN6cjiZeo0EXCiLWWKXOzmLViD/eM7Mr9F3d3OpJ4kQpdJExYa/njJ5uYuXQ3d5zfmYdG9cQYzc8SSlToImHAWsufF2zhtSXZTByeyqNjeqvMQ5CuQxcJcdZanluwhZe/3sENQzry+JVpKvMQpUIXCWHWWp7+ZBOvL8nmhiEdeXp8X5V5CFOhi4Qot7vm0XFvLt3NxOGpOjIPAyp0kRDkdlse/Wg9s1fmMOmCLjxyeS+VeRhQoYuEGJfbMvn9dXywOpd7R3bjwVE9VOZhQoUuEkKqXW4efC+Tf67dxwOX9uCXus48rKjQRUJEeZWLe99Zw5eb8pg8uqfuAA1DKnSREHCkvIqfz8xg5a5Cnhrfh5uGpTodSRygQhcJcgVHK7hl+gq25R9l6vUDGTegrdORxCEqdJEgllNYyk2vLyfvSAWv3XIOF/ZIcTqSOEiFLhKkthw4yk2vL6ei2s0/7hjCoI7NnY4kDlOhiwShlbsKuX3GSuJiInnvrmH0aJXgdCQJACp0kSAzL3MfD76XSftmccy8bTAdWsQ7HUkChApdJEhYa3ll0U6e/Wwz56Q2Z9pN6TRvHON0LAkgKnSRIFDtcvPY3CzeWb6HKwe05c/X9ic2OtLpWBJgVOgiAa6kopp731nN11sK+MWIrjw0qicREbqVX35MhS4SwPYWlfHzmRlszTvKn67qxw1DOjodSQKYCl0kQGXsKuTOt1ZRWe3m9VvSGdGzpdORJMDV+wg6Y8x0Y0y+MWZDHcuNMeZFY8x2Y8w6Y8wg78cUCS9zVuYw4e/LaBoXzUf3nKsyF4948kzRGcDoUyy/HOhe+2cS8HLDY4mEp2qXmyfmZTH5g3UM7ZLEx784l24tmzgdS4JEvYVurV0MFJ5ilfHAm7bGMqCZMaaNtwKeJA/rc4t99fIijikqreTWGSt549td3HZuZ96YeA6J8dFOx5Ig4skRen3aATnHvZ9b+28+8d6qXMa9tIS/frEVl9v6ajMifrVhbzFX/G0Jy3Ye4rlr+vPYlWlERXrj21PCiTe+Yk52/dRJm9YYM8kYk2GMySgoKDijjV3Zvy1XD2zPiwu3MfGNFRQeqzyj1xEJFO+u3MPVL3+Hy22Zc+cwrjung9ORJEh5o9BzgeO/AtsD+062orV2mrU23VqbnpJyZrPCxcVE8vxP+vPM1f1Ynl3IFS9+w5o9h8/otUScVF7lYvL7mTz8wXqGdG7B/PvOY6Am2JIG8EahzwVurr3aZShQbK3d74XXrZMxhgmDO/LBXcOJiDBc9+pS3ly6C2s1BCPBYc+hUq55+TvmZORy30XdmHHrYJKaNHI6lgS5eq9DN8bMAkYAycaYXOBxIBrAWvsK8CkwBtgOlAK3+irsifq1T2T+fefxwJxMHvtnFiuyC/njVf1IjNOJJAlc8zL38eiH6zEGpk9M56JerZyOJCHCOHVUm56ebjMyMrzyWm635eVFO/jrF1tp3TSWFyecxdmdWnjltUW8pbSymifmbuTdjBwGdmzGi9cP1EyJctqMMausteknWxYSp9EjIgz3jOzGe3cNIyICrnt1GS8u3KarYCRgZO2ruYplzqoc7hnZlTl3DlOZi9eFRKF/b1DH5nzyy/MZ268Nf/1iKzf8fRn7isqcjiVhzFrLG99mc9VL31FSXs0/bh/Cby7rRbQuSRQfCLmvqqax0Uy9/iye/8kA1u8t5vKp3/DPtXt1wlT8bn9xGTdPX8ET8zZyfvdkPvvVBQzvlux0LAlhITk5lzGGa89uz9mdmvPAnLXcP3stn204wNP/1VdXEojPWWv5cPVepszLotpleWp8H24c2gljNOWt+FZInBQ9FZfbMm3xTl74YisJsVH88aq+jO7rs5kJJMwVHK3g0Y/W88XGPM5Jbc7zPxlAp6TGTseSEBLyJ0VPJTLCcPeIrsy77zzaNIvlrrdX86vZaygq1R2m4j3WWuav28eoFxaxaGsBvx/bm9mThqnMxa9CcsjlZHq2TuCjX5zL//17B3/7ahtLth/kD1ekMW5AW/0qLA2yt6iMxz7ewMLN+fRvn8hfrxtAt5YJTseSMBTyQy4ns3HfER75cB2ZucVc0COFp8f3pWOSLiGT0+NyW2Z8t4u/fL4Fa+HBUT2YODxVk2qJT51qyCUsCx1qvhnfXrabPy/YQpXLzf2XdOeO87vocjLxyIa9xTz60XrW5RYzsmcKT47vq+vKxS9U6Kewv7iMKXOzWJCVR89WCUwZ14dhXZOcjiUBqri0ihe+3Mpby3bTPD6GKePSGNuvjYbtxG9U6B74POsAT8zbyN6iMsb0a82jY3rTvrmOuKSGy22ZvXIPzy/YQnFZFT8b0omHRvXUAyjE705V6GFzUrQ+o/q05oIeKby6aCcvL9rOwk353HlBF+4a0ZX4GH2YwtmK7EKmzM1i4/4jDOncginj+tC7TVOnY4n8iI7QT2JfURnP/Gsz8zL30SYxlsmjezJ+QDsiIvRrdTjJPniM5xds4ZP1+2nXLI5Hx/RmTL/WGl4RR2nI5QytyC7kyflZbNh7hF6tE5g8uicje7bUN3SIyz9SztSF23h3ZQ4xURHccX4X7rqwK3ExkU5HE1GhN4TbbZm/fj9/+XwLuw+Vck5qcx4e3Yv0VE3PG2qOlFfx6qIdTF+yiyqXmxuGdOS+i7qTkqDpIiRwqNC9oMrl5t2VOUxduI2CoxVc3Ksl91/Snf7tmzkdTRroaHkVby7dzd+/2UlRaRVXDmjLg5f2IDVZd3lK4FGhe1FpZTVvfLuLaYt3UlxWxYU9Urjvom46Yg9CRaWVTP92FzO+zeZIeTUje6bw4Kie9G2X6HQ0kTqp0H3gaHkVby/bw2vf7OTQsUqGdmnBfRd1Z3jXJI2xB7hDJRW8tiSbt5bupqSimlFprbjvou70a68il8CnQvehskoXs1bs4dXFO8g7UkG/doncdl4qY/u1JSZKd50Gki0HjvLGt9l8tGYvlS43Y/q14d6R3XQJogQVFbofVFS7+GDVXl5fspMdBcdISWjETUM7ccOQjiRrDnbHuN2WRdsKmL4km2+2HaRRVARXD2rP7ed1plvLJk7HEzltKnQ/crst32w/yPQl2SzaWkBMVATjB7RlwpCODOzQTMMxflJ4rJKP1uzlneW72VFwjJYJjbhleCoTBnekReMYp+OJnDEVukO25x/ljW938eHqvZRVuejRqgnXpXfg6kHtVSo+4HZblu48xKwVe/g8K49Kl5uzOjRj4vBUxvRroyEwCQkqdIcdLa9i/rr9vLsyh7U5RURHGkalteaas9txXrcUFU0DZR88xrzMfby/Kpc9haUkxkVz1cB2XD+4A71aa3xcQosKPYBsPnCEd1fm8NGavRSVVpEYF81lfVpxRf+2DO+apLm0PbS3qIxP1u1jbuY+Nuw9gjEwtHMS1w/uwGV9WhMbrbs6JTSp0ANQZbWbb7YVMH/dfr7YmEdJRTUtGsdwWZ/WXJrWkuFdk1VKx7HWsqOghIWb8vl8Yx6rdh8GYECHZlzZvw1j+7ehTWKcwylFfE+FHuDKq1x8vaWA+ev28dXmfEorXcRGR3Bu12Qu6t2SkT1b0rZZ+JVVeZWLjF2HWbg5j4Wb8tlTWApAWpumjO3fhiv7t9WTpiTsqNCDSEW1i+U7C/lqcz4LN+eRU1gGQKekeIZ1SWJY1ySGdUmiZdNYh5N6X2W1m8zcIpbuOMR3Ow6yencRlS43MVERnNs1iYt7t+KiXuH5w03keyr0IGWtZXt+CYu3HWTpjkMszz7E0fJqAFKT4hnQoRkD2jdjQIdm9GnbNKiGaKy15B4uIzO3iLV7ilibU8SGfcWUV7kxpuYofHjXmh9gQ7skaU56kVoq9BDhcls27jvC0p0HWbX7MJk5xRw4Ug5AVIShW8smdG+VQI/v/27VhA4t4h19Tqq1lsOlVewsKGFL3lG2Hjha83deCYXHKgFoFBVB33aJDGjfjMGdWzC0SwuaxeuyTpGT0ROLQkRkhKFf+8T/mHPkQHE5mblFZOYUsWn/EVbvPsy8zH3/8X9aN42lffM4OrSIp33zOFISGpHUOIakJo1o0TiGpMYxxMdEndblky63pazKxdHyKg6VVHKwpIJDJZUcOlbB/uJycgrLyD1cSu7hMkoqqn/4f00aRdG9VRNGpbWiT9umnNWhOb3aJOjh3CJeoEIPcq0TY2md2JrL+rT+4d+OVVSzPb+ErXlH2VNYU6o5haUs2XaQvKPl1PVLWVSEIS46kriYSGKjIzGG/1j3+xIvraymvMpdZ6a46Eg6toinQ4s4hnZJokOLeDonx9OzdVPaJsbqblkRH1Ghh6DGjaJqxtc7/Hiu9spqN4XHao6oC4/VHFEXHquitKKasipXzZ9KF+VVrh/+z/cFbAzEx0QSHxNFXHQk8TGRNImNIrlJI5KbxJDUuBFJTWJo0ihKpS3iAI8K3RgzGpgKRAKvWWv/+4TlnYDpQApQCNxorc31clbxgpioiNqj+tC7SkYk3NU7cGmMiQReAi4H0oAJxpi0E1Z7HnjTWtsfeBJ4xttBRUTk1Dw5EzUY2G6t3WmtrQRmA+NPWCcNWFj79r9PslxERHzMk0JvB+Qc935u7b8dLxO4pvbtq4AEY0zSiS9kjJlkjMkwxmQUFBScSV4REamDJ4V+srNbJ14n8RBwoTFmDXAhsBeo/tF/snaatTbdWpuekpJy2mFFRKRunpwUzQU6HPd+e2Df8StYa/cBVwMYY5oA11hri70VUkRE6ufJEfpKoLsxprMxJga4Hph7/ArGmGRjzPev9Qg1V7yIiIgf1Vvo1tpq4F5gAbAJmGOtzTLGPGmMGVe72ghgizFmK9AK+KOP8oqISB00l4uISBAJyMm5jDEFwO4z/O/JwEEvxgkG2ufwoH0ODw3Z507W2pNeVeJYoTeEMSajrp9QoUr7HB60z+HBV/usKe5EREKECl1EJEQEa6FPczqAA7TP4UH7HB58ss9BOYYuIiI/FqxH6CIicgIVuohIiAjoQjfGjDbGbDHGbDfG/PYkyxsZY96tXb7cGJPq/5Te5cE+P2CM2WiMWWeMWVj7cJGgVt8+H7fetcYYa4wJ+kvcPNlnY8x1tZ/rLGPMO/7O6G0efG13NMb82xizpvbre4wTOb3FGDPdGJNvjNlQx3JjjHmx9uOxzhgzqMEbtdYG5B9qno60A+gCxFAzRW/aCev8Anil9u3rgXedzu2HfR4JxNe+fXc47HPtegnAYmAZkO50bj98nrsDa4Dmte+3dDq3H/Z5GnB37dtpwC6nczdwny8ABgEb6lg+BvgXNTPaDgWWN3SbgXyE7smDNcYDM2vffh+42AT3wyzr3Wdr7b+ttaW17y6jZvbLYObJ5xngKeA5oNyf4XzEk32+A3jJWnsYwFqb7+eM3ubJPlugae3biZwwq2uwsdYupuaRnHUZT82T3qy1dhnQzBjTpiHbDORC9+TBGi5XlyQAAAIYSURBVD+sY2smESsGfvRgjSDiyT4f73ZqfsIHs3r32RgzEOhgrZ3vz2A+5MnnuQfQwxjzrTFmWe1zfYOZJ/s8BbjRGJMLfArc559ojjnd7/d6efSQaId48mANT9YJJh7vjzHmRiCdmgeKBLNT7nPttMwvABP9FcgPPPk8R1Ez7DKCmt/CvjHG9LXWFvk4m694ss8TgBnW2r8YY4YBb9Xus9v38Rzh9f4K5CP0eh+scfw6xpgoan5NO9WvOIHOk33GGHMJ8DtgnLW2wk/ZfKW+fU4A+gJfG2N2UTPWODfIT4x6+rX9T2ttlbU2G9hCTcEHK0/2+XZgDoC1dikQS80kVqHKo+/30xHIhV7vgzVq37+l9u1rga9s7dmGIOXJw0QGAq9SU+bBPq4K9eyztbbYWptsrU211qZSc95gnLU2mOde9uRr+2NqToBjjEmmZghmp19Tepcn+7wHuBjAGNObmkIP5YcPzwVurr3aZShQbK3d36BXdPpMcD1niccAW6k5O/672n97kppvaKj5hL8HbAdWAF2czuyHff4SyAPW1v6Z63RmX+/zCet+TZBf5eLh59kAfwU2AuuB653O7Id9TgO+peYKmLXAKKczN3B/ZwH7gSpqjsZvB+4C7jruc/xS7cdjvTe+rnXrv4hIiAjkIRcRETkNKnQRkRChQhcRCREqdBGREKFCFxEJESp0EZEQoUIXEQkR/w8N3TK/E8mUuQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xx = np.linspace(0, 1, 500)\n",
    "plt.plot(xx, my_quadratic_function(xx))\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "А теперь сгенерируем данные:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(42)\n",
    "m = 20\n",
    "x = np.random.rand(m)\n",
    "y = my_quadratic_function(x) + 0.05*np.random.randn(m)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "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": [
    "xx = np.linspace(0, 1, 500)\n",
    "plt.plot(xx, my_quadratic_function(xx))\n",
    "plt.plot(x, y, 'o')\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Насколько точно мы сможем восстановить параболу по точкам?  \n",
    "\n",
    "Попробуем оба метода: `lstsq()` и `polyfit()`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [],
   "source": [
    "X = np.stack((np.ones(m), x, x**2)).T\n",
    "beta, _, _, _ = linalg.lstsq(X, y)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.01701246, -0.74509807,  1.11942245])"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta0, beta1, beta2 = beta\n",
    "beta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "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": [
    "xx = np.linspace(0, 1, 500)\n",
    "plt.plot(xx, my_quadratic_function(xx), label = 'Original')\n",
    "plt.plot(x, y, 'o', label = 'Data')\n",
    "plt.plot(xx, beta0 + beta1*xx + beta2*xx**2, label = 'Restored')\n",
    "plt.legend()\n",
    "plt.grid()\n",
    "pass"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "То же самое можно проделать, используя функцию `polyfit()`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 1.11942245, -0.74509807,  1.01701246])"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "beta = np.polyfit(x, y, 2)\n",
    "beta"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Получили те же коэффициенты."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Выводы\n",
    "\n",
    "1. Линейная задача наименьших квадратов - это задача поиска коэффициентов $\\beta_1,\\dots,\\beta_n$, доставляющих минимум функции\n",
    "$$\n",
    " \\sum_{i=1}^m \\left(\\beta_1 f_1(x_i) + \\dots + \\beta_n f_n(x_i) - y_i \\strut\\right)^2 \\to \\min_{\\beta_1, \\dots, \\beta_n},\n",
    "$$\n",
    "где $(x_1, y_1), (x_2, y_2), \\dots, (x_m, y_m)$ - данные, а $f_1(x), f_2(x), \\dots, f_n(x)$ - известные функции.\n",
    "\n",
    "2. Линейная задача наименьших квадратов может быть сформулирована как задача поиска псевдорешения системы $X\\beta = y$, т.е. поиска вектора $\\beta$, минимизирующего евклидову норму невязки.\n",
    "\n",
    "3. Решение линейной задачи наименьших квадратов $X\\beta \\approx y$ сводится к решению системы нормальных линейных уравнений\n",
    "$X^{\\rm T} X \\beta = X^{\\rm T} y$.\n",
    "\n"
   ]
  }
 ],
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  "kernelspec": {
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   "language": "python",
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