{
"cells": [
{
"cell_type": "markdown",
"id": "94f0a60c",
"metadata": {},
"source": [
"# Регрессия"
]
},
{
"cell_type": "code",
"execution_count": 100,
"id": "8aa79ab0",
"metadata": {},
"outputs": [],
"source": [
"import os\n",
"\n",
"import numpy as np\n",
"import pandas as pd\n",
"from matplotlib import pyplot as plt\n",
"import seaborn as sns\n",
"plt.rcParams.update({\"font.size\": 16})\n",
"\n",
"\n",
"def generate_problem(function, a, b, randomize_x=False, noise=None, n=50): \n",
" x = np.random.uniform(low=a, high=b, size=n) if randomize_x else np.linspace(a, b, n)\n",
" y = function(x)\n",
" if noise:\n",
" y += np.random.normal(loc=0., scale=noise, size=n)\n",
" return pd.DataFrame({\n",
" \"x\": x,\n",
" \"y\": y\n",
" })\n",
"\n",
"def plot_regression_problem(ax, data, regressor=None, test_data=None):\n",
" # dataset column\n",
" if test_data is None:\n",
" data = data.copy()\n",
" data[\"dataset\"] = \"train\"\n",
" else:\n",
" data = pd.concat([data, test_data], keys=[\"train\", \"test\"])\n",
" data = data.reset_index(level=0).rename(columns={\"level_0\": \"dataset\"})\n",
" \n",
" # scatter\n",
" ax.set_ylim([\n",
" data[\"y\"].min() - 0.1,\n",
" data[\"y\"].max() + 0.1\n",
" ])\n",
" sns.scatterplot(data=data, x=\"x\", y=\"y\", hue=\"dataset\", ax=ax)\n",
" \n",
" # regressor\n",
" if regressor: \n",
" x_min, x_max = data[\"x\"].min(), data[\"x\"].max()\n",
" x = np.linspace(x_min, x_max, 100)\n",
" y = regressor(x)\n",
" sns.lineplot(x=x, y=y, color=\"green\", ax=ax)"
]
},
{
"cell_type": "markdown",
"id": "c98edb4d",
"metadata": {},
"source": [
"## Аппроксимация полиномами\n",
"\n",
"Рассмотрим задачу аппроксимации неизвестной функции одного аргумента. Пусть имеется набор из $N$ точек $(x_i, y_i) \\in\\mathbb{R}, \\, i=1,\\ldots, N$ и перед нами стоит цель найти функцию $f\\colon \\mathbb{R}\\to\\mathbb{R}$, которая бы хорошо аппроксимировала имеющиеся точки. Для этого можно поставить задачу поиска функции $f$ из класса $F$ с наименьшей среднеквадратичной ошибкой\n",
"\n",
"$$\n",
"\\dfrac1N\\sum_{i=1}^N(f(x_i) - y_i)^2 \\sim \\min\\limits_{f\\in F}.\n",
"$$\n",
"\n",
"В ячейке ниже генерируется набор таких точек.\n",
"- значения $x_i$ генерируются на равномерной сетке на отрезке $[0, 4]$;\n",
"- значения $y_i$ генерируются в качестве значения полинома $y_i = f(x_i) = 1 - \\dfrac{x_i^2}{2!} + \\dfrac{x_i^4}{4!}$."
]
},
{
"cell_type": "code",
"execution_count": 74,
"id": "b2afc2e8",
"metadata": {},
"outputs": [
{
"data": {
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