268 lines
8 KiB
Text
268 lines
8 KiB
Text
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{
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"cells": [
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{
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"cell_type": "markdown",
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"id": "dff037b7-7d71-49c2-8a47-48017c073f81",
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"metadata": {},
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"source": [
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"$R^2$ für ein polynomielles Modell"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 51,
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"id": "8b1ff6ff-f80e-4cc3-b266-0ad417911d1d",
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"metadata": {},
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"outputs": [],
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"source": [
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"import numpy as np\n",
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"import pandas as pd\n",
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"# plotting settings\n",
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"pd.plotting.register_matplotlib_converters()\n",
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"import matplotlib.pyplot as plt\n",
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"%matplotlib inline\n",
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"import seaborn as sns\n",
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"from tqdm.notebook import tqdm"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 52,
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"id": "f698283d-7346-4618-9b87-60a3de061a98",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"Index(['Suburb', 'Address', 'Rooms', 'Type', 'Price', 'Method', 'SellerG',\n",
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" 'Date', 'Distance', 'Postcode', 'Bedroom2', 'Bathroom', 'Car',\n",
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" 'Landsize', 'BuildingArea', 'YearBuilt', 'CouncilArea', 'Lattitude',\n",
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" 'Longtitude', 'Regionname', 'Propertycount'],\n",
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" dtype='object')"
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]
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},
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"execution_count": 52,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"melbourne_file_path = 'data/melb_data.csv'\n",
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"melbourne_data = pd.read_csv(melbourne_file_path)\n",
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"melbourne_data = melbourne_data.dropna(axis=0) # entfernen von Daten mit fehlenden Werten\n",
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"melbourne_data.columns # Spaltennamen der Tabelle (potentielle Features)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 53,
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"id": "dac65c52-f2ce-47b6-ba65-3c6bd915dfe8",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/html": [
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"<div>\n",
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"<style scoped>\n",
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" .dataframe tbody tr th:only-of-type {\n",
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" vertical-align: middle;\n",
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" }\n",
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"\n",
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" .dataframe tbody tr th {\n",
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" vertical-align: top;\n",
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" }\n",
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"\n",
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" .dataframe thead th {\n",
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" text-align: right;\n",
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" }\n",
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"</style>\n",
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"<table border=\"1\" class=\"dataframe\">\n",
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" <thead>\n",
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" <tr style=\"text-align: right;\">\n",
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" <th></th>\n",
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" <th>Rooms</th>\n",
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" <th>BuildingArea</th>\n",
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" <th>Lattitude</th>\n",
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" <th>Price</th>\n",
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" <th>price_per_area</th>\n",
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" </tr>\n",
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" </thead>\n",
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" <tbody>\n",
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" <tr>\n",
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" <th>1</th>\n",
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" <td>2</td>\n",
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" <td>79.0</td>\n",
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" <td>-37.8079</td>\n",
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" <td>1035000.0</td>\n",
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" <td>161460000.0</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>2</th>\n",
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" <td>3</td>\n",
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" <td>150.0</td>\n",
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" <td>-37.8093</td>\n",
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" <td>1465000.0</td>\n",
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" <td>196310000.0</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>4</th>\n",
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" <td>4</td>\n",
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" <td>142.0</td>\n",
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" <td>-37.8072</td>\n",
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" <td>1600000.0</td>\n",
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" <td>192000000.0</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>6</th>\n",
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" <td>3</td>\n",
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" <td>210.0</td>\n",
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" <td>-37.8024</td>\n",
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" <td>1876000.0</td>\n",
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" <td>459620000.0</td>\n",
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" </tr>\n",
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" <tr>\n",
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" <th>7</th>\n",
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" <td>2</td>\n",
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" <td>107.0</td>\n",
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" <td>-37.8060</td>\n",
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" <td>1636000.0</td>\n",
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" <td>418816000.0</td>\n",
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" </tr>\n",
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" </tbody>\n",
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"</table>\n",
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"</div>"
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],
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"text/plain": [
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" Rooms BuildingArea Lattitude Price price_per_area\n",
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"1 2 79.0 -37.8079 1035000.0 161460000.0\n",
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"2 3 150.0 -37.8093 1465000.0 196310000.0\n",
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"4 4 142.0 -37.8072 1600000.0 192000000.0\n",
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"6 3 210.0 -37.8024 1876000.0 459620000.0\n",
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"7 2 107.0 -37.8060 1636000.0 418816000.0"
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]
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},
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"execution_count": 53,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"#features = ['BuildingArea', Rooms', 'Bathroom', 'Landsize', 'Lattitude', 'Longtitude', 'YearBuilt', 'Distance']\n",
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"features = ['Rooms', 'BuildingArea', 'Lattitude']\n",
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"data = melbourne_data[features + ['Price']]\n",
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"data = data.assign(price_per_area = melbourne_data['Price'] * melbourne_data[\"Landsize\"])\n",
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"data.describe()\n",
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"data.head()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 54,
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"id": "afe5a08a-abec-4164-85c4-1d3ac8398a62",
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"metadata": {},
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"outputs": [],
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"source": [
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"def h(x, w):\n",
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" \"\"\"x und w sind numpy arrays; x kann auch die komplette Feature-Matrix sein\"\"\"\n",
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" # Diese Form erlaubt es für x eine ganze (Feature-)Matrix zu übergeben. Die Matrix enthält\n",
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" # zeilenweise je einen Datenpunkt, für den h berechnet werden soll.\n",
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" # w @ x.T ist dann ein Vektor mit je einem Ergebnis in den Komponenten des Vektors pro Zeile\n",
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" # der übergebenen (Feature-)Matrix.\n",
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" return x @ w\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 55,
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"id": "c2e256a3-3575-45c4-a99c-d41c3c56e1c3",
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"metadata": {},
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"outputs": [],
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"source": [
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"# Definition der Kostenfunktion\n",
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"def J(w, X, y):\n",
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" \"\"\"\n",
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" w, X, y müssen numpy arrays sein\n",
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" X: Feature-Matrix aller Trainingsdaten inkl. Spalte mit 1; Dimension: n x (d+1)\n",
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" y: Vektor aller Targets zu X\n",
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" \"\"\"\n",
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" errors = y - h(x=X, w=w)\n",
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" mse = 1.0/(2.0*len(y)) * ( errors @ errors )\n",
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" return mse\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 56,
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"id": "b41b9c03-0c1d-4a6b-80e8-d7e8775b69c0",
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"metadata": {},
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"outputs": [],
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"source": [
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"def feature_matrix_from_data(data):\n",
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" # hier erzeugen wir die Matrix mit unseren Input-Daten (Features) inklusive der Spalte mit \"1\"\n",
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" return np.hstack((np.ones((len(data),1)), data.to_numpy(copy=True)))\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 57,
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"id": "2c631e17-eb36-43d0-97b1-59add1c93dd9",
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"metadata": {},
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"outputs": [],
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"source": [
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"# hier erzeugen wir die Matrix mit unseren Input-Daten (Features) inklusive der Spalte mit \"1\"\n",
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"#X = np.hstack((np.ones((len(data),1)), data[features].to_numpy(copy=True)))\n",
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"X = feature_matrix_from_data(data[features])\n",
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"# und ausserdem den Vektor der Targets\n",
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"y = data.Price.to_numpy(copy=True)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 58,
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"id": "8301eb74-9aae-446c-ad46-924811b99777",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Die 4 Parameter der linearen Regression:\n",
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"[-6.97461781e+07 2.41559504e+05 2.31456611e+03 -1.84562537e+06]\n",
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"Kostenfunktion J(w_ana): 137899453867.5851\n",
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"CPU times: user 475 μs, sys: 38 μs, total: 513 μs\n",
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"Wall time: 500 μs\n"
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]
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}
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],
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"source": [
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"%%time\n",
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"w_ana = np.linalg.solve(X.T @ X, X.T @ y)\n",
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"print('Die {} Parameter der linearen Regression:\\n{}'.format(len(w_ana), w_ana))\n",
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"J_ana = J(w=w_ana, X=X, y=y)\n",
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"print('Kostenfunktion J(w_ana): {}'.format(J_ana))\n"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python (ki)",
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"language": "python",
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"name": "myenv"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.8"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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