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01: i tried to do it all, but this is all crap
2025-01-20 15:05:31 +01:00
{
"cells": [
{
"cell_type": "markdown",
"id": "9496e038",
"metadata": {},
"source": [
"# Lineare Regression mit 1 Feature ($d=1$)"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "5754d665",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import pandas as pd\n",
"# plotting settings\n",
"pd.plotting.register_matplotlib_converters()\n",
"import matplotlib.pyplot as plt\n",
"%matplotlib inline\n",
"import seaborn as sns"
]
},
{
"cell_type": "markdown",
"id": "282549b7",
"metadata": {},
"source": [
"Wir verwenden hier beispielhaft den Datensatz [Melbourne Housing Snapshot](https://www.kaggle.com/datasets/dansbecker/melbourne-housing-snapshot). Diesen finden Sie auch im Moodle unter `data/melb_data.csv`."
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "cfe20800",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Index(['Suburb', 'Address', 'Rooms', 'Type', 'Price', 'Method', 'SellerG',\n",
" 'Date', 'Distance', 'Postcode', 'Bedroom2', 'Bathroom', 'Car',\n",
" 'Landsize', 'BuildingArea', 'YearBuilt', 'CouncilArea', 'Lattitude',\n",
" 'Longtitude', 'Regionname', 'Propertycount'],\n",
" dtype='object')"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"melbourne_file_path = 'data/melb_data.csv'\n",
"melbourne_data = pd.read_csv(melbourne_file_path)\n",
"melbourne_data = melbourne_data.dropna(axis=0) # entfernen von Daten mit fehlenden Werten\n",
"melbourne_data.columns # Spaltennamen der Tabelle (potentielle Features)"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "e13b23ac",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Suburb</th>\n",
" <th>Address</th>\n",
" <th>Rooms</th>\n",
" <th>Type</th>\n",
" <th>Price</th>\n",
" <th>Method</th>\n",
" <th>SellerG</th>\n",
" <th>Date</th>\n",
" <th>Distance</th>\n",
" <th>Postcode</th>\n",
" <th>...</th>\n",
" <th>Bathroom</th>\n",
" <th>Car</th>\n",
" <th>Landsize</th>\n",
" <th>BuildingArea</th>\n",
" <th>YearBuilt</th>\n",
" <th>CouncilArea</th>\n",
" <th>Lattitude</th>\n",
" <th>Longtitude</th>\n",
" <th>Regionname</th>\n",
" <th>Propertycount</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>Abbotsford</td>\n",
" <td>25 Bloomburg St</td>\n",
" <td>2</td>\n",
" <td>h</td>\n",
" <td>1035000.0</td>\n",
" <td>S</td>\n",
" <td>Biggin</td>\n",
" <td>4/02/2016</td>\n",
" <td>2.5</td>\n",
" <td>3067.0</td>\n",
" <td>...</td>\n",
" <td>1.0</td>\n",
" <td>0.0</td>\n",
" <td>156.0</td>\n",
" <td>79.0</td>\n",
" <td>1900.0</td>\n",
" <td>Yarra</td>\n",
" <td>-37.8079</td>\n",
" <td>144.9934</td>\n",
" <td>Northern Metropolitan</td>\n",
" <td>4019.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>Abbotsford</td>\n",
" <td>5 Charles St</td>\n",
" <td>3</td>\n",
" <td>h</td>\n",
" <td>1465000.0</td>\n",
" <td>SP</td>\n",
" <td>Biggin</td>\n",
" <td>4/03/2017</td>\n",
" <td>2.5</td>\n",
" <td>3067.0</td>\n",
" <td>...</td>\n",
" <td>2.0</td>\n",
" <td>0.0</td>\n",
" <td>134.0</td>\n",
" <td>150.0</td>\n",
" <td>1900.0</td>\n",
" <td>Yarra</td>\n",
" <td>-37.8093</td>\n",
" <td>144.9944</td>\n",
" <td>Northern Metropolitan</td>\n",
" <td>4019.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>Abbotsford</td>\n",
" <td>55a Park St</td>\n",
" <td>4</td>\n",
" <td>h</td>\n",
" <td>1600000.0</td>\n",
" <td>VB</td>\n",
" <td>Nelson</td>\n",
" <td>4/06/2016</td>\n",
" <td>2.5</td>\n",
" <td>3067.0</td>\n",
" <td>...</td>\n",
" <td>1.0</td>\n",
" <td>2.0</td>\n",
" <td>120.0</td>\n",
" <td>142.0</td>\n",
" <td>2014.0</td>\n",
" <td>Yarra</td>\n",
" <td>-37.8072</td>\n",
" <td>144.9941</td>\n",
" <td>Northern Metropolitan</td>\n",
" <td>4019.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>6</th>\n",
" <td>Abbotsford</td>\n",
" <td>124 Yarra St</td>\n",
" <td>3</td>\n",
" <td>h</td>\n",
" <td>1876000.0</td>\n",
" <td>S</td>\n",
" <td>Nelson</td>\n",
" <td>7/05/2016</td>\n",
" <td>2.5</td>\n",
" <td>3067.0</td>\n",
" <td>...</td>\n",
" <td>2.0</td>\n",
" <td>0.0</td>\n",
" <td>245.0</td>\n",
" <td>210.0</td>\n",
" <td>1910.0</td>\n",
" <td>Yarra</td>\n",
" <td>-37.8024</td>\n",
" <td>144.9993</td>\n",
" <td>Northern Metropolitan</td>\n",
" <td>4019.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7</th>\n",
" <td>Abbotsford</td>\n",
" <td>98 Charles St</td>\n",
" <td>2</td>\n",
" <td>h</td>\n",
" <td>1636000.0</td>\n",
" <td>S</td>\n",
" <td>Nelson</td>\n",
" <td>8/10/2016</td>\n",
" <td>2.5</td>\n",
" <td>3067.0</td>\n",
" <td>...</td>\n",
" <td>1.0</td>\n",
" <td>2.0</td>\n",
" <td>256.0</td>\n",
" <td>107.0</td>\n",
" <td>1890.0</td>\n",
" <td>Yarra</td>\n",
" <td>-37.8060</td>\n",
" <td>144.9954</td>\n",
" <td>Northern Metropolitan</td>\n",
" <td>4019.0</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"<p>5 rows × 21 columns</p>\n",
"</div>"
],
"text/plain": [
" Suburb Address Rooms Type Price Method SellerG \\\n",
"1 Abbotsford 25 Bloomburg St 2 h 1035000.0 S Biggin \n",
"2 Abbotsford 5 Charles St 3 h 1465000.0 SP Biggin \n",
"4 Abbotsford 55a Park St 4 h 1600000.0 VB Nelson \n",
"6 Abbotsford 124 Yarra St 3 h 1876000.0 S Nelson \n",
"7 Abbotsford 98 Charles St 2 h 1636000.0 S Nelson \n",
"\n",
" Date Distance Postcode ... Bathroom Car Landsize BuildingArea \\\n",
"1 4/02/2016 2.5 3067.0 ... 1.0 0.0 156.0 79.0 \n",
"2 4/03/2017 2.5 3067.0 ... 2.0 0.0 134.0 150.0 \n",
"4 4/06/2016 2.5 3067.0 ... 1.0 2.0 120.0 142.0 \n",
"6 7/05/2016 2.5 3067.0 ... 2.0 0.0 245.0 210.0 \n",
"7 8/10/2016 2.5 3067.0 ... 1.0 2.0 256.0 107.0 \n",
"\n",
" YearBuilt CouncilArea Lattitude Longtitude Regionname \\\n",
"1 1900.0 Yarra -37.8079 144.9934 Northern Metropolitan \n",
"2 1900.0 Yarra -37.8093 144.9944 Northern Metropolitan \n",
"4 2014.0 Yarra -37.8072 144.9941 Northern Metropolitan \n",
"6 1910.0 Yarra -37.8024 144.9993 Northern Metropolitan \n",
"7 1890.0 Yarra -37.8060 144.9954 Northern Metropolitan \n",
"\n",
" Propertycount \n",
"1 4019.0 \n",
"2 4019.0 \n",
"4 4019.0 \n",
"6 4019.0 \n",
"7 4019.0 \n",
"\n",
"[5 rows x 21 columns]"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"melbourne_data.head()"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "8680d0c9",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"<Axes: xlabel='BuildingArea', ylabel='Price'>"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sns.scatterplot(x=melbourne_data['BuildingArea'], y=melbourne_data['Price'])"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "da3b8409",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"1 79.00\n",
"2 150.00\n",
"4 142.00\n",
"6 210.00\n",
"7 107.00\n",
" ... \n",
"12205 149.00\n",
"12206 115.00\n",
"12207 35.64\n",
"12209 61.60\n",
"12212 388.50\n",
"Name: BuildingArea, Length: 6196, dtype: float64"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"melbourne_data['BuildingArea']"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "c1172236",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>BuildingArea</th>\n",
" <th>Price</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>79.0</td>\n",
" <td>1035000.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>150.0</td>\n",
" <td>1465000.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>142.0</td>\n",
" <td>1600000.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>6</th>\n",
" <td>210.0</td>\n",
" <td>1876000.0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>7</th>\n",
" <td>107.0</td>\n",
" <td>1636000.0</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" BuildingArea Price\n",
"1 79.0 1035000.0\n",
"2 150.0 1465000.0\n",
"4 142.0 1600000.0\n",
"6 210.0 1876000.0\n",
"7 107.0 1636000.0"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# wählen für unser Beispiel einen kleinen Ausschnitt aus den Daten\n",
"max_area = 400\n",
"max_datapoints = 100\n",
"data = melbourne_data[melbourne_data['BuildingArea'] < max_area][:max_datapoints][['BuildingArea', 'Price']]\n",
"data.head()"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "8f9dec63",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"100"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"len(data)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "f1293084",
"metadata": {},
"outputs": [
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"ax = sns.scatterplot(x=data['BuildingArea'], y=data['Price'])"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "00dc4dee",
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"BuildingArea 79.0\n",
"Price 1035000.0\n",
"Name: 1, dtype: float64\n",
"[[1, 79.0]]\n"
]
}
],
"source": [
"X = []\n",
"Y = []\n",
"for _, row in data.iterrows():\n",
" X.append([1] + [row['BuildingArea']])\n",
" Y.append(row['Price'])\n",
" break\n",
"X = np.array(X)\n",
"Y = np.array(Y)\n",
"print(X[:5], Y[:5])"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "efecad93",
"metadata": {},
"outputs": [],
"source": [
"def h_w(x, w):\n",
" return w[0] + w[1]*x"
]
},
{
"cell_type": "markdown",
"id": "e0577f21",
"metadata": {},
"source": [
"## Analytische Lösung der linearen Regression\n",
"\n",
"`np.linalg.solve(A, b)` berechnet $w$ im linearen Gleichungssystem\n",
"\n",
"$ A w = b $\n",
"\n",
"$A$ - Matrix,\n",
"$w$ - Vektor (unsere unbekannten),\n",
"$b$ - Vektor.\n",
"\n",
"Wir suchen die Lösung $w$ im folgenden Gleichungssystem:\n",
"\n",
"$$ X^T X w = X^T Y $$\n",
"\n",
"Mit $A = X^TX$ und $b = X^T Y$ berechnet `np.linalg.solve(A, b)` unsere gesuchten Paramter für die lineare Regression."
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "35a78137",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"CPU times: user 42 µs, sys: 7 µs, total: 49 µs\n",
"Wall time: 51.3 µs\n"
]
}
],
"source": [
"%%time\n",
"w_ana = np.linalg.solve(X.T @ X, X.T @ Y)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "9a6041bd",
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[441524.42083181 6024.22929588]\n"
]
}
],
"source": [
"print(w_ana)"
]
},
{
"cell_type": "markdown",
"id": "f51a85af",
"metadata": {},
"source": [
"Plot der analytischen Lösung"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "6486ec38",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<Axes: xlabel='BuildingArea', ylabel='Price'>"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAioAAAHACAYAAACMB0PKAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjguMiwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy8g+/7EAAAACXBIWXMAAA9hAAAPYQGoP6dpAABKrUlEQVR4nO3de3yT9d0//lfS8ynpIU1poUChpSmUQhVllXPLFKbeeJhzTHejuO2hgtPpdiM7ON1U8Hhv7r7nNicyf7eicxP8yaboWgpykAG2HKQtbUVaOTQ9Jj0mbXJ9/wgNTZumbZrk+iR5PR8PHg+bK7366WWa65XP4f1RSJIkgYiIiEhASrkbQERERDQcBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYTGoEBERkbAYVIiIiEhYDCpEREQkLAYVIiIiElbABJW9e/fixhtvRFpaGhQKBXbs2DHmc0iShOeffx4zZsxAREQEJk6ciKeeesrzjSUiIqJRCZW7AZ7S2dmJOXPmYO3atbjlllvcOseDDz6Ijz76CM8//zxmz56NlpYWtLS0eLilRERENFqKQNyUUKFQYPv27bjpppvsj5lMJvzsZz/Dtm3b0NbWhtzcXDzzzDNYunQpAKCiogJ5eXk4efIksrOz5Wk4EREROQiYoZ+RrF+/HgcPHsRbb72F48eP47bbbsOKFStQXV0NAHj//fcxbdo07Ny5ExkZGZg6dSq+973vsUeFiIhIRkERVOrq6vDaa6/hnXfewaJFizB9+nT8+Mc/xsKFC/Haa68BAL744gucPXsW77zzDl5//XVs3boVR48exTe/+U2ZW09ERBS8AmaOiisnTpyAxWLBjBkzHB43mUxISkoCAFitVphMJrz++uv257366qu48sorUVVVxeEgIiIiGQRFUOno6EBISAiOHj2KkJAQh2OxsbEAgNTUVISGhjqEmZycHAC2HhkGFSIiIt8LiqCSn58Pi8UCvV6PRYsWOX3OggUL0NfXh9raWkyfPh0AcPr0aQDAlClTfNZWIiIiuixgVv10dHSgpqYGgC2YvPjii1i2bBkSExMxefJk3Hnnndi/fz9eeOEF5Ofno7GxEcXFxcjLy8P1118Pq9WKq666CrGxsfjNb34Dq9WKdevWQaVS4aOPPpL5tyMiIgpOARNUSktLsWzZsiGPr1mzBlu3bkVvby+efPJJvP766zh37hw0Gg2+9rWv4YknnsDs2bMBAOfPn8cDDzyAjz76CDExMVi5ciVeeOEFJCYm+vrXISIiIgRQUCEiIqLAExTLk4mIiMg/MagQERGRsPx61Y/VasX58+cRFxcHhUIhd3OIiIhoFCRJQnt7O9LS0qBUuu4z8eugcv78eaSnp8vdDCIiInJDfX09Jk2a5PI5fh1U4uLiANh+UZVKJXNriIiIaDSMRiPS09Pt93FX/Dqo9A/3qFQqBhUiIiI/M5ppG5xMS0RERMJiUCEiIiJhMagQERGRsBhUiIiISFgMKkRERCQsBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYfl1CX0iokBl6DKjqcMMY08vVFFh0MSEQx0dLneziHyOQYWISDDn27qx4e/H8Ul1k/2xxVkabL41D2nxUTK2jMj3OPRDRCQQQ5d5SEgBgL3VTXj078dh6DLL1DIieTCoEBEJpKnDPCSk9Ntb3YSmDgYVCi4MKkREAjH29Lo83j7CcaJAw6BCRCQQVWSYy+NxIxwnCjQMKkREAtHEhmNxlsbpscVZGmhiufKHgguDChGRQNTR4dh8a96QsLI4S4Nnbs3jEmUKOlyeTEQkmLT4KPxudT6aOsxo7+lFXGQYNLGso0LBiUGFiEhA6mgGEyKAQz9EREQkMAYVIiIiEhaDChEREQmLQYWIiIiExaBCREREwmJQISIiImExqBAREZGwGFSIiIhIWAwqREREJCwGFSIiIhIWgwoREREJi0GFiIiIhMWgQkRERMJiUCEiIiJhMagQERGRsBhUiIiISFgMKkRERCQsBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYTGoEBERkbAYVIiIiEhYDCpEREQkLAYVIiIiEhaDChEREQmLQYWIiIiExaBCREREwmJQISIiImExqBAREZGwGFSIiIhIWAwqREREJCwGFSIiIhIWgwoREREJi0GFiIiIhMWgQkRERMJiUCEiIiJhMagQERGRsBhUiIiISFgMKkRERCQsBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYQkTVDZv3gyFQoGHHnpI7qYQERGRIIQIKocPH8Yf//hH5OXlyd0UIiIiEojsQaWjowN33HEHXnnlFSQkJMjdHCIiIhKI7EFl3bp1uP7667F8+XK5m0JERESCCZXzh7/11lv47LPPcPjw4VE932QywWQy2b82Go3eahoREREJQLYelfr6ejz44IN44403EBkZOarv2bRpE9Rqtf1fenq6l1tJREREclJIkiTJ8YN37NiBm2++GSEhIfbHLBYLFAoFlEolTCaTwzHAeY9Keno6DAYDVCqVz9pORERE7jMajVCr1aO6f8s29FNUVIQTJ044PHb33XdDp9Nhw4YNQ0IKAERERCAiIsJXTSQiIiKZyRZU4uLikJub6/BYTEwMkpKShjxOREREwUn2VT9EREREw5F11c9gpaWlcjeBiIiIBMIeFSIiIhIWgwoREREJi0GFiIiIhMWgQkRERMJiUCEiIiJhMagQERGRsBhUiIiISFgMKkRERCQsBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYTGoEBERkbAYVIiIiEhYDCpEREQkLAYVIiIiEhaDChEREQmLQYWIiIiExaBCREREwmJQISIiImExqBAREZGwGFSIiIhIWAwqREREJCwGFSIiIhIWgwoREREJi0GFiIiIhMWgQkRERMJiUCEiIiJhMagQERGRsBhUiIiISFgMKkRERCQsBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYTGoEBERkbAYVIiIiEhYDCpEREQkLAYVIiIiEhaDChEREQmLQYWIiIiExaBCREREwmJQISIiImExqBAREZGwGFSIiIhIWAwqREREJCwGFSIiIhIWgwoREREJi0GFiIiIhMWgQkRERMJiUCEiIiJhMagQERGRsBhUiIiISFgMKkRERCQsBhUiIiISFoMKERERCYtBhYiIiITFoEJERETCYlAhIiIiYTGoEBERkbAYVIiIiEhYDCpEREQkLAYVIiIiEhaDChEREQkrVO4GEJG4DF1mNHWYYezphSoqDJqYcKijw+VuFhEFEVl7VF5++WXk5eVBpVJBpVKhoKAAH3zwgZxNIqJLzrd1Y/22MhS9uAc3//4Ail7Ygwe2leF8W7fcTSOiICJrUJk0aRI2b96Mo0eP4siRIygsLMSqVavw+eefy9ksoqBn6DJjw9+P45PqJofH91Y34dG/H4ehyyxTy4go2Mg69HPjjTc6fP3UU0/h5ZdfxqeffopZs2bJ1CoiauowDwkp/fZWN6Gpw8whICLyCWHmqFgsFrzzzjvo7OxEQUGB0+eYTCaYTCb710aj0VfNIwoqxp5el8fbRzhOROQpsq/6OXHiBGJjYxEREYF7770X27dvx8yZM50+d9OmTVCr1fZ/6enpPm4tUXBQRYa5PB43wnEiIk+RPahkZ2ejvLwchw4dwn333Yc1a9bg1KlTTp+7ceNGGAwG+7/6+noft5YoOGhiw7E4S+P02OIsDTSxHPYhIt9QSJIkyd2IgZYvX47p06fjj3/844jPNRqNUKvVMBgMUKlUPmgdUfA439aNR/9+HHsHzFVZnKXBM7fmITU+SsaWEZG/G8v9W5g5Kv2sVqvDPBQikkdafBR+tzofTR1mtPf0Ii4yDJpY1lEhIt+SNahs3LgRK1euxOTJk9He3o4333wTpaWl2LVrl5zNIqJL1NEMJkQkL1mDil6vx3/+53/iwoULUKvVyMvLw65du/D1r39dzmYRERGRIGQ
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"ax = sns.scatterplot(x=data['BuildingArea'], y=data['Price'])\n",
"\n",
"xplot = [min(data['BuildingArea']), max(data['BuildingArea'])]\n",
"yplot = [h_w(x, w_ana) for x in xplot]\n",
"sns.lineplot(x=xplot, y=yplot, ax=ax)"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "aab92a40",
"metadata": {},
"outputs": [],
"source": [
"# Definition der Kostenfunktion\n",
"def J(w, x, y):\n",
" \"\"\"w, x, y müssen numpy arrays sein\"\"\"\n",
" errors = y - h_w(x=x, w=w)\n",
" mse = 1.0/(2.0*len(errors)) * ( errors @ errors )\n",
" return mse"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "7ef64eb2",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Kosten der analytischen Lösung: 200141433273.1325\n"
]
}
],
"source": [
"x = data['BuildingArea'].to_numpy(copy=True)\n",
"y = data['Price'].to_numpy(copy=True)\n",
"J_ana = J(w=w_ana, x=x, y=y)\n",
"print('Kosten der analytischen Lösung: {}'.format(J_ana))"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "0272e5ad",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([441524.42083181, 6024.22929588])"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"w_ana"
]
},
{
"cell_type": "markdown",
"id": "217f80c5",
"metadata": {},
"source": [
"## Numerische Lösung mit Gradient Descent"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "6cda3066",
"metadata": {},
"outputs": [],
"source": [
"def grad_desc_upd(w, alpha, x, y):\n",
" \"\"\"y, x sind Vektoren (numpy-arrays)\"\"\"\n",
" errors = y - h_w(x=x, w=w)\n",
" w_0_upd = w[0] + alpha / len(x) * sum(errors)\n",
" \n",
" errors_x_x1 = errors @ x\n",
" w_1_upd = w[1] + alpha / len(x) * errors_x_x1\n",
" return [w_0_upd, w_1_upd]"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "b349e5ab",
"metadata": {},
"outputs": [],
"source": [
"def grad_desc(w, alpha, x, y, n_iterations):\n",
" J_all = [J(w=w, x=x, y=y)]\n",
" for it in range(n_iterations):\n",
" w = grad_desc_upd(w=w, alpha=alpha, x=x, y=y)\n",
" J_all.append(J(w=w, x=x, y=y))\n",
" return w, J_all"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "b87084ca",
"metadata": {},
"outputs": [],
"source": [
"w_tmp, J_tmp = grad_desc(w=[1e5, 1000.], alpha=1e-9, x=data['BuildingArea'].to_numpy(), y=data['Price'].to_numpy(), n_iterations=1)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "a129a532",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"0.9999715711803561"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"J_tmp[1]/J_tmp[0]"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "fd1bb601",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"w_gd_1e4: [915045.6766397628, 2959.2952860626924]\n",
"Vergleich zu Startkosten: 0.8899738097177349\n",
"Vergleich zu analytischer Lösung: 1.0924465228987312\n",
"(w0_gd - w0_ana)/w0_ana: 1.0724690039021088\n",
"(w1_gd - w1_ana)/w1_ana: -0.5087678206259784\n",
"CPU times: user 256 ms, sys: 8.18 ms, total: 264 ms\n",
"Wall time: 90.6 ms\n"
]
}
],
"source": [
"%%time\n",
"n_iterations = 10000\n",
"alpha = 0.0001 # mit alpha experimentieren\n",
"w_init = [1e6, 1000.]\n",
"x = data['BuildingArea'].to_numpy()\n",
"y = data['Price'].to_numpy()\n",
"w_gd_1e4, J_all_1e4 = grad_desc(w=w_init, alpha=alpha, x=x, y=y, n_iterations=n_iterations)\n",
"\n",
"print('w_gd_1e4: {}'.format(w_gd_1e4))\n",
"print('Vergleich zu Startkosten: {}'.format(J_all_1e4[-1]/J_all_1e4[0]))\n",
"print('Vergleich zu analytischer Lösung: {}'.format(J_all_1e4[-1]/J_ana))\n",
"print('(w0_gd - w0_ana)/w0_ana: {}'.format((w_gd_1e4[0]-w_ana[0])/w_ana[0]))\n",
"print('(w1_gd - w1_ana)/w1_ana: {}'.format((w_gd_1e4[1]-w_ana[1])/w_ana[1]))"
]
},
{
"cell_type": "code",
"execution_count": 22,
"id": "1c26fde8",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"w_gd_1e5: [548748.7304152894, 5330.2046262550075]\n",
"Vergleich zu Startkosten: 0.8185228834109243\n",
"Vergleich zu analytischer Lösung: 1.004740216095697\n",
"(w0_gd - w0_ana)/w0_ana: 0.24285023551238794\n",
"(w1_gd - w1_ana)/w1_ana: -0.1152055533638727\n",
"CPU times: user 958 ms, sys: 20.8 ms, total: 978 ms\n",
"Wall time: 825 ms\n"
]
}
],
"source": [
"%%time\n",
"n_iterations = 100000\n",
"alpha = 0.0001 # mit alpha experimentieren\n",
"w_init = [1e6, 1000.]\n",
"x = data['BuildingArea'].to_numpy()\n",
"y = data['Price'].to_numpy()\n",
"w_gd_1e5, J_all_1e5 = grad_desc(w=w_init, alpha=alpha, x=x, y=y, n_iterations=n_iterations)\n",
"\n",
"print('w_gd_1e5: {}'.format(w_gd_1e5))\n",
"print('Vergleich zu Startkosten: {}'.format(J_all_1e5[-1]/J_all_1e5[0]))\n",
"print('Vergleich zu analytischer Lösung: {}'.format(J_all_1e5[-1]/J_ana))\n",
"print('(w0_gd - w0_ana)/w0_ana: {}'.format((w_gd_1e5[0]-w_ana[0])/w_ana[0]))\n",
"print('(w1_gd - w1_ana)/w1_ana: {}'.format((w_gd_1e5[1]-w_ana[1])/w_ana[1]))"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "ebff7a0b",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"w_gd_3e5: [445476.78736763657, 5998.647038552499]\n",
"Vergleich zu Startkosten: 0.8146664601191632\n",
"Vergleich zu analytischer Lösung: 1.000006440595892\n",
"(w0_gd - w0_ana)/w0_ana: 0.008951637439164814\n",
"(w1_gd - w1_ana)/w1_ana: -0.0042465610235696925\n",
"CPU times: user 2.4 s, sys: 53.3 ms, total: 2.45 s\n",
"Wall time: 2.46 s\n"
]
}
],
"source": [
"%%time\n",
"n_iterations = 300000\n",
"alpha = 0.0001 # mit alpha experimentieren\n",
"w_init = [1e6, 1000.]\n",
"x = data['BuildingArea'].to_numpy()\n",
"y = data['Price'].to_numpy()\n",
"w_gd_3e5, J_all_3e5 = grad_desc(w=w_init, alpha=alpha, x=x, y=y, n_iterations=n_iterations)\n",
"\n",
"print('w_gd_3e5: {}'.format(w_gd_3e5))\n",
"print('Vergleich zu Startkosten: {}'.format(J_all_3e5[-1]/J_all_3e5[0]))\n",
"print('Vergleich zu analytischer Lösung: {}'.format(J_all_3e5[-1]/J_ana))\n",
"print('(w0_gd - w0_ana)/w0_ana: {}'.format((w_gd_3e5[0]-w_ana[0])/w_ana[0]))\n",
"print('(w1_gd - w1_ana)/w1_ana: {}'.format((w_gd_3e5[1]-w_ana[1])/w_ana[1]))"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "a1b5db98",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"w_gd_1e6: [441524.45883596264, 6024.229049890118]\n",
"Vergleich zu Startkosten: 0.8146612132155007\n",
"Vergleich zu analytischer Lösung: 1.0000000000000007\n",
"(w0_gd - w0_ana)/w0_ana: 8.60748531445252e-08\n",
"(w1_gd - w1_ana)/w1_ana: -4.0832990952070806e-08\n",
"CPU times: user 7.99 s, sys: 155 ms, total: 8.14 s\n",
"Wall time: 8.22 s\n"
]
}
],
"source": [
"%%time\n",
"n_iterations = 1000000\n",
"alpha = 0.0001 # mit alpha experimentieren\n",
"w_init = [1e6, 1000.]\n",
"x = data['BuildingArea'].to_numpy()\n",
"y = data['Price'].to_numpy()\n",
"w_gd_1e6, J_all_1e6 = grad_desc(w=w_init, alpha=alpha, x=x, y=y, n_iterations=n_iterations)\n",
"\n",
"print('w_gd_1e6: {}'.format(w_gd_1e6))\n",
"print('Vergleich zu Startkosten: {}'.format(J_all_1e6[-1]/J_all_1e6[0]))\n",
"print('Vergleich zu analytischer Lösung: {}'.format(J_all_1e6[-1]/J_ana))\n",
"print('(w0_gd - w0_ana)/w0_ana: {}'.format((w_gd_1e6[0]-w_ana[0])/w_ana[0]))\n",
"print('(w1_gd - w1_ana)/w1_ana: {}'.format((w_gd_1e6[1]-w_ana[1])/w_ana[1]))"
]
},
{
"cell_type": "markdown",
"id": "f35b62d4",
"metadata": {},
"source": [
"### Kosten J als Funktion von Gradient Descent Schritten"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "b18c5272",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"<Axes: >"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "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
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"sns.lineplot(x=list(range(len(J_all_1e6))), y=J_all_1e6)"
]
},
{
"cell_type": "markdown",
"id": "5ec059f6",
"metadata": {},
"source": [
"### Plotten der Ergebnisse und Vergleich zwischen analytischer und numerischer Lösung\n",
"Nach $10^4$ Schritten des Gradient Descent Algorithmus weicht der lineare Fit noch sichtbar von der analytischen Lösung ab. Nach $10^5$ Schritten ist der Unterschied im Plot kaum zu erkennen.\n",
"Die numerische Lösung war in diesem Beispiel deutlich langsamer als die analytische. Allerdings haben wir für die analytische Lösung auch eine effiziente numpy-Implementierung genutzt und für die numerische unoptimierten Python-Code."
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "21c941e4",
"metadata": {},
"outputs": [
{
"data": {
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"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# plot\n",
"xplot = [min(data['BuildingArea']), max(data['BuildingArea'])]\n",
"yplot_ana = [h_w(x, w_ana) for x in xplot]\n",
"yplot_gd_1e4 = [h_w(x, w_gd_1e4) for x in xplot]\n",
"yplot_gd_1e5 = [h_w(x, w_gd_1e5) for x in xplot]\n",
"# yplot_gd_3e5 = [h_w(x, w_gd_3e5) for x in xplot]\n",
"yplot_gd_1e6 = [h_w(x, w_gd_1e6) for x in xplot]\n",
"ax = sns.scatterplot(x=data['BuildingArea'], y=data['Price'])\n",
"ax = sns.lineplot(x=xplot, y=yplot_ana, ax=ax)\n",
"ax = sns.lineplot(x=xplot, y=yplot_gd_1e4, color='red', ax=ax)\n",
"ax = sns.lineplot(x=xplot, y=yplot_gd_1e5, color='grey', ax=ax)\n",
"# ax = sns.lineplot(x=xplot, y=yplot_gd_3e5, color='green', linestyle='dotted', ax=ax)\n",
"ax = sns.lineplot(x=xplot, y=yplot_gd_1e6, color='pink', linestyle='--', ax=ax)"
]
},
{
"cell_type": "markdown",
"id": "60bc96a1",
"metadata": {},
"source": [
"## Vorhersagen unseres Modells\n",
"\n",
"Man kann die Vorhersagen des Modells entweder im Plot oben auf der Geraden ablesen. Zu jedem Wert von `BuildingArea` (x-Achse des Plots) kann so der `Preis` auf der y-Achse abgelesen werden.\n",
"\n",
"Alternativ können wir die von uns oben definierte Funktion `h_w(x, w)` aufrufen. Der Parameter `w` ist die gefunden Lösung und `x` die `BuildingArea` für die wir einen Preis vorhersagen wollen."
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "e13003c8",
"metadata": {
"scrolled": true
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Preis laut analytischem Modell: 2170478.23\n",
"Preis laut Gradient Descent Modell nach 10^5 Iterationen: 2078517.46\n",
"Preis laut Gradient Descent Modell nach 3*10^5 Iterationen: 2167088.49\n",
"Preis laut Gradient Descent Modell nach 1*10^6 Iterationen: 2170478.20\n"
]
}
],
"source": [
"# Beispiel: Vorhersage unseres Modells für ein Haus mit Wohnfläche 287:\n",
"# wir machen je eine Vorhersage mit\n",
"# 1. den analytisch gefundenen Paramtern\n",
"# 2. den mit Gradient Descent nach 10^5 Iterationen gefundenen Parametern\n",
"# 3. den mit Gradient Descent nach 3*10^5 Iterationen gefundenen Parametern\n",
"building_area_new = 287\n",
"price_ana = h_w(x=building_area_new, w=w_ana)\n",
"price_1e5 = h_w(x=building_area_new, w=w_gd_1e5)\n",
"price_3e5 = h_w(x=building_area_new, w=w_gd_3e5)\n",
"price_1e6 = h_w(x=building_area_new, w=w_gd_1e6)\n",
"print('Preis laut analytischem Modell: {:.2f}'.format(price_ana))\n",
"print('Preis laut Gradient Descent Modell nach 10^5 Iterationen: {:.2f}'.format(price_1e5))\n",
"print('Preis laut Gradient Descent Modell nach 3*10^5 Iterationen: {:.2f}'.format(price_3e5))\n",
"print('Preis laut Gradient Descent Modell nach 1*10^6 Iterationen: {:.2f}'.format(price_1e6))"
]
},
{
"cell_type": "markdown",
"id": "fca62677",
"metadata": {},
"source": [
"## $R^2$"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "f1703c7f",
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"erklärte Varianz (R^2): 0.22971025499088604\n"
]
}
],
"source": [
"x = data['BuildingArea'].to_numpy(copy=True)\n",
"y = data['Price'].to_numpy(copy=True)\n",
"J_ana = J(w=w_ana, x=x, y=y)\n",
"MSE = 2*J_ana\n",
"mu_y = sum(y)/len(y)\n",
"sigma_y_quadrat = ( (y - mu_y) @ (y - mu_y) ) / len(y)\n",
"R2 = 1 - MSE/sigma_y_quadrat\n",
"print('erklärte Varianz (R^2): {}'.format(R2))"
]
}
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