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"source": [
"<img width=\"800px\" src=\"../fidle/img/header.svg\"></img>\n",
"\n",
"# <!-- TITLE --> [POLR1] - Complexity Syndrome\n",
"<!-- DESC --> Illustration of the problem of complexity with the polynomial regression\n",
"<!-- AUTHOR : Jean-Luc Parouty (CNRS/SIMaP) -->\n",
"\n",
"## Objectives :\n",
" - Visualizing and understanding under and overfitting\n",
" \n",
"## What we're going to do :\n",
"\n",
"We are looking for a polynomial function to approximate the observed series : \n",
"$ y = a_n\\cdot x^n + \\dots + a_i\\cdot x^i + \\dots + a_1\\cdot x + b $ \n",
"\n",
"\n",
"## Step 1 - Import and init"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"import math\n",
"import random\n",
"import matplotlib\n",
"import matplotlib.pyplot as plt\n",
"import sys\n",
"import fidle\n",
"\n",
"# Init Fidle environment\n",
"run_id, run_dir, datasets_dir = fidle.init('POLR1')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Step 2 - Dataset generation"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"# ---- Parameters\n",
"\n",
"n = 100\n",
"\n",
"xob_min = -5\n",
"xob_max = 5\n",
"\n",
"deg = 7\n",
"a_min = -2\n",
"a_max = 2\n",
"\n",
"noise = 2000\n",
"\n",
"# ---- Train data\n",
"# X,Y : data\n",
"# X_norm,Y_norm : normalized data\n",
"\n",
"X = np.random.uniform(xob_min,xob_max,(n,1))\n",
"# N = np.random.uniform(-noise,noise,(n,1))\n",
"N = noise * np.random.normal(0,1,(n,1))\n",
"\n",
"a = np.random.uniform(a_min,a_max, (deg,))\n",
"fy = np.poly1d( a )\n",
"\n",
"Y = fy(X) + N\n",
"\n",
"# ---- Data normalization\n",
"#\n",
"X_norm = (X - X.mean(axis=0)) / X.std(axis=0)\n",
"Y_norm = (Y - Y.mean(axis=0)) / Y.std(axis=0)\n",
"\n",
"# ---- Data visualization\n",
"\n",
"width = 12\n",
"height = 6\n",
"nb_viz = min(2000,n)\n",
"\n",
"def vector_infos(name,V):\n",
" m=V.mean(axis=0).item()\n",
" s=V.std(axis=0).item()\n",
" print(\"{:8} : mean={:+12.4f} std={:+12.4f} min={:+12.4f} max={:+12.4f}\".format(name,m,s,V.min(),V.max()))\n",
"\n",
"\n",
"fidle.utils.display_md('#### Generator :')\n",
"print(f\"Nomber of points={n} deg={deg} bruit={noise}\")\n",
"\n",
"fidle.utils.display_md('#### Datasets :')\n",
"print(f\"{nb_viz} points visibles sur {n})\")\n",
"plt.figure(figsize=(width, height))\n",
"plt.plot(X[:nb_viz], Y[:nb_viz], '.')\n",
"plt.tick_params(axis='both', which='both', bottom=False, left=False, labelbottom=False, labelleft=False)\n",
"plt.xlabel('x axis')\n",
"plt.ylabel('y axis')\n",
"fidle.scrawler.save_fig(\"01-dataset\")\n",
"plt.show()\n",
"\n",
"fidle.utils.display_md('#### Before normalization :')\n",
"vector_infos('X',X)\n",
"vector_infos('Y',Y)\n",
"\n",
"fidle.utils.display_md('#### After normalization :') \n",
"vector_infos('X_norm',X_norm)\n",
"vector_infos('Y_norm',Y_norm)\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Step 3 - Polynomial regression with NumPy\n",
"### 3.1 - Underfitting"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"def draw_reg(X_norm, Y_norm, x_hat,fy_hat, size, save_as):\n",
" plt.figure(figsize=size)\n",
" plt.plot(X_norm, Y_norm, '.')\n",
"\n",
" x_hat = np.linspace(X_norm.min(), X_norm.max(), 100)\n",
"\n",
" plt.plot(x_hat, fy_hat(x_hat))\n",
" plt.tick_params(axis='both', which='both', bottom=False, left=False, labelbottom=False, labelleft=False)\n",
" plt.xlabel('x axis')\n",
" plt.ylabel('y axis')\n",
" fidle.scrawler.save_fig(save_as)\n",
" plt.show()"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"reg_deg=1\n",
"\n",
"a_hat = np.polyfit(X_norm.reshape(-1,), Y_norm.reshape(-1,), reg_deg)\n",
"fy_hat = np.poly1d( a_hat )\n",
"\n",
"print(f'Nombre de degrés : {reg_deg}')\n",
"draw_reg(X_norm[:nb_viz],Y_norm[:nb_viz], X_norm,fy_hat, (width,height), save_as='02-underfitting')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3.2 - Good fitting"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"reg_deg=5\n",
"\n",
"a_hat = np.polyfit(X_norm.reshape(-1,), Y_norm.reshape(-1,), reg_deg)\n",
"fy_hat = np.poly1d( a_hat )\n",
"\n",
"print(f'Nombre de degrés : {reg_deg}')\n",
"draw_reg(X_norm[:nb_viz],Y_norm[:nb_viz], X_norm,fy_hat, (width,height), save_as='03-good_fitting')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 3.3 - Overfitting"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"reg_deg=24\n",
"\n",
"a_hat = np.polyfit(X_norm.reshape(-1,), Y_norm.reshape(-1,), reg_deg)\n",
"fy_hat = np.poly1d( a_hat )\n",
"\n",
"print(f'Nombre de degrés : {reg_deg}')\n",
"draw_reg(X_norm[:nb_viz],Y_norm[:nb_viz], X_norm,fy_hat, (width,height), save_as='04-over_fitting')"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {},
"outputs": [],
"source": [
"fidle.end()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"---\n",
"<img width=\"80px\" src=\"../fidle/img/logo-paysage.svg\"></img>"
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