{ "cells": [ { "cell_type": "markdown", "id": "6cf3da8a", "metadata": {}, "source": [ "# Modul 4: Teori" ] }, { "cell_type": "markdown", "id": "1a43b5f1", "metadata": {}, "source": [ "## Pyodide-bemærkning (Python direkte i Browseren)" ] }, { "cell_type": "markdown", "id": "fdf208a4", "metadata": {}, "source": [ "Koden i denne Notebook er beregnet til at blive kørt lokalt på din egen computer og ikke direkte i browseren via Pyodide. Årsagen er, at vi bruger `scikit-learn` til at downloade hele MNIST-datasættet. Dette er en data-tung operation, som involverer download af en stor fil, hvilket ikke umiddelbart understøttes eller er praktisk i et browser-baseret miljø som Pyodide." ] }, { "cell_type": "markdown", "id": "bd249e1e", "metadata": {}, "source": [ "## Digitale billeder" ] }, { "cell_type": "code", "execution_count": 10, "id": "0c3d4815", "metadata": {}, "outputs": [], "source": [ "import matplotlib.pyplot as plt\n", "import numpy as np\n", "from sklearn.datasets import fetch_openml\n", "# Hent MNIST (784 = 28x28 pixels)\n", "mnist = fetch_openml('mnist_784', version=1, as_frame=False)\n", "X, y = mnist.data, mnist.target.astype('int64')" ] }, { "cell_type": "code", "execution_count": 2, "id": "cacd749c", "metadata": {}, "outputs": [ { "data": { "image/png": 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", "text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "plt.figure(figsize=(12, 12))\n", "plt.subplot(221)\n", "plt.imshow(X[0].reshape(28, 28), cmap='gray')\n", "plt.title(f\"Digit: {y[0]}\")\n", "plt.subplot(222)\n", "plt.imshow(X[1].reshape(28, 28), cmap='gray')\n", "plt.title(f\"Digit: {y[1]}\")\n", "plt.subplot(223)\n", "plt.imshow(X[2].reshape(28, 28), cmap='gray')\n", "plt.title(f\"Digit: {y[2]}\")\n", "plt.subplot(224)\n", "plt.imshow(X[3].reshape(28, 28), cmap='gray')\n", "plt.title(f\"Digit: {y[3]}\")\n", "# vis grafen\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "af74256c", "metadata": {}, "source": [ "## Vektorfunktioner" ] }, { "cell_type": "markdown", "id": "1232974e", "metadata": {}, "source": [ "Lad $d,k \\in \\mathbb{N}$. En vektorfunktion af flere variable er en funktion af formen\n", "\n", "\\begin{equation*}\n", " \\pmb{f} \\colon \\operatorname{dom}(\\pmb{f}) \\to \\mathbb{R}^k, \\text{\\; hvor }\\operatorname{dom}(\\pmb{f}) \\subseteq \\mathbb{R}^d.\n", "\\end{equation*}\n", "\n", "Altså har en vektorfunktion $\\pmb{f} = \\pmb{x} \\mapsto \\pmb{f}(\\pmb{x})$:\n", "* Input (domænet): vektorer $\\pmb{x}$ i $\\mathbb{R}^d$\n", "* Output (kodomænet): vektorer $\\pmb{f}(\\pmb{x})$ i $\\mathbb{R}^k$" ] }, { "cell_type": "markdown", "id": "e6cdfc9f", "metadata": {}, "source": [ "### Input $\\pmb{x}_0$" ] }, { "cell_type": "code", "execution_count": 3, "id": "a16d61da", "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/plain": [ "
" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "# Vis det tredje billede fra træningssættet\n", "x0 = X[2].reshape(28, 28)\n", "plt.figure(figsize=(7, 7))\n", "plt.imshow(x0, cmap='gray')\n", "plt.axis('off')\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "6162841e", "metadata": {}, "source": [ "### Output $\\pmb{f}(\\pmb{x}_0)$" ] }, { "cell_type": "markdown", "id": "8b379ec1", "metadata": {}, "source": [ "* Output: \n", "\n", "

\"Dette er cifret 4\"

\n", "\n", "---\n", "\n", "\n", "* Ideelt output:\n", "\n", "\\begin{equation*}\n", " \\pmb{f}(\\pmb{x}_0) = \\begin{bmatrix} 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\\\ 0 \\end{bmatrix} \n", "\\end{equation*}\n", "\n", "

Dette betyder: \"100% sikker på, at dette er cifret 4\"

\n", "\n", "---\n", "\n", "* Mere realistisk output:\n", "\n", "\\begin{equation*}\n", " \\pmb{f}(\\pmb{x}_0) = \\begin{bmatrix} 0.01 \\\\ 0.03 \\\\ 0.01 \\\\ 0.01 \\\\ 0.87 \\\\ 0.01 \\\\ 0.01 \\\\ 0.03 \\\\ 0.01 \\\\ 0.01 \\end{bmatrix} \n", "\\end{equation*}\n", "\n", "

Dette betyder: \"87% sikker på, at dette er cifret 4. Lille sandsynlighed (3%) for, at det er 1 eller 7.\"

\n", "\n", "---\n", "\n", "* Under alle omstændigheder: **Outputtet er en vektor af sandsynligheder i $\\mathbb{R}^{10}$. Derfor er $\\operatorname{co\\text{-}dom}(\\pmb{f})=\\mathbb{R}^{10}$**." ] }, { "cell_type": "markdown", "id": "d0d60287", "metadata": {}, "source": [ "### Hvad med inputtet?" ] }, { "cell_type": "markdown", "id": "7af8838b", "metadata": {}, "source": [ "Det er faktisk en matrix på størrelse 28x28:\n", "\n", "$$\n", "%\\tiny\n", "\\scriptsize\n", "\\begin{bmatrix}\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 67 & 232 & 39 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 62 & 81 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 120 & 180 & 39 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 126 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 153 & 210 & 40 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 220 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27 & 254 & 162 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 222 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 183 & 254 & 125 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 46 & 245 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 198 & 254 & 56 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 120 & 254 & 163 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 23 & 231 & 254 & 29 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 159 & 254 & 120 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 163 & 254 & 216 & 16 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 159 & 254 & 67 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 14 & 86 & 178 & 248 & 254 & 91 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 159 & 254 & 85 & 0 & 0 & 0 & 47 & 49 & 116 & 144 & 150 & 241 & 243 & 234 & 179 & 241 & 252 & 40 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 150 & 253 & 237 & 207 & 207 & 207 & 253 & 254 & 250 & 240 & 198 & 143 & 91 & 28 & 5 & 233 & 250 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 119 & 177 & 177 & 177 & 177 & 177 & 98 & 56 & 0 & 0 & 0 & 0 & 0 & 102 & 254 & 220 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 137 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 57 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 57 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 255 & 94 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 96 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 254 & 153 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 169 & 255 & 153 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 96 & 254 & 153 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n", "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\n", "\\end{bmatrix}\n", "$$\n", "\n", "Men det er ikke umiddelbart en (søjle)vektor!\n", "\n", "Vi kan gøre den om til en vektor ved at stakke billedets rækker oven på hinanden. Dette kaldes *flattening* (udfladning) af billedet i Python." ] }, { "cell_type": "code", "execution_count": 4, "id": "573cd0ac", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Søjle-repræsentation af billedet (784 tal):\n" ] }, { "data": { "text/plain": [ "array([[ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 0],\n", " [ 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"print(\"\\nSøjle-repræsentation af billedet (784 tal):\")\n", "x0.reshape(784,1)" ] }, { "cell_type": "markdown", "id": "bae1772e", "metadata": {}, "source": [ "Så vi kan tænke på inputtet som vektorer i $\\mathbb{R}^{784}$! Det vil sige" ] }, { "cell_type": "markdown", "id": "334a4498", "metadata": {}, "source": [ "### En AI funktion" ] }, { "cell_type": "markdown", "id": "4196afcf", "metadata": {}, "source": [ "**AI'en er en vektorfunktion med $d=784$, $k=10$.**\n", "\n", "I maskinlæring kaldes funktionen for en model, og den specifikke AI-funktion afhænger typisk af tusindvis eller millioner af parametre (kaldet vægte). For hvert sæt af parametre/vægte får vi en ny AI-funktion. Sådanne funktioner (også dybe neurale netværk) er ikke særligt komplicerede, men ofte lange at skrive eksplicit. De er bygget af\n", "\n", "* Sammensætning af simple vektorfunktioner $\\pmb{g} \\circ \\pmb{h}$\n", "* De simple funktioner er normalt kun:\n", " 1. Affine vektorfunktioner $\\pmb{x} \\mapsto A \\pmb{x} + \\pmb{b}$ (elementerne i matrixen $A$ og vektoren $\\pmb{b}$ er parametrene/vægtene)\n", " 2. En ikke-lineær aktiveringsfunktion, fx ReLU.\n", "\n", "Antallet af sammensætninger $\\pmb{g}_1 \\circ \\pmb{g}_2 \\circ \\pmb{g}_3 \\circ \\cdots \\circ \\pmb{g}_N$ beskriver netværkets **dybde**." ] }, { "cell_type": "markdown", "id": "ff1aa3a8", "metadata": {}, "source": [ "## Neurale netværk" ] }, { "cell_type": "markdown", "id": "dd40506f", "metadata": {}, "source": [ "Hvordan ser disse \"AI-funktioner\" ud?" ] }, { "cell_type": "markdown", "id": "a88e3b4f", "metadata": {}, "source": [ "### Generel notation for et feedforward ReLU-netværk" ] }, { "cell_type": "markdown", "id": "8787b403", "metadata": {}, "source": [ "Et feedforward-netværk beregner outputtet ved sekventielt at føre input gennem en række lag. For hvert lag $\\ell$ beregnes først en **pre-aktivering** (også kaldet **logits**), $z^{(\\ell)}$, efterfulgt af en **aktivering** (eller **hidden state**), $h^{(\\ell)}$.\n", "\n", "\\begin{equation*}\n", "\\begin{aligned}\n", "&h^{(0)} = x \\in \\mathbb{R}^{n_0} &&\\text{(Input)}\\\\\n", "&z^{(\\ell)} = W_\\ell h^{(\\ell-1)} + b_\\ell, && \\ell = 1,2,\\dots,L &&\\text{(Logits)}\\\\\n", "&h^{(\\ell)} =\n", "\\begin{cases}\n", "\\sigma (z^{(\\ell)}), & \\ell < L, \\\\\n", "z^{(\\ell)}, & \\ell = L,\n", "\\end{cases}\n", "&&\\text{(Activation)}\n", "\\end{aligned}\n", "\\end{equation*}\n", "\n", "hvor\n", "\n", "* $h^{(0)}$ er input-vektoren $x$.\n", "* $W_\\ell \\in \\mathbb{R}^{n_\\ell \\times n_{\\ell-1}}$ er vægtmatricen for lag $\\ell$.\n", "* $b_\\ell \\in \\mathbb{R}^{n_\\ell}$ er bias-vektoren.\n", "* $\\sigma:\\mathbb{R}\\to\\mathbb{R}$ er en ikke-lineær aktiveringsfunktion (anvendt koordinatvis) typisk **ReLU**:\n", "\n", "\\begin{equation*}\n", " \\sigma(z) = \\max(0,z)\n", " \\quad\\text{(ReLU)}\n", "\\end{equation*}\n", "\n", "* $n_0=d$ er inputdimensionen, $n_L=k$ outputdimensionen. \n", "\n", "Vi siger kort at netværket er af **formen** $n_0 \\to n_1 \\to \\cdots \\to n_L$. \n", "\n", "Netværkets samlede funktion $\\Phi: \\mathbb{R}^d \\to \\mathbb{R}^k$ giver det endelige output:\n", "\n", "\\begin{equation*}\n", "\\Phi(x) = z^{(L)} = W_L h^{(L-1)} + b_L,\n", "\\end{equation*}\n", "\n", "idet sidste lag her er lineært (uden ReLU-aktivering).\n", "\n", "![Et neuralt netværk](images/DNN_example_crop.png)" ] }, { "cell_type": "markdown", "id": "cda12e61", "metadata": {}, "source": [ "### Shallow netværk (ét skjult lag med L=2)" ] }, { "cell_type": "markdown", "id": "d6990a90", "metadata": {}, "source": [ "For et **shallow** netværk $\\Phi:\\mathbb{R}^2\\to\\mathbb{R}$ med ét skjult lag af størrelse $n$:\n", "\n", "\\begin{equation*}\n", "\\Phi(x) = W_2 \\, \\sigma(W_1 x + b_1) + b_2,\n", "\\end{equation*}\n", "\n", "hvor\n", "\n", "* $x \\in \\mathbb{R}^2$,\n", "* $W_1 \\in \\mathbb{R}^{n\\times 2},\\ b_1 \\in \\mathbb{R}^{n},$\n", "* $W_2 \\in \\mathbb{R}^{1\\times n},\\ b_2 \\in \\mathbb{R}.$\n", "\n", "\n", "**Illustration af lagene**:\n", "\n", "\\begin{equation*}\n", "h^{(0)} \\xrightarrow{W_1,b_1} z^{(1)} \\xrightarrow{\\sigma} h^{(1)}\n", "\\xrightarrow{W_2,b_2} z^{(2)} \\xrightarrow{\\sigma} h^{(2)} \\xrightarrow{W_3,b_3} z^{(3)} = \\Phi(x)\n", "\\end{equation*}\n", "\n", "Hvert ReLU-lag opdeler rummet i *lineære regioner* bestemt af ligningerne $ (W_\\ell h^{(\\ell-1)} + b_\\ell)_i = 0 $, så $\\Phi$ er en **stykkevist lineær funktion** på $\\mathbb{R}^d$." ] }, { "cell_type": "markdown", "id": "4662f8d2", "metadata": {}, "source": [ "### Generel formel for et ReLU-netværk med $L=3$" ] }, { "cell_type": "markdown", "id": "0605b183", "metadata": {}, "source": [ "Vi betragter en funktion\n", "\n", "\\begin{equation*}\n", "\\Phi:\\mathbb{R}^{n_0}\\to\\mathbb{R}^{n_3}\n", "\\end{equation*}\n", "defineret som et **fuldt forbundet netværk** med to skjulte lag:\n", "\n", "\\begin{equation*}\n", "\\begin{aligned}\n", "h^{(0)} &= x \\in \\mathbb{R}^{n_0}, \\\\[2pt]\n", "z^{(1)} &= W_1 h^{(0)} + b_1,\\\\\n", "h^{(1)} &= \\sigma \\bigl(z^{(1)}\\bigr), \\\\[4pt]\n", "z^{(2)} &= W_2 h^{(1)} + b_2,\\\\\n", "h^{(2)} &= \\sigma \\bigl(z^{(2)}\\bigr), \\\\[4pt]\n", "z^{(3)} &= W_3 h^{(2)} + b_3,\\\\\n", "h^{(3)} &= z^{(3)}.\n", "\\end{aligned}\n", "\\end{equation*}\n", "\n", "hvor $n_0 = d$, $n_3 = k$, og\n", "\n", "* $W_1 \\in \\mathbb{R}^{n_1\\times n_0},\\ b_1\\in\\mathbb{R}^{n_1}$\n", "* $W_2 \\in \\mathbb{R}^{n_2\\times n_1},\\ b_2\\in\\mathbb{R}^{n_2}$\n", "* $W_3 \\in \\mathbb{R}^{n_3\\times n_2},\\ b_3\\in\\mathbb{R}^{n_3}$\n", "\n", "Det samlede funktionsudtryk bliver:\n", "\n", "\\begin{equation*}\n", "\\Phi(x)\n", "= W_3 \\, \\sigma \\bigl(W_2 \\, \\sigma(W_1 x + b_1) + b_2 \\bigr) + b_3.\n", "\\end{equation*}\n", "\n", "\n", "**Eksempel**: For et konkret netværk med to inputvariabler og ét output af formen $2 \\to n_1 \\to n_2 \\to 1$:\n", "\n", "\\begin{equation*}\n", "\\Phi(x_1,x_2)\n", "= W_3 \\,\\sigma \\Big(W_2 \\, \\sigma \\big(W_1\n", "\\begin{bmatrix}x_1 \\\\ x_2\\end{bmatrix} + b_1\\big) + b_2\\Big) + b_3,\n", "\\end{equation*}\n", "\n", "hvor dimensionerne er\n", "\n", "\\begin{equation*}\n", "W_1 \\in \\mathbb{R}^{n_1\\times2},\\quad\n", "W_2 \\in \\mathbb{R}^{n_2\\times n_1},\\quad\n", "W_3 \\in \\mathbb{R}^{1\\times n_2}.\n", "\\end{equation*}" ] }, { "cell_type": "markdown", "id": "2b8e8f05", "metadata": {}, "source": [ "### Netværk og træning direkte i SKLearn" ] }, { "cell_type": "markdown", "id": "2323cd17", "metadata": {}, "source": [ "Vi skal finde en \"AI-funktion\" \n", "\n", "\\begin{equation*}\n", "\\Phi : \\mathbb{R}^{784} \\to \\mathbb{R}^{10}\n", "\\end{equation*}\n", "\n", "der \"bedst\"-muligt kan klassificere et billede af et håndskrevet ciffer.\n", "\n", "I koden nedenfor opbygges denne som et ReLU-netværk med to skjulte lag, hvilket giver en samlet dybde på $L=3$. Formen af netværket er $784 \\to 256 \\to 128 \\to 10$. Altså er vægt-matricerne af størrelse\n", "\n", "\\begin{equation*}\n", "\\begin{aligned}\n", "&W_1 \\in \\mathbb{R}^{256 \\times 784}, \\quad b_1 \\in \\mathbb{R}^{256}, \\\\\n", "&W_2 \\in \\mathbb{R}^{128 \\times 256}, \\quad b_2 \\in \\mathbb{R}^{128}, \\\\\n", "&W_3 \\in \\mathbb{R}^{10 \\times 128}, \\quad b_3 \\in \\mathbb{R}^{10}.\n", "\\end{aligned}\n", "\\end{equation*}" ] }, { "cell_type": "code", "execution_count": 5, "id": "18483aa7", "metadata": {}, "outputs": [], "source": [ "from sklearn.model_selection import train_test_split\n", "from sklearn.neural_network import MLPClassifier\n", "\n", "# Normaliser til [0,1]\n", "X = X / 255.0\n", "\n", "X_train, X_test, y_train, y_test = train_test_split(\n", " X, y, test_size=1/7, random_state=42\n", ")\n", "\n", "# DNN med samme “størrelse” som PyTorch-eksemplet\n", "clf = MLPClassifier(\n", " hidden_layer_sizes=(256, 128), # to skjulte lag\n", " activation='relu',\n", " solver='adam',\n", " batch_size=64,\n", " learning_rate_init=1e-3,\n", " max_iter=6,\n", " verbose=True\n", ")" ] }, { "cell_type": "markdown", "id": "e962ffda", "metadata": {}, "source": [ "#### Netværkets størrelse" ] }, { "cell_type": "markdown", "id": "579a1f51", "metadata": {}, "source": [ "> Hvor mange parametre er der?" ] }, { "cell_type": "markdown", "id": "a54d5c57", "metadata": {}, "source": [ "##### Svar" ] }, { "cell_type": "markdown", "id": "225ea351", "metadata": {}, "source": [ "Det samlede antal parametre for netværket er:\n", "\n", "* Lag 1: (784 input × 256 neuroner) + 256 bias = 200.704 + 256 = 200.960\n", "* Lag 2: (256 input × 128 neuroner) + 128 bias = 32.768 + 128 = 32.896\n", "* Lag 3: (128 input × 10 neuroner) + 10 bias = 1.280 + 10 = 1.290\n", "\n", "I alt: 200.960 + 32.896 + 1.290 = 235.146" ] }, { "cell_type": "markdown", "id": "12059d0a", "metadata": {}, "source": [ "#### Træning via SKLearn" ] }, { "cell_type": "markdown", "id": "18c0ce30", "metadata": {}, "source": [ "Vi finder de optimale værdier for alle parametrene i $Phi-$funktionen ved at træne modellen med `fit`-metoden:" ] }, { "cell_type": "code", "execution_count": 6, "id": "30eb599b", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Iteration 1, loss = 0.22757620\n", "Iteration 2, loss = 0.08920455\n", "Iteration 3, loss = 0.05955793\n", "Iteration 4, loss = 0.04347159\n", "Iteration 5, loss = 0.03397271\n", "Iteration 6, loss = 0.02918372\n" ] }, { "name": "stderr", "output_type": "stream", "text": [ "/home/jakle/Python/envs/intermat/lib/python3.10/site-packages/sklearn/neural_network/_multilayer_perceptron.py:781: ConvergenceWarning: Stochastic Optimizer: Maximum iterations (6) reached and the optimization hasn't converged yet.\n", " warnings.warn(\n" ] }, { "data": { "text/html": [ "
MLPClassifier(batch_size=64, hidden_layer_sizes=(256, 128), max_iter=6,\n",
       "              verbose=True)
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
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" ], "text/plain": [ "MLPClassifier(batch_size=64, hidden_layer_sizes=(256, 128), max_iter=6,\n", " verbose=True)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "# Træn\n", "clf.fit(X_train, y_train)" ] }, { "cell_type": "markdown", "id": "92cc0d42", "metadata": {}, "source": [ "#### Forudsigelse på et enkelt billede" ] }, { "cell_type": "markdown", "id": "beda7585", "metadata": {}, "source": [ "Denne funktion kan vi nu bruge på et enkelt input-billede for at få en forudsigelse. Lad os tage et enkelt billede fra vores testsæt og se, hvad modellen forudsiger." ] }, { "cell_type": "code", "execution_count": 7, "id": "540e552f", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0.61960784 0.90588235 0.14901961 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0.02745098 0.25490196 0.56862745\n", " 0.56862745 0.34509804 0. 0.05490196 0.83137255 0.99215686\n", " 0.28235294 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.09019608 0.14509804\n", " 0.25882353 0.7254902 0.99215686 0.9372549 0.91372549 0.99215686\n", " 0.59607843 0.45490196 0.99215686 0.80784314 0.10980392 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0.06666667 0.87058824 0.99215686 0.99215686 0.9372549\n", " 0.80392157 0.27058824 0.14509804 0.64705882 0.99607843 0.99215686\n", " 0.89803922 0.10980392 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.0745098\n", " 0.90196078 0.99215686 0.99215686 0.48235294 0. 0.\n", " 0. 0.58823529 0.99607843 0.81176471 0.24313725 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.14901961 0.60392157\n", " 0.99215686 0.78823529 0.19215686 0. 0.46666667 0.98431373\n", " 0.98431373 0.30588235 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0.03921569 0.76470588 0.99215686\n", " 0.92941176 0.80784314 0.97647059 0.99215686 0.56470588 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0.10196078 0.50196078 0.92156863 0.99215686\n", " 0.99215686 0.99215686 0.90196078 0.38431373 0.14509804 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0.38039216 0.99215686 0.99215686 0.99215686\n", " 0.99607843 0.99215686 0.9372549 0.58431373 0.12156863 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0.56078431 0.99215686 0.78823529 0.06666667 0.29411765 0.51764706\n", " 0.81176471 0.99215686 0.84705882 0.11372549 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0.1372549 0.90196078 0.99607843\n", " 0.54509804 0. 0. 0. 0.07058824 0.56470588\n", " 0.98431373 0.8627451 0.12156863 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0.80784314 0.99215686 0.09411765 0.\n", " 0. 0. 0. 0. 0.66666667 0.99215686\n", " 0.69019608 0.01568627 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0.50980392 0.99215686 0.38823529 0. 0. 0.\n", " 0. 0. 0.09019608 0.78039216 0.99215686 0.14117647\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.27058824 0.96862745\n", " 0.64705882 0.64313725 0. 0. 0. 0.\n", " 0. 0.28627451 0.99215686 0.45882353 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0.85490196 0.99215686 0.99215686\n", " 0.11764706 0. 0. 0. 0. 0.28627451\n", " 0.99215686 0.61176471 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0.38431373 0.83137255 0.99215686 0.8745098 0.3254902\n", " 0. 0. 0. 0.07058824 0.85098039 0.90588235\n", " 0.07058824 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0.28627451 0.98039216 0.99607843 0.88627451 0.30980392 0.\n", " 0. 0.23529412 0.95686275 0.86666667 0.0627451 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.13333333\n", " 0.99607843 0.99215686 0.83921569 0.30980392 0. 0.28627451\n", " 0.99215686 0.43529412 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0.36470588 0.99215686\n", " 0.99215686 0.97647059 0.78823529 0.81568627 0.99215686 0.14117647\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0.00392157 0.31764706 0.81176471 0.99215686\n", " 0.99215686 0.85490196 0.36078431 0.00784314 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. 0. 0.\n", " 0. 0. 0. 0. ]]\n", "8\n", "[[7.08580512e-05 2.61484819e-04 1.44964793e-05 1.17057312e-03\n", " 1.67150883e-05 5.34671916e-02 4.29566890e-01 8.57747096e-06\n", " 5.14443444e-01 9.79769433e-04]]\n", "[8]\n" ] } ], "source": [ "image_index = 2 # Vælg et billede (input)\n", "input_image = X_test[image_index:image_index+1] # Format (1, 784)\n", "true_label = y_test[image_index]\n", "\n", "probability_vector = clf.predict_proba(input_image) # Få sandsynlighedsvektoren\n", "predicted_label = clf.predict(input_image) # Lav forudsigelse \n", "\n", "print(input_image)\n", "print(true_label)\n", "print(probability_vector)\n", "print(predicted_label)" ] }, { "cell_type": "markdown", "id": "f58aa6ac", "metadata": {}, "source": [ "#### Total forudsigelse" ] }, { "cell_type": "markdown", "id": "212eca12", "metadata": {}, "source": [ "Den samlede procentandel af billeder i testsættet, som modellen klassificerede korrekt kan findes ved:" ] }, { "cell_type": "code", "execution_count": 8, "id": "f45e90fb", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Test accuracy: 0.9778\n" ] } ], "source": [ "# Samlet evaluering\n", "print(\"Test accuracy:\", clf.score(X_test, y_test))" ] }, { "cell_type": "markdown", "id": "d7850f35", "metadata": {}, "source": [ "#### Visualisering af forudsigelser" ] }, { "cell_type": "markdown", "id": "b5afb6a4", "metadata": {}, "source": [ "For at få en bedre fornemmelse af, hvordan modellen opfører sig, kan vi visualisere dens forudsigelser på enkelte billeder fra testsættet. Nedenstående funktion plotter billedet, den korrekte label, den forudsagte label og et søjlediagram over de forudsagte sandsynligheder for hver klasse. Dette er nyttigt for at se, hvornår modellen er sikker, og hvornår den er i tvivl." ] }, { "cell_type": "code", "execution_count": 13, "id": "bcc8937e", "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "def show_images_with_mlp_probabilities(clf, X_test, y_test, X_test_orig, \n", " num_images=5, only_incorrect=False,\n", " image_shape=(8,8)):\n", " # Predict full test set\n", " pred_labels = clf.predict(X_test)\n", " probas = clf.predict_proba(X_test)\n", "\n", " # Select indices\n", " all_indices = np.arange(len(X_test))\n", " if only_incorrect:\n", " indices = all_indices[pred_labels != y_test][:num_images]\n", " else:\n", " indices = all_indices[:num_images]\n", "\n", " plt.figure(figsize=(12, 6))\n", "\n", " for i, idx in enumerate(indices):\n", " # --- Image plot ---\n", " plt.subplot(2, num_images, i + 1)\n", " img = X_test_orig[idx].reshape(image_shape)\n", " plt.imshow(img, cmap='gray')\n", " plt.title(f\"Idx {idx}\\nTrue {y_test[idx]}\\nPred {pred_labels[idx]}\")\n", " plt.axis('off')\n", "\n", " # --- Probability distribution ---\n", " plt.subplot(2, num_images, num_images + i + 1)\n", "\n", " p = probas[idx]\n", "\n", " classes = np.arange(len(p))\n", " colors = [\n", " \"red\" if c == y_test[idx] else\n", " (\"green\" if c == pred_labels[idx] else \"blue\")\n", " for c in classes\n", " ]\n", "\n", " plt.bar(classes, p, color=colors)\n", " plt.xticks(classes)\n", " plt.ylim(0, 1)\n", " plt.xlabel(\"Class\")\n", " plt.ylabel(\"Probability\")\n", "\n", " plt.tight_layout()\n", " plt.show()\n", "\n", "show_images_with_mlp_probabilities(\n", " clf, \n", " X_test, \n", " y_test, \n", " X_test_orig=X_test, \n", " only_incorrect=False,\n", " num_images=5,\n", " image_shape=(28,28)\n", ") " ] }, { "cell_type": "code", "execution_count": 12, "id": "c05364ee", "metadata": {}, "outputs": [ { "data": { "image/png": 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" ] }, "metadata": {}, "output_type": "display_data" } ], "source": [ "show_images_with_mlp_probabilities(\n", " clf, \n", " X_test, \n", " y_test, \n", " X_test_orig=X_test, \n", " only_incorrect=True,\n", " num_images=5,\n", " image_shape=(28,28)\n", ") " ] }, { "cell_type": "code", "execution_count": null, "id": "875bcfdc", "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.10.12" } }, "nbformat": 4, "nbformat_minor": 5 }