forked from pms/uczenie-maszynowe
1298 lines
73 KiB
Plaintext
1298 lines
73 KiB
Plaintext
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"### Uczenie maszynowe\n",
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"# 5. Metody ewaluacji"
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]
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## 5.1. Metodologia testowania"
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]
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
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},
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"source": [
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"W uczeniu maszynowym bardzo ważna jest ewaluacja budowanego modelu. Dlatego dobrze jest podzielić posiadane dane na odrębne zbiory – osobny zbiór danych do uczenia i osobny do testowania. W niektórych przypadkach potrzeba będzie dodatkowo wyodrębnić tzw. zbiór walidacyjny."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### Zbiór uczący a zbiór testowy"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Na zbiorze uczącym (treningowym) uczymy algorytmy, a na zbiorze testowym sprawdzamy ich poprawność.\n",
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"* Zbiór uczący powinien być kilkukrotnie większy od testowego (np. 4:1, 9:1 itp.).\n",
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"* Zbiór testowy często jest nieznany.\n",
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"* Należy unikać mieszania danych testowych i treningowych – nie wolno „zanieczyszczać” danych treningowych danymi testowymi!"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"Czasami potrzebujemy dobrać parametry modelu, np. $\\alpha$ – który zbiór wykorzystać do tego celu?"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### Zbiór walidacyjny"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Do doboru parametrów najlepiej użyć jeszcze innego zbioru – jest to tzw. **zbiór walidacyjny**"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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" * Zbiór walidacyjny powinien mieć wielkość zbliżoną do wielkości zbioru testowego, czyli np. dane można podzielić na te trzy zbiory w proporcjach 3:1:1, 8:1:1 itp."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"### Walidacja krzyżowa"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"Którą część danych wydzielić jako zbiór walidacyjny tak, żeby było „najlepiej”?"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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" * Niech każda partia danych pełni tę rolę naprzemiennie!"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"<img width=\"100%\" src=\"https://chrisjmccormick.files.wordpress.com/2013/07/10_fold_cv.png\"/>\n",
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"Żródło: https://chrisjmccormick.wordpress.com/2013/07/31/k-fold-cross-validation-with-matlab-code/"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### Walidacja krzyżowa\n",
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"\n",
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"* Podziel dane $D = \\left\\{ (x^{(1)}, y^{(1)}), \\ldots, (x^{(m)}, y^{(m)})\\right\\} $ na $N$ rozłącznych zbiorów $T_1,\\ldots,T_N$\n",
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"* Dla $i=1,\\ldots,N$, wykonaj:\n",
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" * Użyj $T_i$ do walidacji i zbiór $S_i$ do trenowania, gdzie $S_i = D \\smallsetminus T_i$. \n",
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" * Zapisz model $\\theta_i$.\n",
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"* Akumuluj wyniki dla modeli $\\theta_i$ dla zbiorów $T_i$.\n",
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"* Ustalaj parametry uczenia na akumulowanych wynikach."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### Walidacja krzyżowa – wskazówki\n",
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"\n",
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"* Zazwyczaj ustala się $N$ w przedziale od $4$ do $10$, tzw. $N$-krotna walidacja krzyżowa (*$N$-fold cross validation*). \n",
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"* Zbiór $D$ warto zrandomizować przed podziałem.\n",
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"* W jaki sposób akumulować wyniki dla wszystkich zbiórow $T_i$?\n",
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"* Po ustaleniu parametrów dla każdego $T_i$, trenujemy model na całych danych treningowych z ustalonymi parametrami.\n",
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"* Testujemy na zbiorze testowym (jeśli nim dysponujemy)."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### _Leave-one-out_\n",
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"\n",
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"Jest to szczególny przypadek walidacji krzyżowej, w której $N = m$."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "fragment"
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}
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},
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"source": [
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"* Jaki jest rozmiar pojedynczego zbioru $T_i$?\n",
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"* Jakie są zalety i wady tej metody?\n",
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"* Kiedy może być przydatna?"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "subslide"
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}
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},
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"source": [
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"### Zbiór walidujący a algorytmy optymalizacji\n",
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"\n",
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"* Gdy błąd rośnie na zbiorze uczącym, mamy źle dobrany parametr $\\alpha$. Należy go wtedy zmniejszyć.\n",
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"* Gdy błąd zmniejsza się na zbiorze trenującym, ale rośnie na zbiorze walidującym, mamy do czynienia ze zjawiskiem **nadmiernego dopasowania** (*overfitting*).\n",
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"* Należy wtedy przerwać optymalizację. Automatyzacja tego procesu to _early stopping_."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"## 5.2. Miary jakości"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
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},
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"source": [
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"Aby przeprowadzić ewaluację modelu, musimy wybrać **miarę** (**metrykę**), jakiej będziemy używać.\n",
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"\n",
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"Jakiej miary użyc najlepiej?\n",
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" * To zależy od rodzaju zadania.\n",
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" * Innych metryk używa się do regresji, a innych do klasyfikacji"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"### Metryki dla zadań regresji\n",
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"\n",
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"Dla zadań regresji możemy zastosować np.:\n",
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" * błąd średniokwadratowy (*root-mean-square error*, RMSE):\n",
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" $$ \\mathrm{RMSE} \\, = \\, \\sqrt{ \\frac{1}{m} \\sum_{i=1}^{m} \\left( \\hat{y}^{(i)} - y^{(i)} \\right)^2 } $$\n",
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" * średni błąd bezwzględny (*mean absolute error*, MAE):\n",
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" $$ \\mathrm{MAE} \\, = \\, \\frac{1}{m} \\sum_{i=1}^{m} \\left| \\hat{y}^{(i)} - y^{(i)} \\right| $$"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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}
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},
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"source": [
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"W powyższych wzorach $y^{(i)}$ oznacza **oczekiwaną** wartości zmiennej $y$ w $i$-tym przykładzie, a $\\hat{y}^{(i)}$ oznacza wartość zmiennej $y$ w $i$-tym przykładzie wyliczoną (**przewidzianą**) przez nasz model."
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {
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"slideshow": {
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"slide_type": "slide"
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}
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},
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"source": [
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"### Metryki dla zadań klasyfikacji\n",
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"\n",
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"Aby przedstawić kilka najpopularniejszych metryk stosowanych dla zadań klasyfikacyjnych, posłużmy się następującym przykładem:"
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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": 34,
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"metadata": {
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"slideshow": {
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"slide_type": "notes"
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|||
|
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}
|
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},
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"outputs": [],
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"source": [
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"# Przydatne importy\n",
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"\n",
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"import ipywidgets as widgets\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"import pandas\n",
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"import random\n",
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"import seaborn\n",
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"\n",
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"%matplotlib inline"
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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": 35,
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"metadata": {
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|||
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"slideshow": {
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|||
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"slide_type": "notes"
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|||
|
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}
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},
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"outputs": [],
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"source": [
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"def powerme(x1,x2,n):\n",
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" \"\"\"Funkcja, która generuje n potęg dla zmiennych x1 i x2 oraz ich iloczynów\"\"\"\n",
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" X = []\n",
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" for m in range(n+1):\n",
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" for i in range(m+1):\n",
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" X.append(np.multiply(np.power(x1,i),np.power(x2,(m-i))))\n",
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" return np.hstack(X)"
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]
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},
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{
|
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"cell_type": "code",
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|||
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"execution_count": 36,
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"metadata": {
|
|||
|
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"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"def plot_data_for_classification(X, Y, xlabel=None, ylabel=None, Y_predicted=[], highlight=None):\n",
|
|||
|
|
" \"\"\"Wykres danych dla zadania klasyfikacji\"\"\"\n",
|
|||
|
|
" fig = plt.figure(figsize=(16*.6, 9*.6))\n",
|
|||
|
|
" ax = fig.add_subplot(111)\n",
|
|||
|
|
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
|
|||
|
|
" X = X.tolist()\n",
|
|||
|
|
" Y = Y.tolist()\n",
|
|||
|
|
" X1n = [x[1] for x, y in zip(X, Y) if y[0] == 0]\n",
|
|||
|
|
" X1p = [x[1] for x, y in zip(X, Y) if y[0] == 1]\n",
|
|||
|
|
" X2n = [x[2] for x, y in zip(X, Y) if y[0] == 0]\n",
|
|||
|
|
" X2p = [x[2] for x, y in zip(X, Y) if y[0] == 1]\n",
|
|||
|
|
" \n",
|
|||
|
|
" if len(Y_predicted) > 0:\n",
|
|||
|
|
" Y_predicted = Y_predicted.tolist()\n",
|
|||
|
|
" X1tn = [x[1] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 0 and yp[0] == 0]\n",
|
|||
|
|
" X1fn = [x[1] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 1 and yp[0] == 0]\n",
|
|||
|
|
" X1tp = [x[1] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 1 and yp[0] == 1]\n",
|
|||
|
|
" X1fp = [x[1] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 0 and yp[0] == 1]\n",
|
|||
|
|
" X2tn = [x[2] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 0 and yp[0] == 0]\n",
|
|||
|
|
" X2fn = [x[2] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 1 and yp[0] == 0]\n",
|
|||
|
|
" X2tp = [x[2] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 1 and yp[0] == 1]\n",
|
|||
|
|
" X2fp = [x[2] for x, y, yp in zip(X, Y, Y_predicted) if y[0] == 0 and yp[0] == 1]\n",
|
|||
|
|
" \n",
|
|||
|
|
" if highlight == 'tn':\n",
|
|||
|
|
" ax.scatter(X1tn, X2tn, c='red', marker='x', s=100, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fn, X2fn, c='gray', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1tp, X2tp, c='gray', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fp, X2fp, c='gray', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" elif highlight == 'fn':\n",
|
|||
|
|
" ax.scatter(X1tn, X2tn, c='gray', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fn, X2fn, c='green', marker='o', s=100, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1tp, X2tp, c='gray', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fp, X2fp, c='gray', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" elif highlight == 'tp':\n",
|
|||
|
|
" ax.scatter(X1tn, X2tn, c='gray', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fn, X2fn, c='gray', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1tp, X2tp, c='green', marker='o', s=100, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fp, X2fp, c='gray', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" elif highlight == 'fp':\n",
|
|||
|
|
" ax.scatter(X1tn, X2tn, c='gray', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fn, X2fn, c='gray', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1tp, X2tp, c='gray', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fp, X2fp, c='red', marker='x', s=100, label='Dane')\n",
|
|||
|
|
" else:\n",
|
|||
|
|
" ax.scatter(X1tn, X2tn, c='red', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fn, X2fn, c='green', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1tp, X2tp, c='green', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1fp, X2fp, c='red', marker='x', s=50, label='Dane')\n",
|
|||
|
|
"\n",
|
|||
|
|
" else:\n",
|
|||
|
|
" ax.scatter(X1n, X2n, c='r', marker='x', s=50, label='Dane')\n",
|
|||
|
|
" ax.scatter(X1p, X2p, c='g', marker='o', s=50, label='Dane')\n",
|
|||
|
|
" \n",
|
|||
|
|
" if xlabel:\n",
|
|||
|
|
" ax.set_xlabel(xlabel)\n",
|
|||
|
|
" if ylabel:\n",
|
|||
|
|
" ax.set_ylabel(ylabel)\n",
|
|||
|
|
" \n",
|
|||
|
|
" ax.margins(.05, .05)\n",
|
|||
|
|
" return fig"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 37,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Wczytanie danych\n",
|
|||
|
|
"import pandas\n",
|
|||
|
|
"import numpy as np\n",
|
|||
|
|
"\n",
|
|||
|
|
"alldata = pandas.read_csv('data-metrics.tsv', sep='\\t')\n",
|
|||
|
|
"data = np.matrix(alldata)\n",
|
|||
|
|
"\n",
|
|||
|
|
"m, n_plus_1 = data.shape\n",
|
|||
|
|
"n = n_plus_1 - 1\n",
|
|||
|
|
"\n",
|
|||
|
|
"X2 = powerme(data[:, 1], data[:, 2], n)\n",
|
|||
|
|
"Y2 = np.matrix(data[:, 0]).reshape(m, 1)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 38,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAm8AAAFmCAYAAAA70X3dAAAABHNCSVQICAgIfAhkiAAAAAlwSFlzAAALEgAACxIB0t1+/AAAADh0RVh0U29mdHdhcmUAbWF0cGxvdGxpYiB2ZXJzaW9uMy4yLjIsIGh0dHA6Ly9tYXRwbG90bGliLm9yZy+WH4yJAAAgAElEQVR4nO3df3Dc913n8ddbieS5rndo7bjFdWKSYk2H2NCS6kKhmiqFJpeIay2LFiVnIDeXOZOjmXHtwMR33EGHH3PAQHzuEcqlpkPL+JrNTSXFUBU3DZRieoUomSS160ulhpC4yiWqXdq1OCQn+74/vt+v/dVqV/pK2t3v97v7fMzsaPfz/X7Xn/16V/vS9/PL3F0AAADIh660KwAAAIDkCG8AAAA5QngDAADIEcIbAABAjhDeAAAAcoTwBgAAkCNXpl2BNFx11VV+7bXXpl0NAACARZ544olvufuW5fbpyPB27bXXanJyMu1qAAAALGJm/7DSPjSbAgAA5AjhDQAAIEdSD29m9gkze8XMTtXZbmb2UTObNrNnzOyG2LZbzezZcNuh1tUaAAAgHamHN0l/LOnWZbbfJqk3vO2T9DFJMrMrJD0Qbr9e0h1mdn1TawoAAJCy1MObu39J0vlldtkt6VMe+Iqk15vZVkk3Spp29+fcfUHSQ+G+AAAAbSv18JbANkkvxh6fDcvqlQMAALStPIQ3q1Hmy5TXfhKzfWY2aWaTs7OzDascAABAK+UhvJ2VdE3s8dWSZpYpr8ndH3T3Pnfv27Jl2bnvAAAAMisP4e24pJ8LR52+U9J33P0lSY9L6jWz68ysR9Lt4b4AqrlLY2PBzyTlAIDMSj28mdmnJf1vSW81s7NmdpeZ3W1md4e7TEh6TtK0pI9L+gVJcvdXJd0j6YSkM5IedvfTLX8BQB6Mj0vDw9KBA5eDmnvweHg42A4AyIXUl8dy9ztW2O6SPlRn24SCcAdgOUND0v790pEjwePDh4PgduRIUD40lG79AACJpR7eALSAWRDYpCCwRSFu//6g3GqN/wEAZJF5B/Z16evrcxamR0dyl7pivSUqFYIbAGSImT3h7n3L7ZN6nzcALRL1cYuL94EDAOQC4Q3oBFFwi/q4VSqX+8AR4AAgV+jzBnSC8fHLwS3q4xbvAzcwIO3Zk24dAQCJEN6ATjA0JI2OBj+jPm5RgBsYYLQpAOQI4Q3oBGa1r6zVKwcAZBZ93gAAAHKE8AYAAJAjhDcAAIAcIbwBAADkCOENAAAgRwhvAAAAOUJ4AwAAyBHCGwAAQI4wSS8AADWU58sqnS5p6tyUejf3amTniIobimlXCyC8AQBQ7eQLJzV4bFAVr2ju4pwK3QUdPHFQE3sn1L+9P+3qocPRbAoAQEx5vqzBY4MqL5Q1d3FOkjR3cU7lhaD8wsKFlGuITkd4AwAgpnS6pIpXam6reEWlU6UW1whYjPAGAEDM1LmpS1fcqs1dnNP0+ekW1whYjPAGAEBM76YdKtiGmtsKtkE7Nn1/i2sELEZ4AwAgZuTvC+r65/ma27r+eV4jz72uxTUCFiO8AQAQU/ypOzQx/wEV56WCB5MyFPxKFeelifkPaONP3ZFyDdHpmCoEAIA4M/X/7sOaOfAhlf78Y5reJO04/6pG3v0ftPHwA5JZ2jVEhzN3T7sOLdfX1+eTk5NpVwMAkGXuUlesgapSIbih6czsCXfvW24fmk0BAKjmLh04sLjswIGgHEgZ4Q0AgLgouB05Iu3fH1xx278/eEyAQwbQ5w0AgLjx8cvB7fDhoKn08OFg25Ej0sCAtGdPunVER8tEeDOzWyUdkXSFpKPu/ltV239J0t7w4ZWSfkDSFnc/b2bPSypLek3Sqyu1EwMAsKyhIWl0NPgZ9XGLAtzAQFAOpCj1AQtmdoWkr0u6WdJZSY9LusPdv1Zn//dJOuDuPx4+fl5Sn7t/K+m/yYAFAACQRXkZsHCjpGl3f87dFyQ9JGn3MvvfIenTLakZAABAxmQhvG2T9GLs8dmwbAkze52kWyV9Jlbskj5vZk+Y2b56/4iZ7TOzSTObnJ2dbUC1AXQ8d2lsbGkH9nrlANAAWQhvtSbNqfcb732S/sbdz8fK3uXuN0i6TdKHzOzdtQ509wfdvc/d+7Zs2bK+GgOAFHRsHx5ePAIxGqk4PBxsB4AGy0J4OyvpmtjjqyXN1Nn3dlU1mbr7TPjzFUljCpphAaD5hoaWTiERn2KCju0AmiALo00fl9RrZtdJ+qaCgPZvqncys++RNCDpZ2JlBUld7l4O798i6ddaUmsAqJ5C4siR4H58igkAaLDUr7y5+6uS7pF0QtIZSQ+7+2kzu9vM7o7tukfS5919Llb2JkknzexpSX8n6bPu/uetqjsALApwEYIbgCbKwpU3ufuEpImqsj+sevzHkv64quw5SW9rcvUAoL56yygR4AA0SepX3gAgt1hGCUAKMnHlDQByiWWUAKSA8AYAa8UySgBSQHgDgLUyq31lrV45ADQAfd4AAAByhPAGAACQI4Q3AACAHCG8AQAA5AjhDQAAIEcIbwAAADlCeAMAAMgRwhsAAECOEN4AAAByhPAGAACQI4Q3AACAHCG8AQAA5AjhDQAAIEcIbwAAADlCeAMAAMgRwhsAAECOEN4AAAByhPAGAACQI4Q3AACAHCG8AQAA5AjhDQAAIEcIbwAAADlCeAMAAMiRTIQ3M7vVzJ41s2kzO1Rj+01m9h0zeyq8/UrSYwEAANrJlWlXwMyukPSApJslnZX0uJkdd/evVe361+7+r9d4LAAAQFvIwpW3GyVNu/tz7r4g6SFJu1twLAAAQO5kIbxtk/Ri7PHZsKzaj5rZ02b2OTPbucpjZWb7zGzSzCZnZ2cbUW8AAICWS73ZVJLVKPOqx09K+j53v2Bmg5LGJfUmPDYodH9Q0oOS1NfXV3MfIOvK82WVTpc0dW5KvZt7NbJzRMUNxbSrBQBooSyEt7OSrok9vlrSTHwHd/9u7P6Emf2BmV2V5FigXZx84aQGjw2q4hXNXZxTobuggycOamLvhPq396ddPQBAi2Sh2fRxSb1mdp2Z9Ui6XdLx+A5m9r1mZuH9GxXU+1ySY4F2UJ4va/DYoMoLZc1dnJMkzV2cU3khKL+wcCHlGgIAWiX18Obur0q6R9IJSWckPezup83sbjO7O9ztA5JOmdnTkj4q6XYP1Dy29a8CaK7S6ZIqXqm5reIVlU6VWlwjAEBastBsKnefkDRRVfaHsfu/L+n3kx4LtJupc1OXrrhVm7s4p+nz0y2uEQAgLalfeQOwst7NvSp0F2puK3QXtGPTjhbXCACQFsIbkAMjO0fUZbU/rl3WpZFdIy2uEQAgLYQ3IAeKG4qa2DuhYk/x0hW4QndBxZ6gfGPPxpRrGHKXxsaCn0nKAQCrlok+bwBW1r+9XzP3zqh0qqTp89PasWmHRnaNZCe4SdL4uDQ8LO3fLx0+LJkFge3AAenIEWl0VNqzJ+1aAkCuEd6AHNnYs1F33XBX2tWob2goCG5HjgSPDx++HNz27w+2AwDWhfAGoHHMgsAmBYEtCnHxK3EAgHWhz1u7ou8R0hIPcBGCGwA0DOGtXUV9jw4cuBzUor5Hw8PBdqAZovdZXPx9CABYF8Jbu4r3PYq+OOl7hGarfp9VKkvfhwCAdaHPW7ui7xHSMD5+ObhF77P4+3BggNGmALBO5h34l3BfX59PTk6mXY3WcJe6YhdYKxWCG5rHPQhwQ0OL32f1ygEAi5jZE+7et9w+NJu2M/oeodXMgitr1QGtXjkAYNUIb+2KvkcAALQl+ry1K/oeAQDQlghv7WpoKFiKKN7HKApwAwOMNgUAIKdoNm1X9D3CajGxMwDkAuENQICJnTtWeb6so08e1X2P3qejTx5Veb6cdpUALINmUwAqz5dV2v4tTd33DvU+ekQjBxZUPPwAEzt3gJMvnNTgsUFVvKK5i3MqdBd08MRBTeydUP/2/rSr11xMbYOcYp43oMMt+fL2K9W18Komjkn9L4iJndtYeb6sbfdvU3lh6ZW2Yk9RM/fOaGPPxhRq1iJjY8F
|
|||
|
|
"text/plain": [
|
|||
|
|
"<Figure size 691.2x388.8 with 1 Axes>"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {
|
|||
|
|
"needs_background": "light"
|
|||
|
|
},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"fig = plot_data_for_classification(X2, Y2, xlabel=r'$x_1$', ylabel=r'$x_2$')"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 39,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"def safeSigmoid(x, eps=0):\n",
|
|||
|
|
" \"\"\"Funkcja sigmoidalna zmodyfikowana w taki sposób, \n",
|
|||
|
|
" żeby wartości zawsze były odległe od asymptot o co najmniej eps\n",
|
|||
|
|
" \"\"\"\n",
|
|||
|
|
" y = 1.0/(1.0 + np.exp(-x))\n",
|
|||
|
|
" if eps > 0:\n",
|
|||
|
|
" y[y < eps] = eps\n",
|
|||
|
|
" y[y > 1 - eps] = 1 - eps\n",
|
|||
|
|
" return y\n",
|
|||
|
|
"\n",
|
|||
|
|
"def h(theta, X, eps=0.0):\n",
|
|||
|
|
" \"\"\"Funkcja hipotezy (regresja logistyczna)\"\"\"\n",
|
|||
|
|
" return safeSigmoid(X*theta, eps)\n",
|
|||
|
|
"\n",
|
|||
|
|
"def J(h,theta,X,y, lamb=0):\n",
|
|||
|
|
" \"\"\"Funkcja kosztu dla regresji logistycznej\"\"\"\n",
|
|||
|
|
" m = len(y)\n",
|
|||
|
|
" f = h(theta, X, eps=10**-7)\n",
|
|||
|
|
" j = -np.sum(np.multiply(y, np.log(f)) + \n",
|
|||
|
|
" np.multiply(1 - y, np.log(1 - f)), axis=0)/m\n",
|
|||
|
|
" if lamb > 0:\n",
|
|||
|
|
" j += lamb/(2*m) * np.sum(np.power(theta[1:],2))\n",
|
|||
|
|
" return j\n",
|
|||
|
|
"\n",
|
|||
|
|
"def dJ(h,theta,X,y,lamb=0):\n",
|
|||
|
|
" \"\"\"Gradient funkcji kosztu\"\"\"\n",
|
|||
|
|
" g = 1.0/y.shape[0]*(X.T*(h(theta,X)-y))\n",
|
|||
|
|
" if lamb > 0:\n",
|
|||
|
|
" g[1:] += lamb/float(y.shape[0]) * theta[1:] \n",
|
|||
|
|
" return g\n",
|
|||
|
|
"\n",
|
|||
|
|
"def classifyBi(theta, X):\n",
|
|||
|
|
" \"\"\"Funkcja predykcji - klasyfikacja dwuklasowa\"\"\"\n",
|
|||
|
|
" prob = h(theta, X)\n",
|
|||
|
|
" return prob"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 40,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"def GD(h, fJ, fdJ, theta, X, y, alpha=0.01, eps=10**-3, maxSteps=10000):\n",
|
|||
|
|
" \"\"\"Metoda gradientu prostego dla regresji logistycznej\"\"\"\n",
|
|||
|
|
" errorCurr = fJ(h, theta, X, y)\n",
|
|||
|
|
" errors = [[errorCurr, theta]]\n",
|
|||
|
|
" while True:\n",
|
|||
|
|
" # oblicz nowe theta\n",
|
|||
|
|
" theta = theta - alpha * fdJ(h, theta, X, y)\n",
|
|||
|
|
" # raportuj poziom błędu\n",
|
|||
|
|
" errorCurr, errorPrev = fJ(h, theta, X, y), errorCurr\n",
|
|||
|
|
" # kryteria stopu\n",
|
|||
|
|
" if abs(errorPrev - errorCurr) <= eps:\n",
|
|||
|
|
" break\n",
|
|||
|
|
" if len(errors) > maxSteps:\n",
|
|||
|
|
" break\n",
|
|||
|
|
" errors.append([errorCurr, theta]) \n",
|
|||
|
|
" return theta, errors"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 41,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"theta = [[ 1.37136167]\n",
|
|||
|
|
" [ 0.90128948]\n",
|
|||
|
|
" [ 0.54708112]\n",
|
|||
|
|
" [-5.9929264 ]\n",
|
|||
|
|
" [ 2.64435168]\n",
|
|||
|
|
" [-4.27978238]]\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"# Uruchomienie metody gradientu prostego dla regresji logistycznej\n",
|
|||
|
|
"theta_start = np.matrix(np.zeros(X2.shape[1])).reshape(X2.shape[1],1)\n",
|
|||
|
|
"theta, errors = GD(h, J, dJ, theta_start, X2, Y2, \n",
|
|||
|
|
" alpha=0.1, eps=10**-7, maxSteps=10000)\n",
|
|||
|
|
"print('theta = {}'.format(theta))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 42,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"def plot_decision_boundary(fig, theta, X):\n",
|
|||
|
|
" \"\"\"Wykres granicy klas\"\"\"\n",
|
|||
|
|
" ax = fig.axes[0]\n",
|
|||
|
|
" xx, yy = np.meshgrid(np.arange(-1.0, 1.0, 0.02),\n",
|
|||
|
|
" np.arange(-1.0, 1.0, 0.02))\n",
|
|||
|
|
" l = len(xx.ravel())\n",
|
|||
|
|
" C = powerme(xx.reshape(l, 1), yy.reshape(l, 1), n)\n",
|
|||
|
|
" z = classifyBi(theta, C).reshape(int(np.sqrt(l)), int(np.sqrt(l)))\n",
|
|||
|
|
"\n",
|
|||
|
|
" plt.contour(xx, yy, z, levels=[0.5], lw=3);"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 43,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"Y_expected = Y2.astype(int)\n",
|
|||
|
|
"Y_predicted = (classifyBi(theta, X2) > 0.5).astype(int)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 44,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Przygotowanie interaktywnego wykresu\n",
|
|||
|
|
"\n",
|
|||
|
|
"dropdown_highlight = widgets.Dropdown(options=['all', 'tp', 'fp', 'tn', 'fn'], value='all', description='highlight')\n",
|
|||
|
|
"\n",
|
|||
|
|
"def interactive_classification(highlight):\n",
|
|||
|
|
" fig = plot_data_for_classification(X2, Y2, xlabel=r'$x_1$', ylabel=r'$x_2$',\n",
|
|||
|
|
" Y_predicted=Y_predicted, highlight=highlight)\n",
|
|||
|
|
" plot_decision_boundary(fig, theta, X2)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 45,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"application/vnd.jupyter.widget-view+json": {
|
|||
|
|
"model_id": "165999bd4b234b55b10cdc34e64e8244",
|
|||
|
|
"version_major": 2,
|
|||
|
|
"version_minor": 0
|
|||
|
|
},
|
|||
|
|
"text/plain": [
|
|||
|
|
"interactive(children=(Dropdown(description='highlight', options=('all', 'tp', 'fp', 'tn', 'fn'), value='all'),…"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"text/plain": [
|
|||
|
|
"<function __main__.interactive_classification(highlight)>"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"execution_count": 45,
|
|||
|
|
"metadata": {},
|
|||
|
|
"output_type": "execute_result"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"widgets.interact(interactive_classification, highlight=dropdown_highlight)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Zadanie klasyfikacyjne z powyższego przykładu polega na przypisaniu punktów do jednej z dwóch kategorii:\n",
|
|||
|
|
" 0. <font color=\"red\">czerwone krzyżyki</font>\n",
|
|||
|
|
" 1. <font color=\"green\">zielone kółka</font>\n",
|
|||
|
|
"\n",
|
|||
|
|
"W tym celu zastosowano regresję logistyczną."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"W rezultacie otrzymano model, który dzieli płaszczyznę na dwa obszary:\n",
|
|||
|
|
" 0. <font color=\"red\">na zewnątrz granatowej krzywej</font>\n",
|
|||
|
|
" 1. <font color=\"green\">wewnątrz granatowej krzywej</font>\n",
|
|||
|
|
" \n",
|
|||
|
|
"Model przewiduje klasę <font color=\"red\">0 („czerwoną”)</font> dla punktów znajdujący się w obszarze na zewnątrz krzywej, natomiast klasę <font color=\"green\">1 („zieloną”)</font> dla punktów znajdujących sie w obszarze wewnąrz krzywej."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Wszysktie obserwacje możemy podzielić zatem na cztery grupy:\n",
|
|||
|
|
" * **true positives (TP)** – prawidłowo sklasyfikowane pozytywne przykłady (<font color=\"green\">zielone kółka</font> w <font color=\"green\">wewnętrznym obszarze</font>)\n",
|
|||
|
|
" * **true negatives (TN)** – prawidłowo sklasyfikowane negatywne przykłady (<font color=\"red\">czerwone krzyżyki</font> w <font color=\"red\">zewnętrznym obszarze</font>)\n",
|
|||
|
|
" * **false positives (FP)** – negatywne przykłady sklasyfikowane jako pozytywne (<font color=\"red\">czerwone krzyżyki</font> w <font color=\"green\">wewnętrznym obszarze</font>)\n",
|
|||
|
|
" * **false negatives (FN)** – pozytywne przykłady sklasyfikowane jako negatywne (<font color=\"green\">zielone kółka</font> w <font color=\"red\">zewnętrznym obszarze</font>)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Innymi słowy:\n",
|
|||
|
|
"\n",
|
|||
|
|
"<img width=\"50%\" src=\"https://blog.aimultiple.com/wp-content/uploads/2019/07/positive-negative-true-false-matrix.png\">"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 23,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "skip"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"TP = 5\n",
|
|||
|
|
"TN = 35\n",
|
|||
|
|
"FP = 3\n",
|
|||
|
|
"FN = 6\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"# Obliczmy TP, TN, FP i FN\n",
|
|||
|
|
"\n",
|
|||
|
|
"tp = 0\n",
|
|||
|
|
"tn = 0\n",
|
|||
|
|
"fp = 0\n",
|
|||
|
|
"fn = 0\n",
|
|||
|
|
"\n",
|
|||
|
|
"for i in range(len(Y_expected)):\n",
|
|||
|
|
" if Y_expected[i] == 1 and Y_predicted[i] == 1:\n",
|
|||
|
|
" tp += 1\n",
|
|||
|
|
" elif Y_expected[i] == 0 and Y_predicted[i] == 0:\n",
|
|||
|
|
" tn += 1\n",
|
|||
|
|
" elif Y_expected[i] == 0 and Y_predicted[i] == 1:\n",
|
|||
|
|
" fp += 1\n",
|
|||
|
|
" elif Y_expected[i] == 1 and Y_predicted[i] == 0:\n",
|
|||
|
|
" fn += 1\n",
|
|||
|
|
" \n",
|
|||
|
|
"print('TP =', tp)\n",
|
|||
|
|
"print('TN =', tn)\n",
|
|||
|
|
"print('FP =', fp)\n",
|
|||
|
|
"print('FN =', fn)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "skip"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Możemy teraz zdefiniować następujące metryki:"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"#### Dokładność (*accuracy*)\n",
|
|||
|
|
"$$ \\mbox{accuracy} = \\frac{\\mbox{przypadki poprawnie sklasyfikowane}}{\\mbox{wszystkie przypadki}} = \\frac{TP + TN}{TP + TN + FP + FN} $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Dokładność otrzymujemy przez podzielenie liczby przypadków poprawnie sklasyfikowanych przez liczbę wszystkich przypadków:"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 24,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Accuracy: 0.8163265306122449\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"accuracy = (tp + tn) / (tp + tn + fp + fn)\n",
|
|||
|
|
"print('Accuracy:', accuracy)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"**Uwaga:** Nie zawsze dokładność będzie dobrą miarą, zwłaszcza gdy klasy są bardzo asymetryczne!\n",
|
|||
|
|
"\n",
|
|||
|
|
"*Przykład:* Wyobraźmy sobie test na koronawirusa, który **zawsze** zwraca wynik negatywny. Jaką przydatność będzie miał taki test w praktyce? Żadną. A jaka będzie jego *dokładność*? Policzmy:\n",
|
|||
|
|
"$$ \\mbox{accuracy} \\, = \\, \\frac{\\mbox{szacowana liczba osób zdrowych na świecie}}{\\mbox{populacja Ziemi}} \\, \\approx \\, \\frac{7\\,700\\,000\\,000 - 600\\,000}{7\\,700\\,000\\,000} \\, \\approx \\, 0.99992 $$\n",
|
|||
|
|
"(zaokrąglone dane z 27 marca 2020)\n",
|
|||
|
|
"\n",
|
|||
|
|
"Powyższy wynik jest tak wysoki, ponieważ zdecydowana większość osób na świecie nie jest zakażona, więc biorąc losowego Ziemianina możemy w ciemno strzelać, że nie ma koronawirusa.\n",
|
|||
|
|
"\n",
|
|||
|
|
"W tym przypadku duża różnica w liczności obu zbiorów (zakażeni/niezakażeni) powoduje, że *accuracy* nie jest dobrą metryką.\n",
|
|||
|
|
"\n",
|
|||
|
|
"Dlatego dysponujemy również innymi metrykami:"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"#### Precyzja (*precision*)\n",
|
|||
|
|
"$$ \\mbox{precision} = \\frac{TP}{TP + FP} $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 25,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Precision: 0.625\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"precision = tp / (tp + fp)\n",
|
|||
|
|
"print('Precision:', precision)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Precyzja określa, jaka część przykładów sklasyfikowanych jako pozytywne to faktycznie przykłady pozytywne."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"#### Pokrycie (czułość, *recall*)\n",
|
|||
|
|
"$$ \\mbox{recall} = \\frac{TP}{TP + FN} $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 26,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"Recall: 0.45454545454545453\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"recall = tp / (tp + fn)\n",
|
|||
|
|
"print('Recall:', recall)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Pokrycie mówi nam, jaka część przykładów pozytywnych została poprawnie sklasyfikowana."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"#### *$F$-measure* (*$F$-score*)\n",
|
|||
|
|
"$$ F = \\frac{2 \\cdot \\mbox{precision} \\cdot \\mbox{recall}}{\\mbox{precision} + \\mbox{recall}} $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 27,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"name": "stdout",
|
|||
|
|
"output_type": "stream",
|
|||
|
|
"text": [
|
|||
|
|
"F-score: 0.5263157894736842\n"
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"fscore = (2 * precision * recall) / (precision + recall)\n",
|
|||
|
|
"print('F-score:', fscore)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"$F$-_measure_ jest kompromisem między precyzją a pokryciem (a ściślej: jest średnią harmoniczną precyzji i pokrycia)."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"$F$-_measure_ jest szczególnym przypadkiem ogólniejszej miary:"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"*$F_\\beta$-measure*:\n",
|
|||
|
|
"$$ F_\\beta = \\frac{(1 + \\beta) \\cdot \\mbox{precision} \\cdot \\mbox{recall}}{\\beta^2 \\cdot \\mbox{precision} + \\mbox{recall}} $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "fragment"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Dla $\\beta = 1$ otrzymujemy:\n",
|
|||
|
|
"$$ F_1 \\, = \\, \\frac{(1 + 1) \\cdot \\mbox{precision} \\cdot \\mbox{recall}}{1^2 \\cdot \\mbox{precision} + \\mbox{recall}} \\, = \\, \\frac{2 \\cdot \\mbox{precision} \\cdot \\mbox{recall}}{\\mbox{precision} + \\mbox{recall}} \\, = \\, F $$"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "slide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"## 5.3. Obserwacje odstające"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"**Obserwacje odstające** (*outliers*) – to wszelkie obserwacje posiadające nietypową wartość.\n",
|
|||
|
|
"\n",
|
|||
|
|
"Mogą być na przykład rezultatem błędnego pomiaru albo pomyłki przy wprowadzaniu danych do bazy, ale nie tylko.\n",
|
|||
|
|
"\n",
|
|||
|
|
"Obserwacje odstające mogą niekiedy znacząco wpłynąć na parametry modelu, dlatego ważne jest, żeby takie obserwacje odrzucić zanim przystąpi się do tworzenia modelu."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"W poniższym przykładzie można zobaczyć wpływ obserwacji odstających na wynik modelowania na przykładzie danych dotyczących cen mieszkań zebranych z ogłoszeń na portalu Gratka.pl: tutaj przykładem obserwacji odstającej może być ogłoszenie, w którym podano cenę w tys. zł zamiast ceny w zł."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 28,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Przydatne funkcje\n",
|
|||
|
|
"\n",
|
|||
|
|
"def h_linear(Theta, x):\n",
|
|||
|
|
" \"\"\"Funkcja regresji liniowej\"\"\"\n",
|
|||
|
|
" return x * Theta\n",
|
|||
|
|
"\n",
|
|||
|
|
"def linear_regression(theta):\n",
|
|||
|
|
" \"\"\"Ta funkcja zwraca funkcję regresji liniowej dla danego wektora parametrów theta\"\"\"\n",
|
|||
|
|
" return lambda x: h_linear(theta, x)\n",
|
|||
|
|
"\n",
|
|||
|
|
"def cost(theta, X, y):\n",
|
|||
|
|
" \"\"\"Wersja macierzowa funkcji kosztu\"\"\"\n",
|
|||
|
|
" m = len(y)\n",
|
|||
|
|
" J = 1.0 / (2.0 * m) * ((X * theta - y).T * (X * theta - y))\n",
|
|||
|
|
" return J.item()\n",
|
|||
|
|
"\n",
|
|||
|
|
"def gradient(theta, X, y):\n",
|
|||
|
|
" \"\"\"Wersja macierzowa gradientu funkcji kosztu\"\"\"\n",
|
|||
|
|
" return 1.0 / len(y) * (X.T * (X * theta - y)) \n",
|
|||
|
|
"\n",
|
|||
|
|
"def gradient_descent(fJ, fdJ, theta, X, y, alpha=0.1, eps=10**-5):\n",
|
|||
|
|
" \"\"\"Algorytm gradientu prostego (wersja macierzowa)\"\"\"\n",
|
|||
|
|
" current_cost = fJ(theta, X, y)\n",
|
|||
|
|
" logs = [[current_cost, theta]]\n",
|
|||
|
|
" while True:\n",
|
|||
|
|
" theta = theta - alpha * fdJ(theta, X, y)\n",
|
|||
|
|
" current_cost, prev_cost = fJ(theta, X, y), current_cost\n",
|
|||
|
|
" if abs(prev_cost - current_cost) > 10**15:\n",
|
|||
|
|
" print('Algorithm does not converge!')\n",
|
|||
|
|
" break\n",
|
|||
|
|
" if abs(prev_cost - current_cost) <= eps:\n",
|
|||
|
|
" break\n",
|
|||
|
|
" logs.append([current_cost, theta]) \n",
|
|||
|
|
" return theta, logs\n",
|
|||
|
|
"\n",
|
|||
|
|
"def plot_data(X, y, xlabel, ylabel):\n",
|
|||
|
|
" \"\"\"Wykres danych (wersja macierzowa)\"\"\"\n",
|
|||
|
|
" fig = plt.figure(figsize=(16*.6, 9*.6))\n",
|
|||
|
|
" ax = fig.add_subplot(111)\n",
|
|||
|
|
" fig.subplots_adjust(left=0.1, right=0.9, bottom=0.1, top=0.9)\n",
|
|||
|
|
" ax.scatter([X[:, 1]], [y], c='r', s=50, label='Dane')\n",
|
|||
|
|
" \n",
|
|||
|
|
" ax.set_xlabel(xlabel)\n",
|
|||
|
|
" ax.set_ylabel(ylabel)\n",
|
|||
|
|
" ax.margins(.05, .05)\n",
|
|||
|
|
" plt.ylim(y.min() - 1, y.max() + 1)\n",
|
|||
|
|
" plt.xlim(np.min(X[:, 1]) - 1, np.max(X[:, 1]) + 1)\n",
|
|||
|
|
" return fig\n",
|
|||
|
|
"\n",
|
|||
|
|
"def plot_regression(fig, fun, theta, X):\n",
|
|||
|
|
" \"\"\"Wykres krzywej regresji (wersja macierzowa)\"\"\"\n",
|
|||
|
|
" ax = fig.axes[0]\n",
|
|||
|
|
" x0 = np.min(X[:, 1]) - 1.0\n",
|
|||
|
|
" x1 = np.max(X[:, 1]) + 1.0\n",
|
|||
|
|
" L = [x0, x1]\n",
|
|||
|
|
" LX = np.matrix([1, x0, 1, x1]).reshape(2, 2)\n",
|
|||
|
|
" ax.plot(L, fun(theta, LX), linewidth='2',\n",
|
|||
|
|
" label=(r'$y={theta0:.2}{op}{theta1:.2}x$'.format(\n",
|
|||
|
|
" theta0=float(theta[0][0]),\n",
|
|||
|
|
" theta1=(float(theta[1][0]) if theta[1][0] >= 0 else float(-theta[1][0])),\n",
|
|||
|
|
" op='+' if theta[1][0] >= 0 else '-')))"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 29,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Wczytanie danych (mieszkania) przy pomocy biblioteki pandas\n",
|
|||
|
|
"\n",
|
|||
|
|
"alldata = pandas.read_csv('data_flats_with_outliers.tsv', sep='\\t',\n",
|
|||
|
|
" names=['price', 'isNew', 'rooms', 'floor', 'location', 'sqrMetres'])\n",
|
|||
|
|
"data = np.matrix(alldata[['price', 'sqrMetres']])\n",
|
|||
|
|
"\n",
|
|||
|
|
"m, n_plus_1 = data.shape\n",
|
|||
|
|
"n = n_plus_1 - 1\n",
|
|||
|
|
"Xn = data[:, 0:n]\n",
|
|||
|
|
"\n",
|
|||
|
|
"Xo = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n + 1)\n",
|
|||
|
|
"yo = np.matrix(data[:, -1]).reshape(m, 1)\n",
|
|||
|
|
"\n",
|
|||
|
|
"Xo /= np.amax(Xo, axis=0)\n",
|
|||
|
|
"yo /= np.amax(yo, axis=0)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 30,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "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
|
|||
|
|
"text/plain": [
|
|||
|
|
"<Figure size 691.2x388.8 with 1 Axes>"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {
|
|||
|
|
"needs_background": "light"
|
|||
|
|
},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"fig = plot_data(Xo, yo, xlabel=u'metraż', ylabel=u'cena')\n",
|
|||
|
|
"theta_start = np.matrix([0.0, 0.0]).reshape(2, 1)\n",
|
|||
|
|
"theta, logs = gradient_descent(cost, gradient, theta_start, Xo, yo, alpha=0.01)\n",
|
|||
|
|
"plot_regression(fig, h_linear, theta, Xo)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Na powyższym przykładzie obserwacja odstająca jawi sie jako pojedynczy punkt po prawej stronie wykresu. Widzimy, że otrzymana krzywa regresji zamiast odwzorowywać ogólny trend, próbuje „dopasować się” do tej pojedynczej obserwacji.\n",
|
|||
|
|
"\n",
|
|||
|
|
"Dlatego taką obserwację należy usunąć ze zbioru danych (zobacz ponizej)."
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 31,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"# Odrzućmy obserwacje odstające\n",
|
|||
|
|
"alldata_no_outliers = [\n",
|
|||
|
|
" (index, item) for index, item in alldata.iterrows() \n",
|
|||
|
|
" if item.price > 100 and item.sqrMetres > 10]\n",
|
|||
|
|
"\n",
|
|||
|
|
"alldata_no_outliers = alldata.loc[(alldata['price'] > 100) & (alldata['sqrMetres'] > 100)]"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 32,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [],
|
|||
|
|
"source": [
|
|||
|
|
"data = np.matrix(alldata_no_outliers[['price', 'sqrMetres']])\n",
|
|||
|
|
"\n",
|
|||
|
|
"m, n_plus_1 = data.shape\n",
|
|||
|
|
"n = n_plus_1 - 1\n",
|
|||
|
|
"Xn = data[:, 0:n]\n",
|
|||
|
|
"\n",
|
|||
|
|
"Xo = np.matrix(np.concatenate((np.ones((m, 1)), Xn), axis=1)).reshape(m, n + 1)\n",
|
|||
|
|
"yo = np.matrix(data[:, -1]).reshape(m, 1)\n",
|
|||
|
|
"\n",
|
|||
|
|
"Xo /= np.amax(Xo, axis=0)\n",
|
|||
|
|
"yo /= np.amax(yo, axis=0)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "code",
|
|||
|
|
"execution_count": 33,
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "subslide"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"outputs": [
|
|||
|
|
{
|
|||
|
|
"data": {
|
|||
|
|
"image/png": "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
|
|||
|
|
"text/plain": [
|
|||
|
|
"<Figure size 691.2x388.8 with 1 Axes>"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
"metadata": {
|
|||
|
|
"needs_background": "light"
|
|||
|
|
},
|
|||
|
|
"output_type": "display_data"
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"source": [
|
|||
|
|
"fig = plot_data(Xo, yo, xlabel=u'metraż', ylabel=u'cena')\n",
|
|||
|
|
"theta_start = np.matrix([0.0, 0.0]).reshape(2, 1)\n",
|
|||
|
|
"theta, logs = gradient_descent(cost, gradient, theta_start, Xo, yo, alpha=0.01)\n",
|
|||
|
|
"plot_regression(fig, h_linear, theta, Xo)"
|
|||
|
|
]
|
|||
|
|
},
|
|||
|
|
{
|
|||
|
|
"cell_type": "markdown",
|
|||
|
|
"metadata": {
|
|||
|
|
"slideshow": {
|
|||
|
|
"slide_type": "notes"
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"source": [
|
|||
|
|
"Na powyższym wykresie widać, że po odrzuceniu obserwacji odstających otrzymujemy dużo bardziej „wiarygodną” krzywą regresji."
|
|||
|
|
]
|
|||
|
|
}
|
|||
|
|
],
|
|||
|
|
"metadata": {
|
|||
|
|
"celltoolbar": "Slideshow",
|
|||
|
|
"kernelspec": {
|
|||
|
|
"display_name": "Python 3.10.6 64-bit",
|
|||
|
|
"language": "python",
|
|||
|
|
"name": "python3"
|
|||
|
|
},
|
|||
|
|
"language_info": {
|
|||
|
|
"codemirror_mode": {
|
|||
|
|
"name": "ipython",
|
|||
|
|
"version": 3
|
|||
|
|
},
|
|||
|
|
"file_extension": ".py",
|
|||
|
|
"mimetype": "text/x-python",
|
|||
|
|
"name": "python",
|
|||
|
|
"nbconvert_exporter": "python",
|
|||
|
|
"pygments_lexer": "ipython3",
|
|||
|
|
"version": "3.10.6"
|
|||
|
|
},
|
|||
|
|
"livereveal": {
|
|||
|
|
"start_slideshow_at": "selected",
|
|||
|
|
"theme": "white"
|
|||
|
|
},
|
|||
|
|
"vscode": {
|
|||
|
|
"interpreter": {
|
|||
|
|
"hash": "916dbcbb3f70747c44a77c7bcd40155683ae19c65e1c03b4aa3499c5328201f1"
|
|||
|
|
}
|
|||
|
|
}
|
|||
|
|
},
|
|||
|
|
"nbformat": 4,
|
|||
|
|
"nbformat_minor": 4
|
|||
|
|
}
|