From c5864ab9ad57ae994a952b890f6450caa3a0975b Mon Sep 17 00:00:00 2001 From: Filip Gralinski Date: Tue, 27 Apr 2021 19:00:36 +0200 Subject: [PATCH] =?UTF-8?q?wyk=C5=82ad=208?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- wyk/08_Regresja_liniowa.ipynb | 214 ++++++++++++++++++++++++ wyk/08_files/linregr1.pdf | Bin 0 -> 8165 bytes wyk/08_files/linregr1.png | Bin 0 -> 103329 bytes wyk/08_files/linregr1.r | 8 + wyk/08_files/linregr2.pdf | Bin 0 -> 8783 bytes wyk/08_files/linregr2.png | Bin 0 -> 127797 bytes wyk/08_files/linregr2.r | 8 + wyk/08_files/mieszkania.tsv | 121 ++++++++++++++ wyk/08_files/morskieoko.jpg | Bin 0 -> 297583 bytes wyk/08_files/regresja-liniowa-tekst.png | Bin 0 -> 62071 bytes 10 files changed, 351 insertions(+) create mode 100644 wyk/08_Regresja_liniowa.ipynb create mode 100644 wyk/08_files/linregr1.pdf create mode 100644 wyk/08_files/linregr1.png create mode 100644 wyk/08_files/linregr1.r create mode 100644 wyk/08_files/linregr2.pdf create mode 100644 wyk/08_files/linregr2.png create mode 100644 wyk/08_files/linregr2.r create mode 100644 wyk/08_files/mieszkania.tsv create mode 100644 wyk/08_files/morskieoko.jpg create mode 100644 wyk/08_files/regresja-liniowa-tekst.png diff --git a/wyk/08_Regresja_liniowa.ipynb b/wyk/08_Regresja_liniowa.ipynb new file mode 100644 index 0000000..4e01af8 --- /dev/null +++ b/wyk/08_Regresja_liniowa.ipynb @@ -0,0 +1,214 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "continent-intermediate", + "metadata": {}, + "source": [ + "# Regresja liniowa\n", + "\n", + "Regresja liniowa jest prosta...\n", + "\n", + "![Ceny mieszkań](./08_files/linregr1.png)\n", + "\n", + "... dosłownie — dopasuj prostą $y = ax + b$ do punktów\n", + "\n", + "Należy odgadnąć $a$ i $b$ tak, aby zminimalizować błąd\n", + "kwadratowy, tj. wartość:\n", + "\n", + "$$\\sum_{i=1}^n (y_i - (ax_i + b))^2$$\n", + "\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "id": "original-speed", + "metadata": {}, + "source": [ + "Regresje liniowa (jednej zmiennej) jest łatwa do rozwiązania — wystarczy podstawić do wzoru!\n", + "\n", + "$$\\hat{b} = \\frac{ \\sum_{i=1}^{n}{x_i y_i} - \\frac{1}{n} \\sum_{i=1}^n x_i\n", + " \\sum_{j=1}^n y_j}{ \\sum_{i=1}^n {x_i^2} - \\frac{1}{n} (\\sum_{i=1}^n\n", + " x_i)^2 }$$\n", + "\n", + "$$\\hat{a} = \\bar{y} - \\hat{b}\\,\\bar{x}$$\n", + "\n", + "\n", + "Na przykład dla mieszkań: $b =$ -30809.203 zł, $a =$ 5733.693 zł/m$^2$.\n", + "\n", + "![Ceny mieszkań](./08_files/linregr2.png)\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "id": "significant-relaxation", + "metadata": {}, + "source": [ + "## Regresja wielu zmiennych\n", + "\n", + "W praktyce mamy do czynienia z **wielowymiarową** regresją\n", + "liniową.\n", + "\n", + "Cena mieszkań może być prognozowana na podstawie:\n", + "\n", + "* powierzchni ($x_1 = 32.3$) \n", + "\n", + "* liczby pokoi ($x_2 = 3$)\n", + " \n", + "* piętra ($x_3 = 4$)\n", + "\n", + "* wieku ($x_4 = 13$)\n", + "\n", + "* odległości od Dworca Centralnego w Warszawie ($x_5 = 371.3$)\n", + "\n", + "* cech zerojedynkowych:\n", + "\n", + " * czy wielka płyta? ($x_6 = 0$)\n", + "\n", + " * czy jest jacuzzi? ($x_7 = 1$)\n", + "\n", + " * czy jest grzyb? ($x_8 = 0$)\n", + "\n", + " * czy to Kielce? ($x_9 = 1$)\n", + "\n", + "* ...\n", + "\n", + "... więc uogólniamy na wiele ($k$) wymiarów:\n", + "\n", + "$$ y = w_0 + w_1x_1 + \\ldots + w_kx_k = w_0 + \\sum_{j=1}^{k} w_jx_j $$\n", + "\n", + "gdzie:\n", + "\n", + "* $x_1,\\dots,x_k$ -- zmienne, na podstawie których zgadujemy\n", + "\n", + "* $w_0, w_1,\\dots,w_k$ -- wagi modelu (do wymyślenia na\n", + " podstawie przykładów)\n", + "\n", + "* $y$ -- odgadywana wartość\n", + "\n", + "Też istnieje wzór ładny wzór na wyliczenie wektora wag!\n", + "\n", + "$$\\mathbf{w} = (\\mathbf{X}^{\\rm T}\\mathbf{X})^{-1} \\mathbf{X}^{\\rm T}\\mathbf{y}$$\n", + "\n", + "... niestety odwracanie macierzy nie jest tanie :(" + ] + }, + { + "cell_type": "markdown", + "id": "ordinary-appendix", + "metadata": {}, + "source": [ + "## Kilka sporzeżeń\n", + "\n", + "Regresja liniowa to najprostszy możliwy model:\n", + "\n", + "* im czegoś więcej na wejściu, tym proporcjonalnie (troszkę) więcej/mniej na wyjściu\n", + "\n", + "* nic prostszego nie da się wymyślić (funkcja stała??)\n", + "\n", + "* niestety model liniowy czasami kompletnie nie ma sensu (np. wzrost człowieka w\n", + " stosunku do wieku)\n" + ] + }, + { + "cell_type": "markdown", + "id": "egyptian-austria", + "metadata": {}, + "source": [ + "## Uczenie\n", + "\n", + "A jak nauczyć się wag z przykładów?\n", + "\n", + "* wzór (z odwracaniem macierzy) — problematyczny\n", + "\n", + "### Metoda gradientu prostego\n", + "\n", + "![Morskie Oko - Krzysztof Dudzik](08_files/morskieoko.jpg)\n", + "\n", + "Schodź wzdłuż lokalnego spadku funkcji błędu.\n", + "\n", + "Tak więc w praktyce zamiast podstawiać do wzoru lepiej się uczyć iteracyjnie -\n", + " metodą **gradientu prostego** (ang. _gradient descent_).\n", + "\n", + "1. Zacznij od byle jakich wag $w_i$ (np. wylosuj)\n", + "2. Weź losowy przykład uczący $x_1,\\dots,x_n$, $y$.\n", + "3. Oblicz wyjście $\\hat{y}$ na podstawie $x_1,\\dots,x_n$.\n", + "4. Oblicz funkcję błędu między $y$ a $\\hat{y}$.\n", + "5. Zmodyfikuj lekko wagi $(w_i)$ w kierunku spadku funkcji błędu.\n", + "6. Jeśli błąd jest duży, idź do 2.\n", + "\n", + "Modyfikacja wag:\n", + "\n", + "$$w_i := w_i - x_i (\\hat{y} - y) \\eta$$\n", + "\n", + "gdzie $\\eta$ to **współczynnik uczenia** _learning rate_.\n" + ] + }, + { + "cell_type": "markdown", + "id": "exact-train", + "metadata": {}, + "source": [ + "## Ewaluacja regresji\n", + "\n", + "To miary błędu (im mniej, tym lepiej!)}\n", + "\n", + "### Błąd bezwzględny (Mean Absolute Error, MAE)\n", + "\n", + "$$\\frac{1}{n}\\sum_{i=1}^n |\\hat{y}_i - y_i| $$\n", + "\n", + "### Mean Squared Error (MSE)\n", + "\n", + "$$\\frac{1}{n}\\sum_{i=1}^n (\\hat{y}_i - y_i)^2$$\n", + "\n", + "### Root Mean Squared Error (RMSE)\n", + "\n", + "$$\\sqrt{\\frac{1}{n}\\sum_{i=1}^n (\\hat{y}_i - y_i)^2}$$\n", + "\n" + ] + }, + { + "cell_type": "markdown", + "id": "selective-agriculture", + "metadata": {}, + "source": [ + "## Regresja liniowa dla tekstu\n", + "\n", + "Czym jest wektor $\\vec{x} = (x_1,\\dots,x_n)$? Wiemy, np. reprezentacja tf-idf (być z trikiem z haszowaniem, Word2vec etc.).\n", + "\n", + "![schemat regresji liniowej](08_files/regresja-liniowa-tekst.png)\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "numerous-limitation", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "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": "3.9.2" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} diff --git a/wyk/08_files/linregr1.pdf b/wyk/08_files/linregr1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..9773332ee475ccad434c047b0892372673111d97 GIT binary patch literal 8165 zcmb7Jc|4Tc|0kE2HcOIp^`sOPXXcr`Qbfp-HDt?}8BC3tF-uvp6%q1nkCG*!<+>>q z5^-(G8dBO*H%d}z5vt!AjB&sB_xt1L^>SY4`ON2h-sd^bdA~pB`8>MzX66PsLjp!O zx;y%1bW3!*Cm(}Dun5<8J7(=#3^3(0X+jp4V@4A)5k0fjC>BS-lCUHcOCX~ZeGIVW zGXBrsoLNFP9OZ~u@M*jN7F~eYiG)0n5Dv5Fa~UE!lP~@iZ%8!6LfaS?S8US9tXqd+ zav0)Spp&yffO80si2zfYkjCcvBfy^K&lDhd=-`M*2ax|YK

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