From 340299b7ace63f11136fe6315a23adce75cc2120 Mon Sep 17 00:00:00 2001 From: manzerbredes Date: Sat, 13 Feb 2021 14:39:28 +0100 Subject: [PATCH] Add linear regression --- .gitignore | 1 + data/polynomial.csv | 8 ++++ linear_regression/polynomial.gif | Bin 0 -> 421935 bytes linear_regression/polynomial.org | 33 +++++++++++++++ linear_regression/polynomial.pdf | Bin 0 -> 112220 bytes linear_regression/polynomial.py | 69 +++++++++++++++++++++++++++++++ 6 files changed, 111 insertions(+) create mode 100644 .gitignore create mode 100644 data/polynomial.csv create mode 100644 linear_regression/polynomial.gif create mode 100644 linear_regression/polynomial.org create mode 100644 linear_regression/polynomial.pdf create mode 100755 linear_regression/polynomial.py diff --git a/.gitignore b/.gitignore new file mode 100644 index 0000000..01564af --- /dev/null +++ b/.gitignore @@ -0,0 +1 @@ +*.tex diff --git a/data/polynomial.csv b/data/polynomial.csv new file mode 100644 index 0000000..e339b1c --- 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A common approach is to use the *Mean Square Error* +cost function: +\begin{equation}\label{eq:cost} + J(w) = \frac{1}{2n} \sum_{i=0}^n (h_w(x^{(i)}) - \hat{y}^{(i)})^2 +\end{equation} + +Note that in Equation \ref{eq:cost} we average by $2n$ and not $n$. This is because it get simplify +while doing the partial derivatives as we will see below. This is a pure cosmetic approach which do +not impact the gradient decent (see [[https://math.stackexchange.com/questions/884887/why-divide-by-2m][here]] for more informations). The next step is to $min_w J(w)$ +for each weight $w_i$ (performing the gradient decent). Thus we compute each partial derivatives: +\begin{align} + \frac{\partial J(w)}{\partial w_1}&=\frac{\partial J(w)}{\partial h_w(x)}\frac{\partial h_w(x)}{\partial w_1}\nonumber\\ + &= \frac{1}{n} \sum_{i=0}^n (h_w(x^{(i)}) - \hat{y}^{(i)})\\ + \text{similarly:}\nonumber\\ + \frac{\partial J(w)}{\partial w_2}&= \frac{1}{n} \sum_{i=0}^n x(h_w(x^{(i)}) - \hat{y}^{(i)})\\ + \frac{\partial J(w)}{\partial w_3}&= \frac{1}{n} \sum_{i=0}^n x^2(h_w(x^{(i)}) - \hat{y}^{(i)}) +\end{align} + + diff --git a/linear_regression/polynomial.pdf b/linear_regression/polynomial.pdf new file mode 100644 index 0000000000000000000000000000000000000000..ecf2f285fbaaec717c8deeaf256efdddcf375910 GIT binary patch literal 112220 zcma&MLy#_9&~90_tIj!P+qP}nwr$(CZ5wadwr$(GUw8E1(Gj-?caL&Ua+HxPBA-Pf zFDy#KK+6I{GPg9m2E&F=k8fva3B%0|Lnm!wYvyc@&%n;YjQ@W>7&=i4YiAQjd^%BU z17{Oq6C*og6Bu4z7$;{(69XF<_l;`}t-1|SB;QlDxA}5|%bq$a5z8V{g~h0iBW`6$ z#c^QBY7kx)B*!10-kvr6Mr?KpvWQAFDu9i$oGR$n}I>F|{DOJo`*} 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z%ID~=h2&s_gz}D1EzB!qP!s$tIOWYWFJC!#(7iv)9zth-;o!=}%hk)9PeZ|8S)L1j@au3%vH||^ZIKjcyV5AS3HF%*HTGY%0DX1{at8hj?-C^$a1J%SLd^f(VDMAca}QHo-)CyQM;_s>tL>}Xor!qaKo`6H+-9~ zU0c`bYpFGRp1XLCLxB19<9=|-Na#tql1(kjv#{fVje+picGHb($(Ux{ua=*%uW4F{ zaBsQQ{6$s2-kj<~RWs%Fh-0RF89# zmBk$T>E~TJiZ|~^Iu4!FGo0d4exWkW1iE52Td zmYU-)lj-UeZPgL&FYz25fz5I_>y3!c<~)MJ&c_MLN+^!{VD!I(IXxZQ0_Xo!{4d8) g4F6A`IJ+1*xp+8zdx43GiGzy;mW)hHK^*pf0ZTmym;e9( literal 0 HcmV?d00001 diff --git a/linear_regression/polynomial.py b/linear_regression/polynomial.py new file mode 100755 index 0000000..a62474a --- /dev/null +++ b/linear_regression/polynomial.py @@ -0,0 +1,69 @@ +#!/usr/bin/env python + +import pandas as pd +import matplotlib.pyplot as plt +from matplotlib.animation import FuncAnimation +import numpy as np + +# Load the data +csv="../data/polynomial.csv" +data=pd.read_csv(csv) +x=np.array(data["x"]) +y=np.array(data["y"]) + +# Define the weight +w1=w2=w3=10 + +# Define our model +def h(x): + return(w1+w2*x+w3*(x**2)) + +# Define all partial derivative +def dh1(): + return(1/len(x)*np.sum(h(x)-y)) +def dh2(): + return(1/len(x)*np.sum((h(x)-y)*x)) +def dh3(): + return(1/len(x)*np.sum((h(x)-y)*(x**2))) + +# Perform the gradient decent +fig, ax = plt.subplots() +frame=0 # Current frame (plot animation) +alpha=0.005 # Proportion of the gradient to take into account +accuracy=0.000001 # Accuracy of the decent +done=False +def decent(i): + global w1,w2,w3,x,y,frame + while True: + w1_old=w1 + w1_new=w1-alpha*dh1() + w2_old=w2 + w2_new=w2-alpha*dh2() + w3_old=w3 + w3_new=w3-alpha*dh3() + w1=w1_new + w2=w2_new + w3=w3_new + + if abs(w1_new-w1_old) <= accuracy and abs(w2_new-w2_old) <= accuracy and abs(w2_new-w2_old) <= accuracy: + done=True + frame+=1 + if frame >=1000: + frame=0 + ax.clear() + ax.set_xlim([0, 7]) + ax.set_ylim([0, 5]) + ax.plot(x,y,"ro") + ax.plot(x,h(x)) + break + +def IsDone(): + global done + i = 0 + while not done: + i += 1 + yield i + +anim=FuncAnimation(fig,decent,frames=IsDone,repeat=False) +anim.save('polynomial.gif',dpi=80,writer="imagemagick") +