From 14856930a79b4246bff951330e56200ba043d34d Mon Sep 17 00:00:00 2001 From: manzerbredes Date: Sat, 20 Feb 2021 20:36:33 +0100 Subject: [PATCH] Add more details to the logistic regression --- logistic_regression/binary.org | 79 ++++++++++++++++++++++++++++++++ logistic_regression/binary.pdf | Bin 0 -> 191996 bytes logistic_regression/binary.py | 6 ++- logistic_regression/sigmoid.png | Bin 0 -> 21312 bytes 4 files changed, 84 insertions(+), 1 deletion(-) create mode 100644 logistic_regression/binary.org create mode 100644 logistic_regression/binary.pdf create mode 100644 logistic_regression/sigmoid.png diff --git a/logistic_regression/binary.org b/logistic_regression/binary.org new file mode 100644 index 0000000..59be860 --- /dev/null +++ b/logistic_regression/binary.org @@ -0,0 +1,79 @@ +#+TITLE: Binary Logistic Regression +#+AUTHOR: Loic Guegan + +#+OPTIONS: toc:nil + +#+LATEX_HEADER: \usepackage{fullpage}\usepackage{cancel} +#+latex_header: \hypersetup{colorlinks=true,linkcolor=blue} + +Binary logistic regression are used to predict an binary issue (win/loss, true/false) according to +various parameters. First, we have to choose a polynomial function $h_w(x)$ according to the data +complexity (see \textit{data/binary\_logistic.csv}). In our case, we want to predict our issue (1 +or 0) according to two parameters. Thus: +\begin{equation} +h_w(x_1,x_2) = w_1 + w_2x_1 + w_3x_2 +\end{equation} +However, the function we are looking for should return a *binary* result! To achieve this goal, we +can use a sigmoid (or logistic) with the following property $\mathbb{R} \to ]0;1[$ with the following +form: +#+BEGIN_SRC python :results file :exports none :session + import numpy as np + import matplotlib.pyplot as plt + + x=np.arange(-5,5,0.1) + plt.xlabel("X") + plt.ylabel("Y") + plt.title("Sigmoid Function") + plt.text(-3,0.5,r'$\frac{1}{1+e^{-x}}$',fontsize=30) + plt.plot(x,1/(1+np.exp(-x))) + plt.grid() + plt.savefig("sigmoid.png") + "sigmoid.png" + plt.close() +#+END_SRC +#+ATTR_LATEX: :width 10cm +[[file:sigmoid.png]] + +To this end, we can define the following function: +\begin{equation}\label{eq:cost} + g_w(x_1,x_2) = \frac{1}{1+e^{-h_w(x_1,x_2)}} +\end{equation} + +The next step is to define a cost function. A common approach in binary logistic function is to use +the *Cross-Entropy* loss function. It is much more convenient than the classical Mean Square Error +used in polynomial regression. Indeed, the gradient is stronger even for small error (see [[https://www.youtube.com/watch?v=gIx974WtVb4&t=110s][here]] for +more informations). Thus, it looks like the following: +\begin{equation}\label{eq:cost} + J(w) = -\frac{1}{n} \sum_{i=0}^n \left[y^{(i)}log(g_w(x_1^{(i)},x_2^{(i)})) + (1-y^{(i)})log(1-g_w(x_1^{(i)},x_2^{(i)}))\right] +\end{equation} + +With $n$ the number of observations, $x_j^{(i)}$ is the value of the $j^{th}$ independant variable +associated with the observation $y^{(i)}$. The next step is to $min_w J(w)$ for each weight $w_i$ +(performing the gradient decent, see [[https://towardsdatascience.com/gradient-descent-demystified-bc30b26e432a][here]]). Thus we compute each partial derivatives: +\begin{align*} + \frac{\partial J(w)}{\partial w_1}&=\frac{\partial J(w)}{\partial g_w(x_1,x_2)}\frac{\partial g_w(x_1,x_2)}{\partial h_w(x_1,x_2)}\frac{\partial h_w(x_1,x_2)}{\partial w_1}\nonumber\\ + \frac{\partial J(w)}{\partial g_w(x_1,x_2)}&=-\frac{1}{n} \sum_{i=0}^n \left[y^{(i)}\frac{1}{g_w(x_1^{(i)},x_2^{(i)})} + (1-y^{(i)})\times\frac{1}{1-g_w(x_1^{(i)},x_2^{(i)})}\times (-1)\right]\nonumber\\ + &=-\frac{1}{n} \sum_{i=0}^n \left[\frac{y^{(i)}}{g_w(x_1^{(i)},x_2^{(i)})} - \frac{1-y^{(i)}}{1-g_w(x_1^{(i)},x_2^{(i)})}\right]\nonumber\\ + &=-\frac{1}{n} \sum_{i=0}^n \left[\frac{y^{(i)}(1-g_w(x_1^{(i)},x_2^{(i)}))}{g_w(x_1^{(i)},x_2^{(i)})(1-g_w(x_1^{(i)},x_2^{(i)}))} - \frac{g_w(x_1^{(i)},x_2^{(i)})(1-y^{(i)})}{g_w(x_1^{(i)},x_2^{(i)})(1-g_w(x_1^{(i)},x_2^{(i)}))}\right]\nonumber\\ + &=-\frac{1}{n} \sum_{i=0}^n \left[\frac{y^{(i)}\cancel{-y^{(i)}g_w(x_1^{(i)},x_2^{(i)})} -g_w(x_1^{(i)},x_2^{(i)})\cancel{+y^{(i)}g_w(x_1^{(i)},x_2^{(i)})}}{g_w(x_1^{(i)},x_2^{(i)})(1-g_w(x_1^{(i)},x_2^{(i)}))}\right]\nonumber\\ + &=\frac{1}{n} \sum_{i=0}^n \left[\frac{-y^{(i)} +g_w(x_1^{(i)},x_2^{(i)})}{g_w(x_1^{(i)},x_2^{(i)})(1-g_w(x_1^{(i)},x_2^{(i)}))}\right]\nonumber\\ + \frac{\partial g_w(x_1,x_2)}{\partial h_w(x_1,x_2)}&=\frac{\partial (1+e^{-h_w(x_1,x_2)})^{-1}}{\partial h_w(x_1,x_2)}=-(1+e^{-h_w(x_1,x_2)})^{-2}\times \frac{\partial (1+e^{-h_w(x_1,x_2)})}{\partial h_w(x_1,x_2)}\nonumber\\ + &=-(1+e^{-h_w(x_1,x_2)})^{-2}\times -e^{-h_w(x_1,x_2)}=\frac{e^{-h_w(x_1,x_2)}}{(1+e^{-h_w(x_1,x_2)})^2}\nonumber\\ + &=\frac{e^{-h_w(x_1,x_2)}}{(1+e^{-h_w(x_1,x_2)})(1+e^{-h_w(x_1,x_2)})}=\frac{1}{(1+e^{-h_w(x_1,x_2)})}\frac{e^{-h_w(x_1,x_2)}}{(1+e^{-h_w(x_1,x_2)})}\nonumber\\ + &=\frac{1}{(1+e^{-h_w(x_1,x_2)})}\frac{e^{-h_w(x_1,x_2)}+1-1}{(1+e^{-h_w(x_1,x_2)})}=\frac{1}{(1+e^{-h_w(x_1,x_2)})}\left(1+\frac{-1}{(1+e^{-h_w(x_1,x_2)})}\right)\nonumber\\ + &=g_w(x_1,x_2)(1-g_w(x_1,x_2))\nonumber\\ + \frac{\partial h_w(x_1,x_2)}{\partial w_1}=1\nonumber\\ + \text{Finally:}\\ + \frac{\partial J(w)}{\partial w_1}&=\frac{1}{n} \sum_{i=0}^n \left[\frac{-y^{(i)}+g_w(x_1^{(i)},x_2^{(i)})}{\cancel{g_w(x_1^{(i)},x_2^{(i)})(1-g_w(x_1^{(i)},x_2^{(i)}))}} \times \cancel{g_w(x_1^{(i)},x_2^{(i)})(1-g_w(x_1^{(i)},x_2^{(i)}))} \right]\nonumber\\ + &=\frac{1}{n} \sum_{i=0}^n \left[-y^{(i)}+g_w(x_1^{(i)},x_2^{(i)})\right] +\end{align*} +\begin{align*} + \text{Similarly:}\\ + \frac{\partial J(w)}{\partial w_2}&=\frac{1}{n} \sum_{i=0}^n x_1\left[-y^{(i)}+g_w(x_1^{(i)},x_2^{(i)})\right]\\ + \frac{\partial J(w)}{\partial w_1}&=\frac{1}{n} \sum_{i=0}^n x_2\left[-y^{(i)}+g_w(x_1^{(i)},x_2^{(i)})\right]\\ +\end{align*} + +For more informations on binary logistic regression, here are usefull links: +- [[https://ml-cheatsheet.readthedocs.io/en/latest/logistic_regression.html][Logistic Regression -- ML Glossary documentation]] +- [[https://math.stackexchange.com/questions/2503428/derivative-of-binary-cross-entropy-why-are-my-signs-not-right][Derivative of the Binary Cross Entropy]] + diff --git a/logistic_regression/binary.pdf b/logistic_regression/binary.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a69a04f2e65c7be3c0ca9881bac6916893f5579e GIT binary patch literal 191996 zcmd>mWmuG7*RP5qB_I+aAT0_aEe#4N2uO#tNOyOP2#AVEcS%Wu(mevwB_)z0-3;9f zXWxVVAD{PquIs$#!@16f!zZ1&?;UHe_^sbsdl(fY9#^!}N zj*t)zo2-$wiK8hE2R}bI4f+pmYx_$*-5!rq`1~$n`B7f*kRnM@YWOk2Y-tf{NQ&~Rn%93a!BH)v zI1`qN6KwqWWDQkS%B^L@?cw}b{#c}rPY}lZtQl`owrypKbNc;NR<3-K(3;kJQN!`a zcC{>{J;jyJ>0rZoO9h2ws72LM^W1{m`0m5K9zrhgrGkom7Hy7u50i8}D(a zqX)liau%Gf6s~mhJ{|H|Q0l9(*&94Cs$68|GeKq8e=!{~s@1w=V5)?*`aac=JwE*5 z*73Bg8DX*YZX%+BdDLEOhot>S=Rtve4ONI z#UtUjr1kSd^|{WnPp%o%cQvw>SiZ|((B5oaSB@hlW1g_Zmy%F?SaD}6qn`I8x8U_W zJ@pBV22C;A3Z@CoDB)*vripBEtAn2oRrXAO^>`CBA#<2bKeptN&wi64kEKs^pw>OL zjEj9y8LV5hAo7X9MD{998Py#u+LZL}WTX(%7=c;eUeI94xkN};am4~dH{fH_BP6e) 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