From a136feac0734c72ad39e681386e0df27cd133b64 Mon Sep 17 00:00:00 2001 From: manzerbredes Date: Sun, 21 Feb 2021 10:07:19 +0100 Subject: [PATCH] Add decision boundary --- logistic_regression/binary.org | 18 ++++++++++++++++++ logistic_regression/binary.pdf | Bin 191996 -> 208668 bytes logistic_regression/binary.py | 8 ++++++-- 3 files changed, 24 insertions(+), 2 deletions(-) diff --git a/logistic_regression/binary.org b/logistic_regression/binary.org index 59be860..009b134 100644 --- a/logistic_regression/binary.org +++ b/logistic_regression/binary.org @@ -77,3 +77,21 @@ 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]] +* Desision Boundary + +The method used here is similar to the one used [[https://scipython.com/blog/plotting-the-decision-boundary-of-a-logistic-regression-model/][here]]. In binary logistic regression, decision +boundary is located where:\\ \[g_w(x_1,x_2)=0.5 \implies h_w(x_1,x_2)=0\] +In addition we now that our decision boundary has the following form \[x_2=ax_1+b\] +Thus, we can easily deduce b since if $x_1=0$ we have $x_2=a\times 0 + b \implies x_2=b$. Thus: +\begin{equation} +h_w(0,x_2)=w_1 + w_3x_2=0\\ +\implies x_2=\frac{-w_1}{w_3} +\end{equation} +To deduce the a coefficient, it is slighly more complicated. If we know two points $(x_1^a,x_2^a)$ and $(x_1^b,x_2^b)$ +on the decision boundary line, we know that $a=\frac{x_2^b-x_2^a}{x_1^b-x_1^a}$. thus if we compute: +\begin{align*} +h_w(X_1^b,x_2^b)-h_w(X_1^a,x_2^a)&=\cancel{w_1}+w_2x_1^b+w_3x_2^b\cancel{-w_1}-w_2x_1^a-w_3x_2^a = 0 \\ +&\implies w_2(x_1^b-x_1^a)+w_3(x_2^b-x_2^a) = 0 \implies \frac{w_2}{-w_3}=\frac{(x_1^b-x_1^a)}{(x_2^b-x_2^a)}=a +\end{align*} +Thus we have the decision boundary defined as follow: +\[ d(x) = \frac{w_2}{-w_3} x - \frac{w_1}{w_3} \] diff --git a/logistic_regression/binary.pdf b/logistic_regression/binary.pdf index a69a04f2e65c7be3c0ca9881bac6916893f5579e..8b1282e1ad4744acb0ed696b81c2e2e0675efa0e 100644 GIT binary patch delta 90329 zcmV)AK*YcN+Y6lD3=t(zL`E$!E;R}z(A34)3NbU05cvUnvkCzS2LvR#dqS7NhyfFm zDF|}`HM5-v>OTQBlPL&iv+qcY0RcF(drHI%7d9X;Aa7!73OqatFHB`_XLM*FGBq+Z zlaWd(e_LCR+qe~epI@Oz8I2k8@SY+-k#@64cKguoJQPUwK^c3-LOgP6doszt@1dw0 zW!jR)&Y)=b$udQg=lY!sX`0+6H2L|8R-bRKuU@_dn=nc&wJf>5O@IPXTqT-XLIn;j 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