mhjensen
584 lines
19 KiB
Plaintext
584 lines
19 KiB
Plaintext
{
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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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"source": [
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"# Logistic Regression\n",
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"\n",
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"## Introduction\n",
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"In linear regression our main interest was centered on learning the\n",
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"coefficients of a functional fit (say a polynomial) in order to be\n",
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"able to predict the response of a continuous variable on some unseen\n",
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"data. The fit to the continuous variable $y_i$ is based on some\n",
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"independent variables $\\hat{x}_i$. Linear regression resulted in\n",
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"analytical expressions for standard ordinary Least Squares or Ridge\n",
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"regression (in terms of matrices to invert) for several quantities,\n",
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"ranging from the variance and thereby the confidence intervals of the\n",
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"parameters $\\hat{\\beta}$ to the mean squared error. If we can invert\n",
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"the product of the design matrices, linear regression gives then a\n",
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"simple recipe for fitting our data.\n",
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"\n",
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"\n",
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"Classification problems, however, are concerned with outcomes taking\n",
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"the form of discrete variables (i.e. categories). We may for example,\n",
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"on the basis of DNA sequencing for a number of patients, like to find\n",
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"out which mutations are important for a certain disease; or based on\n",
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"scans of various patients' brains, figure out if there is a tumor or\n",
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"not; or given a specific physical system, we'd like to identify its\n",
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"state, say whether it is an ordered or disordered system (typical\n",
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"situation in solid state physics); or classify the status of a\n",
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"patient, whether she/he has a stroke or not and many other similar\n",
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"situations.\n",
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"\n",
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"The most common situation we encounter when we apply logistic\n",
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"regression is that of two possible outcomes, normally denoted as a\n",
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"binary outcome, true or false, positive or negative, success or\n",
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"failure etc.\n",
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"\n",
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"Logistic regression will also serve as our stepping stone towards\n",
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"neural network algorithms and supervised deep learning. For logistic\n",
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"learning, the minimization of the cost function leads to a non-linear\n",
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"equation in the parameters $\\hat{\\beta}$. The optimization of the\n",
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"problem calls therefore for minimization algorithms. This forms the\n",
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"bottle neck of all machine learning algorithms, namely how to find\n",
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"reliable minima of a multi-variable function. This leads us to the\n",
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"family of gradient descent methods. The latter are the working horses\n",
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"of basically all modern machine learning algorithms.\n",
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"\n",
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"We note also that many of the topics discussed here on logistic \n",
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"regression are also commonly used in modern supervised Deep Learning\n",
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"models, as we will see later.\n",
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"\n",
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"\n",
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"\n",
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"## Basics\n",
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"\n",
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"We consider the case where the dependent variables, also called the\n",
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"responses or the outcomes, $y_i$ are discrete and only take values\n",
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"from $k=0,\\dots,K-1$ (i.e. $K$ classes).\n",
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"\n",
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"The goal is to predict the\n",
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"output classes from the design matrix $\\hat{X}\\in\\mathbb{R}^{n\\times p}$\n",
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"made of $n$ samples, each of which carries $p$ features or predictors. The\n",
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"primary goal is to identify the classes to which new unseen samples\n",
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"belong.\n",
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"\n",
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"Let us specialize to the case of two classes only, with outputs\n",
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"$y_i=0$ and $y_i=1$. Our outcomes could represent the status of a\n",
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"credit card user that could default or not on her/his credit card\n",
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"debt. That is"
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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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"source": [
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"$$\n",
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"y_i = \\begin{bmatrix} 0 & \\mathrm{no}\\\\ 1 & \\mathrm{yes} \\end{bmatrix}.\n",
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"$$"
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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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"source": [
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"Before moving to the logistic model, let us try to use our linear\n",
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"regression model to classify these two outcomes. We could for example\n",
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"fit a linear model to the default case if $y_i > 0.5$ and the no\n",
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"default case $y_i \\leq 0.5$.\n",
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"\n",
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"We would then have our \n",
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"weighted linear combination, namely"
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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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"source": [
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"<!-- Equation labels as ordinary links -->\n",
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"<div id=\"_auto1\"></div>\n",
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"\n",
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"$$\n",
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"\\begin{equation}\n",
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"\\hat{y} = \\hat{X}^T\\hat{\\beta} + \\hat{\\epsilon},\n",
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"\\label{_auto1} \\tag{1}\n",
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"\\end{equation}\n",
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"$$"
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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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"source": [
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"where $\\hat{y}$ is a vector representing the possible outcomes, $\\hat{X}$ is our\n",
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"$n\\times p$ design matrix and $\\hat{\\beta}$ represents our estimators/predictors.\n",
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"\n",
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"\n",
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"The main problem with our function is that it takes values on the\n",
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"entire real axis. In the case of logistic regression, however, the\n",
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"labels $y_i$ are discrete variables. A typical example is the credit\n",
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"card data discussed below here, where we can set the state of\n",
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"defaulting the debt to $y_i=1$ and not to $y_i=0$ for one the persons\n",
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"in the data set (see the full example below).\n",
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"\n",
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"One simple way to get a discrete output is to have sign\n",
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"functions that map the output of a linear regressor to values $\\{0,1\\}$,\n",
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"$f(s_i)=sign(s_i)=1$ if $s_i\\ge 0$ and 0 if otherwise. \n",
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"We will encounter this model in our first demonstration of neural networks. Historically it is called the \"perceptron\" model in the machine learning\n",
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"literature. This model is extremely simple. However, in many cases it is more\n",
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"favorable to use a ``soft\" classifier that outputs\n",
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"the probability of a given category. This leads us to the logistic function.\n",
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"\n",
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"\n",
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"\n",
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"## The logistic function\n",
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"\n",
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"The perceptron is an example of a ``hard classification\" model. We\n",
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"will encounter this model when we discuss neural networks as\n",
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"well. Each datapoint is deterministically assigned to a category (i.e\n",
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"$y_i=0$ or $y_i=1$). In many cases, it is favorable to have a \"soft\"\n",
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"classifier that outputs the probability of a given category rather\n",
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"than a single value. For example, given $x_i$, the classifier\n",
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"outputs the probability of being in a category $k$. Logistic regression\n",
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"is the most common example of a so-called soft classifier. In logistic\n",
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"regression, the probability that a data point $x_i$\n",
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"belongs to a category $y_i=\\{0,1\\}$ is given by the so-called logit function (or Sigmoid) which is meant to represent the likelihood for a given event,"
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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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"source": [
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"$$\n",
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"p(t) = \\frac{1}{1+\\mathrm \\exp{-t}}=\\frac{\\exp{t}}{1+\\mathrm \\exp{t}}.\n",
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"$$"
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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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"source": [
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"Note that $1-p(t)= p(-t)$.\n",
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"\n",
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"\n",
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"The following code plots the logistic function, the step function and other functions we will encounter from here and on."
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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": 1,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"%matplotlib inline\n",
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"\n",
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"\"\"\"The sigmoid function (or the logistic curve) is a\n",
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"function that takes any real number, z, and outputs a number (0,1).\n",
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"It is useful in neural networks for assigning weights on a relative scale.\n",
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"The value z is the weighted sum of parameters involved in the learning algorithm.\"\"\"\n",
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"\n",
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"import numpy\n",
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"import matplotlib.pyplot as plt\n",
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"import math as mt\n",
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"\n",
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"z = numpy.arange(-5, 5, .1)\n",
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"sigma_fn = numpy.vectorize(lambda z: 1/(1+numpy.exp(-z)))\n",
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"sigma = sigma_fn(z)\n",
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"\n",
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"fig = plt.figure()\n",
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"ax = fig.add_subplot(111)\n",
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"ax.plot(z, sigma)\n",
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"ax.set_ylim([-0.1, 1.1])\n",
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"ax.set_xlim([-5,5])\n",
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"ax.grid(True)\n",
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"ax.set_xlabel('z')\n",
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"ax.set_title('sigmoid function')\n",
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"\n",
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"plt.show()\n",
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"\n",
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"\"\"\"Step Function\"\"\"\n",
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"z = numpy.arange(-5, 5, .02)\n",
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"step_fn = numpy.vectorize(lambda z: 1.0 if z >= 0.0 else 0.0)\n",
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"step = step_fn(z)\n",
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"\n",
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"fig = plt.figure()\n",
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"ax = fig.add_subplot(111)\n",
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"ax.plot(z, step)\n",
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"ax.set_ylim([-0.5, 1.5])\n",
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"ax.set_xlim([-5,5])\n",
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"ax.grid(True)\n",
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"ax.set_xlabel('z')\n",
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"ax.set_title('step function')\n",
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"\n",
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"plt.show()\n",
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"\n",
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"\"\"\"tanh Function\"\"\"\n",
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"z = numpy.arange(-2*mt.pi, 2*mt.pi, 0.1)\n",
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"t = numpy.tanh(z)\n",
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"\n",
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"fig = plt.figure()\n",
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"ax = fig.add_subplot(111)\n",
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"ax.plot(z, t)\n",
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"ax.set_ylim([-1.0, 1.0])\n",
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"ax.set_xlim([-2*mt.pi,2*mt.pi])\n",
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"ax.grid(True)\n",
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"ax.set_xlabel('z')\n",
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"ax.set_title('tanh function')\n",
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"\n",
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"plt.show()"
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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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"source": [
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"## Two parameters\n",
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"\n",
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"We assume now that we have two classes with $y_i$ either $0$ or $1$. Furthermore we assume also that we have only two parameters $\\beta$ in our fitting of the Sigmoid function, that is we define probabilities"
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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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"source": [
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"$$\n",
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"\\begin{align*}\n",
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"p(y_i=1|x_i,\\hat{\\beta}) &= \\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}},\\nonumber\\\\\n",
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"p(y_i=0|x_i,\\hat{\\beta}) &= 1 - p(y_i=1|x_i,\\hat{\\beta}),\n",
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"\\end{align*}\n",
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"$$"
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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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"source": [
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"where $\\hat{\\beta}$ are the weights we wish to extract from data, in our case $\\beta_0$ and $\\beta_1$. \n",
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"\n",
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"Note that we used"
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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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"source": [
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"$$\n",
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"p(y_i=0\\vert x_i, \\hat{\\beta}) = 1-p(y_i=1\\vert x_i, \\hat{\\beta}).\n",
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"$$"
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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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"source": [
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"## Maximum likelihood\n",
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"\n",
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"In order to define the total likelihood for all possible outcomes from a \n",
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"dataset $\\mathcal{D}=\\{(y_i,x_i)\\}$, with the binary labels\n",
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"$y_i\\in\\{0,1\\}$ and where the data points are drawn independently, we use the so-called [Maximum Likelihood Estimation](https://en.wikipedia.org/wiki/Maximum_likelihood_estimation) (MLE) principle. \n",
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"We aim thus at maximizing \n",
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"the probability of seeing the observed data. We can then approximate the \n",
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"likelihood in terms of the product of the individual probabilities of a specific outcome $y_i$, that is"
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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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"source": [
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"$$\n",
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"\\begin{align*}\n",
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"P(\\mathcal{D}|\\hat{\\beta})& = \\prod_{i=1}^n \\left[p(y_i=1|x_i,\\hat{\\beta})\\right]^{y_i}\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]^{1-y_i}\\nonumber \\\\\n",
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"\\end{align*}\n",
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"$$"
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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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"source": [
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"from which we obtain the log-likelihood and our **cost/loss** function"
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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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"source": [
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"$$\n",
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"\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left( y_i\\log{p(y_i=1|x_i,\\hat{\\beta})} + (1-y_i)\\log\\left[1-p(y_i=1|x_i,\\hat{\\beta}))\\right]\\right).\n",
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"$$"
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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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"source": [
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"Reordering the logarithms, we can rewrite the **cost/loss** function as"
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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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"source": [
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"$$\n",
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"\\mathcal{C}(\\hat{\\beta}) = \\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
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"$$"
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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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"source": [
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"The maximum likelihood estimator is defined as the set of parameters that maximize the log-likelihood where we maximize with respect to $\\beta$.\n",
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"Since the cost (error) function is just the negative log-likelihood, for logistic regression we have that"
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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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"source": [
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"$$\n",
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"\\mathcal{C}(\\hat{\\beta})=-\\sum_{i=1}^n \\left(y_i(\\beta_0+\\beta_1x_i) -\\log{(1+\\exp{(\\beta_0+\\beta_1x_i)})}\\right).\n",
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"$$"
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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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"source": [
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"This equation is known in statistics as the **cross entropy**. Finally, we note that just as in linear regression, \n",
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"in practice we often supplement the cross-entropy with additional regularization terms, usually $L_1$ and $L_2$ regularization as we did for Ridge and Lasso regression.\n",
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"\n",
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"\n",
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"The cross entropy is a convex function of the weights $\\hat{\\beta}$ and,\n",
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"therefore, any local minimizer is a global minimizer. \n",
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"\n",
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"\n",
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"Minimizing this\n",
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"cost function with respect to the two parameters $\\beta_0$ and $\\beta_1$ we obtain"
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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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"source": [
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"$$\n",
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"\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_0} = -\\sum_{i=1}^n \\left(y_i -\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right),\n",
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"$$"
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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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"source": [
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"and"
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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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"source": [
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"$$\n",
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"\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\beta_1} = -\\sum_{i=1}^n \\left(y_ix_i -x_i\\frac{\\exp{(\\beta_0+\\beta_1x_i)}}{1+\\exp{(\\beta_0+\\beta_1x_i)}}\\right).\n",
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"$$"
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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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"source": [
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"Let us now define a vector $\\hat{y}$ with $n$ elements $y_i$, an\n",
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"$n\\times p$ matrix $\\hat{X}$ which contains the $x_i$ values and a\n",
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"vector $\\hat{p}$ of fitted probabilities $p(y_i\\vert x_i,\\hat{\\beta})$. We can rewrite in a more compact form the first\n",
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"derivative of cost function as"
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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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"source": [
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"$$\n",
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"\\frac{\\partial \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}} = -\\hat{X}^T\\left(\\hat{y}-\\hat{p}\\right).\n",
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"$$"
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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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"source": [
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"If we in addition define a diagonal matrix $\\hat{W}$ with elements \n",
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"$p(y_i\\vert x_i,\\hat{\\beta})(1-p(y_i\\vert x_i,\\hat{\\beta})$, we can obtain a compact expression of the second derivative as"
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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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"source": [
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"$$\n",
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"\\frac{\\partial^2 \\mathcal{C}(\\hat{\\beta})}{\\partial \\hat{\\beta}\\partial \\hat{\\beta}^T} = \\hat{X}^T\\hat{W}\\hat{X}.\n",
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"$$"
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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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"source": [
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"Within a binary classification problem, we can easily expand our model to include multiple predictors. Our ratio between likelihoods is then with $p$ predictors"
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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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"source": [
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"$$\n",
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"\\log{ \\frac{p(\\hat{\\beta}\\hat{x})}{1-p(\\hat{\\beta}\\hat{x})}} = \\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p.\n",
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"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"Here we defined $\\hat{x}=[1,x_1,x_2,\\dots,x_p]$ and $\\hat{\\beta}=[\\beta_0, \\beta_1, \\dots, \\beta_p]$ leading to"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"p(\\hat{\\beta}\\hat{x})=\\frac{ \\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}{1+\\exp{(\\beta_0+\\beta_1x_1+\\beta_2x_2+\\dots+\\beta_px_p)}}.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"## Including more classes\n",
|
|
"\n",
|
|
"Till now we have mainly focused on two classes, the so-called binary\n",
|
|
"system. Suppose we wish to extend to $K$ classes. Let us for the sake\n",
|
|
"of simplicity assume we have only two predictors. We have then\n",
|
|
"following model"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"1\n",
|
|
"5\n",
|
|
" \n",
|
|
"<\n",
|
|
"<\n",
|
|
"<\n",
|
|
"!\n",
|
|
"!\n",
|
|
"M\n",
|
|
"A\n",
|
|
"T\n",
|
|
"H\n",
|
|
"_\n",
|
|
"B\n",
|
|
"L\n",
|
|
"O\n",
|
|
"C\n",
|
|
"K"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"\\log{\\frac{p(C=2\\vert x)}{p(K\\vert x)}} = \\beta_{20}+\\beta_{21}x_1,\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"and so on till the class $C=K-1$ class"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"\\log{\\frac{p(C=K-1\\vert x)}{p(K\\vert x)}} = \\beta_{(K-1)0}+\\beta_{(K-1)1}x_1,\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"and the model is specified in term of $K-1$ so-called log-odds or\n",
|
|
"**logit** transformations.\n",
|
|
"\n",
|
|
"\n",
|
|
"\n",
|
|
"In our discussion of neural networks we will encounter the above again\n",
|
|
"in terms of a slightly modified function, the so-called **Softmax** function.\n",
|
|
"\n",
|
|
"The softmax function is used in various multiclass classification\n",
|
|
"methods, such as multinomial logistic regression (also known as\n",
|
|
"softmax regression), multiclass linear discriminant analysis, naive\n",
|
|
"Bayes classifiers, and artificial neural networks. Specifically, in\n",
|
|
"multinomial logistic regression and linear discriminant analysis, the\n",
|
|
"input to the function is the result of $K$ distinct linear functions,\n",
|
|
"and the predicted probability for the $k$-th class given a sample\n",
|
|
"vector $\\hat{x}$ and a weighting vector $\\hat{\\beta}$ is (with two\n",
|
|
"predictors):"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"p(C=k\\vert \\mathbf {x} )=\\frac{\\exp{(\\beta_{k0}+\\beta_{k1}x_1)}}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}}.\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"It is easy to extend to more predictors. The final class is"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"$$\n",
|
|
"p(C=K\\vert \\mathbf {x} )=\\frac{1}{1+\\sum_{l=1}^{K-1}\\exp{(\\beta_{l0}+\\beta_{l1}x_1)}},\n",
|
|
"$$"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"and they sum to one. Our earlier discussions were all specialized to\n",
|
|
"the case with two classes only. It is easy to see from the above that\n",
|
|
"what we derived earlier is compatible with these equations.\n",
|
|
"\n",
|
|
"To find the optimal parameters we would typically use a gradient\n",
|
|
"descent method. Newton's method and gradient descent methods are\n",
|
|
"discussed in the material on [optimization\n",
|
|
"methods](https://compphysics.github.io/MachineLearning/doc/pub/Splines/html/Splines-bs.html)."
|
|
]
|
|
}
|
|
],
|
|
"metadata": {},
|
|
"nbformat": 4,
|
|
"nbformat_minor": 4
|
|
}
|