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Logistic Regression Prof. Navneet Goyal CS & IS BITS, Pilani

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Perceptron

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Logistic Regression In Linear regression, the dependent variable is continuous What if the dependent is dichotomous or binary? A person will vote for Reagan (1) or Carter (0)? A woman will give birth to a low weight baby(1) or not (0)? Does the person have a disease? Yes (1) or No (0) Outcome of a baseball game? Win (1) or loss (0) A linear regression model will not be able solve it with acceptable error Moreover, values above 1 and below 0 do not make any sense Also, we are more interested in probabilites (or odds ratio) than in 0 or 1 output What can be done?

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Logistic Regression Actuary Example: Model P(death|X), the probability that a person X will die within the next 10 years X = {x1=age,x2=M/F,x3= cholestrol level} ∑ w i x i = w T X is not a probability!! Introduce logistic function: σ (a) = 1/(1+e -a )

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Logistic Regression 2-class Multi-class Parameters using Maximum Likelihood Iterative Reweighted Least Squares (IRLS) Probit Regression

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The Logistic Function Symmetry property: Inverse of the logistic sigmoid fn is given by & is called the logit or log odds function because it represents the log of the ratio of the probabilities Sigmoid fn is S-shaped and is also called the squashing fn because it maps the whole real line into a finite interval (maps real a ε (-∞, +∞) to finite (0,1) interval) Plays an important role in many classification algorithms

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Probabilistic Discriminative Models Logistic Regression Example Data Set Response Variable –> Admission to Grad School (Admit) 0 if admitted, 1 if not admitted Predictor Variables GRE Score (gre) Continuous University Prestige (topnotch) 1 if prestigious, 0 otherwise Grade Point Average (gpa) Continuous

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Probabilistic Discriminative Models First 10 Observations of the Data Set ADMITGRETOPNOTCHGPA

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Dot-plot: Data from Table 2

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Logistic regression (2) Table 3 Prevalence (%) of signs of CD according to age group

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Dot-plot: Data from Table 3 Diseased % Age (years)

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Logistic Regression Consider the linear probability model Issue: π (X i ) can take on values less than 0 or greater than 1 Issue: Predicted probability for some subjects fall outside of the [0,1] range.

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Logistic Regression Consider the logistic regression model GLM with binomial random component and identity link g( μ ) = logit( μ ) Range of values for π (X i ) is 0 to 1

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Logistic Regression Consider the logistic regression model And the linear probability model Then the graph of the predicted probabilities for different grade point averages:

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What is Logistic Regression? In a nutshell: A statistical method used to model dichotomous or binary outcomes (but not limited to) using predictor variables. Used when the research method is focused on whether or not an event occurred, rather than when it occurred (time course information is not used).

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What is Logistic Regression? What is the “Logistic” component? Instead of modeling the outcome, Y, directly, the method models the log odds(Y) using the logistic function.

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Logistic Regression Simple logistic regression = logistic regression with 1 predictor variable Multiple logistic regression = logistic regression with multiple predictor variables Multiple logistic regression = Multivariable logistic regression = Multivariate logistic regression

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Logistic Regression predictor variables is the log(odds) of the outcome. dichotomous outcome

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Logistic Regression intercept is the log(odds) of the outcome. model coefficients

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Odds & Probability

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Maximum Likelihood Flipped a fair coin 10 times: T, H, H, T, T, H, H, T, H, H What is the Pr(Heads) given the data? 1/100? 1/5? 1/2? 6/10?

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T, H, H, T, T, H, H, T, H, H What is the Pr(Heads) given the data? Most reasonable data-based estimate would be 6/10. In fact, is the ML estimator of p. Maximum Likelihood

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Discrete distribution, finite parameter space How biased an unfair coin is?unfair coin Call the probability of tossing a HEAD p. Determine p. Toss the coin 80 times Outcome is 49 HEADS and 31 TAILS, Suppose the coin was taken from a box containing three coins: one which gives HEADS with probability p = 1/3, one which gives HEADS with probability p = 1/2 and another which gives HEADS with probability p = 2/3. NO labels on these coins Using maximum likelihood estimation the coin that has the largest likelihood can be found, given the data that were observed. By using the probability mass function of the binomial distribution with sample size equal to 80, number successes equal to 49 but different values of p (the "probability of success"), the likelihood function (defined below) takes one of three values:probability mass functionbinomial distribution Maximum Likelihood: Example

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The likelihood is maximized when p = 2/3, and so this the maximum likelihood estimate for p. Maximum Likelihood: Example Discrete distribution, finite parameter space

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Discrete distribution, continuous parameter space Now suppose that there was only one coin but its p could have been any value 0 ≤ p ≤ 1. The likelihood function to be maximized is: and the maximization is over all possible values 0 ≤ p ≤ 1. differentiatingdifferentiating with respect to p (solutions p = 0, p = 1, and p = 49/80) The solution which maximizes the likelihood is clearly p = 49/80 Thus the maximum likelihood estimator for p is 49/80. Maximum Likelihood: Example

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Continuous distribution, continuous parameter space Do it for Gaussian Distribution yourself! Two parameters, μ & σ Maximum Likelihood: Example Its expectation value is equal to the parameter μ of the given distribution,expectation value which means that the maximum-likelihood estimator μ is unbiased. This means that the estimator is biased. However, is consistent. In this case it could be solved individually for each parameter. In general, it may not be the case.

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The method of maximum likelihood estimation chooses values for parameter estimates (regression coefficients) which make the observed data “maximally likely.” Standard errors are obtained as a by-product of the maximization process Maximum Likelihood

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The Logistic Regression Model intercept is the log(odds) of the outcome. model coefficients

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Maximum Likelihood We want to choose β ’s that maximizes the probability of observing the data we have: Assumption: independent y’s

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Obvious possibility is to use traditional linear regression model But this has problems Distribution of dependent variable hardly normal Predicted probabilities cannot be less than 0, greater than 1 Linear probability model

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Linear probability model predictions

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Instead, use logistic transformation (logit) of probability, log of the odds Logistic regression model

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Logistic regression model predictions

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Least-squares no longer best way of estimating parameters of logistic regression model Instead use maximum likelihood estimation Finds values of parameters that have greatest probability, given data Estimation of logistic regression model

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Data on 24 space shuttle launches prior to Challenger Dependent variable, whether shuttle flight experienced thermal distress incident Independent variables Date – whether shuttle changes or age has effect Temperature – whether joint temperature on booster has effect Space shuttle data

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Dependent variable Any thermal distress on launch Independent variable Date (days since 1/1/60) SPSS procedure Regression, Binary logistic First model—date as single independent variable

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Predicted probability of thermal distress using date

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Exponential of B as change in odds

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Odds is the ratio of probability of success to probability of failure Like odds on horse races Even odds, odds = 1, implies probability equals 0.5 Odds = 2 means 2 to 1 in favor of success, implies probability of Odds = 0.5 means 1 to 2 in favor (or 2 to 1 against) success, implies probability of What does “odds” mean?

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Logistic regression can be extended to use multiple independent variables exactly like linear regression Multiple logistic regression

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Dependent variable Any thermal distress on launch Independent variables Date (days since 1/1/60) Joint temperature, degrees F Adding joint temperature to the logistic regression model

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Probabilistic Discriminative Models Find posterior class probabilities directly Use functional form of generalized linear model and determine its parameters directly by using maximum likelihood principle Iterative Reweighted Least Squares (IRLS) Maximize a likelihood fn defined through the conditional distribution p(Ck|x), which represents a form of discriminative training Advantage of discriminative approach – fewer no. of adaptive parameters to be determined (linear in M) Improved predictive performance particularly when the class-conditional density assumptions give a poor approximation of the true distributions

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Probabilistic Discriminative Models Classification methods work directly with the original input vector All such algorithms are still applicable if we first make a fixed NL transformation of the inputs using a vector of basis fns ϕ(x) Decision boundaries linear in the feature space ϕ In linear models of regression, one of the basis fn is typically set to a constant say, so that the corresponding parameter plays the role of bias. A fixed basis fn transformation ϕ(x) will be used in

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Probabilistic Discriminative Models Original Input Space (x1,x2) Feature Space ( φ 1, φ 2) Although we use linear classification models Linear-separability in feature space does not imply linear-separability in input space

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