Logistic Regression

Logistic RegressionInstructor: Ping LiDepartment of Statistics and BiostatiticsDepartment of Computer ScienceRutgers University20151 Calculus Review: DerivativesSimple derivatives:[logx] =1x,[xn] =nxn 1,[ex] =ex,[ax] =axlogaChain rule:[f(h(x))] =f (h(x))h (x) log ax2+e2x =1(ax2+e2x) ax2+e2x =2ax+ 2e2x(ax2+ex)Multivariate derivatives:f(x, y) =ax+xny+cy2, f(x, y) x=axloga+nxn 1y, f(x, y) y=xn+ 2cy2 Quick Review of Numerical OptimizationSlides 4 - 15 are for reviewing some basic stuff about numerical optimization, which isessential in modern Likelihood Estimation (MLE)Observationsxi,i= 1ton, are samples from a distribution with probability densityfunctionfX(x; 1, 2, .., k),where j,j= 1tok, are parameters to be maximum likelihood estimator seeks the to maximize the joint likelihood =argmax nYi=1fX(xi; )Or, equivalently, to maximize thelogjoint likelihood =argmax nXi=1logfX(xi; )This is aconvexoptimization Example: Normal DistributionIfX N , 2 , thenfX x; , 2 =1 2 e (x )22 2 Fix 2= 1,x= x; , 2 logfX

Logistic Regression Logistic regression is one of the most widely used statistical tools for predicting cateogrical outcomes. General setup for binary logistic regression n observations: {xi,yi},i = 1 to n. xi can be a vector. yi ∈ {0,1}. For example, “1” = “YES” and “0” = “NO”. Define p(xi) = Pr(yi = 1|xi) = π(xi)

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  Logistics, Binary, Binary logistic

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