Regression Usman Roshan CS 698 Machine Learning

Regression Same problem as classification except that the target variable y i is continuous. Popular solutions – Linear regression (perceptron) – Support vector regression – Logistic regression (for regression)

Linear regression Suppose target values are generated by a function y i = f(x i ) + e i We will estimate f(x i ) by g(x i,θ). Suppose each e i is being generated by a Gaussian distribution with 0 mean and σ 2 variance (same variance for all e i ). Now we can ask what is the probability of y i given the input x i and variables θ (denoted as p(x i |y i,θ) This is normally distributed with mean g(x i,θ) and variance σ 2.

Linear regression Apply maximum likelihood to estimate g(x, θ) Assume each (x i,y i ) i.i.d. Then probability of data given model (likelihood) is P(X|θ) = p(x 1,y 1 )p(x 2,y 2 )…p(x n,y n ) Each p(x i,y i )=p(y i |x i )p(x i ) Maximizing the log likelihood gives us least squares (linear regression)

Logistic regression Similar to linear regression derivation Minimize sum of squares between predicted and actual value However – predicted is given by sigmoid function and – y i is constrained in the range [0,1]

Support vector regression Makes no assumptions about probability distribution of the data and output (like support vector machine). Change the loss function in the support vector machine problem to the e-sensitive loss to obtain support vector regression

Support vector regression Solved by applying Lagrange multipliers like in SVM Solution w is given by a linear combination of support vectors (like in SVM) The solution w can also be used for ranking features. From regularized risk minimization the loss would be

Application Prediction of continuous phenotypes in mice from genotype (Predicting unobserved phen…)Predicting unobserved phen Data are vectors x i where each feature takes on values 0, 1, and 2 to denote number of alleles of a particular single nucleotide polymorphism (SNP) Output y i is a phenotype value. For example coat color (represented by integers), chemical levels in blood

Mouse phenotype prediction from genotype Rank SNPs by Wald test – First perform linear regression y = wx + w 0 – Calculate p-value on w using t-test t-test: (w-w null )/stderr(w)) w null = 0 T-test: w/stderr(w) stderr(w) given by Σ i (y i -wx i -w 0 ) 2 /(x i -mean(x i )) – Rank SNPs by p-values – OR by Σ i (y i -wx i -w 0 ) Rank SNPs by support vector regression (w vector in SVR) Perform linear regression on top k ranked SNP under cross- validation.

Prediction of MCH in mouse

Prediction of CD8 in mouse