Spline and Kernel method Gaussian Processes Recitation 3 April 30 Spline and Kernel method Gaussian Processes

Penalized Cubic Regression Splines gam() in library “mgcv” gam( y ~ s(x, bs=“cr”, k=n.knots) , knots=list(x=c(…)), data = dataset) By default, the optimal smoothing parameter selected by GCV R Demo 1

Kernel Method Nadaraya-Watson locally constant model locally linear polynomial model How to define “local”? By Kernel function, e.g. Gaussian kernel R Demo 1 R package: “locfit” Function: locfit(y~x, kern=“gauss”, deg= , alpha= ) Bandwidth selected by GCV: gcvplot(y~x, kern=“gauss”, deg= , alpha= bandwidth range)

Gaussian Processes Distribution on functions f ~ GP(m,κ) m: mean function κ: covariance function p(f(x1), . . . , f(xn)) ∼ Nn(μ, K) μ = [m(x1),...,m(xn)] Kij = κ (xi,xj) Idea: If xi, xj are similar according to the kernel, then f(xi) is similar to f(xj) Just as a quick tutorial. A Gaussian process is a distribution over functions. In particular, the process is uniquely defined by a mean function and a covariance function. By definition of the GP, for any set of n locations, the function is normally distributed with mean given by evaluating the mean function at these locations and covariance given by defining the following kernel matrix. For example, a standard covariance function is the squared-exponential or RBF kernel leading to smooth, continually differentiable functions. This function has two parameters, a variance parameter that determines how much the function can vary from it’s mean and a length-scale parameter that determines the smoothness.

Gaussian Processes – Noise free observations Example task: learn a function f(x) to estimate y, from data (x, y) A function can be viewed as a random variable of infinite dimensions GP provides a distribution over functions.

Gaussian Processes – Noise free observations Model (x, f) are the observed locations and values (training data) (x*, f*) are the test or prediction data locations and values. After observing some noise free data (x, f), Length-scale R Demo 2

Gaussian Processes – Noisy observations (GP for Regression) Model (x, y) are the observed locations and values (training data) (x*, f*) are the test or prediction data locations and values. After observing some noisy data (x, y), R Demo 3

Reference Chapter 2 from Gaussian Processes for Machine Learning Carl Edward Rasmussen and Christopher K. I. Williams 527 lecture notes by Emily Fox