plot the curve xx np arange ( , + , reshape - , np zeros_like (xxqx np hstack (np ones_like (xx)xx) np zeros_like (xxn np shape (xx)[ kx np zeros (( , )for in range ( )for in range ( )kx[ ,jk( [ ]xx[ ] np hstack (kx tqx)ad plt ylim (( , )plt plot(xx glabel ' {}format ( )linewidth plt plot( , ' 'markersize = plt xlabel ('$ 'plt ylabel ('$ 'plt legend ( gaussian process regression another application of the kernel machinery is to gaussian process regression gaussian process (gpon space is stochastic process { xwherefor any choice of indices xn the vector [ xn ]has multivariate gaussian distribution as suchthe distribution of gp is completely specified by its mean and covariance functions and rrespectively the covariance function is finite positive semidefinite functionand hencein view of theorem can be viewed as reproducing kernel on as for ordinary regressionthe objective of gp regression is to learn regression function that predicts response (xfor each feature vector this is done in bayesian fashionby establishing ( prior pdf for and ( the likelihood of the datafor given from these two we then derivevia bayesformulathe posterior distribution of given the data we refer to section for the general bayesian framework simple bayesian model for gp regression is as follows firstthe prior distribution of is taken to be the distribution of gp with some known mean function and covariance function (that iskernelk most often is taken to be constantand for simplicity of expositionwe take it to be the gaussian kernel ( is often used for the covariance function for radial basis function kernels (including the gaussian kernel)points that are closer will be more highly correlated or "similar[ ]independent of translations in space secondsimilar to standard regressionwe view the observed feature vectors xn as fixed and the responses yn as outcomes of random variables yn specificallygiven gwe model the {yi as yi (xi ei ( gaussian process