- figure the gaussian kde (solid lineis the equally weighted mixture of normal pdfs centered around the data and with standard deviation (dashed for fixed the gaussian kernel function ( - ) ( : pt is the solution to fourier' heat equation ( tf ( ) rt with initial condition ( ( (the dirac function at show this as consequencethe gaussian kde is the solution to the same heat equationbut now with initial condition ( - ni= ( xi this was the motivation for the theta kde [ ]which is solution to the same heat equation but now on bounded interval show that the ward linkage given in ( is equal to dward (ij| |jkxi xj | | carry out the agglomerative hierarchical clustering of example via the linkage method from scipy cluster hierarchy show that the linkage matrices are the same give scatterplot of the datacolor coded into clusters suppose that we have the data tn { xn in and decide to train the twocomponent gaussian mixture model ( ) ( ) exp exp ( thw ps ps where the parameter vector th [ ]belongs to the set th {th [ ]ui rsi isuppose that the training is via the maximum likelihood in ( show that sup ln (xi ththth = in other wordsfind sequence of values for th th such that the likelihood grows without bound how can we restrict the set th to ensure that the likelihood remains bounded