Statistics using R (spring of 2017) Computer lab: Multivariate Plots, Principal Component Analysis and Discriminant Analysis

Consider a data matrix X of real numbers with n rows and k columns. We view the rows X1, . . . ,Xn as independent samples from a multivariate distribution with mean vector μ and covariance matrix Σ. Can we visualize X in some easy way? One possibility is to use heat plots (also known as heat maps). These are constructed as follows: think of a rectangular matrix in which column (or row) standardized observation numbers are substituted by colored squares in a continuous color scale, e.g. intensely green color for very negative values, intensely red color for very positive values and some suitable interpolation for the numbers in between. By sorting both rows and columns in various ways, one can get a convenient visual overview of all the vectors. Observe that the data structure and information after such sorting is kept fully intact. In genomics such plots were introduced by Eisen et al. (1998), and in this branch they are often called Eisen-plots after him. An interesting historical sketch is given in (Wilkinson and Friendly, 2009). There are many options available for hierarchical clustering. It can be performed in a sequential manner from the root of a tree (divisive methods) or it can start by merging leaves in the tree (agglomerative methods). The measures of similarity or distance between objects and the groups of objects that are clustered can be varied in many different ways. A quite pedagogical account of traditional clustering terminology and ideas is given by Jain et al. (1999). One option for the sorting is to use hierarchical clustering and display the result as trees in which observation vectors that are similar are displayed near to each other in the row sorting and highly correlated column patterns are also sorted together.