Clustering methods are used to place items in natural patterns or convenient groups. They can be used to place genes into clusters to have similar expression patterns across the tissue samples of interest. They can also be used to cluster tissues into groups on the basis of their gene profiles. Examples of the methods used are hierarchical agglomerative clustering, k-means clustering, self organizing maps, and model-based methods. The focus of this chapter is on using mixtures of multivariate normal distributions to provide model-based clusterings of tissue samples and of genes.
Some Heuristic Clustering Methods
In cluster analysis, we wish to group a number (n) of entities into a smaller number (g) of groups on the basis of measurements of some variables associated with each entity. We let yj = (y1j, …, ypj)T be the observation or feature vector of p measurements y1j, …, ypj made on the jth entity (j = 1, …, n) to be clustered. In discriminant analysis the data belong to g known classes and we wish to create an allocation rule to allow us to assign an unclassified entity to one of these classes on the basis of its feature vector.
Key Terms in this Chapter
Likelihood: The likelihood function is found by evaluating the joint density of the random variables in the model defining the random phenomenon under study at their observed values.
Multivariate: A multivariate problem one has more than one response variables.
EM Algorithm: A method for calculating maximum likelihood estimates of parameters in statistical models in situations where the observed data can be usefully viewed as being incomplete. It proceeds by consideration of the complete data log likelihood, which is formed on the basis of the complete data. The latter comprises the observed data and the ‘missing data’. It is implemented iteratively by alternating two steps known as the expectation (E) step and the maximization (M) step. On the E-step the Q-function is calculated by averaging the complete data log likelihood over the conditional distribution of the complete data given the observed data, using the current value for the parameter vector. This is followed by the M-step in which the current estimate of the parameter vector is updated to that value which globally maximizes the Q-function.
Microarray: A slide which contains a grid consisting of a large number of microscopic spots of different DNAs, each of which will hybridize with a particular target RNA or DNA sequence. The target RNA or DNA is generally attached to a fluorescent marker. When the target RNA or DNA binds to the complementary DNA on the slide this binds the fluorescent marker to the slide. The measurements taken are of the intensity of the fluorescence from these markers.
Factor Analysis: Factor analysis is a statistical technique in which the correlation between the variables is approximated by the linear dependence of the latter on a set of unobservable (latent) variables.
Heuristic: An empirical method of solving a problem which does not necessarily reflect the underlying nature of the problem.
Principal Components: The principal components of data set are the projections of the data vectors onto new coordinate axes that result from a rotation of the centered data set. This rotation is done in such a way that the first principal component (the projection onto the first coordinate axis) has the largest possible variance, the second principal component has the next largest, and so on.