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Interpretable and fast dimension reduction of multivariate data
The main objective of this thesis is to propose new techniques to simplify the interpretation of newly formed 'variables' or components, while reducing the dimensionality of multivariate data. Most attention is given to the interpretation of principal components, although one chapter is devoted to that of factors in factor analysis. Sparse principal components are proposed, in which some of the component loadings are made exactly zero. One approach is to make use of the idea of correlation biplots, where orthogonal matrix of sparse loadings is obtained from computing the biplot factors of the product of principal component loading matrix and functions of their variances. Other approaches in volve clustering of variables as a pre-processings tep, so that sparse components are computed from the data or correlation matrix of each cluster. New clustering techniques are proposed for this purpose. In addition, a penalized varimax approach is proposed for simplifying the interpretation of factors in factor analysis, especially for factor solutions with considerably different sum of squares. This is done by adding a penalty term to the ordinary varimax criterion. Data sets of varying sizes, both synthetic and real, are used to illustrate the proposed methods, and the results are compared with those of existing ones. In the case of principal component analysis, the resulting sparse components are found to be more interpretable (sparser) and explain higher cumulative percentage of adjusted variance compared to their counterparts from other techniques. The penalized varimax approach contributes in finding a factor solution with simple structures which are not revealed by the standard varimax solution. The proposed methods are very simple to understand and involve fast algorithms compared to some of the existing methods. They contribute much to the interpretation of components in a reduced dimension while dealing with dimensionality reduction of multivariate data
Detection of Infectious Disease Outbreaks From Laboratory Data With Reporting Delays
Many statistical surveillance systems for the timely detection of outbreaks of infectious disease operate on laboratory data. Such data typically incur reporting delays between the time at which a specimen is collected for diagnostic purposes, and the time at which the results of the laboratory analysis become available. Statistical surveillance systems currently in use usually make some ad hoc adjustment for such delays, or use counts by time of report. We propose a new statistical approach that takes account of the delays explicitly, by monitoring the number of specimens identified in the current and past m time units, where m is a tuning parameter. Values expected in the absence of an outbreak are estimated from counts observed in recent years (typically 5 years). We study the method in the context of an outbreak detection system used in the United Kingdom and several other European countries. We propose a suitable test statistic for the null hypothesis that no outbreak is currently occurring. We derive its null variance, incorporating uncertainty about the estimated delay distribution. Simulations and applications to some test datasets suggest the method works well, and can improve performance over ad hoc methods in current use. Supplementary materials for this article are available online