Biplot graphical display techniques

Includes bibliography.The thesis deals with graphical display techniques based on the singular value decomposition. These techniques, known as biplots, are used to find low dimensional representations of multidimensional data matrices. The aim of the thesis is to provide a review of biplots for a practical statistician who is not familiar with the area. It therefore focuses on the underlying theory, assuming a standard statisticians' knowledge of matrix algebra, and on the interpretation of the various plots. The topic falls in the realm of descriptive statistics. As such, the methods are chiefly exploratory. They are a means of summarising the data. The data matrix is represented in a reduced number of dimensions, usually two, for simplicity of display. The aim is to summarise the information in the matrix and to present a visual representation of this information. The aim in using graphical display techniques is that the "gain in interpretability far exceeds the loss in information" (Greenacre, 1984). A graphical description is often more easy to understand than a numerical one. Histograms and pie charts are familiar forms of data representation to many people with no other, or very rudimentary, statistical understanding. These are applicable to univariate data. For multivariate data sets, univariate methods do not reveal interesting relationships in the data set as a whole. In addition, a biplot can be presented in a manner which can be readily understood by non-statistically minded individuals. Greenacre (1984) comments that only in recent years has the value of statistical graphics been recognised. Young (1989) notes that recently there has been a shift in emphasis, among statisticians towards exploratory data analysis methods. This school of thought was given momentum by the publication of the book "Exploratory Data Analysis" (Tukey, 1977). The trend has been facilitated by advances in computer technology which have increased both the power and the accessibility of computers. Biplot techniques include the popular correspondence analysis. The original proponents of correspondence analysis (among them Benzecri) reject probabilistic modelling. At the other extreme, some view graphical display techniques as a mere preliminary to the more traditional statistical approaches. Under the latter view, graphical display techniques are used to suggest models and hypotheses. The emphasis in exploratory data techniques such as graphical displays is on 'getting a feel' for the data rather than on building models and testing hypotheses. These methods do not replace model building and hypothesis testing, but supplement them. The essence of the philosophy is that models are suggested by the data, rather than the frequently followed route of first fitting a model. Some work has gone into developing inferential methods, with hypothesis tests and associated p-values for biplot-type techniques (Lebart et al, 1984, Greenacre, 1984). However, this aspect is not important if the techniques are viewed merely as exploratory. Chapter Two provides the mathematical concepts necessary for understanding biplots. Chapter Three explains exactly what a biplot is, and lays the theoretical framework for the biplot techniques that follow. The goal of this chapter is to provide a framework in which biplot techniques can be classified and described. Correlation biplots are described in Chapter Four. Chapter Five discusses the principal component biplot, and the link between these and principal component analysis is drawn. In Chapter Six, correspondence analysis is presented. In Chapter Seven practical issues such as choice of centre are discussed. Practical examples are presented in Chapter Eight. The aim is that these examples illustrate techniques commonly applicable in practice. Evaluation and choice of biplot is discussed in Chapter Nine