Niël le Roux

Measures of fit in principal component and canonical variate analyses

Treating principal component analysis (PCA) and canonical variate analysis (CVA) as methods for approximating tables, we develop measures, collectively termed predictivity, that assess the quality of fit independently for each variable and for all dimensionalities. We illustrate their use with data from aircraft development, the African timber industry and copper froth measurements from the mining industry. Similar measures are described for assessing the predictivity associated with the individual samples (in the case of PCA and CVA) or group means (in the case of CVA). For these measures to be meaningful, certain essential orthogonality conditions must hold that are shown to be satisfied by predictivity.

An emerging market perspective on peer group selection based on valuation fundamentals

The developed market literature suggests that peer group selection based on a careful selection of valuation fundamentals may improve the valuation accuracy of multiples. However, the literature does not offer an emerging market perspective in this regard. In this article the valuation performances of 16 equity multiples are investigated, based on three individual valuation fundamentals and three different combinations of these valuation fundamentals. The valuation performance of these 16 multiples is assessed in the equity valuation of South African companies listed on the JSE Securities Exchange over the period 2001 to 2010. The empirical results revealed, among other findings, that peer group selection based on a careful selection of valuation fundamentals could, on average, increase valuation accuracy of multiples by as much as 37.88%.

Spline-based nonlinear biplots

Biplots are helpful tools to establish the relations between samples and variables in a single plot. Most biplots use a projection interpretation of sample points onto linear lines representing variables. These lines can have marker points to make it easy to find the reconstructed value of the sample point on that variable. For classical multivariate techniques such as principal components analysis, such linear biplots are well established. Other visualization techniques for dimension reduction, such as multidimensional scaling, focus on an often nonlinear mapping in a low dimensional space with emphasis on the representation of the samples. In such cases, the linear biplot can be too restrictive to properly describe the relations between the samples and the variables. In this paper, we propose a simple nonlinear biplot that represents the marker points of a variable on a curved line that is governed by splines. Its main attraction is its simplicity of interpretation: the reconstructed value of a sample point on a variable is the value of the closest marker point on the smooth curved line representing the variable. The proposed spline-based biplot can never lead to a worse overall sample fit of the variable as it contains the linear biplot as a special case.

The Canonical Analysis of Distance

Canonical Variate Analysis (CVA) is one of the most useful of multivariate methods. It is concerned with separating between and within group variation among N samples from K populations with respect to p measured variables. Mahalanobis distance between the K group means can be represented as points in a (K - 1) dimensional space and approximated in a smaller space, with the variables shown as calibrated biplot axes. Within group variation may also be shown, together with circular confidence regions and other convex prediction regions, which may be used to discriminate new samples. This type of representation extends to what we term Analysis of Distance (AoD), whenever a Euclidean inter-sample distance is defined. Although the N × N distance matrix of the samples, which may be large, is required, eigenvalue calculations are needed only for the much smaller K × K matrix of distances between group centroids. All the ancillary information that is attached to a CVA analysis is available in an AoD analysis. We outline the theory and the R programs we developed to implement AoD by presenting two examples.

Extensions of Biplot Methodology to Discriminant Analysis

AbstractIn this paper we show how biplot methodology can be combined with various forms of discriminant analyses leading to highly informative visual displays of the respective class separations. It is demonstrated that the concept of distance as applied to discriminant analysis provides a unified approach to a wide variety of discriminant analysis procedures that can be accommodated by just changing to an appropriate distance metric. These changes in the distance metric are crucial for the construction of appropriate biplots. Several new types of biplots viz. quadratic discriminant analysis biplots for use with heteroscedastic stratified data, discriminant subspace biplots and flexible discriminant analysis biplots are derived and their use illustrated. Advantages of the proposed procedures are pointed out. Although biplot methodology is in particular well suited for complementing J > 2 classes discrimination problems its use in 2-class problems is also illustrated.

The analysis of distance of grouped data with categorical variables

We use generalised biplots to develop the important special case of (i) when all variables are categorical and (ii) the samples fall into K recognised groups. We term this Categorical Canonical Variate Analysis (CatCVA), because it has similar characteristics to Raos Canonical Variate Analysis (CVA), especially its visual aspects. It allows centroids of groups to be exhibited in increasing numbers of dimensions, together with information on within-group sample variation. Variables are represented by category-level-points (CLPs) which are a counterpart of numerically calibrated biplot axes for quantitative variables. Mechanisms are provided for relating the samples to their category levels, for giving convex regions to help predict categories, and for adding new samples. Inter-sample distance may be measured by any Euclidean embeddable distance. Computation is minimised by working in the K - 1 dimensional space containing the group centroids.The methodology is illustrated by an example with three groups and 37 samples but the number of samples size is not a serious limitation. The visualisation of group structure is the main focus of this paper; computational efficiency is a bonus.

Biplot Methodology for Discriminant Analysis Based upon Robust Methods and Principal Curves

Biplots not only are useful graphical representations of multidimensional data,but formulating discriminant analysis in terms of biplot methodology can lead to several novel extensions. In this paper it is shown that incorporating both principal curves and robust canonical variate analysis algorithms in biplot methodology often leads to superior classification.

Chemical alterations induced by Pycnoporus sanguineus/Aspergillus flavipes co-cultures in wood from different tree species

Chemical alterations following inoculation of Acacia mearnsii, Eucalyptus dunnii, E. grandis, and E. macarthurii with a Pycnoporus sanguineus/Aspergillus flavipes co-culture were investigated. Several wood chemical parameters were measured using standard methods from the pulp and paper industry. The data were described and analyzed using univariate as well as multivariate statistical techniques. Boxplots and in particular biplots show clearly how the chemical composition of each tree species was differently affected by the co-culture. Lignin content was significantly decreased in A. mearnsii, while E. dunnii showed a decrease in cellulose content. The results, therefore, indicate that the manner in which wood is degraded by a specific fungal co-culture depends on the tree species involved. This phenomenon should be considered when selecting fungi for bio-pulping.

Sub-species of Homopus Areolatus? Biplots and Small Class Inference with Analysis of Distance

A canonical variance analysis (CVA) biplot can visually portray a oneway MANOVA. Both techniques are subject to the assumption of equal class covariance matrices. In the application considered, very small sample sizes resulted in some singular class covariance matrix estimates and furthermore it seemed unlikely that the assumption of homogeneity of covariance matrices would hold. Analysis of distance (AOD) is employed as nonparametric inference tool. In particular, AOD biplots are introduced for a visual display of samples and variables, analogous to the CVA biplot.

Modified Biplots for Enhancing Two-Class Discriminant Analysis

When applied to discriminant analysis (DA) biplot methodology leads to useful graphical displays for describing and quantifying multidimensional separation and overlap among classes. The principles of ordinary scatterplots are extended in these plots by adding information of all variables on the plot. However, we show that there are fundamental differences between two-class DA problems and the case J > 2: describing overlap in the two-class situation is relatively straightforward using density estimates but adding information by way of multiple axes to the plot can be ambiguous unless care is taken. Contrary to this, describing overlap for J > 2 classes is relatively more complicated but the fitting of multiple calibrated axes to biplots is well defined.