Gardner S

Discriminating between the end products of the earlier Middle Stone Age sub-stages at Klasies River using biplot methodology

The difference between the culture-stratigraphic entities, the MSA I and MSA II, in the Klasies River sequence is explored by statistical analysis of the end products. Technological analysis of the cores, end products and waste products suggested that the MSA I and MSA II represent distinct technological traditions aimed at producing different end products. To quantify the difference between the end products, points and blades, extensive univariate and multivariate statistical analyses of continuous variables have been undertaken. Biplot methodology is adopted for enhancing the statistical analysis. Canonical variate analysis biplots are constructed and alpha-bags added for visual displays of the overlap and separation among the different groups. It is demonstrated that the platform thickness relative to length is useful in discriminating between the end products of the MSA I and MSA II. Statistical analyses support a clear distinction between the MSA I and MSA II.

A synthesis of canonical variate analysis, generalised canonical correlation and Procrustes analysis

Canonical variate analysis (CVA) is concerned with the analysis of J classes of samples, all described by the same variables. Generalised canonical correlation analysis (GCCA) is concerned with the analysis of K sets of variables, all describing the same samples. A generalised procrustes analysis context is used for data partitioned into J classes of samples and K sets of variables to explore the links between GCCA and CVA. Biplot methodology is used to exploit the visualisation properties of these techniques. This methodology is illustrated by an example of 1425 samples described by three sets of variables (K = 3), the initial analysis of which suggests a grouping of the samples into four classes (J = 4), followed by subsequent more detailed analyses.

Extending a scatterplot for displaying group structure in multivariate data : a case study

The power of canonical variate analysis (CVA) biplots, when regarded as extensions of ordinary scatterplots to describe variation and group structure in multivariate observations, is demonstrated by presenting a case study from the South African wood pulp industry. It is shown how multidimensional standards specified by users of a product may be added to the biplot in the form of acceptance regions such that the roles of the respective variables that influence the product can be ascertained. The case study considers an alternative to CVA and multivariate analysis of variance (MANOVA) when the application of these procedures becomes questionable as a result of dealing with small sample sizes and heterogeneity of covariance matrices. It is explained how analysis of distance (AOD) analogous to analysis of variance may be performed in such cases. Biplots to accompany AOD are provided. The biplots and AOD illustrated in the case study from the wood pulp industry have the potential to be used widely where a primary product, influenced by several variables, is produced and where this product is of importance to various secondary manufacturers depending on which set of multidimensional specifications are met.

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.

Biplots for displaying multidimensional financial performance data graphically

Invariably financial performance of an enterprise is not judged on a single criterion but is described in terms of various related financial performance indicators. Financial decision makers are thus typically confronted with multidimensional data. This paper demonstrates to financial managers several uses of the biplot for displaying multidimensional data graphically. It is shown that biplots not only serve in understanding multidimensional data better but also might reveal features of such data not perceptible by graphing each variable on its own. Two different types of biplots are discussed: One for describing variation in multidimensional data and the other for separating different groups optimally. The first of these types of biplots is applied to a data set originating in risk management and the second to a data set where, on the basis of various financial ratio indicators, the aim is to separate enterprises in danger of financial failure within a year from those not in such danger. In the latter example a distinction is also made between companies classified as manufacturing enterprises and those classified as trading enterprises.

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.

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.

Discolouration during kiln drying of South African Pinus elliottii

Abstract The effect of different parameters on the surface discolouration (yellow stain and kiln brown stain) and thermal discolouration of Pinus elliottii during kiln drying was investigated. Boards were dried with different kiln schedules and discolouration was assessed at different depth levels from the surface of each board. The discolouration data were analysed using notched boxplots, multivariate analysis of variance and canonical variate analysis biplots. Thermal and surface discolouration were distinguishable by the distribution of colour data point plots in the CIE-L*a*b* colour system. Thermal discolouration was less intense and less varied than surface discolouration. The greatest degree of thermal discolouration occurred in one of the higher temperature (Tdb≥90°C during first drying phase) schedules that was run for an excessively long period due to low air velocity. Discolouration progressively decreased as the boards were planed to a greater depth. Higher temperature schedules yielded 77.5-82.5% discoloured replicates at the 2 mm depth level. Lower temperature (Tdb≤71°C during first drying phase) schedules yielded no discoloured wood at the surface or at the centre of the boards.

Discriminant Analysis With Categorical Variables: A Biplot Based Approach

(1996) provide a new perspective on the traditional biplot of (1971) by viewing biplots as multivariate analogues of ordinary scatterplots. It is demonstrated by (2001) that extending biplot methodology to discriminant analysis is not only extremely useful for visual displays to accompany discriminant analysis but how the process of classification can be performed using biplot methodology. In particular, a method developed for the inclusion of categorical predictors is discussed. Specific attention is devoted to what is termed a ‘reversal’ when dealing with two binary (categorical) predictor variables. A proposal using biplot methodology is made for dealing with this problem.