Special Properties in Selection Performance of Qualitative Variables in Discriminant Analysis

The paper deals with the optimal Bayes discriminant rule for qualitative variables. The performance of variable selection is investigated under strong assumptions like the restriction to dichotomous variables, which are assumed to be independent or dependent with fixed dependence structure, and all parameters known. Differences in comparison with normal variables in linear discriminant analysis can be shown. This is a further reason for applying special methods of discriminant analysis in the case of qualitative variables.     

Goodness of Fit of Discriminant Functions from the Vector Space of Fixed Variates

Alternative proofs of some of KSHIRSAGAR's (1971) results on testing discriminant functions or canonical variables in the vector space of fixed variates are given. These results are derived in terms of the original variates unlike KSHIRSAGAR (1971) who derives the results by using random orthogonal transformations and triangular decompositions of the original matrix variates.     

New Procedure for Selection of Variables in Location Model for Mixed Variable Discrimination

A new method for the choice of variables with the greatest discriminatory power in the location model for mixed variable discriminant analysis is presented in the paper. The procedure based on the multivariate discriminatory measure enables a simultaneous reduction of the number of discrete and continuous variables. The introduced criterion can be used for both optimal or step-wise selection of variable subset. As an example the results of the stepwise variable selection for some medical data are presented in the paper.     

A numerical phytoclimatic classification of China

The relationship is examined between vegetation and climate using climatic variables collected from 644 meteorological stations located throughout China. Multivariate methods are applied directly to the raw climatic data in order to define climatic clusters and to examine the relationship between the clusters and vegetation types. This approach is based on the concept of multidimensional climatic space defined by the combination of climatic variables. Phytoclimatic classes are defined on the basis of the distribution of vegetation types in climatic clusters and a new phytoclimatic classification of China is proposed. Patterns of climatic changes between neighbouring phytoclimatic classes are described. Two indexes of the influence of climate on vegetation are proposed based on discriminant analysis.     

Distinction between Mulloidichthys flavolineatus juveniles from Reunion Island and Mauritius Island (south-west Indian Ocean) based on otolith morphometrics

Sagittal otoliths of the yellowstripe goatfish Mulloidichthys flavolineatus were analysed in order to compare Reunion Island fish stocks with those of Mauritius (south-west Indian Ocean). Conventional otolith morphometric variables (area, perimeter, length and width), shape indices (form factor, roundness, circularity, rectangularity, ellipticity and eccentricity) and Fourier shape analysis were compared between three sites; two in Reunion Island and one in Mauritius. Regional and site-specific differences were found for all the conventional otolith morphometric features. Regarding the shape indices, the differences between sites were best described by form factor, roundness, circularity and rectangularity. A classification by canonical discriminant analysis indicated significant differences between the three sampling sites. The combined use of morphometric variables (size and shape) and external outlines (shape analysis through Fourier series) showed the importance of otolith shape for intraspecific discrimination.     

太行山猕猴下颌骨变量相关性研究

本研究测量了１８例（雄４,雌１４）成年太行山猕猴下颌骨的１２项变量。通过对下颌骨变量的性别判别分析和相关分析。结果显示：太行山猕猴下颌骨变量有其自身特征,与其它猕猴下颌骨资料比较表现出一定差异;用Ｏｔｔｅｓｔａｔ方法对有关变量建立性别判别函数式,其判别率达８６．８６％,下颌骨变量相关性研究表明,各变量之间有正相关,也有负相关,大多数变量之间相互程度较差,不同变量之间其相关程度存在一定差异。     

Sexual dimorphism in human skulls. A comparison of sexual dimorphism in different populations

Application and comparison of sex discriminant functions in different populations led to the conclusion that a certain combination and weighting of a few sex dimorphism variables (in this study we only used craniometric variables) can give a good discrimination between male and female individuals, independent of the racial group to which this function is applied. In our study, the sex-discriminatory power of five discriminant functions which were based on different ordination and selection procedures (e.g. professional knowledge, stepwise discriminant analysis, literature) of the cranial variables is compared. These discriminant functions were applied to three different data sets, the first being skull measurements from an Amsterdam series (Europids), the second skull measurements of a Zulu series (Negrids) and the third skull measurements of a Japan series (Mongolids). Our decision as to whether a function is a good or less good sex-discriminating function is determined by the Dt values (these values give an idea about the discriminatory value of the discriminant function when applied to a new test sample), the number of variables necessary to obtain this Dt and the location of the sectioning point (i.e. comparison between the estimation of the sectioning point and the ”real” sectioning point). These discriminant functions were compared withGiles Elliot's (1962, 1963) “race-independent” sex function.     

A generalized discriminant rule when training population and test population differ on their descriptive parameters

Standard discriminant analysis methods make the assumption that both the labeled sample used to estimate the discriminant rule and the nonlabeled sample on which this rule is applied arise from the same population. In this work, we consider the case where the two populations are slightly different. In the multinormal context, we establish that both populations are linked through linear mapping. Estimation of the nonlabeled sample discriminant rule is then obtained by estimating parameters of this linear relationship. Several models describing this relationship are proposed and associated estimated parameters are given. An experimental illustration is also provided in which sex of birds that differ morphometrically over their geographical range is to be deterrmined and a comparison with the standard allocation rule is performed. Extension to a partially labeled sample is also discussed.     

Prediction of clinical outcome with microarray data: a partial least squares discriminant analysis (PLS-DA) approach

Partial least squares discriminant analysis (PLS-DA) is a partial least squares regression of a set Y of binary variables describing the categories of a categorical variable on a set X of predictor variables. It is a compromise between the usual discriminant analysis and a discriminant analysis on the significant principal components of the predictor variables. This technique is specially suited to deal with a much larger number of predictors than observations and with multicollineality, two of the main problems encountered when analysing microarray expression data. We explore the performance of PLS-DA with published data from breast cancer (Perou et al. 2000). Several such analyses were carried out: (1) before vs after chemotherapy treatment, (2) estrogen receptor positive vs negative tumours, and (3) tumour classification. We found that the performance of PLS-DA was extremely satisfactory in all cases and that the discriminant cDNA clones often had a sound biological interpretation. We conclude that PLS-DA is a powerful yet simple tool for analysing microarray data.     

Evaluating the use of macroscale variables as proxies for local aquatic variables and to model stream fish distributions

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Conditional Quadratic Discrimination in the Identification of Biological Markers for Disease Screening

Conditional multivariate normal density functions are used to construct conditional quadratic discriminant functions that adjust for covariate differences between disease groups. An expected actual error rate for the conditional discriminant function is defined. The purpose of this paper is to use the conditional quadratic discriminant function and its misolassification error rate in order to help determine if a set of discriminators is a good biological marker for disease screening. The conditional quadratic discriminant analysis is illustrated using data from two alcoholism classification problems. It is shown how the discriminant functions can identify a set of variables that can be used as biological markers.     

Quantitative digital and palmar dermatoglyphics among Sardinian linguistic groups

Discriminant analyses of 23 digital and 15 palmar quantitative dermatoglyphic variables of 1364 Sardinians, 689 males and 675 females, were performed to identify biological relationships among five Sardinian linguistic groups of both sexes. By various subsets of dermatoglyphic variables (23 and 20 digital, 15 and 14 palmar, 4 summary traits) MANOVA revealed high intergroup heterogeneity among the groups of both sexes and within each sex. In the latter case the males are an exception when 15 and 14 (MLI removed) palmar variables are used. Standard discriminant analysis of the 23 digital variables, i.e. the radial and ulnar ridge counts on each of the 10 fingers plus total finger ridge count (TFRC), absolute finger ridge count (AFRC) and pattern intensity (PI), resulted in imperfect separation of males and females and an unclear picture of the biological relationships among the groups. In contrast, standard discriminant analysis of 20 digital variables (TFRC, AFRC and PI were removed from the analysis) resulted in separation of the sexes and a pattern of relationships among the populations consistent with their ethno-historical backgrounds. Standard discriminant analysis of 15 palmar dermatoglyphic variables failed to provide separation of the sexes and produced a pattern of relationships in disagreement with both the linguistic and ethno-historical backgrounds, even removing MLI (Main Line Index). Standard discriminant analysis of 4 summary dermatoglyphic variables (TFRC, AFRC, PI and MLI) yielded imperfect separation of males and females and an unclear pattern of relationships. By stepwise discriminant analysis with p < or = 0.01 as F-to-enter and p < or = 0.05 as F-to-remove, only 4 of the 38 digital and palmar variables were in the model (URC R5, RRC L5, RRC R5, URC R4). The pattern of inter-population biological relationships was conceptually similar to the one produced by the 20 digital variables. It showed a clear separation of the Gallurian group (both males and females), which speaks an Italian dialect. The properly Sardinian linguistic groups (Campidanian and Logudorian), the Sassarian group (which speaks an Italian dialect) and the Alghero group (which speaks Catalan) were close to one another. This picture agrees with the ethno-historical background rather than with the linguistic one.     

Robust discriminant analysis and its application to identify protein coding regions of rice genes

Identification of protein coding regions is fundamentally a statistical pattern recognition problem. Discriminant analysis is a statistical technique for classifying a set of observations into predefined classes and it is useful to solve such problems. It is well known that outliers are present in virtually every data set in any application domain, and classical discriminant analysis methods (including linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA)) do not work well if the data set has outliers. In order to overcome the difficulty, the robust statistical method is used in this paper. We choose four different coding characters as discriminant variables and an approving result is presented by the method of robust discriminant analysis.     

Selecting predictors for discriminant analysis of species performance: an example from an amphibious softwater plant

Selecting an appropriate variable subset in linear multivariate methods is an important methodological issue for ecologists. Interest often exists in obtaining general predictive capacity or in finding causal inferences from predictor variables. Because of a lack of solid knowledge on a studied phenomenon, scientists explore predictor variables in order to find the most meaningful (i.e. discriminating) ones. As an example, we modelled the response of the amphibious softwater plant Eleocharis multicaulis using canonical discriminant function analysis. We asked how variables can be selected through comparison of several methods: univariate Pearson chi-square screening, principal components analysis (PCA) and step-wise analysis, as well as combinations of some methods. We expected PCA to perform best. The selected methods were evaluated through fit and stability of the resulting discriminant functions and through correlations between these functions and the predictor variables. The chi-square subset, at P < 0.05, followed by a step-wise sub-selection, gave the best results. In contrast to expectations, PCA performed poorly, as so did step-wise analysis. The different chi-square subset methods all yielded ecologically meaningful variables, while probable noise variables were also selected by PCA and step-wise analysis. We advise against the simple use of PCA or step-wise discriminant analysis to obtain an ecologically meaningful variable subset; the former because it does not take into account the response variable, the latter because noise variables are likely to be selected. We suggest that univariate screening techniques are a worthwhile alternative for variable selection in ecology.     

Relationships of overstory to understory cover variables in a Ponderosa pine/Gambel oak ecosystem

Many studies have indicated relationships between individual species, but none have related combinations of overstory variables to understory herbaceous vegetation in a Ponderosa pine/Gambel oak ecosystem. Our objective was to determine not only the general relationships between the two sets of variables, but also identify the hyghest contributing variables. We used canonical correlation analysis to relate overstory variables (canopy cover, basal cover and density) to herbaceous vegetation cover variables. Canopy, basal, and ground cover were measured by the line intercept method using a 12.2 m tape as a sample unit. Tree density was measured by the Point-Center-Quarter method. The analysis was made with selected overstory variables and 5 understory herbaceous cover variables. This analysis revealed a significant canonical correlation between the two canonical variables (r=0.69). The analysis showed that among herbaceous cover variables, Oregon grape, Kentucky bluegrass, sedge, and foxtail barley; and among overstory variables, the density and the basal cover of Ponderosa pine indicated the highest positive contribution to the correlation of the two linear combinations while the density and canopy of Gambel oak negatively affected the canonical correlation.     

Identification of the Lake Malawi Oreochromis (Nyasalapia) spp. using multivariate morphometric techniques

With the aim of reliably distinguishing these commercially important species on the basis of external characteristics alone, morphometric techniques were employed on a sample of the four species of Oreochromis ( Nyasalapia ) described from Lake Malawi: O. karongae, O. lidole, O. saka and O. squamipinnis . Univariate analysis of variance on the ratios of 23 variables to standard length indicated many differences among all species, but there was considerable individual variation, and consequent overlap. Residuals from a regression of each variable on length were employed for multivariate analysis. Cluster analysis on the means of the residuals was used to construct a phenogram which formed the basis for denning a series of dichotomous discriminant analyses. In each discriminant analysis, variables were successively eliminated in the reverse order of the magnitude of their correlation with the discriminant function. The combination of variables producing 95% accuracy of classification was selected, and the discriminant function equations for each step calculated. Some further variables were eliminated by checking for redundancy through analysis of correlations. The resulting equations enable O. karongae to be separated using eight measurements, O. lidole using 10, and O. saka and O. squamipinnis to be distinguished by a combination of 13 measurements.     

Quantitative estimation of peat,brown coal and lignite humic acids using chemical parameters, 1H-NMR and DTA analyses

Humic acids extracted from peats (P), brown coals (BC) and lignites (L), were characterized using different (chemical, 1H-nuclear magnetic resonance spectroscopy and differential thermal analysis) techniques. Fourteen variables were obtained from these analyses and only five were selected because uncorrelated in multiple partial correlation. The chosen variables were C concentration, aliphatic and aromatic components and the heat of reaction of the second exothermic peak. The multivariate discriminant analysis was performed on these variables and a discriminant function was obtained which was able to efficiently separate the P, BC and L. This function enables simple predictions on samples of unknown origin. The straightforward method proposed and the results obtained are discussed.     

Sex-diagnosis of isolated Os coxae by discriminant functions

After methodical preparatory work on the selection of the metric characters and their technical definition, 16 variables were measured on every right hip bone in our standard series (Weisbach series). The discriminant functions performed on our standard series proved to be very sharply discriminant. The methodical foundation gained from the Weisbach series was applied to a control series (Tyrol series), whose sex had not been determined. The application of the discriminant analyses enabled a complete and plausible sex diagnosis to be made.     

Nonlinear biological batch process monitoring and fault identification based on kernel fisher discriminant analysis

A novel nonlinear biological batch process monitoring and fault identification approach based on kernel Fisher discriminant analysis (kernel FDA) is proposed. This method has a powerful ability to deal with nonlinear data and does not need to predict the future observations of variables. So it is more sensitive to fault detection. In order to improve the monitoring performance, variable trajectories of the batch processes are separated into several blocks. Then data in the original space is mapped into high-dimensional feature space via nonlinear kernel function and the optimal kernel Fisher feature vector and discriminant vector are extracted to perform process monitoring and fault identification. The key to the proposed approach is to calculate the distance of block data which are projected to the optimal kernel Fisher discriminant vector between new batch and reference batch. Through comparing distance with the predefined threshold, it can be considered whether the batch is normal or abnormal. Similar degree between the present discriminant vector and the optimal discriminant vector of fault in historical data set is used to perform fault diagnosis. The proposed method is applied to the process of fed-batch penicillin fermentation simulator benchmark and shows that it can effectively capture nonlinear relationships among process variables and is more efficient than MPCA approach.     

Discriminative features of type I and type III secreted proteins from Gram-negative bacteria

The amino acid composition of sequences and structural attributes (α-helices, β-sheets) of C-and N-terminal fragments (50 amino acids) were compared to annotated (SWISS-PROT/ TrEMBL) type I (20 sequences) and type III (22 sequences) secreted proteins of Gram-negative bacteria. The discriminant analysis together with the stepwise forward and backward selection of variables revealed the frequencies of the residues Arg, Glu, Gly, Ile, Met, Pro, Ser, Tyr, Val as a set of strong (1-P < 0.001) predictor variables to discriminate between the sequences of type I and type III secreted proteins with a cross-validated accuracy of 98.6–100 %. The internal and external validity of discriminant analysis was confirmed by multiple (15 repeats) test-retest procedures using a randomly split original set of proteins; this validation method demonstrated an accuracy of 100 % for 191 non-selected (retest) sequences. The discriminant analysis was also applied using selected variables from the propensities for β-sheets and polarity of C-terminal fragments. This approach produced the next highest and comparable cross-validated classification accuracy for randomly selected and retest proteins (85.4–86.0 % and 82.4–84.5 %, respectively). The proposed sets of predictor variables could be used to assess the compatibility between secretion substrates and secretion pathways of Gram-negative bacteria by means of discriminant analysis.