Discriminant Analysis: Description of Group Separation

There are some problems in the reporting of many numerical analyses—this includes reporting in newspapers, on television, in journal articles, and in books. In this article, the authors focus on problems in reporting results of predictive discriminant analysis (PDA) and descriptive discriminant analysis (DDA) in journal articles. The main issue addressed pertains to information that should be reported for PDA or DDA studies, and what results may be deleted with little loss to hamper judgment calls about the research quality and validity of conclusions. Twenty applications of discriminant analyses that appeared in 19 behavioral science journals (1998-2000) were scrutinized with respect to quality and quantity of reported results. The conclusion is not very positive. PDA and DDA results were very often mixed, which may not be too surprising because study purposes were typically not very explicit. Results specific to PDA and to DDA were very seldom reported with any meaningful interpretations.

Fisher proposes Fisher’s linear discriminant function (LDF) based on maximization of variance ratio. If data satisfies Fisher’s assumption, the same LDF is easily derived by variance-covariance matrix. If variance-covariance matrices of two classes are not same, quadratic discriminant function (QDF) is proposed. These discriminant functions have three problems. First, all discriminant functions except for Revised IP-OLDF can’t discriminate the cases xi on the discriminant hyper plane correctly (the unresolved problem of discriminant analysis). Only Revised IP-OLDF can solve this problem in theoretically. Second, LDF and QDF can’t recognize linear separable data in many cases, and the numbers of misclassifications (NMs) of these functions are usually higher than logistic regression that is used in many users. Third, these are not obtained if the values of some independent variable are constant, because inverse matrix can’t be calculated. JMP prepare the regularized discriminant function if QDF can’t analyze the difficult data. These facts mean that LDF and QDF shouldn’t be used for an important discrimination. On the contrary, Revised IP-OLDF based on Minimum Number of Misclassifications (MNM) criterion resolves these problems. In addition to this, means of error rates of Revised IP-OLDF are less than those of LDF and logistic regression and S-SVM by 100-fold cross validation in many cases.

This paper proposes a new mathematical programming approach to represent the dimensions of the discriminant space for the multiple-group classification problem. Few papers have investigated generalizations of two-group mathematical programming approaches for the classification of multiple groups. While several papers have proposed mathematical programming models for separating groups of observations, the issue of considering the classification problem by finding discriminant linear functions to describe the groups in fewer dimensions has not been addressed. The new mathematical programming approach proposed in this paper first solves the multiple-group problem using a single discriminant function, which essentially represents the separation of the groups in one dimension. Then the multiple-group problem is successively solved using single discriminant functions with the requirement that successive linear discriminant functions have a sample covariance equal to zero. An algorithm is proposed to classify observations from multiple groups using the linear discriminant functions from the mathematical programming approach in a reduced number of dimensions.

Using discriminant function analysis to estimate the impact of a therapeutic intervention in burn care

1) Using the data on Japanese, Pima Indians, Australian Aborigines and American Negroes, the discriminant function coefficients for sexing were computed on the basis of the mesiodistal crown diameters of 7 maxillary and 7 mandibular permanent teeth excluding the third molars.2) Between-sex differences in the mean values were tested in each population (Tables 1 and 2). Although the significant differences were found in few teeth, the discriminant function coefficients were computed on the basis of all the teeth measured. The reason of this procedure was the fact that there were more or less within-group correlations so that the teeth without significant difference could not be excluded from the computation.3) As a result of a direct method of computation, the effectiveness of 76% was obtained in Japanese (Tables 3 and 6).4) A stepwise method employing Wilks' lambda as a statistical standard selected 6 variables, and gave the effectiveness of 77% in Japanese (Tables 4 and 5). These 6 variables agreed quite well with the teeth showing higher M/F ratio reported by GARN et al. (1964), 5) Effectiveness of sex discrimination showed considerably large differences among 4 populations compared. The effectiveness was apparently higher in the populations which showed larger sexual dimorphism in the crown diameters (cf. HANIHARA, 1978), and vice versa.6) Using P(Gκ|ys) which shows the probability that the individual s belongs to the male or female group sexs, the of the individual s can be more effectively discriminated. In Japanese, the discrimination is almost certain when P(Gκ|ys)>0.7. This probability is computed by the formula shown in the text, in which Pi is a prior probability that the individuals belongs to the group k, ys is a discriminant function value for individual s, yk is a mean of the discriminant function values in the group k, and Σy is a pooled covariance matrix.7) A point of advantage to use the dental crown diameters for sexing is its effectiveness on the younger individuals. The discriminant function coefficients so far reported for the skeleton are effective only on the adults, but the coefficients reported in this article can be applied on the individuals whose second molars were erupted, or the individuals over around 12 years of age.The authors are deeply indebted to Dr. A. A. DAHLBERG of the University of Chicago, Dr. L. A. ALTEMUs of Howard University, and the late Dr. M. J. Barrett of the University of Adelaide who kindly permitted one of us (K. H.) to measure the excellent dental casts.

Early gastric cancer can be treated by endoscopic excision or simple wedge surgical resection. Standard gastrectomy often is advised if submucosal invasion is found, even though only 15-25% of these patients have lymph node metastases. In this study, the risk of lymph node involvement was examined by multivariate analysis to develop a simple discriminant function for surgical decision making in this setting. The authors determined factors significantly correlated with lymph node involvement in a retrospective review of 196 patients with gastric adenocarcinoma invading into, but not beyond, the submucosa. Depth and horizontal spread of cancer in the submucosa were evaluated in addition to ordinary pathologic factors. Discriminant analysis for lymph node involvement was performed using explanatory variables chosen from the results of the univariate analyses. Lymph node involvement correlated significantly with larger tumor size; greater dimension of submucosal invasion; deeper submucosal invasion; gross appearance of Type I, IIc + III or IIa + IIc; and severity of vessel invasion. Of the variables, the amount of lymphatic invasion, macroscopic appearance, and maximum dimension of submucosal infiltration were selected as effective predictors of lymph node involvement according to discriminant analysis. A correct discrimination of 74.8% was obtained with a linear discriminant function using these variables. Lymph node involvement was observed in 50% of the cases with a discriminant score less than -1 and in 25% of those with a score between -1 and 0, whereas no lymph node involvement was observed in those with a score greater than 2. Discriminant function as used in this study provided a useful criterion for additional surgery for patients with gastric carcinoma invading the submucosa who were treated initially by localized excision. Prophylactic lymph node dissection may not be necessary for a discriminant score greater than 2, whereas extended lymphadenectomy would be recommended for a score less than -1.

The economic crisis since mid 1997 has internally affected the performance of manufacture companies. Short term foreign loans in foreign currency that can be prolonged have increased the demand for foreign currency especially the US dollar, and has resulted a depreciation of rupiah value more than 80%. This further yielded the performance failure and financial bangkruptcy potentials.This study aimed at testing the usage offinancial ratios to determine whether a company belongs to the financial bangkruptcy potential group or otherwise. Further-more, considering tehe supposedly different condition and performance of manufacture industries before and during the crisis, this study also aimed at determining the performance of a company with and that without thefinancial bangkruptcy potential.Bangkruptcy has been investigated and can be predicted with reference to the company financial peiformance indicator, namely thefinancial ratios. However, considering that the tight competetion of the business world, the size of performance has developed with the entrance of market competetion ratio. Ratio, in this case is the capability of a company to survive in an industrial environment, which is quantified by the market share yield compared with the market size of his competitors.This study compared two periods of financial bangkruptcy potential predictions, namely that prior to the economic crisis in lndonesia (1994 - 1996) and that during the economic crisis (1997 - 1999). The factor, cluster and discriminant analysis were applied to the go public food and beverage industries at the Jakarta Stock Exhange. The discriminant analysis or the Z-score method proved effective in measuring the peiformance in multivariate way and in grouping it. By the SPSS social statistical version 10.0, the discriminant analysis arranging the ratios into given discriminant functions, and base on the function, the sample cases were put in the given groups. if the final ratio multiplication showed higher Z-score the case was categorized having no bangkruptcy potential. Otherwise, the case was grouped as having the bangkruptcy potential.The discriminant analysis showed that the first hypothesis was accepted. Prior to the crisis (1994 1996), factor debt ratio and net profit margin developed the discriminant function and simultaneously determined the classification of as a group with or without bangkruptcy potential. Jn this function, net profit margin factor is dominant part that determinate a grouping of cases, and it showed that a company survival were determined by the management in measure sales producing net profit. The hit ratio of discriminant function was 98 %, showing that the precision of the classification was relatively high.During crisis period (1997 1999), the important factors in the discriminant function were debt ratio, followed current ratio, total asset turn over, and market share. It showed that during crisis, the survival of a company was determinated by management tofind the fund added to invest oppurtunity. The discriminant function had hit ratio 94,1%, that showed the classification accurancy 94,1%. From the ratio or factor comparison use in the both fun ction of the two analysis period, the two hypothesis showed a difference in ratio role to determinate the bangkruptcy potential classification for two different periods.It was also revealed that in the a both analysis periods the cutting score or Z score was null. This means that in term of discriminant function score a case was negative or smaller than Z score, the case was categorized as having the bangkruptcy potential. Othenvise, if the discriminant junction was positi ve or bigger than Z score, the case will be categorized as having no bangkruptcy potential.

Research was conducted to investigate the use of optical, air-born acoustic and plasma charge signals for detecting defects in laser welds. Full penetration, partial penetration and gapped-full penetration butt-joint welds were made in 1.4mm cold-rolled steel sheet. A 3 kW CO2 laser was used to produce these three classes of welds while laser power, travel speed and joint fitup were varied to produce the various conditions. Visible light, audible sound and electrical charge near the plume were measured simultaneously in real-time using a silicon photodiode, a condensor microphone and a plasma charge sensing gas nozzle, respectively. The spectra of the sensor signals were analyzed by a statistical technique known as Linear Discriminant Analysis (LDA). Stepwise LDA was used to identify the most important frequency bands and canonical discriminant analysis was employed to develop discriminant functions. Reliability analysis of the training samples revealed that this LDA method classified laser welds with reliability ranging from 85% to 96%, depending on the welds selected for inclusion in the training set. When the linear discriminant function was validated using new (untrained) samples, it distinguished the three different classes of laser welds with approximately 87% accuracy. Among the three signals, sound produced better separation between the classes of welds, although a discriminator based on all three signals combined and one that used only plasma charge and acoustic signals were of almost equal reliability.Research was conducted to investigate the use of optical, air-born acoustic and plasma charge signals for detecting defects in laser welds. Full penetration, partial penetration and gapped-full penetration butt-joint welds were made in 1.4mm cold-rolled steel sheet. A 3 kW CO2 laser was used to produce these three classes of welds while laser power, travel speed and joint fitup were varied to produce the various conditions. Visible light, audible sound and electrical charge near the plume were measured simultaneously in real-time using a silicon photodiode, a condensor microphone and a plasma charge sensing gas nozzle, respectively. The spectra of the sensor signals were analyzed by a statistical technique known as Linear Discriminant Analysis (LDA). Stepwise LDA was used to identify the most important frequency bands and canonical discriminant analysis was employed to develop discriminant functions. Reliability analysis of the training samples revealed that this LDA method classified laser welds with rel...

A mixed integer programming model is proposed for multiple-class discriminant and classification analysis. When multiple discriminant functions, one for each class, are constructed with the mixed integer programming model, the number of misclassified observations in the sample is minimized. This model is an extension of the linear programming models for multiple-class discriminant analysis but may be considered as a generalization of mixed integer programming formulations for two-class classification analysis. Properties of the model are studied. The model is immune from any difficulties of many mathematical programming formulations for two-class classification analysis, such as nonexistence of optimal solutions, improper solutions, and instability under linear data transformation. In addition, meaningful discriminant functions can be generated under conditions where other techniques fail. Examples are provided. Results on publically accessible datasets show that this model is very effective in generating powerful discriminant functions.

Bats are the second most speciose order of mammals and are under significant threat throughout the world. Survey and monitoring of bats for conservation are severely hampered by the lack of a reliable and user-friendly method of identifying bats from their echolocation calls. We recorded and described time-expanded echolocation calls from 23 bat species in the National Park of Dadia-Lefkimi-Soufli, Greece. We compared the performance of quadratic and linear discriminant function analysis (DFA) of calls as a means of identifying species. Quadratic rather than linear DFA has been used by several researchers because of the violation of the method's basic assumption (homogeneity of variance-covariance matrices). However, when linear DFA was applied for the classification of recorded species in this study, correct classification rate was identical to the quadratic functions (82.4%) and linear models did not misclassify bats to the species with the greatest dispersion, the main problem caused by violation of the homogeneity assumption. The advantage of linear DFA is that it provides discriminant function coefficients. The linear combination of these coefficients and parameters from calls from unidentified bats can be used for species identification without access to the original data sets, an option not provided by quadratic analysis. When separate models were developed for Myotis species and for FM/QCF species, correct classification rates increased to 84.8% and 93.4%, respectively. DF coefficients thus provide a reliable identification tool, but intraspecific geographic variation must be taken into account.

New linear programming approaches are proposed as nonparametric procedures for multiple-class discriminant and classification analysis. A new MSD model minimizing the sum of the classification errors is formulated to construct discriminant functions. This model has desirable properties because it is versatile and is immune to the pathologies of some of the earlier mathematical programming models for two-class classification. It is also purely systematic and algorithmic and no user ad hoc and trial judgment is required. Furthermore, it can be used as the basis to develop other models, such as a multiple-class support vector machine and a mixed integer programming model, for discrimination and classification. A MMD model minimizing the maximum of the classification errors, although with very limited use, is also studied. These models may also be considered as generalizations of mathematical programming formulations for two-class classification. By the same approach, other mathematical programming formulations for two-class classification can be easily generalized to multiple-class formulations. Results on standard as well as randomly generated test datasets show that the MSD model is very effective in generating powerful discriminant functions.

The accuracy of sex determinations based on visual assessments of the mental eminence, orbital margin, glabellar area, nuchal area, and mastoid process was tested on a series of 304 skulls of known age and sex from people of European American, African American, and English ancestry as well as on an ancient Native American sample of 156 individuals whose sex could be reliably determined based on pelvic morphology. Ordinal scores of these sexually dimorphic traits were used to compute sex determination discriminant functions. Linear, kth-nearest-neighbor, logistic, and quadratic discriminant analysis models were evaluated based on their capacity to minimize both misclassifications and sex biases in classification errors. Logistic regression discriminant analysis produced the best results: a logistic model containing all five cranial trait scores correctly classified 88% of the modern skulls with a negligible sex bias of 0.1%. Adding age at death, birth year, and population affinity to the model did not appreciably improve its performance. For the ancient Native American sample, the best logistic regression model assigned the correct pelvic sex to 78% of the individuals with a sex bias of only 0.2%. Similar cranial trait frequency distributions were found in same-sex comparisons of the modern African American, European American, and English samples. The sexual dimorphism of these modern people contrasts markedly with that of the ancient Native Americans. Because of such population differences, discriminant functions like those presented in this paper should be used with caution on populations other than those for which they were developed.

In this paper, we present a convex discriminant analysis formulation, which is extended to solve multi-label classification problems. The original Linear Discriminant Analysis energy optimization function is turned into another form as a convex formulation (namely, convex Approximate LDA, denoted as “convexALDA” for short) using the generalized eigen-decomposition. We give applications by incorporating convexALDA as a regularizer into discriminant regression analysis. Extensive experimental results on multi-label classification tasks and an extensive application scenario on communication characteristics of imperial examination system are provided. In this way we have a brand-new comprehension for it, and a new idea and method was also put forward for studying the system.

We examined the relationship of size to sex, hybridity, and ploidy in five central Ontario populations of the Ambystoma laterale-jeffersonianum complex. For one extensively sampled population, we also considered time-related changes in size for each class of biotype during the breeding period. In addition, we compared the efficiency of visual and mathematical discrimination between hybrid and non- hybrid biotypes via linear discriminant function and subsequent analysis of error rates. Our results dem- onstrate that: (1) there is no significant sexual dimorphism in A. laterale; (2) there are some size differences among populations in both A. laterale and hybrids; (3) changes in mean size in all classes of biotypes are correlated with timing of breeding; (4) rates of maturation of wild polyploids are identical to those of lab-reared forms; and (5) hybrid: non-hybrid ratios can be estimated reliably for populations, either by eye or by discriminant function analysis of three simple morphological variables. In an earlier paper (Lowcock et al., 1991), we reported on the inter-relationships of ploidy structure, sex ratio, and breeding dynamics in six bisexual-unisexual communities of the Am- bystoma laterale-jeffersonianum complex of sala- manders in central Ontario, Canada. We found that unisexuals affect population structure and breeding dynamics of syntopic bisexuals, and that the biotype composition of breeding mi- grants changes over time. We suggested that similar effects may obtain for other ecological considerations. One such variable is individual

We propose a new algorithm for the allocation of an individual to one of several possible groups or populations. The algorithm enables us to define a finite partition over the sample space, based on distance function. This partition is used, jointly with the application of a standard Bayesian decision rule, to allocate individuals to the populations. The algorithm also provides a measure of the allocation confidence for each individual, in a similar manner to that of logistic regression. The error rates for classification are also computed using the leave-one-out method. Results are compared with those obtained with other discriminant analysis techniques previously reported: Fisher's linear discriminant function, the quadratic discriminant function, logistic discrimination, and others.