The Geometry of Nonlinear Embeddings in Kernel Discriminant Analysis

Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a nonlinear feature mapping. We study the geometry of nonlinear embeddings in discriminant analysis with polynomial kernels and Gaussian kernel by identifying the population-level discriminant function that depends on the data distribution and the kernel. In order to obtain the discriminant function, we solve a generalized eigenvalue problem with between-class and within-class covariance operators. The polynomial discriminants are shown to capture the class difference through the population moments explicitly. For approximation of the Gaussian discriminant, we use a particular representation of the Gaussian kernel by utilizing the exponential generating function for Hermite polynomials. We also show that the Gaussian discriminant can be approximated using randomized projections of the data. Our results illuminate how the data distribution and the kernel interact in determination of the nonlinear embedding for discrimination, and provide a guideline for choice of the kernel and its parameters

Discriminant analysis under the common principal components model

For two or more populations of which the covariance matrices have a common set of eigenvectors, but different sets of eigenvalues, the common principal components (CPC) model is appropriate. Pepler et al. (2015) proposed a regularised CPC covariance matrix estimator and showed that this estimator outperforms the unbiased and pooled estimators in situations where the CPC model is applicable. This paper extends their work to the context of discriminant analysis for two groups, by plugging the regularised CPC estimator into the ordinary quadratic discriminant function. Monte Carlo simulation results show that CPC discriminant analysis offers significant improvements in misclassification error rates in certain situations, and at worst performs similar to ordinary quadratic and linear discriminant analysis. Based on these results, CPC discriminant analysis is recommended for situations where the sample size is small compared to the number of variables, in particular for cases where there is uncertainty about the population covariance matrix structures

Influence of observations on the misclassification probability in quadratic discriminant analysis.

In this paper it is analyzed how observations in the training sample affect the misclassification probability of a quadratic discriminant rule. An approach based on partial influence functions is followed. It allows to quantify the effect of observations in the training sample on the quality of the associated classification rule. Focus is more on the effect on the future misclassification rate, than on the influence on the parameters of the quadratic discriminant rule. The expression for the influence function is then used to construct a diagnostic tool for detecting influential observations. Applications on real data sets are provided.Applications; Classification; Data; Diagnostics; Discriminant analysis; Functions; Influence function; Misclassification probability; Outliers; Partial influence functions; Probability; Quadratic discriminant analysis; Quality; Robust covariance estimation; Robust regression; Training;

Influence of observations on the misclassification probability in quadratic discriminant analysis.

In this paper it is studied how observations in the training sample affect the misclassification probability of a quadratic discriminant rule. An approach based on partial influence functions is followed. It allows to quantify the effect of observations in the training sample on the performance of the associated classification rule. Focus is on the effect of outliers on the misclassification rate, merely than on the estimates of the parameters of the quadratic discriminant rule. The expression for the partial influence function is then used to construct a diagnostic tool for detecting influential observations. Applications on real data sets are provided.Applications; Classification; Data; Diagnostics; Discriminant analysis; Estimator; Functions; Influence function; Misclassification probability; Outliers; Parameters; Partial influence functions; Performance; Principal components; Probability; Quadratic discriminant analysis; Tool; Training;

Analyze of Classification Accaptence Subsidy Food Using Kernel Discriminant

Subsidy food is government program for social protection to poor households. The aims of this program are to effort households from starve and to decrease poverty. Less precisely target of this program has negative impact. So that to successful program, it’s important to know accuracy classification of admission subsidy food. The variables classification are number of household members, number of household member in work, average expenditure capita, weighted household, and floor area. Discriminant analysis is a multivariate statistical technique which can be used to classify the new observation into a specific group. Kernel discriminant analysis is a non-parametric method which is flexible because it does not have to concern about assumption from certain distribution and equal variance matrices as in parametric discriminant analysis. The classification using the kernel discriminant analysis with the normal kernel function with optimum bandwidth 0.6 gives accurate classification 75.35%

Classification of Several Skin Cancer Types Based on Autofluorescence Intensity of Visible Light to Near Infrared Ratio

Skin cancer is a Malignant growth on the skin caused by many factors. The most common skin cancers are Basal Cell Cancer (BCC) and Squamous Cell Cancer (SCC). This research uses a discriminant analysis to classify some tissues of skin cancer based on criterion number of independent variables. An independent variable is variation of excitation light sources (LED lamp), filters, and sensors to measure autofluorescence intensity (IAF) of visible light to near infrared (VIS/NIR) ratio of paraffin embedded tissue biopsy from BCC, SCC, and Lipoma. From the result of discriminant analysis, it is known that the discriminant function is determined by 4 (four) independent variables i.e., blue LED-red filter, blue LED-yellow filter, UV LED-blue filter, and UV LED-yellow filter. The accuracy of discriminant in classifying the analysis of three skin cancer tissues is 100%

The Discriminant Analysis Used by the IRS to Predict Profitable Individual Tax Return Audits

This paper discusses past and current methods the IRS uses to determine which individual income tax returns to audit. The IRS currently uses the discriminant function to give all individual tax returns two scores; one based on whether it should be audited or not and one based on if the return is likely to have unreported income. The discriminant function is determined by the IRS’s National Research Program, which takes a sample of returns and ensures their accuracy. Previously, the function was determined by the IRS’s Taxpayer Compliance Measurement Program. However, this was too burdensome and time consuming for taxpayers. The data mining techniques of decision trees, regression, and neural networks were researched to determine if the IRS should change its method. Unfortunately IRS tax data were not obtainable due to their confidentiality; therefore credit data from a German bank was used to compare discriminant analysis results to the three new methods. All of the methods were run to predict creditworthiness and were compared based on misclassification rates. The neural network had the best classification rate closely followed by regression, the decision tree, and then discriminant analysis. Since this comparison is not based on IRS tax data, no conclusion can be made whether the IRS should change its method or not, but because all methods had very close classification rates, it would be worthwhile for the IRS to look into them

Discriminant analysis – simplified

Background: Discriminant function analysis is the statistical analysis used to analyze data when the dependent variable or outcome is categorical and independent variable or predictor variable is parametric. Discriminant function analysis is used to find out the accuracy of a given classification system or predictor variable in predicting the sample into a particular group. Discriminant function analysis includes the development of discriminant functions for each sample and deriving a cutoff score. The cutoff score is used for classifying the samples into different groups. Aim: The aim of this review article is to simplify and explain the discriminant function analysis so that it can be used by medical and dental researchers whenever it is applicable. Conclusion: Discriminant function analysis is a statistical analysis used to find out the accuracy of a given classification system or predictor variables. This paper explains the basics of discriminant analysis and how to interpret the results along with one simple example of mandibular canine index for gender identification. Clinical significance: Whenever a new classification system is introduced or any predictor variable is identified, discriminant function analysis can be used to find out the accuracy with which the classification system or predictor variable can differentiate a sample into different groups. Thus, it is a very useful tool in dental and medical research

Discriminant Analysis: An Analysis of Its Predictship Function

Discriminant analysis as a multivariate statistical method was reviewed in this paper.  Alongside this was the general goal of discriminate analysis which is to predict membership from a set of predictor as well as to classify individuals into one of two or more alternative groups on the basis of a set of measurements. Discriminant analysis builds a predictive model for group membership. This model is composed of a discriminant function (or, for more than two groups, a set of discriminant functions). The discriminate function coefficients gives the contribution of each variable to the function. In order to derive more substantive "meaningful" labels for the discriminant functions, one can also examine the factor structure matrix with the correlations between the variables and the discriminant functions as well as other statistics associated with discriminate analysis. Keywords: Discriminate analysis, discriminate function, prediction, classification discriminate function coefficients. DOI: 10.7176/JEP/10-5-0