Ideal bootstrap estimation of expected prediction error for k-nearest neighbor classifiers: Applications for classification and error assessment

Euclidean distance k-nearest neighbor (k-NN) classifiers are simple nonparametric classification rules. Bootstrap methods, widely used for estimating the expected prediction error of classification rules, are motivated by the objective of calculating the ideal bootstrap estimate of expected prediction error. In practice, bootstrap methods use Monte Carlo resampling to estimate the ideal bootstrap estimate because exact calculation is generally intractable. In this article, we present analytical formulae for exact calculation of the ideal bootstrap estimate of expected prediction error for k-NN classifiers and propose a new weighted k-NN classifier based on resampling ideas. The resampling-weighted k-NN classifier replaces the k-NN posterior probability estimates by their expectations under resampling and predicts an unclassified covariate as belonging to the group with the largest resampling expectation. A simulation study and an application involving remotely sensed data show that the resampling-weighted k-NN classifier compares favorably to unweighted and distance-weighted k-NN classifiers.     

Forward and backward stepping in variable selection

For stepwise regression and discriminant analysis the parameters F _in and F _out govern the inclusion and deletion of variables. The candidate variable with the biggest F—ratio is included if this exceeds F _inthe included variable with the smallest F—ratio is deleted if this is less than F _in If F _in ≧F _out; then return to a previous subset size implies improvement in the criterion measure. This result also holds for a generalization, stepwise multivariate analysis, which includes stepwise regression and discriminant analysis as special cases

Eliminations do not occur if forward regression and backward elimination yield the same sequence of subsets. Conversely, there is a more liberal stepping rule which always eliminates if the two sequences differ.     

Sparse discriminant analysis based on estimation of posterior probabilities

ABSTRACT

Fisher's linear discriminant analysis (FLDA) is known as a method to find a discriminative feature space for multi-class classification. As a theory of extending FLDA to an ultimate nonlinear form, optimal nonlinear discriminant analysis (ONDA) has been proposed. ONDA indicates that the best theoretical nonlinear map for maximizing the Fisher's discriminant criterion is formulated by using the Bayesian a posterior probabilities. In addition, the theory proves that FLDA is equivalent to ONDA when the Bayesian a posterior probabilities are approximated by linear regression (LR). Due to some limitations of the linear model, there is room to modify FLDA by using stronger approximation/estimation methods. For the purpose of probability estimation, multi-nominal logistic regression (MLR) is more suitable than LR. Along this line, in this paper, we develop a nonlinear discriminant analysis (NDA) in which the posterior probabilities in ONDA are estimated by MLR. In addition, in this paper, we develop a way to introduce sparseness into discriminant analysis. By applying L1 or L2 regularization to LR or MLR, we can incorporate sparseness in FLDA and our NDA to increase generalization performance. The performance of these methods is evaluated by benchmark experiments using last_exam17 standard datasets and a face classification experiment.     

An Optimal Semiparametric Method for Two‐group Classification

In the classical discriminant analysis, when two multivariate normal distributions with equal variance–covariance matrices are assumed for two groups, the classical linear discriminant function is optimal with respect to maximizing the standardized difference between the means of two groups. However, for a typical case‐control study, the distributional assumption for the case group often needs to be relaxed in practice. Komori et al. (Generalized t ‐statistic for two‐group classification. Biometrics 2015, 71: 404–416) proposed the generalized t ‐statistic to obtain a linear discriminant function, which allows for heterogeneity of case group. Their procedure has an optimality property in the class of consideration. We perform a further study of the problem and show that additional improvement is achievable. The approach we propose does not require a parametric distributional assumption on the case group. We further show that the new estimator is efficient, in that no further improvement is possible to construct the linear discriminant function more efficiently. We conduct simulation studies and real data examples to illustrate the finite sample performance and the gain that it produces in comparison with existing methods.     

The rank transformation as a method of discrimination with some examples

The procedure of statistical discrimination Is simple in theory but so simple in practice. An observation x₀possibly uiultivariate, is to be classified into one of several populations π₁,…,π_k which have respectively, the density functions f₁(x), ? ? ? , f_k(x). The decision procedure is to evaluate each density function at X₀ to see which function gives the largest value f_i(X₀) , and then to declare that X₀ belongs to the population corresponding to the largest value. If these den-sities can be assumed to be normal with equal covariance matricesthen the decision procedure is known as Fisher’s linear discrimi-nant function (LDF) method. In the case of unequal covariance matrices the procedure is called the quadratic discriminant func-tion (QDF) method. If the densities cannot be assumed to be nor-mal then the LDF and QDF might not perform well. Several different procedures have appeared in the literature which offer discriminant procedures for nonnormal data. However, these pro-cedures are generally difficult to use and are not readily available as canned statistical programs.

Another approach to discriminant analysis is to use some sortof mathematical trans format ion on the samples so that their distribution function is approximately normal, and then use the convenient LDF and QDF methods. One transformation that:applies to all distributions equally well is the rank transformation. The result of this transformation is that a very simple and easy to use procedure is made available. This procedure is quite robust as is evidenced by comparisons of the rank transform results with several published simulation studies.     

Kernel naive Bayes discrimination for high‐dimensional pattern recognition

Kernel discriminant analysis translates the original classification problem into feature space and solves the problem with dimension and sample size interchanged. In high‐dimension low sample size (HDLSS) settings, this reduces the ‘dimension’ to that of the sample size. For HDLSS two‐class problems we modify Mika's kernel Fisher discriminant function which – in general – remains ill‐posed even in a kernel setting; see Mika et al. (1999). We propose a kernel naive Bayes discriminant function and its smoothed version, using first‐ and second‐degree polynomial kernels. For fixed sample size and increasing dimension, we present asymptotic expressions for the kernel discriminant functions, discriminant directions and for the error probability of our kernel discriminant functions. The theoretical calculations are complemented by simulations which show the convergence of the estimators to the population quantities as the dimension grows. We illustrate the performance of the new discriminant rules, which are easy to implement, on real HDLSS data. For such data, our results clearly demonstrate the superior performance of the new discriminant rules, and especially their smoothed versions, over Mika's kernel Fisher version, and typically also over the commonly used naive Bayes discriminant rule.     

Characterizing persistent disturbing behavior using longitudinal and multivariate techniques

Persistent disturbing behavior (PDB) refers to a chronic condition in therapy-resistant psychiatric patients. Since these patients are highly unstable and difficult to maintain in their natural living environment and even in hospital wards, it is important to properly characterize this group. Previous studies in the Belgian province of Limburg indicated that the size of this group was larger than anticipated. Here, using a score calculated from longitudinal psychiatric registration data in 611 patients, we characterize the difference between PDB patients and a set of control patients. These differences are studied both at a given point in time, using discriminant analysis, as well as in terms of the evolution of the score over time, using longitudinal data analysis methods. Further, using clustering techniques, the group of PDB patients is split into two subgroups, characterized in terms of a number of ordinal scores. Such findings are useful from a scientific as well as from an organizational point of view.     

A comparison of regularization methods applied to the linear discriminant function with high-dimensional microarray data

Classification of gene expression microarray data is important in the diagnosis of diseases such as cancer, but often the analysis of microarray data presents difficult challenges because the gene expression dimension is typically much larger than the sample size. Consequently, classification methods for microarray data often rely on regularization techniques to stabilize the classifier for improved classification performance. In particular, numerous regularization techniques, such as covariance-matrix regularization, are available, which, in practice, lead to a difficult choice of regularization methods. In this paper, we compare the classification performance of five covariance-matrix regularization methods applied to the linear discriminant function using two simulated high-dimensional data sets and five well-known, high-dimensional microarray data sets. In our simulation study, we found the minimum distance empirical Bayes method reported in Srivastava and Kubokawa [Comparison of discrimination methods for high dimensional data, J. Japan Statist. Soc. 37(1) (2007), pp. 123–134], and the new linear discriminant analysis reported in Thomaz, Kitani, and Gillies [A Maximum Uncertainty LDA-based approach for Limited Sample Size problems – with application to Face Recognition, J. Braz. Comput. Soc. 12(1) (2006), pp. 1–12], to perform consistently well and often outperform three other prominent regularization methods. Finally, we conclude with some recommendations for practitioners.     

Discriminant analyses of peanut allergy severity scores

Peanut allergy is one of the most prevalent food allergies. The possibility of a lethal accidental exposure and the persistence of the disease make it a public health problem. Evaluating the intensity of symptoms is accomplished with a double blind placebo-controlled food challenge (DBPCFC), which scores the severity of reactions and measures the dose of peanut that elicits the first reaction. Since DBPCFC can result in life-threatening responses, we propose an alternate procedure with the long-term goal of replacing invasive allergy tests. Discriminant analyses of DBPCFC score, the eliciting dose and the first accidental exposure score were performed in 76 allergic patients using 6 immunoassays and 28 skin prick tests. A multiple factorial analysis was performed to assign equal weights to both groups of variables, and predictive models were built by cross-validation with linear discriminant analysis, k-nearest neighbours, classification and regression trees, penalized support vector machine, stepwise logistic regression and AdaBoost methods. We developed an algorithm for simultaneously clustering eliciting dose values and selecting discriminant variables. Our main conclusion is that antibody measurements offer information on the allergy severity, especially those directed against rAra-h1 and rAra-h3. Further independent validation of these results and the use of new predictors will help extend this study to clinical practices.     

Time Series Classification Based on Spectral Analysis

For time series data with obvious periodicity (e.g., electric motor systems and cardiac monitor) or vague periodicity (e.g., earthquake and explosion, speech, and stock data), frequency-based techniques using the spectral analysis can usually capture the features of the series. By this approach, we are able not only to reduce the data dimensions into frequency domain but also utilize these frequencies by general classification methods such as linear discriminant analysis (LDA) and k-nearest-neighbor (KNN) to classify the time series. This is a combination of two classical approaches. However, there is a difficulty in using LDA and KNN in frequency domain due to excessive dimensions of data. We overcome the obstacle by using Singular Value Decomposition to select essential frequencies. Two data sets are used to illustrate our approach. The classification error rates of our simple approach are comparable to those of several more complicated methods.     

Discriminant coordinates analysis for multivariate functional data

Abstract

One of the basic statistical methods of dimensionality reduction is analysis of discriminant coordinates given by Fisher (1936 Fisher, R. A. 1936. The use of multiple measurements in taxonomic problem. Annals of Eugenics 7 (2):179–88. doi:10.1111/j.1469-1809.1936.tb02137.x.[Crossref] , [Google Scholar]) and Rao (1948). The space of discriminant coordinates is a space convenient for presenting multidimensional data originating from multiple groups and for the use of various classification methods (methods of discriminant analysis). In the present paper, we adapt the classical discriminant coordinates analysis to multivariate functional data. The theory has been applied to analysis of textural properties of apples of six varieties, measured over a period of 180?days, stored in two types of refrigeration chamber.     

An asymptotic approximation for EPMC in linear discriminant analysis based on monotone missing data

In this paper, we propose an asymptotic approximation for the expected probabilities of misclassification (EPMC) in the linear discriminant function on the basis of k-step monotone missing training data for general k. We derive certain relations of the statistics in order to obtain the approximation. Finally, we perform Monte Carlo simulation to evaluate the accuracy of our result and to compare it with existing approximations.     

A classification updating procedure motivated by high-content screening data

The current paradigm for the identification of candidate drugs within the pharmaceutical industry typically involves the use of high-throughput screens. High-content screening (HCS) is the term given to the process of using an imaging platform to screen large numbers of compounds for some desirable biological activity. Classification methods have important applications in HCS experiments, where they are used to predict which compounds have the potential to be developed into new drugs. In this paper, a new classification method is proposed for batches of compounds where the rule is updated sequentially using information from the classification of previous batches. This methodology accounts for the possibility that the training data are not a representative sample of the test data and that the underlying group distributions may change as new compounds are analysed. This technique is illustrated on an example data set using linear discriminant analysis, k-nearest neighbour and random forest classifiers. Random forests are shown to be superior to the other classifiers and are further improved by the additional updating algorithm in terms of an increase in the number of true positives as well as a decrease in the number of false positives.     

Optimal design for classification of functional data

We study the design problem for the optimal classification of functional data. The goal is to select sampling time points so that functional data observed at these time points can be classified accurately. We propose optimal designs that are applicable to either dense or sparse functional data. Using linear discriminant analysis, we formulate our design objectives as explicit functions of the sampling points. We study the theoretical properties of the proposed design objectives and provide a practical implementation. The performance of the proposed design is evaluated through simulations and real data applications. The Canadian Journal of Statistics 48: 285–307; 2020 © 2019 Statistical Society of Canada     

Estimation of relaxation time distributions in magnetic resonance imaging

Magnetic resonance imaging techniques can be used to measure some biophysical properties of tissue. In this context, the T₂ relaxation time is an important parameter for soft‐tissue contrast. The authors develop a new technique to estimate the integral of the distribution of T₂ relaxation time without imposing any constraint other than the monotonicity of the underlying cumulative relaxation time distribution. They explore the properties of the estimation and its applications for the analysis of breast tissue data. As they show, an extension of linear discriminant analysis is found to distinguish well between two classes of breast tissue.     

A study on discriminant analysis techniques applied to multivariate lognormal data

The purpose of this paper is to examine the multiple group (>2) discrimination problem in which the group sizes are unequal and the variables used in the classification are correlated with skewed distributions. Using statistical simulation based on data from a clinical study, we compare the performances, in terms of misclassification rates, of nine statistical discrimination methods. These methods are linear and quadratic discriminant analysis applied to untransformed data, rank transformed data, and inverse normal scores data, as well as fixed kernel discriminant analysis, variable kernel discriminant analysis, and variable kernel discriminant analysis applied to inverse normal scores data. It is found that the parametric methods with transformed data generally outperform the other methods, and the parametric methods applied to inverse normal scores usually outperform the parametric methods applied to rank transformed data. Although the kernel methods often have very biased estimates, the variable kernel method applied to inverse normal scores data provides considerable improvement in terms of total nonerror rate.     

The Dantzig Discriminant Analysis with High Dimensional Data

It is well known that linear discriminant analysis (LDA) works well and is asymptotically optimal under fixed-p-large-n situations. But Bickel and Levina (2004 Bickel, P.J., Levina, E. (2004). Some theory for Fishers linear discriminant function, naive Bayes, and some alternatives when there are many more variables than observations. Bernoulli 10:989–1010.[Crossref], [Web of Science ®] , [Google Scholar]) showed that the LDA is as bad as random guessing when p > n. This article studies the sparse discriminant analysis via Dantzig penalized least squares. Our method avoids estimating the high-dimensional covariance matrix and does not need the sparsity assumption on the inverse of the covariance matrix. We show that the new discriminant analysis is asymptotically optimal theoretically. Simulation and real data studies show that the classifier performs better than the existing sparse methods.     

判别分析统计检验体系的探讨

判别分析已越来越受到人们的重视并取得了重要的应用成果，但应用中存在着简单套用的情况，对判别分析的适用性、判别效果的显著性、判别变量的判别能力以及判别函数的判别能力的检验等问题重视不够。为了更好地应用判别分析，就应对判别分析进行统计检验并建立统计检验体系，统计检验体系应包括：判别分析适用性检验，判别效果显著性检验，判别变量的判别能力检验和判别函数的判别能力检验。     

Comparison of algorithms for replacing missing data in discriminant analysis

We examined the impact of different methods for replacing missing data in discriminant analyses conducted on randomly generated samples from multivariate normal and non-normal distributions. The probabilities of correct classification were obtained for these discriminant analyses before and after randomly deleting data as well as after deleted data were replaced using: (1) variable means, (2) principal component projections, and (3) the EM algorithm. Populations compared were: (1) multivariate normal with covariance matrices ∑₁=∑₂, (2) multivariate normal with ∑₁≠∑₂ and (3) multivariate non-normal with ∑₁=∑₂. Differences in the probabilities of correct classification were most evident for populations with small Mahalanobis distances or high proportions of missing data. The three replacement methods performed similarly but all were better than non - replacement.     

Predicting early educational program placement with discrete discriminant analysis

The purpose of this study was to predict placement and nonplacement outcomes for mildly handicapped three through five year old children given knowledge of developmental screening test data. Discrete discriminant analysis (Anderson, 1951; Cochran & Hopkins, 1961; Goldstein & Dillon, 1978) was used to classify children into either a placement or nonplacement group using developmental information retrieved from longitudinal Child Find records (1982-89). These records were located at the Florida Diagnostic and Learning Resource System (FDLRS) in Sarasota, Florida and provided usable data for 602 children. The developmental variables included performance on screening test activities from the Comprehensive Identification Process (Zehrbach, 1975), and consisted of: (a) gross motor skills, (b) expressive language skills, and (c) social-emotional skills. These three dichotomously scored developmental variables generated eight mutually exclusive and exhaustive combinations of screening data. Combined with one of three different types of cost-of-misclassification functions, each child in a random cross-validation sample of 100 was classified into one of the two outcome groups minimizing the expected cost of misclassification based on the remaining 502 children. For each cost function designed by the researchers a comparison was made between classifications from the discrete discriminant analysis procedure and actual placement outcomes for the 100 children. A logit analysis and a standard discriminant analysis were likewise conducted using the 502 children and compared with results of the discrete discriminant analysis for selected cost functions.