Corrected small-sample estimation of the Bayes error

MOTIVATION: A major problem of pattern classification is estimation of the Bayes error when only small samples are available. One way to estimate the Bayes error is to design a classifier based on some classification rule applied to sample data, estimate the error of the designed classifier, and then use this estimate as an estimate of the Bayes error. Relative to the Bayes error, the expected error of the designed classifier is biased high, and this bias can be severe with small samples. RESULTS: This paper provides a correction for the bias by subtracting a term derived from the representation of the estimation error. It does so for Boolean classifiers, these being defined on binary features. Although the general theory applies to any Boolean classifier, a model is introduced to reduce the number of parameters. A key point is that the expected correction is conservative. Properties of the corrected estimate are studied via simulation. The correction applies to binary predictors because they are mathematically identical to Boolean classifiers. In this context the correction is adapted to the coefficient of determination, which has been used to measure nonlinear multivariate relations between genes and design genetic regulatory networks. An application using gene-expression data from a microarray experiment is provided on the website http://gspsnap.tamu.edu/smallsample/ (user:'smallsample', password:'smallsample)').     

Small sample issues for microarray-based classification

In order to study the molecular biological differences between normal and diseased tissues, it is desirable to perform classification among diseases and stages of disease using microarray-based gene-expression values. Owing to the limited number of microarrays typically used in these studies, serious issues arise with respect to the design, performance and analysis of classifiers based on microarray data. This paper reviews some fundamental issues facing small-sample classification: classification rules, constrained classifiers, error estimation and feature selection. It discusses both unconstrained and constrained classifier design from sample data, and the contributions to classifier error from constrained optimization and lack of optimality owing to design from sample data. The difficulty with estimating classifier error when confined to small samples is addressed, particularly estimating the error from training data. The impact of small samples on the ability to include more than a few variables as classifier features is explained.     

What should be expected from feature selection in small-sample settings

MOTIVATION: High-throughput technologies for rapid measurement of vast numbers of biological variables offer the potential for highly discriminatory diagnosis and prognosis; however, high dimensionality together with small samples creates the need for feature selection, while at the same time making feature-selection algorithms less reliable. Feature selection must typically be carried out from among thousands of gene-expression features and in the context of a small sample (small number of microarrays). Two basic questions arise: (1) Can one expect feature selection to yield a feature set whose error is close to that of an optimal feature set? (2) If a good feature set is not found, should it be expected that good feature sets do not exist? RESULTS: The two questions translate quantitatively into questions concerning conditional expectation. (1) Given the error of an optimal feature set, what is the conditionally expected error of the selected feature set? (2) Given the error of the selected feature set, what is the conditionally expected error of the optimal feature set? We address these questions using three classification rules (linear discriminant analysis, linear support vector machine and k-nearest-neighbor classification) and feature selection via sequential floating forward search and the t-test. We consider three feature-label models and patient data from a study concerning survival prognosis for breast cancer. With regard to the two focus questions, there is similarity across all experiments: (1) One cannot expect to find a feature set whose error is close to optimal, and (2) the inability to find a good feature set should not lead to the conclusion that good feature sets do not exist. In practice, the latter conclusion may be more immediately relevant, since when faced with the common occurrence that a feature set discovered from the data does not give satisfactory results, the experimenter can draw no conclusions regarding the existence or nonexistence of suitable feature sets. AVAILABILITY: http://ee.tamu.edu/~edward/feature_regression/     

Strong feature sets from small samples.

For small samples, classifier design algorithms typically suffer from overfitting. Given a set of features, a classifier must be designed and its error estimated. For small samples, an error estimator may be unbiased but, owing to a large variance, often give very optimistic estimates. This paper proposes mitigating the small-sample problem by designing classifiers from a probability distribution resulting from spreading the mass of the sample points to make classification more difficult, while maintaining sample geometry. The algorithm is parameterized by the variance of the spreading distribution. By increasing the spread, the algorithm finds gene sets whose classification accuracy remains strong relative to greater spreading of the sample. The error gives a measure of the strength of the feature set as a function of the spread. The algorithm yields feature sets that can distinguish the two classes, not only for the sample data, but for distributions spread beyond the sample data. For linear classifiers, the topic of the present paper, the classifiers are derived analytically from the model, thereby providing an enormous savings in computation time. The algorithm is applied to cancer classification via cDNA microarrays. In particular, the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the algorithm is used to find gene sets whose expressions can be used to classify BRCA1 and BRCA2 tumors.     

Incorporating prior knowledge induced from stochastic differential equations in the classification of stochastic observations

In classification, prior knowledge is incorporated in a Bayesian framework by assuming that the feature-label distribution belongs to an uncertainty class of feature-label distributions governed by a prior distribution. A posterior distribution is then derived from the prior and the sample data. An optimal Bayesian classifier (OBC) minimizes the expected misclassification error relative to the posterior distribution. From an application perspective, prior construction is critical. The prior distribution is formed by mapping a set of mathematical relations among the features and labels, the prior knowledge, into a distribution governing the probability mass across the uncertainty class. In this paper, we consider prior knowledge in the form of stochastic differential equations (SDEs). We consider a vector SDE in integral form involving a drift vector and dispersion matrix. Having constructed the prior, we develop the optimal Bayesian classifier between two models and examine, via synthetic experiments, the effects of uncertainty in the drift vector and dispersion matrix. We apply the theory to a set of SDEs for the purpose of differentiating the evolutionary history between two species.     

The ties problem resulting from counting-based error estimators and its impact on gene selection algorithms

MOTIVATION: Feature selection approaches, such as filter and wrapper, have been applied to address the gene selection problem in the literature of microarray data analysis. In wrapper methods, the classification error is usually used as the evaluation criterion of feature subsets. Due to the nature of high dimensionality and small sample size of microarray data, however, counting-based error estimation may not necessarily be an ideal criterion for gene selection problem. RESULTS: Our study reveals that evaluating genes in terms of counting-based error estimators such as resubstitution error, leave-one-out error, cross-validation error and bootstrap error may encounter severe ties problem, i.e. two or more gene subsets score equally, and this in turn results in uncertainty in gene selection. Our analysis finds that the ties problem is caused by the discrete nature of counting-based error estimators and could be avoided by using continuous evaluation criteria instead. Experiment results show that continuous evaluation criteria such as generalised the absolute value of w2 measure for support vector machines and modified Relief's measure for k-nearest neighbors produce improved gene selection compared with counting-based error estimators. AVAILABILITY: The companion website is at http://www.ntu.edu.sg/home5/pg02776030/wrappers/ The website contains (1) the source code of all the gene selection algorithms and (2) the complete set of tables and figures of experiments.     

The illusion of distribution-free small-sample classification in genomics

Classification has emerged as a major area of investigation in bioinformatics owing to the desire to discriminate phenotypes, in particular, disease conditions, using high-throughput genomic data. While many classification rules have been posed, there is a paucity of error estimation rules and an even greater paucity of theory concerning error estimation accuracy. This is problematic because the worth of a classifier depends mainly on its error rate. It is common place in bio-informatics papers to have a classification rule applied to a small labeled data set and the error of the resulting classifier be estimated on the same data set, most often via cross-validation, without any assumptions being made on the underlying feature-label distribution. Concomitant with a lack of distributional assumptions is the absence of any statement regarding the accuracy of the error estimate. Without such a measure of accuracy, the most common one being the root-mean-square (RMS), the error estimate is essentially meaningless and the worth of the entire paper is questionable. The concomitance of an absence of distributional assumptions and of a measure of error estimation accuracy is assured in small-sample settings because even when distribution-free bounds exist (and that is rare), the sample sizes required under the bounds are so large as to make them useless for small samples. Thus, distributional bounds are necessary and the distributional assumptions need to be stated. Owing to the epistemological dependence of classifiers on the accuracy of their estimated errors, scientifically meaningful distribution-free classification in high-throughput, small-sample biology is an illusion.     

Probabilistic classifiers with high-dimensional data

For medical classification problems, it is often desirable to have a probability associated with each class. Probabilistic classifiers have received relatively little attention for small n large p classification problems despite of their importance in medical decision making. In this paper, we introduce 2 criteria for assessment of probabilistic classifiers: well-calibratedness and refinement and develop corresponding evaluation measures. We evaluated several published high-dimensional probabilistic classifiers and developed 2 extensions of the Bayesian compound covariate classifier. Based on simulation studies and analysis of gene expression microarray data, we found that proper probabilistic classification is more difficult than deterministic classification. It is important to ensure that a probabilistic classifier is well calibrated or at least not "anticonservative" using the methods developed here. We provide this evaluation for several probabilistic classifiers and also evaluate their refinement as a function of sample size under weak and strong signal conditions. We also present a cross-validation method for evaluating the calibration and refinement of any probabilistic classifier on any data set.     

Improved prediction of fungal effector proteins from secretomes with EffectorP 2.0

Plant‐pathogenic fungi secrete effector proteins to facilitate infection. We describe extensive improvements to EffectorP, the first machine learning classifier for fungal effector prediction. EffectorP 2.0 is now trained on a larger set of effectors and utilizes a different approach based on an ensemble of classifiers trained on different subsets of negative data, offering different views on classification. EffectorP 2.0 achieves an accuracy of 89%, compared with 82% for EffectorP 1.0 and 59.8% for a small size classifier. Important features for effector prediction appear to be protein size, protein net charge as well as the amino acids serine and cysteine. EffectorP 2.0 decreases the number of predicted effectors in secretomes of fungal plant symbionts and saprophytes by 40% when compared with EffectorP 1.0. However, EffectorP 1.0 retains value, and combining EffectorP 1.0 and 2.0 results in a stringent classifier with a low false positive rate of 9%. EffectorP 2.0 predicts significant enrichments of effectors in 12 of 13 sets of infection‐induced proteins from diverse fungal pathogens, whereas a small cysteine‐rich classifier detects enrichment in only seven of 13. EffectorP 2.0 will fast track the prioritization of high‐confidence effector candidates for functional validation and aid in improving our understanding of effector biology. EffectorP 2.0 is available at http://effectorp.csiro.au .     

Feature selection based on empirical-risk function to detect lesions in vascular computed tomography

ObjectiveThe overall goal of the study is to detect coronary artery lesions regardless their nature, calcified or hypo-dense. To avoid explicit modelling of heterogeneous lesions, we adopted an approach based on machine learning and using unsupervised or semi-supervised classifiers. The success of the classifiers based on machine learning strongly depends on the appropriate choice of features differentiating between lesions and regular appearance. The specific goal of this article is to propose a novel strategy devised to select the best feature set for the classifiers used, out of a given set of candidate features.Materials and methodsThe features are calculated in image planes orthogonal to the artery centerline, and the classifier assigns to each of these cross-sections a label “healthy” or “diseased”. The contribution of this article is a feature-selection strategy based on the empirical risk function that is used as a criterion in the initial feature ranking and in the selection process itself. We have assessed this strategy in association with two classifiers based on the density-level detection approach that seeks outliers from the distribution corresponding to the regular appearance. The method was evaluated using a total of 13,687 cross-sections extracted from 53 coronary arteries in 15 patients.ResultsUsing the feature subset selected by the risk-based strategy, balanced error rates achieved by the unsupervised and semi-supervised classifiers respectively were equal to 13.5% and 15.4%. These results were substantially better than the rates achieved using feature subsets selected by supervised strategies. The unsupervised and semi-supervised methods also outperformed supervised classifiers using feature subsets selected by the corresponding supervised strategies.DiscussionSupervised methods require large data sets annotated by experts, both to select the features and to train the classifiers, and collecting these annotations is time-consuming. With these methods, lesions whose appearance differs from the training data may remain undetected. Lesion-detection problem is highly imbalanced, since healthy cross-sections usually are much more numerous than the diseased ones. Training the classifiers based on the density-level detection approach needs a small number of annotations or no annotations at all. The same annotations are sufficient to compute the empirical risk and to perform the selection. Therefore, our strategy associated with an unsupervised or semi-supervised classifier requires a considerably smaller number of annotations as compared to conventional supervised selection strategies. The approach proposed is also better suited for highly imbalanced problems and can detect lesions differing from the training set.ConclusionThe risk-based selection strategy, associated with classifiers using the density-level detection approach, outperformed other strategies and classifiers when used to detect coronary artery lesions. It is well suited for highly imbalanced problems, where the lesions are represented as low-density regions of the feature space, and it can be used in other anomaly detection problems interpretable as a binary classification problem where the empirical risk can be calculated.     

A Bayesian method for comparing and combining binary classifiers in the absence of a gold standard

ABSTRACT: BACKGROUND: Many problems in bioinformatics involve classification based on features such as sequence, structure or morphology. Given multiple classifiers, two crucial questions arise: how does their performance compare, and how can they best be combined to produce a better classifier? A classifier can be evaluated in terms of sensitivity and specificity using benchmark, or gold standard, data, that is, data for which the true classification is known. However, a gold standard is not always available. Here we demonstrate that a Bayesian model for comparing medical diagnostics without a gold standard can be successfully applied in the bioinformatics domain, to genomic scale data sets. We present a new implementation, which unlike previous implementations is applicable to any number of classifiers. We apply this model, for the first time, to the problem of finding the globally optimal logical combination of classifiers. RESULTS: We compared three classifiers of protein subcellular localisation, and evaluated our estimates of sensitivity and specificity against estimates obtained using a gold standard. The method overestimated sensitivity and specificity with only a small discrepancy, and correctly ranked the classifiers. Diagnostic tests for swine flu were then compared on a small data set. Lastly, classifiers for a genome-wide association study of macular degeneration with 541094 SNPs were analysed. In all cases, run times were feasible, and results precise. The optimal logical combination of classifiers was also determined for all three data sets. Code and data are available from http://bioinformatics.monash.edu.au/downloads/. CONCLUSIONS: The examples demonstrate the methods are suitable for both small and large data sets, applicable to the wide range of bioinformatics classification problems, and robust to dependence between classifiers. In all three test cases, the globally optimal logical combination of the classifiers was found to be their union, according to three out of four ranking criteria. We propose as a general rule of thumb that the union of classifiers will be close to optimal.     

Optimality driven nearest centroid classification from genomic data

Nearest-centroid classifiers have recently been successfully employed in high-dimensional applications, such as in genomics. A necessary step when building a classifier for high-dimensional data is feature selection. Feature selection is frequently carried out by computing univariate scores for each feature individually, without consideration for how a subset of features performs as a whole. We introduce a new feature selection approach for high-dimensional nearest centroid classifiers that instead is based on the theoretically optimal choice of a given number of features, which we determine directly here. This allows us to develop a new greedy algorithm to estimate this optimal nearest-centroid classifier with a given number of features. In addition, whereas the centroids are usually formed from maximum likelihood estimates, we investigate the applicability of high-dimensional shrinkage estimates of centroids. We apply the proposed method to clinical classification based on gene-expression microarrays, demonstrating that the proposed method can outperform existing nearest centroid classifiers.     

Learning cluster-based classification systems with ant colony optimization algorithms

Classification is a data mining task the goal of which is to learn a model, from a training dataset, that can predict the class of a new data instance, while clustering aims to discover natural instance-groupings within a given dataset. Learning cluster-based classification systems involves partitioning a training set into data subsets (clusters) and building a local classification model for each data cluster. The class of a new instance is predicted by first assigning the instance to its nearest cluster and then using that cluster’s local classification model to predict the instance’s class. In this paper, we present an ant colony optimization (ACO) approach to building cluster-based classification systems. Our ACO approach optimizes the number of clusters, the positioning of the clusters, and the choice of classification algorithm to use as the local classifier for each cluster. We also present an ensemble approach that allows the system to decide on the class of a given instance by considering the predictions of all local classifiers, employing a weighted voting mechanism based on the fuzzy degree of membership in each cluster. Our experimental evaluation employs five widely used classification algorithms: naïve Bayes, nearest neighbour, Ripper, C4.5, and support vector machines, and results are reported on a suite of 54 popular UCI benchmark datasets.     

Predicting Classifier Performance with Limited Training Data: Applications to Computer-Aided Diagnosis in Breast and Prostate Cancer

Clinical trials increasingly employ medical imaging data in conjunction with supervised classifiers, where the latter require large amounts of training data to accurately model the system. Yet, a classifier selected at the start of the trial based on smaller and more accessible datasets may yield inaccurate and unstable classification performance. In this paper, we aim to address two common concerns in classifier selection for clinical trials: (1) predicting expected classifier performance for large datasets based on error rates calculated from smaller datasets and (2) the selection of appropriate classifiers based on expected performance for larger datasets. We present a framework for comparative evaluation of classifiers using only limited amounts of training data by using random repeated sampling (RRS) in conjunction with a cross-validation sampling strategy. Extrapolated error rates are subsequently validated via comparison with leave-one-out cross-validation performed on a larger dataset. The ability to predict error rates as dataset size increases is demonstrated on both synthetic data as well as three different computational imaging tasks: detecting cancerous image regions in prostate histopathology, differentiating high and low grade cancer in breast histopathology, and detecting cancerous metavoxels in prostate magnetic resonance spectroscopy. For each task, the relationships between 3 distinct classifiers (k-nearest neighbor, naive Bayes, Support Vector Machine) are explored. Further quantitative evaluation in terms of interquartile range (IQR) suggests that our approach consistently yields error rates with lower variability (mean IQRs of 0.0070, 0.0127, and 0.0140) than a traditional RRS approach (mean IQRs of 0.0297, 0.0779, and 0.305) that does not employ cross-validation sampling for all three datasets.     

Optimal approach for classification of acute leukemia subtypes based on gene expression data

The classification of cancer subtypes, which is critical for successful treatment, has been studied extensively with the use of gene expression profiles from oligonucleotide chips or cDNA microarrays. Various pattern recognition methods have been successfully applied to gene expression data. However, these methods are not optimal, rather they are high-performance classifiers that emphasize only classification accuracy. In this paper, we propose an approach for the construction of the optimal linear classifier using gene expression data. Two linear classification methods, linear discriminant analysis (LDA) and discriminant partial least-squares (DPLS), are applied to distinguish acute leukemia subtypes. These methods are shown to give satisfactory accuracy. Moreover, we determined optimally the number of genes participating in the classification (a remarkably small number compared to previous results) on the basis of the statistical significance test. Thus, the proposed method constructs the optimal classifier that is composed of a small size predictor and provides high accuracy.     

An evolutionary approach for gene selection and classification of microarray data based on SVM error-bound theories

Microarrays have thousands to tens-of-thousands of gene features, but only a few hundred patient samples are available. The fundamental problem in microarray data analysis is identifying genes whose disruption causes congenital or acquired disease in humans. In this paper, we propose a new evolutionary method that can efficiently select a subset of potentially informative genes for support vector machine (SVM) classifiers. The proposed evolutionary method uses SVM with a given subset of gene features to evaluate the fitness function, and new subsets of features are selected based on the estimates of generalization error of SVMs and frequency of occurrence of the features in the evolutionary approach. Thus, in theory, selected genes reflect to some extent the generalization performance of SVM classifiers. We compare our proposed method with several existing methods and find that the proposed method can obtain better classification accuracy with a smaller number of selected genes than the existing methods.     

LS Bound based gene selection for DNA microarray data

MOTIVATION: One problem with discriminant analysis of DNA microarray data is that each sample is represented by quite a large number of genes, and many of them are irrelevant, insignificant or redundant to the discriminant problem at hand. Methods for selecting important genes are, therefore, of much significance in microarray data analysis. In the present study, a new criterion, called LS Bound measure, is proposed to address the gene selection problem. The LS Bound measure is derived from leave-one-out procedure of LS-SVMs (least squares support vector machines), and as the upper bound for leave-one-out classification results it reflects to some extent the generalization performance of gene subsets. RESULTS: We applied this LS Bound measure for gene selection on two benchmark microarray datasets: colon cancer and leukemia. We also compared the LS Bound measure with other evaluation criteria, including the well-known Fisher's ratio and Mahalanobis class separability measure, and other published gene selection algorithms, including Weighting factor and SVM Recursive Feature Elimination. The strength of the LS Bound measure is that it provides gene subsets leading to more accurate classification results than the filter method while its computational complexity is at the level of the filter method. AVAILABILITY: A companion website can be accessed at http://www.ntu.edu.sg/home5/pg02776030/lsbound/. The website contains: (1) the source code of the gene selection algorithm; (2) the complete set of tables and figures regarding the experimental study; (3) proof of the inequality (9). CONTACT: ekzmao@ntu.edu.sg.     

Feature Selection for Wearable Smartphone-Based Human Activity Recognition with Able bodied,Elderly, and Stroke Patients

Human activity recognition (HAR), using wearable sensors, is a growing area with the potential to provide valuable information on patient mobility to rehabilitation specialists. Smartphones with accelerometer and gyroscope sensors are a convenient, minimally invasive, and low cost approach for mobility monitoring. HAR systems typically pre-process raw signals, segment the signals, and then extract features to be used in a classifier. Feature selection is a crucial step in the process to reduce potentially large data dimensionality and provide viable parameters to enable activity classification. Most HAR systems are customized to an individual research group, including a unique data set, classes, algorithms, and signal features. These data sets are obtained predominantly from able-bodied participants. In this paper, smartphone accelerometer and gyroscope sensor data were collected from populations that can benefit from human activity recognition: able-bodied, elderly, and stroke patients. Data from a consecutive sequence of 41 mobility tasks (18 different tasks) were collected for a total of 44 participants. Seventy-six signal features were calculated and subsets of these features were selected using three filter-based, classifier-independent, feature selection methods (Relief-F, Correlation-based Feature Selection, Fast Correlation Based Filter). The feature subsets were then evaluated using three generic classifiers (Naïve Bayes, Support Vector Machine, j48 Decision Tree). Common features were identified for all three populations, although the stroke population subset had some differences from both able-bodied and elderly sets. Evaluation with the three classifiers showed that the feature subsets produced similar or better accuracies than classification with the entire feature set. Therefore, since these feature subsets are classifier-independent, they should be useful for developing and improving HAR systems across and within populations.     

SlimPLS: A Method for Feature Selection in Gene Expression-Based Disease Classification

A major challenge in biomedical studies in recent years has been the classification of gene expression profiles into categories, such as cases and controls. This is done by first training a classifier by using a labeled training set containing labeled samples from the two populations, and then using that classifier to predict the labels of new samples. Such predictions have recently been shown to improve the diagnosis and treatment selection practices for several diseases. This procedure is complicated, however, by the high dimensionality if the data. While microarrays can measure the levels of thousands of genes per sample, case-control microarray studies usually involve no more than several dozen samples. Standard classifiers do not work well in these situations where the number of features (gene expression levels measured in these microarrays) far exceeds the number of samples. Selecting only the features that are most relevant for discriminating between the two categories can help construct better classifiers, in terms of both accuracy and efficiency. In this work we developed a novel method for multivariate feature selection based on the Partial Least Squares algorithm. We compared the method''s variants with common feature selection techniques across a large number of real case-control datasets, using several classifiers. We demonstrate the advantages of the method and the preferable combinations of classifier and feature selection technique.