Incorporating prior knowledge of gene functional groups into regularized discriminant analysis of microarray data

MOTIVATION: Discriminant analysis for high-dimensional and low-sample-sized data has become a hot research topic in bioinformatics, mainly motivated by its importance and challenge in applications to tumor classifications for high-dimensional microarray data. Two of the popular methods are the nearest shrunken centroids, also called predictive analysis of microarray (PAM), and shrunken centroids regularized discriminant analysis (SCRDA). Both methods are modifications to the classic linear discriminant analysis (LDA) in two aspects tailored to high-dimensional and low-sample-sized data: one is the regularization of the covariance matrix, and the other is variable selection through shrinkage. In spite of their usefulness, there are potential limitations with each method. The main concern is that both PAM and SCRDA are possibly too extreme: the covariance matrix in the former is restricted to be diagonal while in the latter there is barely any restriction. Based on the biology of gene functions and given the feature of the data, it may be beneficial to estimate the covariance matrix as an intermediate between the two; furthermore, more effective shrinkage schemes may be possible. RESULTS: We propose modified LDA methods to integrate biological knowledge of gene functions (or variable groups) into classification of microarray data. Instead of simply treating all the genes independently or imposing no restriction on the correlations among the genes, we group the genes according to their biological functions extracted from existing biological knowledge or data, and propose regularized covariance estimators that encourages between-group gene independence and within-group gene correlations while maintaining the flexibility of any general covariance structure. Furthermore, we propose a shrinkage scheme on groups of genes that tends to retain or remove a whole group of the genes altogether, in contrast to the standard shrinkage on individual genes. We show that one of the proposed methods performed better than PAM and SCRDA in a simulation study and several real data examples.     

Improved centroids estimation for the nearest shrunken centroid classifier

MOTIVATION: The nearest shrunken centroid (NSC) method has been successfully applied in many DNA-microarray classification problems. The NSC uses 'shrunken' centroids as prototypes for each class and identifies subsets of genes that best characterize each class. Classification is then made to the nearest (shrunken) centroid. The NSC is very easy to implement and very easy to interpret, however, it has drawbacks. RESULTS: We show that the NSC method can be interpreted in the framework of LASSO regression. Based on that, we consider two new methods, adaptive L(infinity)-norm penalized NSC (ALP-NSC) and adaptive hierarchically penalized NSC (AHP-NSC), with two different penalty functions for microarray classification, which improve over the NSC. Unlike the L(1)-norm penalty used in LASSO, the penalty terms that we consider make use of the fact that parameters belonging to one gene should be treated as a natural group. Numerical results indicate that the two new methods tend to remove irrelevant genes more effectively and provide better classification results than the L(1)-norm approach. AVAILABILITY: R code for the ALP-NSC and the AHP-NSC algorithms are available from authors upon request.     

Multiclass cancer classification and biomarker discovery using GA-based algorithms

MOTIVATION: The development of microarray-based high-throughput gene profiling has led to the hope that this technology could provide an efficient and accurate means of diagnosing and classifying tumors, as well as predicting prognoses and effective treatments. However, the large amount of data generated by microarrays requires effective reduction of discriminant gene features into reliable sets of tumor biomarkers for such multiclass tumor discrimination. The availability of reliable sets of biomarkers, especially serum biomarkers, should have a major impact on our understanding and treatment of cancer. RESULTS: We have combined genetic algorithm (GA) and all paired (AP) support vector machine (SVM) methods for multiclass cancer categorization. Predictive features can be automatically determined through iterative GA/SVM, leading to very compact sets of non-redundant cancer-relevant genes with the best classification performance reported to date. Interestingly, these different classifier sets harbor only modest overlapping gene features but have similar levels of accuracy in leave-one-out cross-validations (LOOCV). Further characterization of these optimal tumor discriminant features, including the use of nearest shrunken centroids (NSC), analysis of annotations and literature text mining, reveals previously unappreciated tumor subclasses and a series of genes that could be used as cancer biomarkers. With this approach, we believe that microarray-based multiclass molecular analysis can be an effective tool for cancer biomarker discovery and subsequent molecular cancer diagnosis.     

Eigengene-based linear discriminant model for tumor classification using gene expression microarray data

MOTIVATION: The nearest shrunken centroids classifier has become a popular algorithm in tumor classification problems using gene expression microarray data. Feature selection is an embedded part of the method to select top-ranking genes based on a univariate distance statistic calculated for each gene individually. The univariate statistics summarize gene expression profiles outside of the gene co-regulation network context, leading to redundant information being included in the selection procedure. RESULTS: We propose an Eigengene-based Linear Discriminant Analysis (ELDA) to address gene selection in a multivariate framework. The algorithm uses a modified rotated Spectral Decomposition (SpD) technique to select 'hub' genes that associate with the most important eigenvectors. Using three benchmark cancer microarray datasets, we show that ELDA selects the most characteristic genes, leading to substantially smaller classifiers than the univariate feature selection based analogues. The resulting de-correlated expression profiles make the gene-wise independence assumption more realistic and applicable for the shrunken centroids classifier and other diagonal linear discriminant type of models. Our algorithm further incorporates a misclassification cost matrix, allowing differential penalization of one type of error over another. In the breast cancer data, we show false negative prognosis can be controlled via a cost-adjusted discriminant function. AVAILABILITY: R code for the ELDA algorithm is available from author upon request.     

Construction of robust prognostic predictors by using projective adaptive resonance theory as a gene filtering method

MOTIVATION: For establishing prognostic predictors of various diseases using DNA microarray analysis technology, it is desired to find selectively significant genes for constructing the prognostic model and it is also necessary to eliminate non-specific genes or genes with error before constructing the model. RESULTS: We applied projective adaptive resonance theory (PART) to gene screening for DNA microarray data. Genes selected by PART were subjected to our FNN-SWEEP modeling method for the construction of a cancer class prediction model. The model performance was evaluated through comparison with a conventional screening signal-to-noise (S2N) method or nearest shrunken centroids (NSC) method. The FNN-SWEEP predictor with PART screening could discriminate classes of acute leukemia in blinded data with 97.1% accuracy and classes of lung cancer with 90.0% accuracy, while the predictor with S2N was only 85.3 and 70.0% or the predictor with NSC was 88.2 and 90.0%, respectively. The results have proven that PART was superior for gene screening. AVAILABILITY: The software is available upon request from the authors. CONTACT: honda@nubio.nagoya-u.ac.jp     

Classification of microarrays to nearest centroids

MOTIVATION: Classification of biological samples by microarrays is a topic of much interest. A number of methods have been proposed and successfully applied to this problem. It has recently been shown that classification by nearest centroids provides an accurate predictor that may outperform much more complicated methods. The 'Prediction Analysis of Microarrays' (PAM) approach is one such example, which the authors strongly motivate by its simplicity and interpretability. In this spirit, I seek to assess the performance of classifiers simpler than even PAM. RESULTS: I surprisingly show that the modified t-statistics and shrunken centroids employed by PAM tend to increase misclassification error when compared with their simpler counterparts. Based on these observations, I propose a classification method called 'Classification to Nearest Centroids' (ClaNC). ClaNC ranks genes by standard t-statistics, does not shrink centroids and uses a class-specific gene-selection procedure. Because of these modifications, ClaNC is arguably simpler and easier to interpret than PAM, and it can be viewed as a traditional nearest centroid classifier that uses specially selected genes. I demonstrate that ClaNC error rates tend to be significantly less than those for PAM, for a given number of active genes. AVAILABILITY: Point-and-click software is freely available at http://students.washington.edu/adabney/clanc.     

Application of Discriminant Analysis in Distinguishing Plant Photosynthetic Types-A Case Study in Northeast China Transect (NECT) Area

Discriminant analysis is an important method in multivariate statistic analysis to distinguish whatever type an individual should belong to. Based on the field actual photosynthetic data obtained from the research platform--Northeast China Transect (NECT), the concept and principle of discriminant analysis were used to distinguish the different plant photosynthetic types. A number of indices related to plant photosynthetic rate measured by a LCA4 photosynthesis system were selected to build the discriminant model. In this case study, 15 plant species from C4 plant functional groups and 51 from C3 plant functional groups were selected to build a discriminant model. The rate of accuracy, of returned classification using methods of squared Mahalanobis distances from group centroids and posterior probabilities, reached to 98.48 %. With the help of this model, any plants' photosynthetic types could be distinguished simply by using their four related parameters, viz., photosynthetic rate, transpiration, stomatal conductance and the temperature difference between leaf surface and atmosphere.     

A discriminant analysis of amino acid frequency in collagen-like proteins.

The amino acid composition of collagen-like proteins from seven vertebrate and invertebrate animal phyla was used to construct a set of multiple discriminant classification functions. A stepwise procedure based on conditional F ratios was employed to identify those amino acids whose frequencies varied most consistently among the phyla. Out of 19 possible amino acids, proline, aspartate, lysine, alanine, phenylalanine, threonine, tyrosine, serine and hydroxyproline were selected to become independent variables of the discriminant functions. The reduction in the number of independent variables from 19 to nine is one of the advantages of this method. Variation among phylum centroids on the derived set of five discriminant functions (axes) was statistically highly significant and these five axes accounted for 98·11% of the total variance. All of the possible pairwise comparisons of phylum centroids on these five axes revealed differences which would occur by chance less than 1% of the time. By these criteria, the seven phyla studied were successfully distinguished one from the other on the basis of the amino acid composition of their collagen. Using the classification functions, the most probable phylum affinities for 46 species were calculated and all of these species were identified with their correct phylum. This offers the possibility that this method might be useful in defining membership in more refined taxa.     

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.     

Linear regression and two-class classification with gene expression data

MOTIVATION: Using gene expression data to classify (or predict) tumor types has received much research attention recently. Due to some special features of gene expression data, several new methods have been proposed, including the weighted voting scheme of Golub et al., the compound covariate method of Hedenfalk et al. (originally proposed by Tukey), and the shrunken centroids method of Tibshirani et al. These methods look different and are more or less ad hoc. RESULTS: We point out a close connection of the three methods with a linear regression model. Casting the classification problem in the general framework of linear regression naturally leads to new alternatives, such as partial least squares (PLS) methods and penalized PLS (PPLS) methods. Using two real data sets, we show the competitive performance of our new methods when compared with the other three methods.     

On sparse Fisher discriminant method for microarray data analysis

One of the applications of the discriminant analysis on microarray data is to classify patient and normal samples based on gene expression values. The analysis is especially important in medical trials and diagnosis of cancer subtypes. The main contribution of this paper is to propose a simple Fisher-type discriminant method on gene selection in microarray data. In the new algorithm, we calculate a weight for each gene and use the weight values as an indicator to identify the subsets of relevant genes that categorize patient and normal samples. A l(2) - l(1) norm minimization method is implemented to the discriminant process to automatically compute the weights of all genes in the samples. The experiments on two microarray data sets have shown that the new algorithm can generate classification results as good as other classification methods, and effectively determine relevant genes for classification purpose. In this study, we demonstrate the gene selection's ability and the computational effectiveness of the proposed algorithm. Experimental results are given to illustrate the usefulness of the proposed model.     

A Critical Assessment of Feature Selection Methods for Biomarker Discovery in Clinical Proteomics

In this paper, we compare the performance of six different feature selection methods for LC-MS-based proteomics and metabolomics biomarker discovery—t test, the Mann–Whitney–Wilcoxon test (mww test), nearest shrunken centroid (NSC), linear support vector machine–recursive features elimination (SVM-RFE), principal component discriminant analysis (PCDA), and partial least squares discriminant analysis (PLSDA)—using human urine and porcine cerebrospinal fluid samples that were spiked with a range of peptides at different concentration levels. The ideal feature selection method should select the complete list of discriminating features that are related to the spiked peptides without selecting unrelated features. Whereas many studies have to rely on classification error to judge the reliability of the selected biomarker candidates, we assessed the accuracy of selection directly from the list of spiked peptides. The feature selection methods were applied to data sets with different sample sizes and extents of sample class separation determined by the concentration level of spiked compounds. For each feature selection method and data set, the performance for selecting a set of features related to spiked compounds was assessed using the harmonic mean of the recall and the precision (f-score) and the geometric mean of the recall and the true negative rate (g-score). We conclude that the univariate t test and the mww test with multiple testing corrections are not applicable to data sets with small sample sizes (n = 6), but their performance improves markedly with increasing sample size up to a point (n > 12) at which they outperform the other methods. PCDA and PLSDA select small feature sets with high precision but miss many true positive features related to the spiked peptides. NSC strikes a reasonable compromise between recall and precision for all data sets independent of spiking level and number of samples. Linear SVM-RFE performs poorly for selecting features related to the spiked compounds, even though the classification error is relatively low.Biomarkers play an important role in advancing medical research through the early diagnosis of disease and prognosis of treatment interventions (1, 2). Biomarkers may be proteins, peptides, or metabolites, as well as mRNAs or other kinds of nucleic acids (e.g. microRNAs) whose levels change in relation to the stage of a given disease and which may be used to accurately assign the disease stage of a patient. The accurate selection of biomarker candidates is crucial, because it determines the outcome of further validation studies and the ultimate success of efforts to develop diagnostic and prognostic assays with high specificity and sensitivity. The success of biomarker discovery depends on several factors: consistent and reproducible phenotyping of the individuals from whom biological samples are obtained; the quality of the analytical methodology, which in turn determines the quality of the collected data; the accuracy of the computational methods used to extract quantitative and molecular identity information to define the biomarker candidates from raw analytical data; and finally the performance of the applied statistical methods in the selection of a limited list of compounds with the potential to discriminate between predefined classes of samples. De novo biomarker research consists of a biomarker discovery part and a biomarker validation part (3). Biomarker discovery uses analytical techniques that try to measure as many compounds as possible in a relatively low number of samples. The goal of subsequent data preprocessing and statistical analysis is to select a limited number of candidates, which are subsequently subjected to targeted analyses in large number of samples for validation.Advanced technology, such as high-performance liquid chromatography–mass spectrometry (LC-MS),¹ is increasingly applied in biomarker discovery research. Such analyses detect tens of thousands of compounds, as well as background-related signals, in a single biological sample, generating enormous amounts of multivariate data. Data preprocessing workflows reduce data complexity considerably by trying to extract only the information related to compounds resulting in a quantitative feature matrix, in which rows and columns correspond to samples and extracted features, respectively, or vice versa. Features may also be related to data preprocessing artifacts, and the ratio of such erroneous features to compound-related features depends on the performance of the data preprocessing workflow (4). Preprocessed LC-MS data sets contain a large number of features relative to the sample size. These features are characterized by their m/z value and retention time, and in the ideal case they can be combined and linked to compound identities such as metabolites, peptides, and proteins. In LC-MS-based proteomics and metabolomics studies, sample analysis is so time consuming that it is practically impossible to increase the number of samples to a level that balances the number of features in a data set. Therefore, the success of biomarker discovery depends on powerful feature selection methods that can deal with a low sample size and a high number of features. Because of the unfavorable statistical situation and the risk of overfitting the data, it is ultimately pivotal to validate the selected biomarker candidates in a larger set of independent samples, preferably in a double-blinded fashion, using targeted analytical methods (1).Biomarker selection is often based on classification methods that are preceded by feature selection methods (filters) or which have built-in feature selection modules (wrappers and embedded methods) that can be used to select a list of compounds/peaks/features that provide the best classification performance for predefined sample groups (e.g. healthy versus diseased) (5). Classification methods are able to classify an unknown sample into a predefined sample class. Univariate feature selection methods such as filters (t test or Wilcoxon–Mann–Whitney tests) cannot be used for sample classification. Other classification methods such as the nearest shrunken centroid method have intrinsic feature selection ability, whereas other classification methods such as principal component discriminant analysis (PCDA) and partial least squares regression coupled with discriminant analysis (PLSDA) should be augmented with a feature selection method. There are classifiers having no feature selection option that perform the classification using all variables, such as support vector machines that use non-linear kernels (6). Classification methods without the ability to select features cannot be used for biomarker discovery, because these methods aim to classify samples into predefined classes but cannot identify the limited number of variables (features or compounds) that form the basis of the classification (6, 7). Different statistical methods with feature selection have been developed according to the complexity of the analyzed data, and these have been extensively reviewed (5, 6, 8, 9). Ways of optimizing such methods to improve sensitivity and specificity are a major topic in current biomarker discovery research and in the many “omics-related” research areas (6, 10, 11). Comparisons of classification methods with respect to their classification and learning performance have been initiated. Van der Walt et al. (12) focused on finding the most accurate classifiers for simulated data sets with sample sizes ranging from 20 to 100. Rubingh et al. (13) compared the influence of sample size in an LC-MS metabolomics data set on the performance of three different statistical validation tools: cross validation, jack-knifing model parameters, and a permutation test. That study concluded that for small sample sets, the outcome of these validation methods is influenced strongly by individual samples and therefore cannot be trusted, and the validation tool cannot be used to indicate problems due to sample size or the representativeness of sampling. This implies that reducing the dimensionality of the feature space is critical when approaching a classification problem in which the number of features exceeds the number of samples by a large margin. Dimensionality reduction retains a smaller set of features to bring the feature space in line with the sample size and thus allow the application of classification methods that perform with acceptable accuracy only when the sample size and the feature size are similar.In this study we compared different classification methods focusing on feature selection in two types of spiked LC-MS data sets that mimic the situation of a biomarker discovery study. Our results provide guidelines for researchers who will engage in biomarker discovery or other differential profiling “omics” studies with respect to sample size and selecting the most appropriate feature selection method for a given data set. We evaluated the following approaches: univariate t test and Mann–Whitney–Wilcoxon test (mww test) with multiple testing correction (14), nearest shrunken centroid (NSC) (15, 16), support vector machine–recursive features elimination (SVM-RFE) (17), PLSDA (18), and PCDA (19). PCDA and PLSDA were combined with the rank-product as a feature selection criterion (20). These methods were evaluated with data sets having three characteristics: different biological background, varying sample size, and varying within- and between-class variability of the added compounds. Data were acquired via LC-MS from human urine and porcine cerebrospinal fluid (CSF) samples that were spiked with a set of known peptides (true positives) at different concentration levels. These samples were then combined in two classes containing peptides spiked at low and high concentration levels. The performance of the classification methods with feature selection was measured based on their ability to select features that were related to the spiked peptides. Because true positives were known in our data set, we compared performance based on the f-score (the harmonic mean of precision and recall) and the g-score (the geometric mean of accuracy).     

Regularization network-based gene selection for microarray data analysis

Microarray data contains a large number of genes (usually more than 1000) and a relatively small number of samples (usually fewer than 100). This presents problems to discriminant analysis of microarray data. One way to alleviate the problem is to reduce dimensionality of data by selecting important genes to the discriminant problem. Gene selection can be cast as a feature selection problem in the context of pattern classification. Feature selection approaches are broadly grouped into filter methods and wrapper methods. The wrapper method outperforms the filter method but at the cost of more intensive computation. In the present study, we proposed a wrapper-like gene selection algorithm based on the Regularization Network. Compared with classical wrapper method, the computational costs in our gene selection algorithm is significantly reduced, because the evaluation criterion we proposed does not demand repeated training in the leave-one-out procedure.     

几何形态测量学方法在小哺乳动物化石分类鉴定中的应用——4种(鼠平)类化石大样本的个案研究

小哺乳动物化石在晚新生代生物地层学和生物年代学研究中具有重要作用,尤其是筛洗法在古生物调查中得到广泛应用后,通常可以采集到数量可观的标本,使得其地位较大型哺乳动物化石更加彰显.因此,小哺乳动物化石标本的分类鉴定也成了一项十分关键的工作.然而,传统形态学方法在对大量标本进行分类鉴定时,往往容易受主观因素影响而将不稳定的细微性状变异作为依据建立新种,或者忽视一些肉眼难于察觉的形态学差异而将两个甚至多个类群合并到一起,导致基于形态学的化石分类鉴定随意性增加,失去客观性.此外,对于不具鉴定意义的非关键性单个牙齿,很难凭借肉眼或显微镜观察进行区分.针对这些问题,本文选取了安徽繁昌人字洞早更新世早期三种(鼠平)类甘肃模鼠Mimomys gansunicus,郑氏异模鼠Heteromimomys zhengi和繁昌维蓝尼鼠Hllanyia fanchangensis的1284件臼齿并以早上新世内蒙古比例克的比例克模鼠Mimomys bilikeensis的163件臼齿作为参考,采用几何形态测量学方法在各个臼齿咬合面上分别选取了7～14个同源landmark对咬合面的形态特征进行了线性判别分析,建立了针对这4个种的臼齿咬合面形态差异判别函数.分析结果表明,根据这些landmark所提供的形态差异信息,人字洞的1284件臼齿标本中的确存在3个可以明显区分且形态学性状稳定的类群,先前的分类鉴定得到了验证.与之不同地点不同时代的比例克模鼠也可以很好区分.因此,建立在大样本基础上的这4种(鼠平)类臼齿咬合面同源形态特征的判别函数可以用来描述这些种类较为稳定的形态特征差异,并用来作为今后对样本较少的标本进行分类鉴定时的判别依据.因为几何形态测量学的方法不仅可以对二维的离散landmark数据以及重要形态学特征的轮廓线进行分析,甚至还可以扩展至三维空间,所以上述方法对于较为容易获得大样本标本的小哺乳物化石分类鉴定具有普遍意义.     

Using uncorrelated discriminant analysis for tissue classification with gene expression data

The classification of tissue samples based on gene expression data is an important problem in medical diagnosis of diseases such as cancer. In gene expression data, the number of genes is usually very high (in the thousands) compared to the number of data samples (in the tens or low hundreds); that is, the data dimension is large compared to the number of data points (such data is said to be undersampled). To cope with performance and accuracy problems associated with high dimensionality, it is commonplace to apply a preprocessing step that transforms the data to a space of significantly lower dimension with limited loss of the information present in the original data. Linear discriminant analysis (LDA) is a well-known technique for dimension reduction and feature extraction, but it is not applicable for undersampled data due to singularity problems associated with the matrices in the underlying representation. This paper presents a dimension reduction and feature extraction scheme, called uncorrelated linear discriminant analysis (ULDA), for undersampled problems and illustrates its utility on gene expression data. ULDA employs the generalized singular value decomposition method to handle undersampled data and the features that it produces in the transformed space are uncorrelated, which makes it attractive for gene expression data. The properties of ULDA are established rigorously and extensive experimental results on gene expression data are presented to illustrate its effectiveness in classifying tissue samples. These results provide a comparative study of various state-of-the-art classification methods on well-known gene expression data sets     

Classification trees as an alternative to linear discriminant analysis

Linear discriminant analysis (LDA) is frequently used for classification/prediction problems in physical anthropology, but it is unusual to find examples where researchers consider the statistical limitations and assumptions required for this technique. In these instances, it is difficult to know whether the predictions are reliable. This paper considers a nonparametric alternative to predictive LDA: binary, recursive (or classification) trees. This approach has the advantage that data transformation is unnecessary, cases with missing predictor variables do not require special treatment, prediction success is not dependent on data meeting normality conditions or covariance homogeneity, and variable selection is intrinsic to the methodology. Here I compare the efficacy of classification trees with LDA, using typical morphometric data. With data from modern hominoids, the results show that both techniques perform nearly equally. With complete data sets, LDA may be a better choice, as is shown in this example, but with missing observations, classification trees perform outstandingly well, whereas commercial discriminant analysis programs do not predict classifications for cases with incompletely measured predictor variables and generally are not designed to address the problem of missing data. Testing of data prior to analysis is necessary, and classification trees are recommended either as a replacement for LDA or as a supplement whenever data do not meet relevant assumptions. It is highly recommended as an alternative to LDA whenever the data set contains important cases with missing predictor variables.     

Probabilistic disease classification of expression-dependent proteomic data from mass spectrometry of human serum.

We have developed an algorithm called Q5 for probabilistic classification of healthy versus disease whole serum samples using mass spectrometry. The algorithm employs principal components analysis (PCA) followed by linear discriminant analysis (LDA) on whole spectrum surface-enhanced laser desorption/ionization time of flight (SELDI-TOF) mass spectrometry (MS) data and is demonstrated on four real datasets from complete, complex SELDI spectra of human blood serum. Q5 is a closed-form, exact solution to the problem of classification of complete mass spectra of a complex protein mixture. Q5 employs a probabilistic classification algorithm built upon a dimension-reduced linear discriminant analysis. Our solution is computationally efficient; it is noniterative and computes the optimal linear discriminant using closed-form equations. The optimal discriminant is computed and verified for datasets of complete, complex SELDI spectra of human blood serum. Replicate experiments of different training/testing splits of each dataset are employed to verify robustness of the algorithm. The probabilistic classification method achieves excellent performance. We achieve sensitivity, specificity, and positive predictive values above 97% on three ovarian cancer datasets and one prostate cancer dataset. The Q5 method outperforms previous full-spectrum complex sample spectral classification techniques and can provide clues as to the molecular identities of differentially expressed proteins and peptides.