jmlr jmlr2010 jmlr2010-62 knowledge-graph by maker-knowledge-mining

62 jmlr-2010-Learning Gradients: Predictive Models that Infer Geometry and Statistical Dependence

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Author: Qiang Wu, Justin Guinney, Mauro Maggioni, Sayan Mukherjee

Abstract: The problems of dimension reduction and inference of statistical dependence are addressed by the modeling framework of learning gradients. The models we propose hold for Euclidean spaces as well as the manifold setting. The central quantity in this approach is an estimate of the gradient of the regression or classiﬁcation function. Two quadratic forms are constructed from gradient estimates: the gradient outer product and gradient based diffusion maps. The ﬁrst quantity can be used for supervised dimension reduction on manifolds as well as inference of a graphical model encoding dependencies that are predictive of a response variable. The second quantity can be used for nonlinear projections that incorporate both the geometric structure of the manifold as well as variation of the response variable on the manifold. We relate the gradient outer product to standard statistical quantities such as covariances and provide a simple and precise comparison of a variety of supervised dimensionality reduction methods. We provide rates of convergence for both inference of informative directions as well as inference of a graphical model of variable dependencies. Keywords: gradient estimates, manifold learning, graphical models, inverse regression, dimension reduction, gradient diffusion maps

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Two quadratic forms are constructed from gradient estimates: the gradient outer product and gradient based diffusion maps. [sent-12, score-1.062]

2 The ﬁrst quantity can be used for supervised dimension reduction on manifolds as well as inference of a graphical model encoding dependencies that are predictive of a response variable. [sent-13, score-0.454]

3 The second quantity can be used for nonlinear projections that incorporate both the geometric structure of the manifold as well as variation of the response variable on the manifold. [sent-14, score-0.437]

4 We relate the gradient outer product to standard statistical quantities such as covariances and provide a simple and precise comparison of a variety of supervised dimensionality reduction methods. [sent-15, score-0.597]

5 Keywords: gradient estimates, manifold learning, graphical models, inverse regression, dimension reduction, gradient diffusion maps 1. [sent-17, score-1.095]

6 The modern reprise of this program has been developed in the broad areas of manifold learning and simultaneous dimension reduction and regression. [sent-29, score-0.438]

7 Simultaneous dimension reduction and regression considers the problem of ﬁnding directions that are informative with respect to predicting the response variable. [sent-33, score-0.429]

8 Both linear projections as well as nonlinear embeddings based on gradient estimates are used for supervised dimension reduction. [sent-41, score-0.452]

9 The rate of convergence depends on a geometric quantity, the intrinsic dimension of the gradient on the manifold supporting the data, rather than the sparsity of the graph. [sent-44, score-0.55]

10 A Statistical Foundation for Learning Gradients The problem of regression can be summarized as estimating the function fr (x) = E(Y | X = x) from data D = {Li = (Yi , Xi )}n where Xi is a vector in a p-dimensional compact metric space i=1 X ∈ X ⊂ R p and Yi ∈ R is a real valued output. [sent-46, score-0.371]

11 In this paper we will explain how inference of the gradient of the regression function ∇ fr = ∂ fr ∂ fr , . [sent-49, score-1.166]

12 The ﬁrst approach is averaging local gradient estimates and motivates the study of the gradient outer product (GOP) matrix. [sent-57, score-0.711]

13 The GOP can also be considered as a covariance matrix and used for inference of conditional dependence between predictive explanatory variables. [sent-59, score-0.381]

14 This motivates the study of gradient based diffusion maps (GDM). [sent-61, score-0.471]

15 This operator can be used for nonlinear supervised dimension reduction by embedding the support of the marginal distribution onto a much lower dimensional manifold that varies with respect to the response. [sent-62, score-0.683]

16 The gradient outer product Γ is a p × p matrix with elements Γi j = ∂ fr ∂ fr , ∂xi ∂x j , 2 Lρ X 2 where ρX is the marginal distribution of the explanatory variables and Lρ is the space of square X integrable functions with respect to the measure ρX . [sent-63, score-1.279]

17 Using the notation a ⊗ b = abT for a, b ∈ R p , we can write Γ = E(∇ fr ⊗ ∇ fr ). [sent-64, score-0.616]

18 The GOP is meaningful in both the Euclidean as well as the manifold setting where the marginal distribution ρX is concentrated on a much lower dimensional manifold M of dimension dM (developed in Section 4. [sent-67, score-0.577]

19 This global summary of the joint distribution is advantageous in statistical analyses where global inferences are desired: global predictive factors comprised of the explanatory variables or global estimates of statistical dependence between explanatory variables. [sent-71, score-0.493]

20 Random walks or diffusions on manifolds and graphs have been used for a variety of nonlinear dimension reduction or manifold embedding procedures (Belkin and Niyogi, 2003, 2004; Szummer and Jaakkola, 2001; Coifman et al. [sent-73, score-0.568]

21 Our basic idea is to use local gradient information to construct a random walk on a graph or manifold based on the ideas of diffusion analysis and diffusion geometry (Coifman and Lafon, 2006; Coifman and Maggioni, 2006). [sent-75, score-0.957]

22 The central quantity in diffusion based approaches is the deﬁnition of a diffusion operator L based on a similarity metric Wi j between two points xi and x j . [sent-76, score-0.59]

23 j Dimension reduction is achieved by projection onto a spectral decomposition of the operator L or powers of the operator Lt which corresponds to running the diffusion for some time t. [sent-78, score-0.477]

24 2177 W U , G UINNEY, M AGGIONI AND M UKHERJEE We propose the following similarity metric called the gradient based diffusion map (GDM) to smoothly paste together local gradient estimates Wi j = W f (xi , x j ) = exp − xi − x j σ1 2 − 1 | 2 (∇ fr (xi ) + ∇ fr (x j )) · (xi − x j )|2 σ2 . [sent-79, score-1.343]

25 In the above equation the ﬁrst term xi −x j 2 encodes the local geometry of the marginal distribution and the second term pastes together gradient estimates between neighboring points. [sent-80, score-0.408]

26 The ﬁrst term is used in unsupervised dimension reduction methods such as Laplacian eigenmaps or diffusion maps (Belkin and Niyogi, 2003). [sent-81, score-0.559]

27 The second term can be interpreted as a diffusion map on the function values, the approximation is due to a ﬁrst order Taylor expansion 1 (∇ fr (xi ) + ∇ fr (x j )) · (xi − x j ) ≈ fr (xi ) − fr (x j ) 2 if xi ≈ x j . [sent-82, score-1.53]

28 1 Illustration of Linear Projections and Nonlinear Embeddings The simple example in this section ﬁxes the differences between linear projections and nonlinear embeddings using either diffusion maps or gradient based diffusion maps. [sent-89, score-0.784]

29 In this example a two dimensional manifold is embedded in R3 and the variation of the response variable can be embedded onto one dimension. [sent-91, score-0.374]

30 This example also shows that linear dimension reduction may still make sense in the manifold setting. [sent-97, score-0.438]

31 In Figure 1(d) the data is embedded using gradient based diffusion maps and we capture the direction θ in which the data varies with respect to the regression function. [sent-98, score-0.534]

32 2 Gradient Outer Product Matrix and Dimension Reduction In the case of linear supervised dimension reduction we assume the following semi-parametric model holds Y = fr (X) + ε = g(bT X, . [sent-100, score-0.576]

33 In the following we develop theory relating the gradient of the regression function to the above model of dimension reduction. [sent-167, score-0.374]

34 r (x) Observe that for a vector v ∈ R p , ∂ f∂v = v · ∇ fr is identically zero if fr does not depends on v and is not zero if fr changes along the direction v. [sent-168, score-0.924]

35 The following lemma relates the gradient outer product matrix to supervised dimension reduction. [sent-169, score-0.683]

36 2179 W U , G UINNEY, M AGGIONI AND M UKHERJEE Lemma 1 Under the assumptions of the semi-parametric model (1), the gradient outer product matrix Γ is of rank at most d. [sent-170, score-0.513]

37 , 2002), or explicitly as in outer product of gradient (OPG) (Xia et al. [sent-184, score-0.461]

38 The gradient outer product matrix is deﬁned globally and its relation to dimension reduction in Section 2. [sent-187, score-0.743]

39 However, since the gradient itself is a local concept we can also study the geometric structure encoded in the gradient outer product matrix from a local point of view. [sent-189, score-0.799]

40 3 Gradient Outer Product Matrix as a Covariance Matrix A central concept used in dimension reduction is the covariance matrix of the inverse regression function ΩX|Y = covY [EX (X | Y )]. [sent-191, score-0.463]

41 The ﬁrst observation from this relation is that the gradient outer product matrix is a covariance matrix with a very particular construction. [sent-197, score-0.626]

42 The second observation is that the gradient outer product matrix contains more information than the covariance of the inverse regression since it captures local information. [sent-198, score-0.732]

43 Suppose there exists a non-overlapping partition of the input space I X= Ri i=1 such that in each region Ri the regression function fr is linear and the noise has zero mean fr (x) = βT x, i E εi = 0 for x ∈ Ri . [sent-209, score-0.679]

44 Assuming that matrices Σi are full rank, the gradient outer product matrix i can be computed in terms of these local quantities I Γ = ∑ ρX (Ri )σ2 1 − i i=1 σ2i ε σ2 i 2 Σ−1 Ωi Σ−1 , i i (5) where ρX (Ri ) is the measure of partition Ri with respect to the marginal distribution ρX . [sent-212, score-0.58]

45 i i Supervised dimension reduction methods based on the covariance of the inverse regression require an elliptic condition on X. [sent-218, score-0.411]

46 This motivates the centrality of the gradient outer product. [sent-224, score-0.405]

47 This derivation of the gradient outer product matrix based on local variation has two potential implications. [sent-225, score-0.596]

48 It provides a theoretical comparison between dimension reduction approaches based on the gradient outer product matrix and inverse regression. [sent-226, score-0.8]

49 In the manifold setting, the input space is a manifold X = M of dimension dM ≪ p. [sent-233, score-0.548]

50 1 In this setting the gradient outer product matrix should be deﬁned in terms of the gradient on the manifold, ∇M fr , Γ = E (dϕ(∇M fr ) ⊗ dϕ(∇M fr )) = E dϕ(∇M fr ⊗ ∇M fr )(dϕ)T . [sent-236, score-2.232]

51 Note that the dM × dM matrix ΓM = ∇M fr ⊗ ∇M fr is the central quantity of interest in this setting. [sent-238, score-0.668]

52 However, we know neither the manifold nor the coordinates on the manifold and are only provided points in the ambient space. [sent-239, score-0.462]

53 However, we can understand its properties by analyzing the gradient outer product matrix Γ in the ambient space, a p × p matrix. [sent-241, score-0.559]

54 Estimating Gradients An estimate of the gradient is required in order to estimate the gradient outer product matrix Γ. [sent-246, score-0.692]

55 The motivation for learning gradients is based on Taylor expanding the regression function fr (u) ≈ fr (x) + ∇ fr (x) · (u − x), for x ≈ u, which can be evaluated at data points (xi )n i=1 fr (xi ) ≈ fr (x j ) + ∇ fr (x j ) · (xi − x j ), for xi ≈ x j . [sent-254, score-2.046]

56 Given data D = {(yi , xi )}n the objective is to simultaneously estimate the regression function fr i=1 by a function fD and the gradient ∇ fr by the p-dimensional vector valued function fD . [sent-255, score-0.913]

57 For the sample D = {(qi , yi )}n ∈ (M × R)n , i=1 xi = ϕ(qi ) ∈ R p , the learning gradients algorithm on M provides estimates ( fD , fD ) := arg min p+1 f , f ∈ HK 1 n s w yi − f (x j ) + f (xi ) · (x j − xi ) n2 i,∑ i, j j=1 where fD is a model for dϕ(∇M fr ) and fD is a model for fr . [sent-281, score-0.839]

58 Dimension Reduction Using Gradient Estimates In this section we study some properties of dimension reduction using the gradient estimates. [sent-298, score-0.409]

59 1 Linear Dimension Reduction The theoretical foundation for linear dimension reduction using the spectral decomposition of the gradient outer production matrix was developed in Section 2. [sent-301, score-0.687]

60 The estimate of the gradient obtained by the kernel models in Section 3 provides the following empirical estimate of the gradient outer product matrix 1 n ˆ Γ := cD K 2 cT = ∑ fD (xi ) ⊗ fD (xi ), D n i=1 where K is the kernel matrix with Ki j = K(xi , x j ) and cD is deﬁned in (6). [sent-303, score-0.744]

61 Proposition 9 justiﬁes linear dimension reduction using consistent gradient estimates from a global point of view. [sent-323, score-0.469]

62 In the next subsection we study the gradient outer product matrix from the local point of view and provide details on the relation between gradient based methods and sliced inverse regression. [sent-324, score-0.832]

63 2 I Γ = ∑ ρX (Ri ) Γi ≈ ∑ ρX (Ri )σ2 Σ−1 Ωi Σ−1 , i i i i i=1 where Γi is the gradient outer product matrix on Ri and Ωi = covY (EX (X ∈ Ri | Y )). [sent-348, score-0.513]

64 In this sense, the gradient covariance matrix Γ can be regarded as the weighted sum of the local covariance matrices of the inverse regression function. [sent-349, score-0.511]

65 Recall that SIR suffers from the possible degeneracy of the covariance matrix of the inverse regression function over the entire input space while the local covariance matrix of the inverse regression function will not be degenerate unless the function is constant. [sent-350, score-0.504]

66 Moreover, in the gradient outer product matrix, the importance of local descriptive directions are also taken into account. [sent-351, score-0.552]

67 Hence the same comparison holds between SIR and other gradient based methods such as mean average variance estimation (MAVE) and outer product of gradients (OPG) developed in Xia et al. [sent-354, score-0.541]

68 2 T HEORETICAL F EASIBILITY OF L INEAR P ROJECTIONS FOR N ONLINEAR M ANIFOLDS In this section we explore why linear projections based on the gradient outer product matrix are feasible and have meaning when the manifold structure is nonlinear. [sent-358, score-0.749]

69 The crux of the analysis will ˆ be demonstrating that the estimated gradient outer product matrix Γ is still meaningful. [sent-359, score-0.513]

70 From a modeling perspective we would like the gradient estimate from data fD to approximate dϕ(∇M fr ) (Mukherjee et al. [sent-361, score-0.487]

71 Instead, the estimate provides the following information about ∇M fr lim (dϕ)∗ fD = ∇M fr , n→∞ where is the dual of dϕ, the differential of ϕ. [sent-364, score-0.616]

72 Hence, it is not meaningful to consider the gradient of ∇ fr in the ambient space R p . [sent-366, score-0.533]

73 Also, we cannot recover directly the gradient of fr on the manifold since we know neither the manifold nor the embedding. [sent-367, score-0.903]

74 Proof Let fλ be the sample limit of fD , that is fλ = arg minp f ∈ HK − e M M x−ξ 2 2s2 2 fr (x) − fr (ξ) + f (x) · (ξ − x) dρM (x)dρM (ξ) + λ f 2 K . [sent-376, score-0.616]

75 This means the DR directions can be still found by the spectral decomposition of the estimated gradient outer product matrix. [sent-382, score-0.514]

76 The basic idea was to use local gradient information to construct a diffusion operator L based on a similarity metric Wi j between two points xi and x j . [sent-385, score-0.564]

77 j Dimension reduction is achieved by projection onto a spectral decomposition of the operator L or powers (I − L)t of the operator (I − L) which corresponds to running the diffusion (I − L) for some time t. [sent-387, score-0.477]

78 The gradient based diffusion map (GDM) was deﬁned as Wi j = W f (xi , x j ) = exp − xi − x j σ1 2 − 1 | 2 (∇ fr (xi ) + ∇ fr (x j )) · (xi − x j )|2 σ2 2187 . [sent-388, score-1.093]

79 (8) W U , G UINNEY, M AGGIONI AND M UKHERJEE In the above equation the ﬁrst term xi −x j 2 encodes the local geometry of the marginal distribution and the second term pastes together gradient estimates between neighboring points. [sent-389, score-0.408]

80 The ﬁrst term is used in unsupervised dimension reduction methods such as Laplacian eigenmaps or diffusion maps (Belkin and Niyogi, 2003). [sent-390, score-0.559]

81 The second term can be interpreted as a ﬁrst order Taylor expansion leading to the following approximation 1 (∇ fr (xi ) + ∇ fr (x j )) · (xi − x j ) ≈ fr (xi ) − fr (x j ). [sent-391, score-1.232]

82 The ﬁrst order Taylor expansion on the manifold results in the following approximation f (xi ) − f (x j ) ≈ ∇M f (xi ) · vi j , for vi j ≈ 0, where vi j ∈ Txi M is the tangent vector such that x j = Expxi (vi j ) where Expxi is the exponential map at xi (see do Carmo, 1992; Mukherjee et al. [sent-410, score-0.455]

83 Proposition 12 The following holds fr (xi ) − fr (x j ) ≈ fD (xi ) · (xi − x j ), for xi ≈ x j . [sent-414, score-0.671]

84 Proof By the fact xi − x j ≈ dϕ(vi j ) we have fD (xi ) · (xi − x j ) ≈ fD (xi ), dϕ(vi j ) = (dϕ)∗ ( fD (xi )), vi j ≈ ∇M fr (xi ), vi j which implies the conclusion. [sent-415, score-0.491]

85 This research program of proving convergence of the eigenvectors of a graph Laplacian to the eigenfunctions of a corresponding Laplace-Beltrami operator is a source of extensive research in diffusion maps (Belkin and Niyogi, 2005; Gin´ and Koltchinskii, 2006). [sent-417, score-0.383]

86 In the semi-supervised setting using the labeled as well as the unlabeled data for dimension reduction followed by ﬁtting a regression model has often increased predictive accuracy. [sent-425, score-0.376]

87 In Section 2 we stated that both the covariance of the inverse regression as well as the gradient outer product matrix provide estimates of the covariance of the explanatory variables conditioned on variation in the response variable. [sent-477, score-1.037]

88 We focus on the inverse of the gradient outer product matrix in this paper since it is of use for both linear and nonlinear functions. [sent-479, score-0.612]

89 ˆ Our proof of the convergence of the estimated conditional dependence matrix (Γ)−1 to the pop−1 relies on the assumption that the gradient outer product ulation conditional dependence matrix Γ matrix being low rank. [sent-486, score-0.801]

90 1 S IMULATED DATA The following simple example clariﬁes the information contained in the covariance matrix as well as the gradient outer product matrix. [sent-506, score-0.574]

91 From this i=1 ˆ data we estimate the covariance matrix of the marginals ΣX and the gradient outer product matrix ˆ From ΣX , Figure 2(a), we see that X1 and X2 covary with each other and X3 , X4 , X5 covary. [sent-513, score-0.626]

92 Taking the response variable into account, we ﬁnd in the gradient outer product matrix, Figure 2(d), the variables X2 and ˆˆ X4 are irrelevant while X1 , X3 , X5 are relevant. [sent-517, score-0.544]

93 (2008) the inverse of the gradient outer product was used to infer the dependence between genes that drive tumor progression in prostate cancer and melanoma. [sent-527, score-0.973]

94 The gradient outer product matrix was used to infer detailed models of gene networks that may drive tumor progression. [sent-530, score-0.576]

95 The objective is to understand the dependence structure between genes that are predictive of progression from benign to malignant prostate cancer. [sent-532, score-0.5]

96 2 1 2 3 4 5 (d) (e) (f) ˆ ˆ Figure 2: (a) Covariance matrix ΣX ; (b) Partial correlation matrix RX ; (c) Graphical model repreˆ sentation of partial correlation matrix; (d) Gradient outer product matrix Γ; (e) Partial ˆ ˆ Γ with respect to Γ; (f) Graphical model representation of RΓ . [sent-564, score-0.471]

97 From this reduced data set we estimated the gradient outer product ˆ ˆ matrix, Γ, and used the pseudo-inverse to compute the conditional independence matrix, Jˆ = (Γ)−1 . [sent-568, score-0.498]

98 Discussion The main contribution of this paper is to describe how inference of the gradient of the regression or classiﬁcation function encodes information about the predictive geometry as well as the predictive conditional dependence in the data. [sent-585, score-0.518]

99 Two methods are introduced gradient based diffusion maps and inference of conditional independence structure given gradient estimates. [sent-586, score-0.687]

100 We ﬁnd this direct link be2194 L EARNING G RADIENTS : P REDICTIVE M ODELS THAT I NFER G EOMETRY tween graphical models and dimension reduction intriguing and suggest that the manifold learning perspective holds potential in the analysis and inference of graphical models. [sent-590, score-0.534]

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Abstract: Microarray gene expressions provide new opportunities for molecular classiﬁcation of heterogeneous diseases. Although various reported classiﬁcation schemes show impressive performance, most existing gene selection methods are suboptimal and are not well-matched to the unique characc 2010 Guoqiang Yu, Yuanjian Feng, David J. Miller, Jianhua Xuan, Eric P. Hoffman, Robert Clarke, Ben Davidson, Ie-Ming Shih and Yue Wang Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG teristics of the multicategory classiﬁcation problem. Matched design of the gene selection method and a committee classiﬁer is needed for identifying a small set of gene markers that achieve accurate multicategory classiﬁcation while being both statistically reproducible and biologically plausible. We report a simpler and yet more accurate strategy than previous works for multicategory classiﬁcation of heterogeneous diseases. Our method selects the union of one-versus-everyone (OVE) phenotypic up-regulated genes (PUGs) and matches this gene selection with a one-versus-rest support vector machine (OVRSVM). Our approach provides even-handed gene resources for discriminating both neighboring and well-separated classes. Consistent with the OVRSVM structure, we evaluated the fold changes of OVE gene expressions and found that only a small number of high-ranked genes were required to achieve superior accuracy for multicategory classiﬁcation. We tested the proposed PUG-OVRSVM method on six real microarray gene expression data sets (ﬁve public benchmarks and one in-house data set) and two simulation data sets, observing signiﬁcantly improved performance with lower error rates, fewer marker genes, and higher performance sustainability, as compared to several widely-adopted gene selection and classiﬁcation methods. The MATLAB toolbox, experiment data and supplement ﬁles are available at http://www.cbil.ece.vt.edu/software.htm. Keywords: microarray gene expression, multiclass gene selection, phenotypic up-regulated gene, multicategory classiﬁcation 1. Background The rapid development of gene expression microarrays provides an opportunity to take a genomewide approach for disease diagnosis, prognosis, and prediction of therapeutic responsiveness (Clarke et al., 2008; Wang et al., 2008). When the molecular signature is analyzed with pattern recognition algorithms, new classes of disease are identiﬁed and new insights into disease mechanisms and diagnostic or therapeutic targets emerge (Clarke et al., 2008). For example, many studies demonstrate that global gene expression proﬁling of human tumors can provide molecular classiﬁcations that reveal distinct tumor subtypes not evident by traditional histopathological methods (Golub et al., 1999; Ramaswamy et al., 2001; Shedden et al., 2003; Wang et al., 2006). While molecular classiﬁcation falls neatly within supervised pattern recognition, high gene dimensionality and paucity of microarray samples pose challenges for, and inspire novel developments in classiﬁer design and gene selection methodologies (Wang et al., 2008). For multicategory classiﬁcation using gene expression data, various classiﬁers have been proposed and have achieved promising performance, including k-Nearest Neighbor Rule (kNN) (Golub et al., 1999), artiﬁcial neural networks (Wang et al., 2006), Support Vector Machine (SVM) (Ramaswamy et al., 2001), Na¨ve Bayes Classiﬁer (NBC) (Liu et al., 2002), Weighted Votes (Tibshirani et al., 2002), and Linı ear Regression (Fort and Lambert-Lacroix, 2005). Many comparative studies show that SVM based classiﬁers outperform other methods on most bench-mark microarray data sets (Li et al., 2004; Statnikov et al., 2005). An integral part of classiﬁer design is gene selection, which can improve both classiﬁcation accuracy and diagnostic economy (Liu et al., 2002; Shi et al., 2008; Wang et al., 2008). Many microarray-based studies suggest that, irrespective of the classiﬁcation method, gene selection is vital for achieving good generalization performance (Statnikov et al., 2005). For multicategory classiﬁcation using gene expression data, the criterion function for gene selection should possess high sensitivity and speciﬁcity, well match the speciﬁc classiﬁers used, and identify gene markers that are both statistically reproducible and biologically plausible (Shi et al., 2008; Wang et al., 2008). There are limitations associated with existing gene selection methods (Li et al., 2004; Statnikov et al., 2005). While wrapper methods consider joint discrimination power of a gene subset, complex clas2142 PUG-OVRSVM siﬁers used in wrapper algorithms for small sample size may overﬁt, producing non-reproducible gene subsets (Li et al., 2004; Shi et al., 2008). Moreover, discernment of the (biologically plausible) gene interactions retained by wrapper methods is often difﬁcult due to the black-box nature of most classiﬁers (Shedden et al., 2003). Conversely, most ﬁltering methods for multicategory classiﬁcation are straightforward extensions of binary discriminant analysis. These methods are devised without well matching to the classiﬁer that is used, which typically leads to suboptimal classiﬁcation performance (Statnikov et al., 2005). Popular multicategory ﬁltering methods (which are extensions of two-class methods) include Signal-to-Noise Ratio (SNR) (Dudoit et al., 2002; Golub et al., 1999), Student’s t-statistics (Dudoit et al., 2002; Liu et al., 2002), the ratio of Between-groups to Within-groups sum of squares (BW) (Dudoit et al., 2002), and SVM based Recursive Feature Elimination (RFE) (Li and Yang, 2005; Ramaswamy et al., 2001; Zhou and Tuck, 2007). However, as pointed out by Loog et al. (2001) in proposing their weighted Fisher criterion (wFC), simple extensions of binary discriminant analysis to multicategory gene selection are suboptimal because they overemphasize large betweenclass distances, that is, these methods choose gene subsets that preserve the distances of (already) well-separated classes, without reducing (and possibly with increase in) the large overlap between neighboring classes. This observation and the application of wFC to multicategory classiﬁcation are further evaluated experimentally by Wang et al. (2006) and Xuan et al. (2007). The work most closely related to our gene selection scheme is that of Shedden et al. (2003). These investigators focused on marker genes that are highly expressed in one phenotype relative to one or more different phenotypes and proposed a tree-based one-versus-rest (OVR) fold change evaluation between mean expression levels. The potential limitation here is that the criterion function considers the “rest of the classes” as a “super class”, and thus may select genes that can distinguish a single class from the remaining super class, yet without giving any beneﬁt in discriminating between classes within the super class. Such genes may compromise multicategory classiﬁcation accuracy, especially when a small gene subset is chosen. It is also important to note that, while univariate or multivariate analysis methods using complex criterion functions may reveal subtle marker effects (Cai et al., 2007; Liu et al., 2005; Xuan et al., 2007; Zhou and Tuck, 2007), they are also prone to overﬁtting. Recent studies have found that for small sample sizes, univariate methods fared comparably to multivariate methods (Lai et al., 2006; Shedden et al., 2003) and simple fold change analysis produced more reproducible marker genes than signiﬁcance analysis of variance-incorporated t-tests (Shi et al., 2008). In this paper, we propose matched design of the gene selection mechanism and a committee classiﬁer for multicategory molecular classiﬁcation using microarray gene expression data. A key feature of our approach is to match a simple one-versus-everyone (OVE) gene selection scheme to the OVRSVM committee classiﬁer (Ramaswamy et al., 2001). We focus on marker genes that are highly expressed in one phenotype relative to each of the remaining phenotypes, namely Phenotypic Up-regulated Genes (PUGs). PUGs are identiﬁed using the fold change ratio computed between the speciﬁed phenotype mean and each of the remaining phenotype means. Thus, we consider a gene to be a marker for the speciﬁed phenotype if the average expression associated with this phenotype is high relative to the average expressions in each of the other phenotypes. To assure evenhanded resources for discriminating both neighboring and well-separated classes, we use a ﬁxed number of PUGs for each phenotypic class and pool all phenotype-speciﬁc PUGs together to form a gene marker subset used by the OVRSVM committee classiﬁer. All PUGs referenced by the committee classiﬁer are individually interpretable as potential markers for phenotypic classes, allowing each 2143 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG gene to inform the classiﬁer in a way that is consistent with its mechanistic role (Shedden, et al., 2003). Since PUGs are the union of subPUGs selected by simple univariate OVE fold change analysis, they are expected to be statistically reproducible (Lai et al., 2006; Shedden et al., 2003; Shi et al., 2008). We tested PUG-OVRSVM on ﬁve publicly available benchmarks and one in-house microarray gene expression data set and on two simulation data sets, observing signiﬁcantly improved performance with lower error rates, fewer marker genes, and higher performance stability, as compared to several widely-adopted gene selection and classiﬁcation methods. The reference gene selection methods are OVRSNR (Golub et al., 1999), OVRt-stat (Liu et al., 2002), pooled BW (Dudoit et al., 2002), and OVRSVM-RFE (Guyon et al., 2002), and the reference classiﬁers are kNN, NBC, and one-versus-one (OVO) SVM. With accuracy estimated by leave-one-out cross-validation (LOOCV) (Hastie et al., 2001), our experimental results show that PUG-OVRSVM outperforms all combinations of the above referenced gene selection and classiﬁcation methods in the two simulation data sets and 5 out of the 6 real microarray gene expression data sets, and produces comparable performance on the one remaining data set. Speciﬁcally, tested on the widely-used benchmark microarray gene expression data set “multicategory human cancers data” (GCM) (Ramaswamy et al., 2001; Statnikov et al., 2005), PUG-OVRSVM produces a lower error rate of 11.05% (88.95% correct classiﬁcation rate) than the best known benchmark error rate of 16.72% (83.28% correct classiﬁcation rate) (Cai et al., 2007; Zhou and Tuck, 2007). 2. Methods In this section, we ﬁrst discuss multicategory classiﬁcation and associated feature selection, with an emphasis on OVRSVM and application to gene selection for the microarray domain. This discussion then naturally leads to our proposed PUG-OVRSVM scheme. 2.1 Maximum a Posteriori Decision Rule Classiﬁcation of heterogeneous diseases using gene expression data can be considered a Bayesian hypothesis testing problem (Hastie et al., 2001). Let xi = [xi1 , ..., xi j , ..., xid ] be the real-valued gene expression proﬁle associated with sample i across d genes for i = 1, . . . , N and j = 1, . . . , d. Assume that the sample points xi come from M classes, and denote the class conditional probability density function and class prior probability by p (xi | ωk ) and P (ωk ), respectively, for k = 1, . . . , M. To minimize the Bayes risk averaged over all classes, the optimum classiﬁer uses the well-known maximum a posteriori (MAP) decision rule (Hastie et al., 2001). Based on Bayes’ rule, the class posterior probability for a given sample xi is P (ωk | xi ) = P (ωk ) p (xi | ωk ) M ∑k′ =1 P (ωk′ ) p (xi | ωk′ ) and is used to (MAP) classify xi to ωk when P (ωk | xi ) > P (ωl | xi ) for all l = k. 2144 (1) PUG-OVRSVM 2.2 Supervised Learning and Committee Classiﬁers Practically, multicategory classiﬁcation using the MAP decision rule can be approximated using parameterized discriminant functions that are trained by supervised learning. Let fk (xi , θ), k = 1, 2, . . . , M, be the M outputs of a machine classiﬁer designed to discriminate between M classes (>2), where θ represents the set of parameters that fully specify the classiﬁer, and with the output values assumed to be in the range [0, 1]. The desired output of the classiﬁer will be “1” for the class to which the sample belongs and “0” for all other classes. Suppose that the classiﬁer parameters are selected based on a training set so as to minimize the mean squared error (MSE) between the outputs of the classiﬁer and the desired (class target) outputs, MSE = 1 M [ fk (xi , θ) − 1]2 + ∑l=k fl2 (xi , θ) . ∑ N k=1 xi∑ k ∈ω (2) Then, it can be shown that the classiﬁer is being trained to approximate the posterior probability for class ωk given the observed xi , that is, the classiﬁer outputs will converge to the true posterior class probabilities fk (xi , θ) → P (ωk | xi ) if we allow the classiﬁer to be arbitrarily complex and if N is made sufﬁciently large. This result is valid for any classiﬁer trained with the MSE criterion, where the parameters of the classiﬁer are adjusted to simultaneously approximate M discriminant functions fk (xi , θ) (Gish, 1990). While there are numerous machine classiﬁers that can be used to implement the MAP decision rule (1) (Hastie et al., 2001), a simple yet elegant way of discriminating between M classes, and which we adopt here, is based on an OVRSVM committee classiﬁer (Ramaswamy et al., 2001; Rifkin and Klautau, 2002; Statnikov et al., 2005). Intuitively, each term within the sum over k in (2) corresponds to an OVR binary classiﬁcation problem and can be effectively minimized by suitable training of a binary classiﬁer (discriminating class k from all other classes). By separately minimizing the MSE associated with each term in (2) via binary classiﬁer training and, thus, effectively minimizing the total MSE, a set of discriminant functions { fk (xi , θk ⊆ θ)} can be constructed which, given a new sample point, apply the decision rule (1), but with fk (xi , θ) playing the role of the posterior probability. Among the great variety of binary classiﬁers that use regularization to control the capacity of the function spaces they operate in, the best known example is the SVM (Hastie et al., 2001; Vapnik, 1998). To carry over the advantages of regularization approaches for binary classiﬁcation tasks to multicategory classiﬁcation, the OVRSVM committee classiﬁer uses M different SVM binary classiﬁers, each one separately trained to distinguish the samples in a single class from the samples in all remaining classes. For classifying a new sample point, the M SVMs are run, and the SVM that produces the largest (most positive) output value is chosen as the “winner” (Ramaswamy et al., 2001). For more detailed discussion, see the critical review and experimental comparison by Rifkin and Klautau (2002). Figure 1 shows an illustrative OVRSVM committee classiﬁer for three classes. The OVRSVM committee classiﬁer has proved highly successful at multicategory classiﬁcation tasks involving ﬁnite or limited amounts of high dimensional data in real-world applications. OVRSVM produces results that are often at least as accurate as other more complicated methods including single machine multicategory schemes (Statnikov et al., 2005). Perhaps more importantly for our purposes, the OVR scheme can be matched with an OVE gene selection method, as we elaborate next. 2145 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG Figure 1: Conceptual illustration of OVR committee classiﬁer for multicategory classiﬁcation (three classes, in this case). The dotted lines are the decision hyperplanes associated with each of the component binary SVMs and the bold line-set represents the ﬁnal decision boundary after the winner-take-all classiﬁcation rule is applied. 2.3 One-Versus-Everyone Fold-change Gene Selection While gene selection is vital for achieving good generalization performance (Guyon et al., 2002; Statnikov et al., 2005), perhaps even more importantly, the identiﬁed genes, if statistically reproducible and biologically plausible, are “markers”, carrying information about the disease phenotype (Wang et al., 2008). We will propose two novel, effective gene selection methods for multicategory classiﬁcation that are well-matched to OVRSVM committee classiﬁers, namely, OVR and OVE fold-change analyses. OVR fold-change based PUG selection follows directly from the OVRSVM scheme. Let Nk be the number of sample points belonging to phenotype k; the geometric mean of the expression levels (on the untransformed scale) for gene j under phenotype k is Nk µ j (k) = ∏i∈ω k xi j j = 1, . . . , d; k = 1, . . . , M. Then, we deﬁne the OVRPUGs as: M M JPUG (k) = JPUG = j µ j (k) ∏l=k µ j (l) M−1 k=1 k=1 τk (3) where {τk } are pre-deﬁned thresholds chosen so as to select a ﬁxed (equal) number of PUGs for each phenotype k. This PUG selection scheme (3) is similar to what has been previously proposed by Shedden et al. (2003): M M JPUG (k) = JPUG = k=1 j k=1 µ j (k) N−Nk ∏i∈ωk xi j / τk . (4) The critical difference between (3) and (4) is that the denominator term in (3) is the overall geometric center of the “geometric centers” associated with each of the remaining phenotypes while 2146 PUG-OVRSVM the denominator term in (4) is the geometric center of all sample points belonging to the remaining phenotypes. When {Nk } are signiﬁcantly imbalanced for different k, the denominator term in (4) will be biased toward the dominant phenotype(s). However, a problem associated with both PUG selection schemes speciﬁed by (3) and (4) (and with the OVRSNR criterion Golub et al., 1999) is that the criterion function considers the remaining classes as a single super class, which is suboptimal because it ignores a gene’s ability to discriminate between classes within the super class. We therefore propose OVE fold-change based PUG selection to fully support the objective of multicategory classiﬁcation. Speciﬁcally, the OVEPUGs are deﬁned as: M M JPUG (k) = JPUG = k=1 j k=1 µ j (k) maxl=k µ j (l) τk (5) where the denominator term is the maximum phenotypic mean expression level over the remaining phenotype classes. This seemingly technical modiﬁcation turns out to have important consequences since it assures that the selected PUGs are highly expressed in one phenotype relative to each of the remaining phenotypes, that is, “high” (up-regulated) in phenotype k and “low” (down-regulated) in all phenotypes l = k. In our experimental results, we will demonstrate that (5) leads to better classiﬁcation accuracy than (4) on a well-known multi-class cancer domain. Adopting the same strategy as in Shedden et al. (2003), to assure even-handed gene resources for discriminating both neighboring and well-separated classes, we select a ﬁxed (common) number of top-ranked phenotype-speciﬁc subPUGs for each phenotype, that is, JPUG (k) = NsubPUG for all k, and pool all these subPUGs together to form the ﬁnal gene marker subset JPUG for the OVRSVM committee classiﬁer. In our experiments, the optimum number of PUGs per phenotype, NsubPUG , is determined by surveying the curve of classiﬁcation accuracy versus NsubPUG and selecting the number that achieves the best classiﬁcation performance. More generally, in practice, NsubPUG can be chosen via a cross validation procedure. Figure 2 shows the geometric distribution of the selected PUGs speciﬁed by (5), where the PUGs (highlighted data points) constitute the lateral-edge points of the convex pyramid deﬁned by the scatter plot of the phenotypic mean expressions (Zhang et al., 2008). Different from the PUG selection schemes given by (3) and (4), the PUGs selected based on (5) are most compact yet informative, since the down-regulated genes that are not differentially expressed between the remaining phenotypes (the genes on the lateral faces of the scatter plot convex pyramid) are excluded. From a statistical point of view, extensive studies on the normalized scatter plot of microarray gene expression data by many groups including our own indicate that the PUGs selected by (5) approximately follow an independent multivariate super-Gaussian distribution (Zhao et al., 2005) where subPUGs are mutually exclusive and phenotypic gene expression patterns deﬁned over the PUGs are statistically independent (Wang et al., 2003). It is worth noting that the PUG selection by (5) also adopts a univariate fold-change evaluation that does not require calculation of either expression variance or of correlation between genes (Shi et al., 2008). For the small sample size case typical of microarray data, multivariate gene selection schemes may introduce additional uncertainty in estimating the correlation structure (Lai et al., 2006; Shedden et al., 2003) and thus may fail to identify true gene markers (Wang et al., 2008). The exclusion of the variance in our criterion is also supported by the variance stabilization theory (Durbin et al., 2002; Huber et al., 2002), because the geometric mean in (5) is equivalent to the arithmetic mean after logarithmic transformation and the gene expression after logarithmic transfor2147 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG Figure 2: Geometric illustration of the selected one-versus-everyone phenotypic upregulated genes (OVEPUGs) associated with three phenotypic classes. Three-dimensional geometric distribution (on the untransformed scale) of the selected OVEPUGs, which reside around the lateral-edges of the phenotypic gene expression scatter plot convex pyramid, is shown in the left subﬁgure. A projected distribution of the selected OVEPUGs together with OVEPDGs is shown in the right cross-sectional plot, where OVEPDGs reside along the face-edges of the cross-sectional triangle. mation approximately has the equal variance across different genes, especially for the up-regulated genes. Corresponding to the deﬁnition of OVEPUGs, the OVEPDGs (which are down-regulated in one class while being up-regulated in all other classes) can be deﬁned by the following criterion: M M JPDG (k) = JPDG = j k=1 k=1 minl=k µ j (l) µ j (k) τk . (6) Furthermore, the combination of PUGs and PDGs can be deﬁned as: M M JPUG+PDG (k) = JPUG+PDG = k=1 j max k=1 minl=k µ j (l) µ j (k) , µ j (k) maxl=k µ j (l) τk . (7) Purely from the machine learning view, PDGs have the theoretical capability of being as discriminating as PUGs. Thus, PDGs merit consideration as candidate genes. However, there are several critical differences, with consequential implications, between lowly-expressed genes and highly-expressed genes, such as the extraordinarily large proportion and relatively large noise of the lowly-expressed genes. We have evaluated the classiﬁcation performance of PUGs, PDGs, and 2148 PUG-OVRSVM PUGs+PDGs, respectively. Experimental results show that PDGs have less discriminatory power than PUGs and the inclusion of PDGs actually worsens classiﬁcation accuracy, compared to just using PUGs. Experiments and further discussion will be given in the results section. 2.4 Review of Relevant Gene Selection Methods Here we brieﬂy review four benchmark gene selection methods that have been previously proposed for multicategory classiﬁcation, namely, OVRSNR (Golub et al., 1999), OVR t-statistic (OVRt-stat) (Liu et al., 2002), BW (Dudoit et al., 2002), and SVMRFE (Guyon et al., 2002). Let µ j,k and µ j,-k be the arithmetic means of the expression levels of gene j associated with phenotype k and associated with the super class of remaining phenotypes, respectively, on the log-transformed scale, with σ j,k and σ j,-k the corresponding standard deviations. OVRSNR gene selection for multicategory classiﬁcation is given by: M M JOVRSNR (k) = JOVRSNR = k=1 j k=1 µ j,k − µ j,-k σ j,k + σ j,-k τk , (8) where τk is a pre-deﬁned threshold (Golub et al., 1999). To assess the statistical signiﬁcance of the difference between µ j,k and µ j,-k , OVRt-stat applies a test of the null hypothesis that the means of two assumed normally distributed measurements are equal. Accordingly, OVRt-stat gene selection is given by Liu et al. (2002):       M  M  µ j,k − µ j,-k τk , (9) j JOVRt-stat (k) = JOVRt-stat =    2 2 k=1  k=1   σ j,k Nk + σ j,-k (N − Nk ) where the p-values associated with each gene may be estimated. As aforementioned, one limitation of the gene selection schemes (8) and (9) is that the criterion function considers the remaining classes as a single group. Another is that they both require variance estimation. Dudoit et al. (2002) proposed a pooled OVO gene selection method based on the BW sum of squares across all paired classes. Speciﬁcally, BW gene selection is speciﬁed by JBW = j ∑N ∑M 1ωk (i) µ j,k − µ j i=1 k=1 2 ∑N ∑M 1ωk (i) xi j − µ j,k i=1 k=1 2 τ , (10) where µ j is the global arithmetic center of gene j over all sample points and 1ωk (i) is the indicator function reﬂecting membership of sample i in class k. As pointed out by Loog et al. (2001), BW gene selection may only preserve the distances of already well-separated classes rather than neighboring classes. From a dimensionality reduction point of view, Guyon et al. (2002) proposed a feature subset ranking criterion for linear SVMs, dubbed the SVMRFE. Here, one ﬁrst trains a linear SVM classiﬁer on the full feature space. Features are then ranked based on the magnitude of their weights and are eliminated in the order of increasing weight magnitude. A widely adopted reduction strategy is to eliminate a ﬁxed or decreasing percentage of features corresponding to the bottom portion of the ranked weights and then to retrain the SVM on the reduced feature space. Application to microarray gene expression data shows that the genes selected matter more than the classiﬁers with which they are paired (Guyon et al., 2002). 2149 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG 3. Results We tested PUG-OVRSVM on ﬁve benchmarks and one in-house real microarray data set, and compared the performance to several widely-adopted gene selection and classiﬁcation methods. 3.1 Description of the Real Data Sets The numbers of samples, phenotypes, and genes, as well as the microarray platforms used to generate these gene expression data sets, are brieﬂy summarized in Supplementary Tables 1∼7. The six data sets are the MIT 14 Global Cancer Map data set (GCM) (Ramaswamy et al., 2001), the NCI 60 cancer cell lines data set (NCI60) (Staunton et al., 2001), the University of Michigan cancer data set (UMich) (Shedden et al., 2003), the Central Nervous System tumors data set (CNS) (Pomeroy et al., 2002), the Muscular Dystrophy data set (MD) (Bakay et al., 2006), and the Norway Ascites data set (NAS). To assure a meaningful and well-grounded comparison, we emphasized data quality and suitability in choosing these test data sets. For example, the data sets cannot be too “simple” (if the classes are well-separated, all methods perform equally well) or too “complex” (no method will then perform reasonably well), and each class should contain sufﬁcient samples to support some form of cross-validation assessment. We also performed several important pre-processing steps widely adopted by other researchers (Guyon et al., 2002; Ramaswamy et al., 2001; Shedden et al., 2003; Statnikov et al., 2005). When the expression levels in the raw data take negative values, probably due to global probe-set calls and/or data normalization procedures, these negative values are replaced by a ﬁxed small quantity (Shedden et al., 2003). On the log-transformed scale, we further conducted a variance-based unsupervised gene ﬁltering operation to remove the genes whose expression standard deviations (across all samples) were less than a pre-determined small threshold; this effectively reduces the number of genes by half (Guyon et al., 2002; Shedden et al., 2003). 3.2 Experiment Design We decoupled the two key steps of multicategory classiﬁcation: 1) selecting an informative subset of marker genes and then 2) ﬁnding an accurate decision function. For the crucial ﬁrst step we implemented ﬁve gene selection methods, including OVEPUG speciﬁed by (5), OVRSNR speciﬁed by (8), OVRt-stat speciﬁed by (9), pooled BW speciﬁed by (10), and SVMRFE described in Ramaswamy et al. (2001). We applied these methods to the six data sets, and for each data set, we selected a sequence of gene subsets with varying sizes, indexed by NsubPUG , the number of genes per class. In our experiments, this number was increased from 2 up to 100. There are several reasons why we do not go beyond 100 subPUGs per class. First, classiﬁcation accuracy may be either ﬂat or monotonically decreasing as the number of features increases beyond a certain point, due to the theoretical bias-variance dilemma. Second, even in some cases where best performance is achieved using all the gene features, the idea of feature selection is to ﬁnd the minimum number of features needed to achieve good (near-optimal) classiﬁcation accuracy. Third, when NsubPUG = 100, the total number of genes used for classiﬁcation is already quite large (this number is maximized if the sets JPUG (k) are mutually exclusive, in which case it is NsubPUG times the number of classes). Fourth, but not least important, a large feature reduction may be necessary not only complexity-wise, but also for interpreting the biological functions and pathway involvement when the selected PUGs are most relevant and statistically reproducible. 2150 PUG-OVRSVM The quality of the marker gene subsets was then assessed by prediction performance on four subsequently trained classiﬁers, including OVRSVM, kNN, NBC, and OVOSVM. In relation to the proposed PUG-OVRSVM approach, we evaluated all combinations of these four different gene selection methods and three different classiﬁers on all six benchmark microarray gene expression data sets. To properly estimate the accuracy of predictive classiﬁcation, a validation procedure must be carefully designed, recognizing limits on the accuracy of estimated performance, in particular for small sample size. Clearly, classiﬁcation accuracy must be assessed on labelled samples ‘unseen’ during training. However, for multicategory classiﬁcation based on small, class-imbalanced data sets, single batch held-out test data may be precluded, as there will be insufﬁcient samples for both accurate classiﬁer training and accurate validation (Hastie et al., 2001). A practical alternative is a sound cross-validation procedure, wherein all the data are used for both training and testing, but with held-out samples in a testing fold not used for any phase of classiﬁer training, including gene selection and classiﬁer design (Wang et al., 2008). In our experiments, we chose LOOCV, wherein a test fold consists of a single sample; the rest of the samples are placed in the training set. Using only the training set, the informative genes are selected and the weights of the linear OVRSVM are ﬁt to the data (Liu et al., 2005; Shedden et al., 2003; Yeang et al., 2001). It is worth noting that LOOCV is approximately unbiased, lessening the likelihood of misestimating the prediction error due to small sample size; however, LOOCV estimates do have considerable variance (Braga-Neto and Dougherty, 2004; Hastie et al., 2001). We evaluated both the lowest “sustainable” prediction error rate and the lowest prediction error rate, where the sequence of sustainable prediction error rates were determined based on a moving-average of error rates along the survey axis of the number of genes used for each class, NsubPUG , with a moving window of width 5. We also report the number of genes per class at which the best sustainable performance was obtained. While the error rate is estimated through LOOCV and the optimum number of PUGs used per class is obtained by the aforementioned surveying strategy, we should point out that a two-level LOOCV could be applied to jointly determine the optimum NsubPUG and estimate the associated error rate; however, such an approach is computationally expensive (Statnikov et al., 2005). For the settings of structural parameters in the classiﬁers, we used C = 1.0 in the SVMs for all experiments (Vapnik, 1998), and chose k = 1, 2, 3 in kNNs under different training sample sizes per class, as recommended by Duda et al. (2001). 3.3 Experimental Results Our ﬁrst comparative study focused on the GCM data widely used for evaluating multicategory classiﬁcation algorithms (Cai et al., 2007; Ramaswamy et al., 2001; Shedden et al., 2003; Zhou and Tuck, 2007). The performance curves of OVRSVM committee classiﬁers trained on the commonly pre-processed GCM data using the ﬁve different gene selection methods (OVEPUG, OVRSNR, OVRt-stat, BW, and SVMRFE) are detailed in Figure 3. It can be seen that our proposed OVEPUG selection signiﬁcantly improved the overall multicategory classiﬁcation when using different numbers of marker genes, as compared to the results produced by the four competing gene selection methods. For example, using as few as 9 genes per phenotypic class (with 126 distinct genes in total, that is, mutually exclusive PUGs for each class), we classiﬁed 164 of 190 (86.32%) of the tumors correctly. Furthermore, using LOOCV on the GCM data set of 190 primary malignant tumors, and using the optimal number of genes (61 genes per phenotypic class or 769 unique genes 2151 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG in total), we achieved the best (88.95% or 169 of 190 tumors) sustainable correct predictions. In contrast, at its optimum performance, OVRSNR gene selection achieved 85.37% sustainable correct predictions using 25 genes per phenotypic class, OVRt-stat gene selection achieved 84.53% sustainable correct predictions using 71 genes per phenotypic class, BW gene selection achieved 80.53% sustainable correct predictions using 94 genes per phenotypic class, and SVMRFE gene selection achieved 84.74% sustainable correct predictions using 96 genes per phenotypic class. In our comparative study, instead of solely comparing the lowest error rates achieved by different gene selection methods, we also emphasized the sustainable correct prediction rates, as potential overﬁtting to the data may produce an (unsustainably) good prediction performance. For our experiments in Figure 3, based on the realistic assumption that the probability of good predictions purely “by chance” over a sequence of consecutive gene numbers is low, we deﬁned the sustainable prediction/error rates based on the moving-averaged prediction/error rates over δ = 5 consecutive gene numbers. Here, δ gives the sustainability requirement. Figure 3: Comparative study on ﬁve gene selection methods (OVEPUG, OVRSNR, OVRt-stat, BW, and SVMRFE) using the GCM benchmark data set. The curves of classiﬁcation error rates were generated by using OVRSVM committee classiﬁers with varying size of the input gene subset. For the purpose of information sharing with readers, based on publicly reported optimal results for different methods, we have summarized in Table 1 the comparative performance achieved by PUG-OVRSVM and eight existing/competing methods on the benchmark GCM data set, along with the gene selection methods used, the chosen classiﬁers, sample sizes, and the chosen crossvalidation schemes. Obviously, since the reported prediction error rates were generated by different algorithms and under different conditions, any conclusions based on simple/direct comparisons of the reported results must be carefully drawn. We have chosen not to independently reproduce results 2152 PUG-OVRSVM by re-implementing the methods listed in Table 1, ﬁrstly because we typically do not have access to public domain code implementing other authors’ methods and secondly because we feel that high reproducibility of previously published results may not be expected without knowing some likely undeclared optimization steps and/or additional control parameters used in the actual computer codes. Nevertheless, many reported prediction error rates on the GCM data set were actually based on the same/similar training sample set (144 ∼ 190 primary tumors) and the LOOCV scheme used in our PUG-OVRSVM experiments; furthermore, it was reported that the prediction error rates estimated by LOOCV and 144/54 split/held-out test were very similar (Ramaswamy et al., 2001). Speciﬁcally, the initial work on GCM by Ramaswamy et al. (2001) reported an achieved 77.78% prediction rate, and some improved performance was later reported by Yeang et al. (2001) and Liu et al. (2002), achieving 81.75% and 79.99% prediction rates, respectively. In the work most closely related to our gene selection scheme by Shedden et al. (2003), using a kNN tree classiﬁer and using OVR fold-change based gene selection speciﬁed by (4), a prediction rate of 82.63% was achieved. In relation to these reported results on GCM, as indicated in Table 1, our proposed PUG-OVRSVM method produced the best sustainable prediction rate of 88.95%. References Gene-select Classiﬁer Sample CV scheme Error rate Ramaswamy et al. (2001) OVRSVM RFE OVRSVM 144&198 LOOCV 144/54 22.22% Yeang et al. (2001) N/A OVRSVM 144 LOOCV 18.75% Ooi and Tan (2003) Genetic algorithm MLHD 198 144/54 18.00% Shedden et al. (2003) OVR fold-change kNN Tree 190 LOOCV 17.37% Liu et al. (2005) Genetic algorithm OVOSVM N/A LOOCV 20.01% Statnikov et al. (2005) No gene selection CS-SVM 308 10-fold 23.40% Zhou and Tuck (2007) CS-SVM RFE OVRSVM 198 4-fold 16.72% Cai et al. (2007) DISC-GS kNN 190 144/46 21.74% PUG-OVRSVM PUG OVRSVM 190 LOOCV 11.05% Table 1: Summary of comparative performances by OVEPUG-OVRSVM and eight competing methods (based on publicly reported optimum results) on the GCM benchmark data set. A more stringent evaluation of the robustness of a classiﬁcation method is to carry out the predictions on multiple data sets and then assess the overall performance (Statnikov et al., 2005). Our second comparative study evaluated the aforementioned ﬁve gene selection methods using the six benchmark microarray gene expression data sets. To determine whether the genes selected matter more than the classiﬁers used (Guyon et al., 2002), we used a common OVRSVM committee classiﬁer and LOOCV scheme in all the experiments, and summarized the corresponding results in Table 2. For each experiment that used a distinct gene selection scheme applied to a distinct data set, we reported both sustainable (with sustainability requirement δ = 5) and lowest (within parentheses) prediction error rates, as well as the number of genes per class that were used to produce these results. Clearly, the selected PUGs based on (5) produced the highest overall sustainable prediction rates as compared to the other four competing gene selection methods. Speciﬁcally, PUG is the consistent winner in 22 of 24 competing experiments (combinations of four gene selection schemes and six testing data sets). It should be noted that although BW and OVRSNR achieved comparably low prediction error rates on the CNS data set (with relatively balanced mixture distributions), they 2153 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG also produced high prediction error rates on the other testing data sets; the other competing gene selection methods also show some level of performance instability across data sets. Gene-select GCM NCI60 UMich CNS MD NAS OVE PUG 11.05% (11.05%) [61 g/class] 27.33% (26.67%) [52 g/class] 1.08% (0.85%) [26 g/class] 7.14% (7.14%) [71 g/class] 19.67% (19.01%) [46 g/class] 13.16% (13.16%) [42 g/class] OVR SNR 14.63% (13.68%) [25 g/class] 31.67% (31.67%) [58 g/class] 1.42% (1.42%) [62 g/class] 7.14% (7.14%) [57 g/class] 23.97% (23.97%) [85 g/class] 16.32% (15.79%) [54 g/class] OVR t-stat 15.47% (15.26%) [71 g/class] 31.67% (31.67%) [56 g/class] 1.70% (1.70%) [45 g/class] 7.62% (7.14%) [92 g/class] 23.47% (22.31%) [56 g/class] 15.79% (15.79%) [74 g/class] BW 19.47% (18.95%) [94 g/class] 31.67% (31.67%) [55 g/class] 1.30% (1.13%) [92 g/class] 7.14% (7.14%) [56 g/class] 19.83% (19.01%) [71 g/class] 21.05% (21.05%) [65 g/class] SVM RFE 15.26% (14.21%) [96 g/class] 29.00% (28.33%) [81 g/class] 1.13% (1.13%) [58 g/class] 14.29% (14.29%) [53 g/class] 29.09% (28.10%) [73 g/class] 32.11% (31.58%) [94 g/class] Table 2: Performance comparison between ﬁve different gene selection methods tested on six benchmark microarray gene expression data sets, where the predictive classiﬁcation error rates for all methods were generated based on OVRSVM committee classiﬁcation and an LOOCV scheme. Both sustainable and lowest (within parentheses) error rates are reported together with number of genes used per class. To give more complete comparisons that also involved different classiﬁers (Statnikov et al., 2005), we further illustrate the superior prediction performance of the matched OVEPUG selection and OVRSVM classiﬁer as compared to the best results produced by combinations of three different classiﬁers (OVOSVM, kNN, NBC) and four gene selection methods (PUG, OVRSNR, OVRt-stat, pooled BW). The optimum experimental results achieved over all combinations of these methods on the six data sets are summarized in Table 3, where we report both sustainable prediction error rates and the corresponding gene selection methods. Again, PUG-OVRSVM outperformed all other methods on all six data sets and was a clear winner in all 15 competing experiments. Our comparative studies also reveal that although gene selection is a critical step of multi-category classiﬁcation, the classiﬁers used do indeed play an important role in achieving good prediction performance. 3.4 Comparison Results on the Realistic Simulation Data Sets To more reliably validate and compare the performance of the different gene selection methods, we have conducted additional experiments involving realistic simulations. The advantage of using synthetic data is that, unlike the real data sets often with small sample size and with LOOCV as the only applicable validation method, large testing samples can be generated to allow an accurate and reliable assessment of a classiﬁer’s generalization performance. Two different simulation approaches were implemented. In both, we modeled the joint distribution for microarray data under each class and generated i.i.d. synthetic data sets consistent both with these distributions and with assumed class priors. In the ﬁrst approach, we chose the class-conditional models consistent with commonly 2154 PUG-OVRSVM GCM NCI60 UMich CNS MD NAS OVR SVM 11.05% (OVEPUG) 27.33% (OVEPUG) 1.08% (OVEPUG) 7.14% (OVEPUG) 19.67% (OVEPUG) 13.16% (OVEPUG) OVO SVM 14.74% (OVEPUG) 33.33% (OVRSNR) 1.70% (OVEPUG) 9.52% (BW) 19.83% (BW) 16.32% (OVRSNR) kNN 21.05% (OVEPUG) 31.67% (OVRt-stat) 2.27% (OVEPUG) 13.33% (OVEPUG) 21.81% (BW) 13.68% (OVRt-stat) NBC 36.00% (OVRSNR) 51.67% (OVRSNR) 2.83% (OVRt-stat) 37.62% (BW) 37.69% (BW) 34.21% (OVEPUG) Table 3: Performance comparison based on the lowest predictive classiﬁcation error rates produced by OVEPUG-OVRSVM and the optimum combinations of ﬁve different gene selection methods and three different classiﬁers, tested on six benchmark microarray gene expression data sets and assessed via the LOOCV scheme. accepted properties of microarray data (few discriminating features, many non-discriminating features, and with small sample size) (Hanczar and Dougherty, 2010; Wang et al., 2002). In the second approach, we directly estimated the class-conditional models based on a real microarray data set and then generated the i.i.d. samples according to the learned models. 3.4.1 D ESIGN I We simulated 5000 genes, with 90 “relevant” and 4910 “irrelevant” genes. Inspired by gene clustering concept in modelling local correlations, we divided the genes into 1000 blocks of size ﬁve, each containing exclusively either relevant or irrelevant genes. Within each block the correlation coefﬁcient is 0.9, with zero correlation across blocks. Irrelevant genes are assumed to follow a (univariate) standard normal distribution, for all classes. Relevant genes also follow a normal distribution with variance 1 for all classes. There are three equally likely classes, A, B and C. The mean vectors of the 90 relevant genes under each class are shown in Table 4. The means were chosen to make the classiﬁcation task neither too easy nor too difﬁcult and to simulate unequal distances between the classes—A and B are relatively close, with C more distant from both A and B. The mean vector µ for each class µA [2.8 2.8 2.8 2.8 2.8 1 1 1 1 1 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 1 1 1 1 1 3 3 3 3 3 0.1 0.1 0.1 0.1 0.1] µB [1 1 1 1 1 2.8 2.8 2.8 2.8 2.8 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 2 2 2 2 2 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0.5 0.5 0.5 0.5 0.5 0 0 0 0 0 1 1 1 1 1 3 3 3 3 3 0.1 0.1 0.1 0.1 0.1] µC [1 1 1 1 1 1 1 1 1 1 14.4 14.4 14.4 14.4 14.4 8.5 8.5 8.5 8.5 8.5 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 10 9 9 9 9 9 10 10 10 10 10 3 3 3 3 3 10 10 10 10 10 10 10 10 10 10 10 10 10 10 8.5 8.5 8.5 8.5 8.5 8 8 8 8 8 9 9 9 9 9 11 11 11 11 11 7.1 7.1 7.1 7.1 7.1] Table 4: The mean vectors of the 90 relevant genes under each of the three classes. We randomly generated 100 synthetic data sets, each partitioned into a small training set of 60 samples (20 per class) and a large testing set of 6000 samples. 2155 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG 3.4.2 D ESIGN II The second approach models each class as a more realistic multivariate normal distribution N (µ, Σ), with the class’s mean vector µ and covariance matrix Σ directly learned from the real microarray data set GCM. Estimation of a covariance matrix is certainly a challenging task, speciﬁcally due to the very high dimensionality of the gene space (p = 15, 927 genes in the GCM data) and only a few dozen samples available for estimating p(p − 1)/2 free covariate parameters per class. It is also computationally prohibitive to generate random vectors based on full covariances on a general desktop computer. To address both of these problems, we applied a factor model (McLachlan and Krishnan, 2008), which can signiﬁcantly reduces the number of free parameters to be estimated while capturing the main correlation structure in the data. In factor analysis, the observed p × 1 vector t is modeled as t = µ + Wx + ε, where µ is the mean vector of observation t, W is a p × q matrix of factor loadings, x is the q × 1 latent variable vector with standard normal distribution N (0, I) and ε is noise with independent multivariate normal distribution N (0, Ψ), Ψ = diag σ2 , . . . , σ2 . The resulting covariance matrix Σ p 1 is Σ = WWT + Ψ. Estimation of Σ reduces to estimating W and Ψ, totaling p(q + 1) parameters. Usually, we have q much less than p. The factor model is learned via the EM algorithm (McLachlan and Krishnan, 2008), initialized by probabilistic principal component analysis (Tipping and Bishop, 1999). In our experiments, we set q = 5, which typically accounted for 60% of the energy. We also tried q = 3 and 7 and observed that the relative performance remained unchanged, although the absolute performance of all methods does change with q. Five phenotypic classes were used in our simulation: breast cancer, lymphoma, bladder cancer, leukemia and CNS. 100 synthetic data sets were generated randomly according to the learned class models from the real data of these ﬁve cancer types. The dimension for each sample is 15,927. For each data set, the training sample size was the same as used in the real data experiments, with 11, 22, 11, 30, and 20 samples in the ﬁve respective classes; and the testing set consisted of 3,000 samples, 600 per class. 3.5 Evaluation of Performance For a given gene-selection method and for each data set (indexed by i = 1, . . . , 100), the classiﬁer Fi is learned. We then evaluate Fi on the i-th testing set, and measure the error rate εi . Since the testing set has quite large sample size, we would expect εi to be close to the true classiﬁcation error rate for ¯ Fi . Over 100 simulation data sets, we then calculated both the average classiﬁcation error ε and the standard deviation σ. Furthermore, let εi,PUG denote the error rate associated with PUGs on testing set i, and similarly, let εi,SNR , εi,t-stat , εi,BW and εi,SV MRFE denote the error rates associated with the four peer gene selection methods. The error rate difference between two methods, for example, PUG and SNR, is deﬁned by Di (PUG, SNR) = εi,PUG − εi,SNR . 2156 PUG-OVRSVM For each synthetic data set, we deﬁne the “winner” as the one with the least testing error rate. For each method, the mean and standard deviation of the error rate and the frequency of winning are examined for performance evaluation. In addition, the histogram of error rate differences between PUG and peer methods are provided. 3.6 Experimental Results on the Simulation Data Sets We tested all gene selection methods using the common OVRSVM classiﬁer. All the experiments were done using the same procedure as on the real data sets, except with LOOCV error estimation replaced by the error estimation using large size independent testing data. Figure 4, analogous to Figure 3 while on the realistic synthetic data whose model was estimated from GCM data set (simulation data under design II), shows the comparative study on ﬁve gene selection methods (OVEPUG, OVRSNR, OVRt-stat, BW, and SVMRFE). Tables 5 and 6 show the average error, standard deviation, and frequency of winning, estimated based on the 100 simulation data sets. PUG has the smallest average error over all competing methods. PUG also is the most stable method (with the smallest standard deviation). Tables 7 and 8 provide the comparison results of the ﬁve competing methods on the ﬁrst ten data sets. Figures 5 and 6 show histograms of the error difference between PUG and other methods, where a negative value of the difference indicates better performance by PUG. The red bar shows the position where the two methods are equal. We can see that the vast majority of differences are negative. Actually, as indicated in Tables 5 and 6, there is no positive difference in the subﬁgures of Figure 5 and at most one positive difference in the subﬁgures of Figure 6. mean std deviation frequency of ‘winner’ PUG 0.0724 0.0052 100 SNR 0.1129 0.0180 0 t-stat 0.1135 0.0188 0 BW 0.1165 0.0177 0 SVMRFE 0.1203 0.0224 0 Table 5: The mean and standard deviation of classiﬁcation error and the frequency of winner based on 100 simulation data sets with design I. mean std deviation frequency of ‘winner’ PUG 0.0712 0.0201 99 SNR 0.1311 0.0447 0 t-stat 0.1316 0.0449 0 BW 0.2649 0.0302 0 SVMRFE 0.0910 0.0244 1 Table 6: The mean and standard deviation of classiﬁcation error and the frequency of winner based on 100 simulation data sets with design II. 3.7 Comparison Between PUGs and PDGs In this experiment, we selected PDGs according to the deﬁnition given in (6) and evaluated gene selection based on PUGs, PDGs, and based on their union, as given in (7). Again, all gene selection methods were coupled with the OVRSVM classiﬁer. Table 9 shows classiﬁcation performance for 2157 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG Figure 4: Comparative study on ﬁve gene selection methods (OVEPUG, OVRSNR, OVRt-stat, BW, and SVMRFE) on one simulation data set under design II. The curves of classiﬁcation error rates were generated by using OVRSVM committee classiﬁers with varying size of the input gene subset. PUG SNR t-stat BW SVMRFE sim 1 0.0864 0.1078 0.1109 0.1127 0.1030 sim 2 0.0773 0.1092 0.1089 0.0995 0.1009 sim 3 0.0697 0.1028 0.1022 0.1049 0.0967 sim 4 0.0681 0.1279 0.1251 0.1271 0.1219 sim 5 0.0740 0.1331 0.1333 0.1309 0.1248 sim 6 0.0761 0.1004 0.0991 0.1107 0.1016 sim 7 0.0740 0.1011 0.1016 0.1044 0.1107 sim 8 0.0721 0.1253 0.1268 0.1291 0.1191 sim 9 sim 10 0.0666 0.0758 0.0817 0.0838 0.0823 0.0832 0.0903 0.0845 0.1198 0.0933 Table 7: Comparison of the classiﬁcation error for the ﬁrst ten simulation data sets with design I. PUGs, PDGs and PUGs+PDGs. Clearly, PDGs have less discriminatory power than PUGs, and the inclusion of PDGs (generally) worsens classiﬁcation accuracy, compared with just using PUGs. sim 1 PUG 0.0694 SNR 0.1559 t-stat 0.1559 BW 0.2373 SVMRFE 0.0906 sim 2 0.0610 0.0659 0.0659 0.2698 0.0739 sim 3 0.0748 0.1142 0.1142 0.2510 0.0864 sim 4 0.0675 0.1211 0.1210 0.2650 0.0852 sim 5 0.0536 0.0508 0.0508 0.3123 0.0426 sim 6 0.0474 0.1937 0.1939 0.2464 0.0776 sim 7 0.0726 0.1568 0.1568 0.3070 0.0863 sim 8 0.0818 0.1464 0.1464 0.2236 0.0973 sim 9 sim 10 0.0560 0.0700 0.0797 0.0711 0.0797 0.0712 0.2800 0.3055 0.0655 0.0730 Table 8: Comparison of the classiﬁcation error for the ﬁrst ten simulation data sets with design II. 2158 PUG-OVRSVM Histogram of the error difference between PUG and SNR Histogram of the error difference between PUG and t−stat 25 20 Frequency 20 15 15 10 10 5 5 0 −0.12 −0.1 −0.08 −0.06 E PUG −0.04 −0.02 0 −0.12 0 −0.1 −0.08 −E −0.06 −0.04 E SNR PUG Histogram of the error difference between PUG and BW −0.02 0 −E t−stat Histogram of the error difference between PUG and SVMRFE 25 25 20 20 15 15 10 10 5 5 0 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 −0.12 0 −0.1 EPUG − EBW −0.08 −0.06 −0.04 −0.02 0 EPUG − ESVMRFE Figure 5: Histogram of the error difference between PUG and other methods with design I. Error Rate GCM NCI60 UMich CNS MD NAS PUG 11.05% 27.33% 1.08% 7.14% 19.67% 13.16% PDG 17.58% 30.33% 1.98% 9.52% 26.28% 25.79% PUG+PDG 14.53% 30.67% 1.13% 7.14% 23.14% 15.79% Table 9: Classiﬁcation comparison of PUG and PDG on the six benchmark data sets. There are several potential reasons that may jointly explain the non-contributing or even negative role of the included PDGs. First, the number of PDGs are much less than that of PUGs, that is, PUGs represent the signiﬁcant majority of informative genes when PUGs and PDGs are jointly considered, as shown in Table 10 (Top PUG+PDGs were selected with 10 genes per class and we counted how many PUGs are included in the total). Second, PDGs are less reliable than PUGs due to the noise characteristics of gene expression data, that is, low gene expressions contain relatively large additive noise after log-transformation (Huber et al., 2002; Rocke and Durbin, 2001). This is further exacerbated by the follow-up one-versus-rest classiﬁer because there are many more samples in the ‘rest’ group than in the ‘one’ group. This practically increases the relative noise/variability associated with PDGs in the ‘one’ group. In addition, PUGs are consistent with the practice of molecular pathology and thus may have broader clinical utility, for example, most currently available disease gene markers are highly expressed (Shedden et al., 2003). 2159 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG Histogram of the error difference between PUG and SNR Histogram of the error difference between PUG and t−stat 25 20 Frequency 20 15 15 10 10 5 5 0 −0.15 −0.1 −0.05 E 0 −0.15 0 −0.1 −E PUG −0.05 E SNR Histogram of the error difference between PUG and BW 0 −E PUG t−stat Histogram of the error difference between PUG and SVMRFE 20 30 25 15 20 10 15 10 5 5 0 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 E 0 −0.08 0 −0.06 −E PUG −0.04 E BW −0.02 0 −E PUG SVMRFE Figure 6: Histogram of the error difference between PUG and other methods with design II. GCM NCI60 UMich CNS MD NAS 65 No. of PUG 113 76 56 33 76 No. of PUG+PDG 140 90 60 50 130 70 % of PUG 80.71% 84.44% 93.33% 66.00% 58.46% 92.86% Table 10: Classiﬁcation comparison of PUG and PDG on the six benchmark data sets. 3.8 Marker Gene Validation by Biological Knowledge We have applied existing biological knowledge to validate biological plausibility of the selected PUG markers for two data sets, GCM and NAS. The full list of genes most highly associated with each of the 14 tumor types in the GCM data set are detailed in the Supplementary Tables 8 and 9. 3.8.1 B IOLOGICAL I NTERPRETATION FOR GCM DATA S ET Prolactin-induced protein, which is regulated by prolactin activation of its receptors, ranks highest among the PUGs associated with breast cancer. Postmenopausal breast cancer risk is strongly associated with elevated prolactin levels (PubMed IDs 15375001, 12373602, 10203283). Interestingly, prolactin release proportionally increases with increasing central fat in obese women (PubMed ID 15356045) and women with this pattern of obesity have an increased risk of breast cancer mortality (PubMed ID 14607804). Other genes of interest that rank among the top 10 breast cancer PUGs include CRABP2, which transports retinoic acid to the nucleus. Retinoids are important regulators of breast cell function and show activity as potential breast cancer chemopreventive agents (PubMed IDs 11250995, 12186376). Mammglobin is primarily expressed in normal breast epithelium and breast cancers (PubMed ID 12793902). Carbonic anhydrase XII is expressed in breast cancers and 2160 PUG-OVRSVM is generally considered a marker of a good prognosis (PubMed ID 12671706). The selective expression and/or function of these genes in breast cancers are consistent with their selection as PUGs in the classiﬁcation scheme. The top 10 PUGs associated with prostate cancer include several genes strongly associated with the prostate including prostate speciﬁc antigen (PSA) and its alternatively spliced form 2, and prostatic secretory protein 57. The role of PSA gene KLK3 and KLK1 as a biomarker of prostate cancer is well established (PubMed ID 19213567). Increased NPY expression is associated with high-grade prostatic intraepithelial neoplasia and poor prognosis in prostate cancers (PubMed ID 10561252). ACPP is another prostate speciﬁc protein biomarker (PubMed ID 8244395). The strong representation of genes that show clear selectivity for expression within the prostate illustrates the potential of the PUGs as bio-markers linked to the biology of the underlying tissues. Several of the selected PUG markers for uterine cancer ﬁt very well with our current biological understanding of this disease. It is well-established that estrogen receptor alpha (ESR1) is expressed or ampliﬁed in human uterine cancer (PubMed IDs 18720455, 17911007, 15251938), while the Hox7 gene (MSX1) contributes to uterine function in cow and mouse models, especially at the onset of pregnancy (PubMed IDs 7908629, 14976223, 19007558). Mammaglobin 2 (SCGB2A1) is highly expressed in a speciﬁc type of well-differentiated uterine cancer (endometrial cancers) (PubMed ID 18021217), and PAM expression in the rat uterus is known to be regulated by estrogen (PubMed IDs 9618561, 9441675). Other PUGs provide novel insights into uterine cancer that are deserving of further study. Our PUG selection ranks HE4 higher than the well-established CA125 marker, which may suggest HE4 as a promising alternative for the clinical management of endometrial cancer. One recent study (PubMed ID 18495222) shows that, at 95% speciﬁcity, the sensitivity of differentiating between controls and all stages of uterine cancer is 44.9% using HE4 versus 25.2% using CA125 (p = 0.0001). Osteopontin (OPN) is an integrin-binding protein that is involved in tumorigenesis and metastasis. OPN levels in the plasma of patients with ovarian cancer are much higher compared with plasma from healthy individuals (PubMed ID 11926891). OPN can increase the survival of ovarian cancer cells under stress conditions in vitro and can promote the late progression of ovarian cancer in vivo, and the survival-promoting functions of OPN are mediated through Akt activation (PubMed ID 19016748). Matrix metalloproteinase 2 (MMP2) is an enzyme degrading collagen type IV and other components of the basement membrane. MMP-2 is expressed by metastatic ovarian cancer cells and functionally regulates their attachment to peritoneal surfaces (PubMed ID 18340378). MMP2 facilitates the transmigration of human ovarian carcinoma cells across endothelial extracellular matrix (PubMed ID 15609323). Glutathione peroxidase 3 (GPX3) is one of several isoforms of peroxidases that reduce hydroperoxides to the corresponding alcohols by means of glutathione (GSH) (PubMed ID 17081103). GPX3 has been shown to be highly expressed in ovarian clear cell adenocarcinoma. Moreover, GPX3 has been associated with low cisplatin sensitivity (PubMed ID 19020706). 3.8.2 B IOLOGICAL I NTERPRETATION FOR NAS DATA S ET Several top-ranking gene products identiﬁed by our computational method have been well established as tumor-type speciﬁc markers and many of them have been used in clinical diagnosis. For example, mucin 16, also known as CA125, is a FDA-approved serum marker to monitor disease progression and recurrence in ovarian cancer patients (PubMed ID 19042984). Likewise, kallikrein 2161 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG family members including KLK6 and KLK8 are known to be ovarian cancer associated markers which can be detected in body ﬂuids in ovarian cancer patients (PubMed ID 17303231). TITF1 (also known as TTF1) has been reported as a relatively speciﬁc marker in lung adenocarcinoma (PubMed ID 17982442) and it has been used to assist differential diagnosis of lung cancer from other types of carcinoma. Fatty acid synthase (FASN) is a well-known gene that is often upregulated in breast cancer (PubMed ID 17631500) and the enzyme is amenable for drug targeting using FASN inhibitors, suggesting that it can be used as a therapeutic target in breast cancer. The above ﬁndings indicate the robustness of our computational method in identifying tumor-type speciﬁc markers and in classifying different types of neoplastic diseases. Such information could be useful in translational studies (PubMed ID 12874019). Metastatic carcinoma of unknown origin is a relatively common presentation in cancer patients and an accurate diagnosis of the tumor type in the metastatic diseases is important to direct appropriate treatment and predict clinical outcome. The distinctive patterns of gene expression characteristic to various types of cancer may help pathologists and clinicians to better manage their patients. 3.9 Gene Comparisons Between Methods It may be informative to provide some initial analysis on how the selected genes compare between methods; however, without deﬁnitive ground truth on cancer markers, the utility of this information is somewhat limited and should, thus, be treated as anecdotal, rather than conclusive. Speciﬁcally, we have now done some assessment of how differentially these gene selection methods rank some known cancer marker genes. The overlap rate is deﬁned as the number of genes commonly selected by two methods over the maximum size of the two selected gene sets. Let G1 and G2 denote the gene sets selected by gene selection methods 1 and 2, respectively, and |G| denote the cardinality (the size) of set G. The overlap rate between G1 and G2 is R= |G1 ∩ G2 | . max (|G1 | , |G2 |) Table 11 shows the overlap rate between methods on the top 100 genes per class. We can see that the overlap rates between methods are generally low except for the pair of SNR and t-stat. BW genes are quite different from the genes selected by all other methods and have only about 15% overlap rate with PUG and SVMRFE. The relatively high overlap rate between SNR and t-stat may be expected since they use quite similar summary statistics in their selection criteria. We have also examined a total of 16 genes with known associations with 4 tumor types. These 16 genes are well-known markers supported by current biological knowledge. The rank of biomedical importance of these genes produced by each method is summarized in Table 12. When a gene is not listed in the top 100 genes by a wrapper method like SVMRFE, we simply assign the rank as ‘>100’. Generally but not uniformly across cancer types, these validated marker genes are highly ranked in the PUGs list as compared to other methods, and thus will be surely selected by PUG criterion. 4. Discussion In this paper, we address several critical yet subtle issues in multicategory molecular classiﬁcation applied to real-world biological and/or clinical applications. We propose a novel gene selection methodology matched to the multicategory classiﬁcation approach (potentially with an unbalanced 2162 PUG-OVRSVM Overlapping Rate PUG SNR t-stat BW SVMRFE PUG 1 0.4117 0.3053 0.1450 0.4057 SNR 0.4117 1 0.7439 0.3307 0.3841 t-stat 0.3053 0.7439 1 0.2907 0.3941 BW 0.1450 0.3307 0.2907 1 0.1307 SVMRFE 0.4057 0.3841 0.3941 0.1307 1 Table 11: The overlapping rate between methods on the top 100 genes per class. Breast Cancer Relevant Genes Prostate Cancer Relevant Genes Rank Rank Gene Symbol Gene Symbol PUG SNR t-stat BW SVMRFE PUG SNR t-stat BW SVMRFE PIP 1 5745 6146 473 >100 KLK3 4 5 11 61 15 CRABP2 4 5965 6244 498 >100 KLK1 5 3 9 76 16 SCGB2A2 6 6693 6773 458 14 NPY 7 18 22 344 30 CA12 9 6586 6647 518 >100 ACPP 3 4 8 71 12 Uterine Cancer Relevant Genes Gene Symbol Ovarian Cancer Relevant Genes Rank Rank Gene Symbol PUG SNR t-stat BW SVMRFE PUG SNR t-stat BW SVMRFE ESR1 1 2 16 130 5 OPN 15 334 517 371 63 Hox7 2 4 52 307 12 MMP2 42 2626 3045 481 >100 SCGB2A1 8 3 19 190 4 GPX3 7 411 >100 PAM HE4 10 3 83 1 281 3 365 99 71 5 812 446 Table 12: Detailed comparison between methods on several validated marker genes. mixture distribution) that is not a straightforward pooled extension of binary (two-class) differential analysis. We emphasize the statistical reproducibility and biological plausibility of the selected gene markers under small sample size, supported by their detailed biological interpretations. We tested our method on six benchmark and in-house real microarray gene expression data sets and compared its performance with that of several existing methods. We imposed a rigorous performance assessment where each and all components of the scheme including gene selection are subjected to cross-validation, for example, held-out/unseen samples in a testing fold are not used for any phase of classiﬁer training. Tested on six benchmark real microarray data sets, the proposed PUG-OVRSVM method outperforms several widely-adopted gene selection and classiﬁcation methods with lower error rates, fewer marker genes, and higher performance sustainability. Moreover, while for some data sets, the absolute gain in classiﬁcation accuracy percentage of PUG-OVRSVM is not dramatically large, it must be recognized that the performance may be approaching the minimum Bayes error rate, in which case PUG-OVRSVM is achieving nearly all the improvement that is theoretically attainable. Furthermore, the improved performance is achieved by correct classiﬁcations on some of the most difﬁcult cases, which is considered signiﬁcant for clinical diagnosis (Ramaswamy et al., 2001). Lastly, although improvements will be data set-dependent, our multi-data set tests have shown that PUG-OVRSVM is the consistent winner as compared to several peer methods. 2163 Y U , F ENG , M ILLER , X UAN , H OFFMAN , C LARKE , DAVIDSON , S HIH AND WANG We note that we have opted for simplicity as opposed to theoretical optimality in designing our method. Our primary goal was to demonstrate that a small number of reproducible phenotypicdependent genes are sufﬁcient to achieve improved multicategory classiﬁcation, that is, small sample sizes and a large number of classes need not preclude a high level of performance. Our studies suggest that using genes’ marginal associations with the phenotypic categories as we do here has the potential to stabilize the learning process, leading to a substantial reduction in performance variability with small sample size; whereas, the current generation of complex gene selection techniques may not be stable or powerful enough to reliably exploit gene interactions and/or variations unless the sample size is sufﬁciently large. We have not explored the full ﬂexibility that this method readily allows, with different numbers of subPUGs used by different classiﬁers. Presumably, equal or better performance could be achieved with fewer genes if more markers were selected for the most difﬁcult classiﬁcations, involving the nearest phenotypes. However, such ﬂexibility could actually degrade performance in practice since it introduces extra design choices and, thus, extra sources of variation in classiﬁcation performance. We may also extend our method to account for variation in fold-changes, with the uncertainty estimated on bootstrap samples judiciously applied to eliminate those PUGs with high variations. Notably, multicategory classiﬁcation is intrinsically a nonlinear classiﬁcation problem, and this method (using one-versus-everyone fold-change based PUG selection, linear kernel SVMs, and the MAP decision rule) is most practically suitable to discriminating unimodal classes. Future work will be required to extend PUG-OVRSVM for multimodal class distributions. 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