nips nips2012 nips2012-351 knowledge-graph by maker-knowledge-mining

351 nips-2012-Transelliptical Component Analysis

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Author: Fang Han, Han Liu

Abstract: We propose a high dimensional semiparametric scale-invariant principle component analysis, named TCA, by utilize the natural connection between the elliptical distribution family and the principal component analysis. Elliptical distribution family includes many well-known multivariate distributions like multivariate Gaussian, t and logistic and it is extended to the meta-elliptical by Fang et.al (2002) using the copula techniques. In this paper we extend the meta-elliptical distribution family to a even larger family, called transelliptical. We prove that TCA can obtain a near-optimal s log d/n estimation consistency rate in recovering the leading eigenvector of the latent generalized correlation matrix under the transelliptical distribution family, even if the distributions are very heavy-tailed, have inﬁnite second moments, do not have densities and possess arbitrarily continuous marginal distributions. A feature selection result with explicit rate is also provided. TCA is further implemented in both numerical simulations and largescale stock data to illustrate its empirical usefulness. Both theories and experiments conﬁrm that TCA can achieve model ﬂexibility, estimation accuracy and robustness at almost no cost. 1

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sentIndex sentText sentNum sentScore

1 edu Abstract We propose a high dimensional semiparametric scale-invariant principle component analysis, named TCA, by utilize the natural connection between the elliptical distribution family and the principal component analysis. [sent-3, score-0.715]

2 Elliptical distribution family includes many well-known multivariate distributions like multivariate Gaussian, t and logistic and it is extended to the meta-elliptical by Fang et. [sent-4, score-0.265]

3 In this paper we extend the meta-elliptical distribution family to a even larger family, called transelliptical. [sent-6, score-0.065]

4 d realizations of a random vector X ∈ Rd with population covariance matrix Σ and correlation matrix Σ0 , the Principal Component Analysis (PCA) aims at recovering the top m leading eigenvectors u1 , . [sent-16, score-0.466]

5 In practice, Σ is unknown and the top m leading eigenvectors u1 , . [sent-20, score-0.172]

6 , um of the Pearson sample covariance matrix are obtained as the estimators. [sent-23, score-0.069]

7 However, because the PCA is well-known to be scale-variant, meaning that changing the measurement scale of variables will make the estimators different, the PCA conducted on the sample correlation matrix is also regular in literatures [2]. [sent-24, score-0.247]

8 It aims at recovering the top m leading eigenvectors θ1 , . [sent-25, score-0.207]

9 , θm of Σ0 using the top m leading eigenvectors θ1 , . [sent-28, score-0.172]

10 Because Σ0 is scale-invariant, we call the PCA aiming at recovering the eigenvectors of Σ0 the scale-invariant PCA. [sent-32, score-0.146]

11 In high dimensional settings, when d scales with n, it has been discussed in [14] that u1 and θ1 are generally not consistent estimators of u1 and θ1 . [sent-33, score-0.065]

12 The elliptical distributions are of special interest in Principal Component Analysis. [sent-45, score-0.447]

13 The study of elliptical distributions and their extensions have been launched in statistics recently by [4]. [sent-46, score-0.447]

14 The elliptical distributions can be characterized by their stochastic representations [5]. [sent-47, score-0.447]

15 Elliptical distribution family includes a variety of famous multivariate distributions: multivariate Gaussian, multivariate Cauchy, Student’s t, logistic, Kotz, symmetric Pearson type-II and type-VII distributions. [sent-54, score-0.275]

16 [4] introduce the term meta-elliptical distribution in extending the continuous elliptical distributions whose densities exist to a wider class of distributions with densities existing. [sent-56, score-0.65]

17 The construction of the meta-elliptical distributions is based on the copula technique and it was initially introduced by [25]. [sent-57, score-0.128]

18 In particular, when the latent elliptical distribution is the multivariate Gaussian, we have the meta-Gaussian or the nonparanormal distributions introduced by [16] and [19]. [sent-58, score-0.604]

19 The elliptical distribution is of special interest in Principal Component Analysis (PCA). [sent-59, score-0.412]

20 It has been shown in a variety of literatures [27, 11, 22, 12, 24] that the PCA conducted on elliptical distributions shares a number of good properties enjoyed by the PCA conducted on the Gaussian distribution. [sent-60, score-0.479]

21 In particular, [11] show that with regard to a range of hypothesis relevant to PCA, tests based on a multivariate Gaussian assumption have the identical power for all elliptical distributions even without second moments. [sent-61, score-0.517]

22 In this paper, a new high dimensional scale-invariant principle component analysis approach is proposed, named Transelliptical Component Analysis (TCA). [sent-63, score-0.099]

23 Firstly, to achieve both the estimation accuracy and model ﬂexibility, we build the model of TCA on the transelliptical distributions. [sent-64, score-0.307]

24 , Xd )T is said to follow a transelliptical distribution if there exists a set of univariate strictly monotone functions f = {fj }d such that f (X) := (f1 (X1 ), . [sent-68, score-0.373]

25 , fd (Xd ))T j=1 follows a continuous elliptical distribution with parameters µ = 0 and Σ0 = [Σ0 ] 0. [sent-71, score-0.679]

26 Transelliptical distributions do not necessarily possess densities and are strict extensions to the meta-elliptical distributions deﬁned in [4]. [sent-73, score-0.228]

27 TCA aims at recovering the top m leading eigenvectors θ1 , . [sent-74, score-0.207]

28 We prove that even though the generating variable ξ is changing and marginal distributions are arbitrarily continuous, Kendall’s tau correlation matrix approximates Σ0 in a parametric rate OP ( log d/n). [sent-79, score-0.495]

29 This key observation makes Kendall’s tau a better estimator than Pearson sample correlation matrix with regard to a much larger distribution family than the Gaussian. [sent-80, score-0.451]

30 Thirdly, in terms of methodology and theory, we analyze the general case that X follows a transelliptical distribution and θ1 is sparse. [sent-81, score-0.332]

31 We ob∗ tain the TCA estimator θ1 of θ1 utilizing the Kendall’s tau correlation matrix. [sent-83, score-0.341]

32 We prove that the TCA can obtain a fast convergence rate in terms of parameter estimation and is of the rate sin ∠(θ1 , θ∞ ) = OP (s log d/n), where θ∞ is the estimator TCA obtains. [sent-84, score-0.131]

33 Let MI· and M·J be the submatrix of M with rows in I and all columns, and the submatrix of M with columns in J and all rows. [sent-91, score-0.06]

34 Elliptical and Transelliptical Distributions This section is devoted to a brief discussion of elliptical and transelliptical distributions. [sent-104, score-0.721]

35 , Xd )T is said to be continuous if the marginal distribution functions are all continuous. [sent-108, score-0.147]

36 1 Elliptical Distributions In this section we shall ﬁrstly provide a deﬁnition of the elliptical distributions following [5]. [sent-111, score-0.447]

37 A random variable in R with continuous marginal distribution function does not necessarily possess density. [sent-119, score-0.179]

38 A well-known set of examples is the cantor distribution, whose support set is the cantor set. [sent-120, score-0.072]

39 We deﬁne Σ0 = [Σ0 ] with Σ0 = Σjk / Σjj Σkk to be the generalized correlation matrix of Z. [sent-150, score-0.186]

40 Σ0 jk jk is the correlation matrix of Z when Z’s second moment exists and still reﬂects the rank dependency even when Z has inﬁnite second moment [13]. [sent-151, score-0.41]

41 2 Transelliptical Distributions To extend the elliptical distribution, we ﬁrstly deﬁne two sets of symmetric matrices: R+ = {Σ ∈ d Rd×d : ΣT = Σ, diag(Σ) = 1, Σ 0}; Rd = {Σ ∈ Rd×d : ΣT = Σ, diag(Σ) = 1, Σ 0}. [sent-154, score-0.387]

42 , Xd )T with continuous marginal distribution functions F1 , . [sent-160, score-0.106]

43 , Fd and density existing is said to follow a meta-elliptical distribution if and only if there exists a continuous elliptically distributed random vector Z ∼ ECd (0, Σ0 , g) with the marginal distribution function Qg and Σ0 ∈ R+ , such that (Q−1 (F1 (X1 )), . [sent-163, score-0.309]

44 g g d In this paper, we generalize the meta-elliptical distribution family to a broader class, named the transelliptical. [sent-167, score-0.092]

45 The transelliptical distributions do not assume that densities exist for both X and Z and are therefore strict extensions to meta-elliptical distributions. [sent-168, score-0.402]

46 , Xd )T is said to follow a transelliptical distribution if and only if there exists a set of strictly monotone functions f = {fj }d and a latent j=1 continuous elliptically distributed random vector Z ∼ ECd (0, Σ0 , ξ) with Σ0 ∈ Rd , such that (f1 (X1 ), . [sent-174, score-0.584]

47 , fd ) and Σ0 the latent generalized correlation matrix. [sent-181, score-0.386]

48 If X follows a meta-elliptical distribution, in other words, X possesses density and has continuous marginal distributions F1 , . [sent-184, score-0.141]

49 In summary, transelliptical ⊃ meta-elliptical = meta-elliptical copula ⊃ elliptical* ⊃ elliptical copula, transelliptical ⊃ meta-Gaussian = nonparanormal. [sent-199, score-1.069]

50 Here elliptical* represents the elliptical distributions which are continuous and possess densities. [sent-200, score-0.543]

51 2 Latent Correlation Matrix Estimation for Transelliptical Distributions We ﬁrstly study the correlation and covariance matrices of elliptical distributions. [sent-202, score-0.528]

52 Given Z ∼ ECd (µ, Σ, ξ) with rank(Σ) = q and ﬁnite second moments and Σ0 the 2 generalized correlation matrix of Z, we have E(Z) = µ, Var(Z) = E(ξ ) Σ, and Cor(Z) = Σ0 . [sent-206, score-0.186]

53 q When the random vector is elliptically distributed with second moment ﬁnite, the sample mean and correlation matrices are element-wise consistent estimators of µ and Σ0 . [sent-207, score-0.307]

54 However, the elliptical distributions are generally very heavy-tailed (multivariate t or Cauchy distributions for example), making Pearson sample correlation matrix a bad estimator. [sent-208, score-0.693]

55 When the distribution family is extended to the transelliptical, the Pearson sample correlation matrix is generally no longer a element-wise consistent estimator of Σ0 . [sent-209, score-0.289]

56 TCA is a two-stage method in estimating the leading eigenvectors of Σ0 . [sent-213, score-0.172]

57 Firstly, we estimate the Kendall’s tau correlation matrix R. [sent-214, score-0.348]

58 1 Rank-based Measures of Associations The main idea of the TCA is to exploit the Kendall’s tau statistic to estimate the generalized correlation matrix Σ0 efﬁciently and robustly. [sent-217, score-0.385]

59 , Xd )T be a d−dimensional random vector with marginal distributions F1 , . [sent-221, score-0.093]

60 The population Spearman’s rho and Kendall’s tau correlation coefﬁcients are given by ρ(Xj , Xk ) = Corr(Fj (Xj ), Fk (Xk )), τ (Xj , Xk ) = P((Xj − Xj )(Xk − Xk ) > 0) − P((Xj − Xj )(Xk − Xk ) < 0), where (Xj , Xk ) is a independent copy of (Xj , Xk ). [sent-225, score-0.303]

61 In particular, for Kendall’s tau, we have the following theorem, which states an explicit relationship between τjk and Σ0 given X ∼ jk T Ed (Σ0 , ξ; f1 , . [sent-226, score-0.112]

62 , fd ), no matter what the generating variable ξ is. [sent-229, score-0.245]

63 , fd ) transelliptically distributed, we have π Σ0 = sin τ (Xj , Xk ) . [sent-236, score-0.292]

64 22 in [5] builds the result only for one sample statistic and cannot be generalized to the statistic of multiple samples, like the Kendall’s tau or Spearman’s rho. [sent-243, score-0.236]

65 On the other hand, when X ∼ jk T Ed (Σ0 , ξ; f1 , . [sent-248, score-0.112]

66 , fd ) with ξ =d 1, [10] proves that: ρ(Xj , Xk ) = 3( arcsin Σ0 jk ) π − 4( arcsin Σ0 3 jk ) . [sent-251, score-0.551]

67 We consider the following rank-based statistic:  2 τ =  jk sign (xij − xi j ) (xik − xi k ) , if j = k n(n − 1) (3. [sent-259, score-0.112]

68 , θd ∈ S are the corresponding eigenvectors of λ1 , . [sent-264, score-0.111]

69 4) where B0 (s) := {v ∈ Rd : v 0 ≤ s} and R is the estimated Kendall’s tau correlation matrix. [sent-269, score-0.303]

70 2 TCA Algorithm Generally we can plug in the Kendall’s tau correlation matrix R to any sparse PCA algorithm listed above. [sent-273, score-0.377]

71 We use the iterative deﬂation method to learn the ﬁrst k instead of the ﬁrst one leading eigenvectors, following the discussions of [21, 15, 28, 29]. [sent-281, score-0.061]

72 In detail, a matrix Γ ∈ Rd×s deﬂates a vector v ∈ Rd and achieves a new matrix Γ : Γ := (I − vv T )Γ(I − vv T ). [sent-282, score-0.138]

73 1 Rank-based Correlation Matrix Estimation This section is devoted to the concentration result of the Kendall sample correlation matrix R to the Pearson correlation matrix Σ0 . [sent-287, score-0.399]

74 , fd ) and letting R be the Kendall tau correlation matrix, we have with probability at least 1 − d−5/2 , R − Σ0 max ≤ 3π log d/n. [sent-297, score-0.522]

75 1 can be proved by realizing that τjk is an unbiased estimator of τ (Xj , Xk ) and is a U-statistic with size 2. [sent-301, score-0.064]

76 We assume that the Model Md (Σ0 , ξ, s; f ) holds and the next theorem provides an upper bound on the angle between the estimated leading ∗ eigenvector θ1 and true leading eigenvector θ1 . [sent-306, score-0.212]

77 For any two vectors v1 ∈ Sd−1 and v2 ∈ Sd−1 , letting | sin ∠(v1 , v2 )| = T 1 − (v1 v2 )2 , then we have ∗ P | sin ∠(θ1 , θ1 )| ≤ 6π ·s λ 1 − λ2 log d n ≥ 1 − d−5/2 . [sent-311, score-0.074]

78 When (n, d) goes to inﬁnity, the two leading eigenvalues λ1 and λ2 will typically go to inﬁnity and will at least be away from zero. [sent-317, score-0.061]

79 1 Numerical Simulations In the simulation study we randomly sample n data points from a certain transelliptical distribution T Ed (Σ0 , ξ; f1 , . [sent-333, score-0.332]

80 To determine the transelliptical distribution, ﬁrstly, we derive Σ0 in the following way: A covariance matrix Σ is ﬁrstly synthesized through the eigenvalue decomposition, where the ﬁrst two eigenvalues are given and the correspondd ing eigenvectors are pre-speciﬁed to be sparse. [sent-338, score-0.463]

81 = ωd = 1, and the ﬁrst two leading eigenvectors of Σ, u1 and u2 , are sparse with the ﬁrst s = 10 entries of u1 and the second s = 10 entries of u2 are nonzero, i. [sent-342, score-0.201]

82 The generalized correlation matrix Σ0 is generated from Σ, with λ1 = 4, λ2 = 2. [sent-347, score-0.186]

83 , λd ≤ 1 and the top two leading eigenvectors sparse: θ1j = − √1 10 0 1 ≤ j ≤ 10 otherwise and θ2j = − √1 10 0 11 ≤ j ≤ 20 . [sent-351, score-0.172]

84 X following a multivariate-t distribution with degree of freedom m, mean 0 and covariance matrix Σ0 (Example 2. [sent-382, score-0.07]

85 , fd } = {h1 , h2 , h3 , h4 , h5 , h1 , h2 , h3 , h4 , h5 , . [sent-393, score-0.219]

86 φ(y)dy This is equivalent to say that X is transelliptically distributed with the latent elliptical distribution Z ∼ M td (3, 0, Σ0 ). [sent-397, score-0.5]

87 (B) Successful matches of the market trend proportions only using the stocks in Ak and Bk . [sent-524, score-0.151]

88 We collected the daily closing prices for J=452 stocks that were consistently in the S&P; 500 index between January 1, 2003 through January 1, 2008. [sent-532, score-0.117]

89 Let St = [Stt,j ] denote by the closing price of stock j on day t. [sent-534, score-0.089]

90 We wish to evaluate the ability of using the only k stocks to represent the trend of the whole stock market. [sent-535, score-0.175]

91 To this end, we run Kendall and Pearson on St and obtain the leading eigenvectors θKendall and θP earson using the tuning parameter k ∈ N. [sent-536, score-0.208]

92 And then we let TtW , TtAk and TtBk denote by the trend of the whole stocks, Ak stocks and Bk stocks in tth day compared with t − 1th date, i. [sent-538, score-0.203]

93 In this way, we can calculate the proportion of successful matches 1 of the market trend using the stocks in Ak and Bk as: ρAk := T t I(TtW = TtAk ) and ρBk := Bk 1 W t I(Tt = Tt ). [sent-540, score-0.151]

94 It can be observed from Figure 1 (B) that Kendall summarizes the trend of the whole stock market constantly better than Pearson. [sent-543, score-0.123]

95 Principal component analysis in sensory analysis: covariance or correlation matrix? [sent-557, score-0.177]

96 Metaelliptical copulas and their use in e frequency analysis of multivariate hydrological data. [sent-581, score-0.07]

97 Tca: Transelliptical principal component analysis for high dimensional non-gaussian data. [sent-592, score-0.161]

98 Generalized power method for sparse e a principal component analysis. [sent-627, score-0.154]

99 Augmented sparse principal component analysis for high dimensional data. [sent-667, score-0.19]

100 Minimax rates of estimation for sparse pca in high dimensions. [sent-686, score-0.121]

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