K.-C. Li. On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma. Journal of American Statistical Association, 87:1025--1039, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....preserves as closely as possible their intrinsic global local metric structure [43, 71, 65] 2. Regression, in which the goal is to reduce the dimension of the predictor vector with the minimum loss in its capacity to infer about the conditional distribution of the univariate response variable [53, 54, 17]. 3. Classification, where we should seek reductions that minimize the lowest attainable classification error in the transformed space, i.e. the Bayes error [13] Such disparate interpretations might thereby cast strong influence on both the design and the choice of an appropriate ....

....at every point x. The main issue of ADE is again how to more accurately estimate the derivatives. Various other methods have been developed to estimate the central (mean) subspace, including ordinary least squares (OLS) sliced inverse regression (SIR, 53] principle Hessian directions (pHd, [54, 16]) graphical regression (GREG) and sliced average variance estimation (SAVE, 17] Readers may refer to, e.g. 17] for a comprehensive introduction. Two observations immediately follow discussions in this section: 1. The di#erence between IDS and MIDS is significant only when the loss matrix ....

K.-C. Li. On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma. Journal of the American Statistical Association, 87(420):1025--1039, 1992. 31

....(K 1) d 2) employed by ALB. In many applications, f is well approximated by a function defined on a lower dimensional projection; i.e. f(x) # f 0 (Bx) where B is a d 0 d matrix with d 0 d. There is a substantial body of literature describing stable methods for dimension reduction; see Li (1991, 1992), Cook (1998a,b) and Ferre (1998) Such methods can be used to estimate d 0 and the column space of B, before applying ALB to the lower dimensional predictor space. The a#ne invariance of ALB implies that the subspace basis chosen to define B will not a#ect the resulting estimator. When a ....

K. C. Li (1992). On principal Hessian directions for data visualization and dimension reduction: another application of Stein's Lemma. Journal of the American Statistical Association, 87, 1025--1039.

....Abstract Dimension reduction in regression analysis reduces the dimension of the predictor vector x without specifying a parametric model and without loss of information about the distribution of y given x. We study three existing methods, SIR (Li, 1991) SAVE (Cook and Weisberg, 1991) and pHd (Li, 1992) in a systematic manner and propose broad classes of dimension reduction target matrices. We propose methods for choosing among target matrices and we investigate linear combinations of di erent target matrices. KEY WORDS: Nonparametric regression; SIR; SAVE; pHd; Regression diagnostics. ....

.... Since Li (1991) proposed an inverse regression method called sliced inverse regression (SIR) for dimension reduction, many related ideas have been introduced, for example, Sliced Average Variance Estimation (SAVE) proposed by Cook and Weisberg (1991) Principle Hessian Directions (pHd) proposed by Li (1992), the graphical methods proposed by Cook (1994a, 1996) and Cook and Wetzel (1993) and many ideas discussed by Cook and Weisberg (1994) and Cook (1998) Section 2 brie y reviews three existing methods, SIR, SAVE and pHd. In section 3, we propose a new perspective on these methods. We introduce ....

[Article contains additional citation context not shown here]

Li, K.-C. (1992), \On principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma," JASA, 87, 1025-1039.

....of projection pursuit with a direct search than the nonparametric PPR with backfitting. 18 4 How to Choose Good Directions In this section we propose some heuristics on how to choose good initial directions based on measurements of the derivatives of a function. Duan and Li [8] and Li [29, 30] proposed sliced inverse regression and principal Hessian directions to estimate projection directions in projection pursuit regression. The Hessian approach used by Li is similar to what is described here. Theoretically, given a smooth function of d variables, what are the best d projection ....

Li, K., "On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma." Journal of the American Statistical Association. 87(420): 1025-1039, 1992.

....design conditions concerning the distribution of x are essential for the application of SIR, cf. Li (1991, Condition 3. 1) this aspect is another main topic in the discussion, cf. Cook and Weisberg (1991) H ardle and Tsybakov (1991) Hall and Li (1993) Cook and Nachtsheim (1994) Carroll and Li (1992) use SIR in a general nonlinear regression model with measurement error in the predictor variables. Aragon and Saracco (1997) consider SIR in the situation of small sample sizes. Bura (1997) uses a multivariate linear model for the inverse regression curve. Two nonparametric methods for testing ....

.... and McKean (1997) SIR is implemented in the package XploRe (H ardle, Klinke and Turlach, 1995) There are also more sophisticated methods for estimating the edr direction which are based on second moments (Cook and Weisberg, 1991) and on properties of so called principal Hessian directions (Li, 1992). One aspect of SIR which has not been treated yet is the sensitivity of this method with respect to outliers in the data. Although Li (1991, p. 319) mentions that [ it would help the analysis if closer examination of the distribution of x can be made so that outliers can be removed [ ....

Li, K.{C. (1992), \On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma," Journal of the American Statistical Association, 87, 1025-1039.

....This procedure reverses the more natural forward regression methods which model Y as a function of x. Instead of such direct data fitting, SIR exploits the conditional distribution of x given Y . The first moment method, based on E(xjY ) has been studied extensively in various contexts, see Li (1990, 1991, 1992a,b) Duan and Li (1991) Carroll and Li (1992) and Hsing and Carroll (1992) The second moment, cov(xjY ) is also useful, see for example the SAVE method of Cook and Weisburg (1991) and the SIR II and pHd methods of Li (1991, 1992b) Although model (1.1) is primarily proposed for ....

..... The maximum likelihood estimate is equal to ( Sigma Gamma1 0 Sigma Gamma1 1 ) Gamma1 ( Sigma Gamma1 0 u 0 Sigma Gamma1 1 u 1 ) which can be interpreted as a matrix weighted average of u 0 and u 1 with the covariance matrices as the weights. This combination method can be applied to b v 0 ; bv 1 which have been shown to be asymptotically normal (Duan and Li 1991) However, some modification is needed because the covariance matrix for b v z ; z = 0; 1, denoted by Sigma z , is always degenerate due to the normalization constraint on the length of each eigenvector jjbv z jj = 1, ....

Li, K. C. (1992b). On principal Hessian directions for data visualization and dimension reduction: another application of Stein's lemma. J. Amer. Stat. Assoc., 87, 10251039.

No context found.

K.-C. Li. On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma. Journal of American Statistical Association, 87:1025--1039, 1992.

No context found.

Li, K.-C. On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma. J. Amer. Stat. Assoc., 87:1025--1039, 1992.

No context found.

Li, K.-C. On principal Hessian directions for data visualization and dimension reduction: Another application of Stein's lemma. J. Amer. Stat. Assoc., 87:1025--1039, 1992.

No context found.

Li K.C. (1992), "On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein's Lemma", Journal of the American Statistical Association, 87, 1025-1039.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC