asa.phys.es.03

Sponsoring Section/Society: ASA-SPES

Session Slot: 10:30-12:20 Tuesday

Estimated Audience Size: 100-150

AudioVisual Request: Two Overheads

Session Title: Dimension Reduction

Theme Session: Yes

Applied Session: Yes

Session Organizer: Lin, Dennis K.J. Penn State University

Address: 314 Beam Building, Penn State University, University Park, PA 16802-1913

Phone: 814-865-0377

Fax: 814-863-2381

Email: DKL5@psu.edu

Session Timing: 110 minutes total (Sorry about format):

First Speaker - 25 minutes Second Speaker - 25 minutes Third Speaker - 25 minutes Fourth Speaker - 25 minutes Floor Discusion - 10 minutes

Session Chair: Li, Ker-Chau UCLA

Address: Department of Mathematics, UCLA, LA. CA 90024

Phone: 310-8254897

Fax: 310-2066673

Email: kcli@math.ucla.edu

1. High Dimensional Data Analysis Via the SIR/PHD Approach

Li, Ker-Chau,   University of California, Los Angeles

Address: Department of Mathematics, UCLA. LA. CA. 90024

Phone: 310-8254897

Fax: 310-2066673

Email: kcli@math.ucla.edu

Abstract: Dimensionality is an issue that can arise in every scientific field. Generally speaking, the difficulty lies on how to visualize a high dimensional function or data set. To investigate this issue, Li(1991, 1992) proposed a dimension-reduction model of the form

$$Y=f(\beta_1^\prime X,\cdots,\beta_k^\prime X,\epsilon)$$

where X denotes the input and Y denotes the output variable. This is general enough to incorporate many regression models as special cases. However, the novel feature about this model is that f is unknown and so is the distribution of the random error $\epsilon$. The goal is to find the $\beta$ vectors so that they can be used to reduce the input dimension when k is small. Two simple methods, sliced inverse regression (SIR) and principal Hessian directions (PHD) have been studied.

In this talk, I will give an overview on some recent results about SIR/PHD and describe some ongoing research in various application areas, including censored regression, error-in-regressor, nonlinear confounding, functional data analysis, nonlinear time series, Monte-Carlo and Bayesian computation, and others.

2. A Systematic Approach to the Analysis of Complex Interaction Patterns in 2-Level Factorial Designs

Filliben, James J.,   National Institute of Standards and Technology

Address: Statistical Engineering Division, NIST North, room 353, Gaithersberg MD 20899

Phone: 301-9752855

Fax: 301-9904127

Email: liu@cam.nist.gov

Li, Ker-Chau, University of California, Los Angeles

Abstract: Analysis of data from two-level full factorial designs often ends up with a final prediction equation which gives only the significant main effect and interaction terms. When the number of interactions is small, simple and useful interpretation of the equation can then be drawn immediately.

This article addresses a different situation where the number of significant interactions may be large so that additional efforts are needed in order to sort out the pattern and the relationship between them. In particular, we bring out a class of models where most interactions can be attributed to just one or two (or a very small number of) factors, and conditional on these factors, the models become essentially linear. We offer a strategy for uncovering this structure by Linear Domain Splitting (LDS) whereby a complicated global model is replaced by a series of local domain-specific linear models. We present a recommended methodology (PHD-principal Hessian direction) for systematically proceeding from the global equation to local split-domain analyses. The net result is that guided tree-structured paths are offered for visiting the source-of-interaction factors in sequence, which appropriately reflects their relative importance and mutual relationship. The final stage modeling is simpler (linear). The quality of the fit can be assessed separately in each region, and the analyst comes away with greater insight as to the sensitivity and robustness of the various factor effects over various regions. Applications in digital electronics testing is illustrated by analyzing a data set collected for studying the conversion error of a digital-to-analog converter.

3. Uncertainty Analysis in Mathematical Modeling via Statistical Dimension Reduction with Application in Groundwater Modeling

Horng, Ming-Jame,   Water Resources Bureau, MOEA, ROC

Address: 10F, 41-3, SEC. 3, Hsin-I Rd Taipei, Taiwan, ROC

Phone: 02-7542080

Fax: 02-7542244

Email: mjhorng@wrb.gov.tw

Li, Ker-Chau, University of California, Los Angeles

Yeh, William W-G,

Abstract: Mathematical modeling is generally applied to describe various physical systems in many scientific fields. A common way of understanding the behavior of a mathematical model is through computer simulation. Such a task often involves model parameters which have to be set properly. Yet some model parameters can be rather hard to determine. One such example is a groundwater system used for predicting water head at a designated well. The crucial model parameters such as the hydraulic conductivity parameters in various aquifer zones can at best be estimated roughly from related geologic and hydrologic information. The uncertainty from such input model parameters often lead to substantial prediction error. This article takes a new approach to uncertainty analysis of mathematical models via a statistical dimension reduction method, Sliced Inverse Regression(SIR). It generalizes differential sensitivity analysis and is able to provide a global view on the nonlinear relationship between the output and the input model parameters.

4. A Three-Way Subclassification Approach to Multiple-Class Discriminant Analysis

Chen, Chun-Houh,   Academia Sinica, ROC

Address: Institute of Statistical Science, Academia Sinica Taipei, Taiwan, ROC

Phone:

Fax: (886)-2-7831523

Email: cchen@stat.sinica.edu.tw

Li, Ker-Chau, University of California, Los Angeles

Abstract: Discriminant analysis problems are much harder to handle when the number of classes k is large. In view of this tendency, a strategy based on three-way subclassification is proposed. The original k-class problem is first decomposed into k(k-1)/2 smaller problems, each involving a 3-way classification between a pair of classes A and B, and a combined third class which is just the union of all other classes. Then the results from each subcalssification will be synthesized under a conditional error rate analysis. Our formulation of 3-way subclassification is ideal for exploring the degree of unanimity among different decisions made in subclassification. This advantage is illustrated well in the context of hand-written digit recognition, using the data set of LeCun et al(1989). A large portion of high quality images in the test set can be easily filtered out by 3-way subclassification - they are predicted with an error rate less than $1\%$.

List of speakers who are nonmembers: None