95JCGS02\P0083-------------------------------------------------------
Tools for Two-dimensional Exploratory Projection Pursuit
Christian Posse
In this paper we examine the current status of exploratory projection
pursuit. This involves a large comparison of all two-dimensional projection
indices and optimization algorithms that have been proposed in the
literature.
We show that the efficacy of the exploration depends to a large extent
on the optimization routine. We also stress the importance of
studying the behavior of the empirical projection indices rather than
the underlying theoretical distances they estimate. Indeed, indices
based on orthonormal polynomial expansions differ greatly in their
behavior from the theoretical performance of the weighted
$L^2$-distances they estimate. This study reveals three universal
indices, namely the Legendre index (Friedman, 1987), the Hermite index
(Hall, 1989) and the chi-squared index (Posse, 1993), which are
sensitive to any kind of departure from normality in the core of the
distribution, and two indices ideal for catching clusters, namely the
Laguerre-Fourier index (Morton, 1989) and the Natural Hermite index
(Cook et al., 1993).
To illustrate the efficacy of exploratory projection pursuit, consider
the 5 measurements made on 200 Australian crabs, most of them belonging
to four groups (Campbell and Mahon, 1974). Without using the information
about the groups, an exploration based on the tools recommended in this
paper is able to reveal a clustered projection close to the one
exhibited in Figure 1, not found by principal component analysis and
similar to the one obtained by multiple discriminant analysis.
Key Words: Exploratory multivariate data analysis; Two-dimensional
projections; Nonnormality; Global optimization.
95JCGS02\P0101-----------------------------------------------------------
The Visual Separability of Plotting Symbols in Scatterplots
Lothar Tremmel
Which symbols should be used to represent different groups of data in
the same scatterplot? Hypotheses are derived to predict which symbol
pairs should lead to good separability, based on the contrast of the
symbols' visual properties of `features'. In two experiments,
experimental scatterplots were shown to subjects on a computer screen;
the dependent variable was the decision time to judge which of the two
presented symbols was the more frequent one. Analyses of the
within-subject effects yielded the following results: (1) Important
feature contrasts are brightness, number of line endings, and curvature.
(2) Symbols that differ simultaneously in two feature dimensions may be
more separable than symbols that differ only in either one. (3) The contrasts
between circular symbols and radial line symbols like the plus sign
or the asterisk are excellent. Practical applications of these
findings are discussed, as well as their contribution to the theory of
visual perception.
Keywords: Plotting symbols; Pattern recognition; Scatterplots;
Rapid visual processing; Visual perception.
95JCGS02\P0134----------------------------------------------------------
Differentiation of the Cholesky Algorithm
S. P. Smith
One way to estimate variance components is by restricted maximum
likelihood. The log-likelihood function is fully defined by the
Cholesky factor of a matrix that is usually large and sparse. In
this paper forward and backward differentiation methods are
developed for calculating the first and second derivatives of the
Cholesky factor and functions of it. These differentiation methods
are general and can be applied to either a full or a sparse matrix.
Moreover, these methods can be used to calculate the derivatives
that are needed for restricted maximum likelihood, resulting in
substantial savings in computation.
Key Words: backward differentiation, determinant, forward
differentiation, recursion, restricted maximum likelihood,
sparse matrix, variance components.
95JCGS02\P0113----------------------------------------------------------
Tail-Specific Linear Approximations for Efficient Bootstrap Simulations
Tim Hesterberg
Two effective variance-reduction techniques for estimating probabilities
and quantiles in the tails of bootstrap distributions---importance sampling
and concomitants of order statistics---are based on linear approximations.
Although these techniques offer potential asymptotic variance reductions
by factors of nine to infinity, in practice the reductions may be only
by a factor of two or smaller because of inaccurate linear
approximations.
We develop tail-specific linear approximations that are more accurate
where the accuracy is important, in the tails of distributions. Our
methods fall into two categories---influence function methods and
regression methods. Both can be applied without problem-specific analytical
calculations, and both have tail-specific versions. We apply the
tail-specific approximations to importance sampling and concomitants
and propose another technique that uses linear approximations,
post-stratification implemented using the saddlepoint. This technique,
shares the same $0(n^{-1/2}B^{-1}$ variance as the concomitants
procedure. Tail-specific approximations improve the performance of
the variance-reduction techniques by a factor of about three in our
simulations.
Key Words: Concomitants of order statistics; Importance sampling;
Jackknife; Stratified sampling; Variance reduction.
95JCGS02\P0148----------------------------------------------------------
Conditioning in Markov Chain Monte Carlo
Charles J. Geyer
The so-called ``Rao-Blackwellized'' estimators proposed by Gelfand and
Smith do not always reduce variance in Markov chain Monte Carlo when
the dependence in the Markov chain is taken into account. An
illustrative example is given, and a theorem characterizing the necessary and
sufficient condition for such an estimator to always reduce variance is
proved.
Key Words: Gibbs sampler; Markov chain Monte Carlo; Rao-Blackwellization.