97JCGS02\P0143-----------------------------------------------------
The Filtered Mode Tree
David J. Marchette and Edward J. Wegman
The mode tree is a useful tool for visualizing the modal structure
of a density. Locations of modes of the density are plotted as a
function of the bandwidth used in the kernel estimate of the
density. Because the mode tree uses a single bandwidth in the
kernel estimator, it exhibits all the drawbacks that a single
bandwidth kernel estimator has, particularly for densities with
large tails or differences in the scales of the modes. A
modification is presented that uses the filtered kernel
estimator, a version of the kernel estimator using a small number
of bandwidths. The two mode trees are compared on some
synthetic data, and on a data set from DNA flow cytometry.
Key Words: Bump hunting; Filtered kernel density estimation;
Graphical methods; Kernel density estimation; Mode
estimation; Multimodality.
97JCGS02\P0160----------------------------------------------------
Graphical Explanation in Belief Networks
David Madigan, Krzysztof Mosurski, and Russell G. Almond
Belief networks provide an important bridge between statistical
modeling and expert systems. This article presents methods for
visualizing probabilistic ``evidence flows'' in belief networks,
thereby enabling belief networks to explain their behavior.
Building on earlier research on explanation in expert systems,
we present a hierarchy of explanations, ranging from simple
colorings to detailed displays. Our approach complements parallel
work on textual explanations in belief networks. Graphical-Belief,
Mathsoft Inc.'s belief network software, implements the methods.
Key Words: Bayesian networks; Dynamic graphics; Visualization
97JCGS02\P0182------------------------------------------------------
A Subpixel Image Restoration Algorithm
John Gavin and Christopher Jennison
In statistical image reconstruction, data are often recorded
on a regular grid of squares, known as pixels, and the
reconstructed image is defined on the same pixel grid.
Thus, the reconstruction of a continuous planar image is
piecewise constant on pixels, and boundaries in the image
consist of horizontal and vertical edges lying between
pixels. This approximation to the true boundary can result
in a loss of information that may be quite noticeable for
small objects, only a few pixels in size. Increasing the
resolution of the sensor may not be a practical alternative.
If some prior assumptions are made about the true image,
however, reconstruction to a greater accuracy than that of
the recording sensor's pixel grid is possible. We adopt a
Bayesian approach, incorporating prior information about
the true image in a stochastic model that attaches higher
probability to images with shorter total edge length. In
reconstructions, pixels may be of a single color or split
between two colors. The model is illustrated using both
real and simulated data.
Key Words: Bayesian statistical image reconstruction; Confocal
microscopy; Deconvolution; Edge detection; Gibbs sampler;
Markov chain Monte Carlo; Metropolis algorithm; Subpixel
resolution.
97JCGS02\P0202------------------------------------------------------
On the Orderings and Groupings of Conditional
Maximizations Within ECM-Type Algorithms
David A. van Dyk and Xiao-Li Meng
The ECM and ECME algorithms are generalizations of the EM
algorithm in which the maximization (M) step is replaced by
several conditional maximization (CM) steps. The order that
the CM-steps are performed is trivial to change and generally
affects how fast the algorithm converges. Moreover, the same
order of CM-steps need not be used at each iteration and in
some applications it is feasible to group two or more CM-steps
into one larger CM-step. These issues also arise when
implementing the Gibbs sampler, and in this article we study
them in the context of fitting log-linear and random-effects
models with ECM-type algorithms. We find that some standard
theoretical measures of the rate of convergence can be of
little use in comparing the computational time required,
and that common strategies such as using a random ordering
may not provide the desired effects. We also develop two
algorithms for fitting random-effects models to illustrate
that with careful selection of CM-steps, ECM-type algorithms
can be substantially faster than the standard EM algorithm.
Key Words: Contingency table; Convergence rate; Data augmentation;
Gibbs sampler; EM algorithm; Incomplete data; IPF; Missing
data; Model reduction; Random-effects models.
97JCGS02\P0224---------------------------------------------------------
Spatial Regression Models, Response Surfaces, and Process Optimization
Michael A. O'Connell and Russell D. Wolfinger
Spatial regression models are developed as a complementary
alternative to second-order polynomial response surfaces in
the context of process optimization. These models provide
estimates of design variable effects and smooth, data-faithful
approximations to the unknown response function over the
design space. The predicted response surfaces are driven
by the covariance structures of the models. Several structures,
isotropic and anisotropic, are considered and connections with
thin plate splines are reviewed. Estimation of covariance
parameters is achieved via maximum likelihood and residual
maximum likelihood. A feature of the spatial regression
approach is the visually appealing graphical summaries that are
produced. These allow rapid and intuitive identification of
process windows on the design space for which the response
achieves target performance. Relevant design issues are
briefly discussed and spatial designs, such as the packing
designs available in Gosset, are suggested as a suitable
design complement. The spatial regression models also perform
well with no global design, for example with data obtained from
series of designs on the same space of design variables. The
approach is illustrated with an example involving the
optimization of components in a DNA amplification assay. A
Monte Carlo comparison of the spatial models with both thin
plate splines and second-order polynomial response for a
scenario motivated by the example is also given. This shows
superior performance of the spatial models to the second-order
polynomials with respect to both prediction over the complete
design space and for cross-validation prediction error in the
region of the optimum. An anisotropic spatial regression model
performs best for a high noise case and both this model and the
thin plate spline for a low noise case. Spatial regression is
recommended for construction of response surfaces in all process
optimization applications.
Key Words: Maximum and minimax design; Mixed models; Response surface
methods; Thin plate splines; Universal kriging.
97JCGS02\P0242-------------------------------------------------------
Robustness of Tube Formula Based Confidence Bands
Clive Loader and Jiayang Sun
Simultaneous confidence bands of a regression curve may be used
to quantify the uncertainty of an estimate of the curve. The
tube formula for volumes of tubular neighborhoods of a manifold
provides a very powerful method for obtaining such bands at a
prescribed level, when errors are Gaussian. This article studies
robustness of the tube formula for non-Gaussian errors. The
formula holds without modification for an error vector with a
spherically symmetric distribution. Simulations are used for a
variety of independent non-Gaussian error distributions. The
results are acceptable for contaminated and heavy tailed error
distributions. The formula can break down in some extreme cases
for discrete and highly skewed errors. Computational issues
involved in applying the tube formula are also discussed.
Key Words: Regression; Simultaneous inference; Tail probability.