94JCGS03\P0235-------------------------------------------------------
Automatic Smoothing Spline Projection Pursuit
Charles B. Roosen and Trevor J. Hastie
A highly flexible nonparametric regression model for predicting a
response $y$ given covariates $\{x_k\}_{k=1}^d$ is the projection
pursuit regression (PPR) model $\hat{y}=h({\bf x})=\beta_0 + \sum_j
\beta_j f_j(\mbox{\boldmath $\alpha_j^T x$})$, where the $f_j$ are general
smooth functions with mean 0 and norm 1, and $\sum_{k=1}^d
\alpha_{kj}^2=1$. The standard PPR algorithm of Friedman and
Stuetzle (1981) estimates the smooth functions $f_j$ using the supersmoother
nonparametric scatterplot smoother. Friedman's algorithm constructs a
model with $M_{\max}$ linear combinations, then prunes back to a
simpler model of size $M \leq M_{\max}$, where $M$ and $M_{\max}$ are
specified by the user. The article discusses an alternative algorithm
in which the smooth functions are estimated using smoothing
splines. The direction coefficients \mbox{\boldmath $\alpha_j$},
the amount of smoothing in each direction, and the number of terms
$M$ and $M_{\max}$ are determined to optimize a single
generalized cross-validation measure.
Key Words: GCV; Multivariate function approximation; Neural networks;
Nonparametric regression.
94JCGS03\P0249-------------------------------------------------------
Automating the Partition of Indexes
James E. Stafford
In this article we show how attention to the structure of a
particular algebraic calculation can lead to the simple implementation
of powerful computer algebra tools. The creation of partitions for
a set of indexes is required for the implementation of many
theoretical structures. This may be difficult to do by hand even
when the number of indexes is only moderately large. These partitions
arise through the action of differentiation and so we mimic
differentiation in a computer algebra package to create partitions
of indexes. The strategies employed are extended to the creation of
complementary set partitions, their reduction to equivalence
classes, and the implementation of Edgeworth expansions and the
exlog relations.
Key Words: Bartlett identities; Complementary set partitions; Computer
algebra; Differentation; Equivalence classes; Exlog relations;
Hermite polynomials; Indexes; Partitions.
94JCGS03\P0261-------------------------------------------------------
On Markov Chain Monte Carlo Acceleration
Alan E. Gelfand and Sujit K. Sahu
Markov chain Monte Carlo (MCMC) methods are currently enjoying a
surge of interest within the statistical community. The goal of this
work is to formalize and support two distinct adaptive strategies that
typically accelerate the convergence of an MCMC algorithm. One approach
is through resampling; the other incorporates adaptive switching
of the transition kernel. Support is both by analytic arguments and
simulation study. Application is envisioned in low-dimensional but
nontrivial problems. Two pathological illustrations are presented.
Connections with reparameterization are discussed as well as possible
difficulties with infinitely often adaptation.
Key Words: Adaptive chains; Gibbs sampler; $L^1$ convergence; Markov
chain Monte Carlo; Metropolis-Hastings algorithm; Rejection method;
Resampling.
94JCGS03\P0277-------------------------------------------------------
Cave Plots: A Graphical Technique for Comparing Time Series
Richard A. Becker, Linda A. Clark, and Diane Lambert
Cave plots are a good way to compare data from two or more time
series, even if the two series are observed at irregular times. Cave
plots can show thousands of measurements, but they are readily
understood, preserve individual values, highlight short-term, long-term,
and periodic fluctuations, and make correlation (or lack of correlation)
between two time series apparent. We have used a page of cave plots
to show millions of measurements on network reliability and tens of
thousands of measurements on air quality. The plots highlight features
of the data that warrant a closer look.
Key Words: Rare events; Reliability; Vertical line graph.
94JCGS03\P0285-------------------------------------------------------
Structure Algorithms for Partially Ordered Isotonic Regression
H. Block, S. Qian, and A. Sampson
An algorithm for isotonic regression is called a structure algorithm
if it searches for a ``solution partition''---that is, a class of sets
on each of which the isotonic regression is a constant. We discuss
structure algorithms for partially ordered isotonic regression. In
this article we provide a new class of structure algorithms called the
isotonic block class (IBC) type algorithms. One of these is called
the isotonic block class with recursion method (IBCR) algorithm, which
works for partially ordered isotonic regression. It is a generalization
of the pooled adjacent violators algorithm and is simpler than the
min-max algorithm. We also give a polynomial time algorithm---the
isotonic block class with stratification (IBCS) algorithm for
matrix-ordered isotonic regression. We demonstrate the efficiency
of our IBCR algorithm by using simulation to estimate the relative
frequencies of the numbers of level sets of isotonic regressions on
certain two-dimensional grids with the matrix order.
Key Words: Constrained optimization problems; Order-retricted statistical
inference; Partial order; Structure algorithm; Worst time complexity.
94JCGS03\P0301-------------------------------------------------------
A Better Confidence Interval for Kappa ($\kappa$) on
Measuring Agreement Between Two Raters With Binary Outcomes
J. Jack Lee and Z. Nora Tu
Although the kappa statistic is widely used in measuring interrater
agreement, it is known that the standard confidence interval estimation
behaves poorly in small samples and for nonzero kappas. Efforts have
been made to improve the estimation through transformation and Edgeworth
expansion (Flack 1987). The results remain unsatisfactory when kappa
is far from 0, however, even with the sample size as large as 100. In
this article we reparameterize the kappa statistic to reveal its
relationship with the marginal probability of agreement. The
reparameterization not only gives a more meaningful interpretation of
kappa but also clearly demonstrates that the range of kappa depends on
the marginal probabilities. Various two- and three-dimensional plots
are shown to illustrate the relationship among these parameters. The
immediate application is to construct a new confidence interval based
on the profile variance and reparameterization. Extensive simulation
studies show that the new confidence interval performs very well in
almost all parameter settings even when other methods fail.
Key Words: Interrater agreement; Monte Carlo studies; Profile variance;
Range constraints; Reparameterization.