6 research outputs found
Unsupervised Learning via Mixtures of Skewed Distributions with Hypercube Contours
Mixture models whose components have skewed hypercube contours are developed
via a generalization of the multivariate shifted asymmetric Laplace density.
Specifically, we develop mixtures of multiple scaled shifted asymmetric Laplace
distributions. The component densities have two unique features: they include a
multivariate weight function, and the marginal distributions are also
asymmetric Laplace. We use these mixtures of multiple scaled shifted asymmetric
Laplace distributions for clustering applications, but they could equally well
be used in the supervised or semi-supervised paradigms. The
expectation-maximization algorithm is used for parameter estimation and the
Bayesian information criterion is used for model selection. Simulated and real
data sets are used to illustrate the approach and, in some cases, to visualize
the skewed hypercube structure of the components
Multivariate, Heteroscedastic Empirical Bayes via Nonparametric Maximum Likelihood
Multivariate, heteroscedastic errors complicate statistical inference in many
large-scale denoising problems. Empirical Bayes is attractive in such settings,
but standard parametric approaches rest on assumptions about the form of the
prior distribution which can be hard to justify and which introduce unnecessary
tuning parameters. We extend the nonparametric maximum likelihood estimator
(NPMLE) for Gaussian location mixture densities to allow for multivariate,
heteroscedastic errors. NPMLEs estimate an arbitrary prior by solving an
infinite-dimensional, convex optimization problem; we show that this convex
optimization problem can be tractably approximated by a finite-dimensional
version.
The empirical Bayes posterior means based on an NPMLE have low regret,
meaning they closely target the oracle posterior means one would compute with
the true prior in hand. We prove an oracle inequality implying that the
empirical Bayes estimator performs at nearly the optimal level (up to
logarithmic factors) for denoising without prior knowledge. We provide
finite-sample bounds on the average Hellinger accuracy of an NPMLE for
estimating the marginal densities of the observations. We also demonstrate the
adaptive and nearly-optimal properties of NPMLEs for deconvolution. We apply
our method to two denoising problems in astronomy, constructing a fully
data-driven color-magnitude diagram of 1.4 million stars in the Milky Way and
investigating the distribution of 19 chemical abundance ratios for 27 thousand
stars in the red clump. We also apply our method to hierarchical linear models,
illustrating the advantages of nonparametric shrinkage of regression
coefficients on an education data set and on a microarray data set
Subspace coverings with multiplicities
We study the problem of determining the minimum number of affine
subspaces of codimension that are required to cover all points of
at least times while covering the
origin at most times. The case is a classic result of Jamison,
which was independently obtained by Brouwer and Schrijver for . The
value of also follows from a well-known theorem of Alon and F\"uredi
about coverings of finite grids in affine spaces over arbitrary fields. Here we
determine the value of this function exactly in various ranges of the
parameters. In particular, we prove that for we have
, while for we have , and also study the
transition between these two ranges. While previous work in this direction has
primarily employed the polynomial method, we prove our results through more
direct combinatorial and probabilistic arguments, and also exploit a connection
to coding theory.Comment: 15 page
Space programs summary no. 37-49, volume 3 for the period December 1, 1967 to January 30, 1968. Supporting research and advanced development
Space program research projects on systems analysis and engineering, telecommunications, guidance and control, propulsion, and data system