27,885 research outputs found
Some Asymptotic Theory for Functional Regression and Classification
Exploiting an expansion for analytic functions of operators, the asymptotic distribution of an estimator of the functional regression parameter is obtained in a rather simple way; the result is applied to testing linear hypotheses. The expansion is also used to obtain a quick proof for the asymptotic optimality of a functional classification rule, given Gaussian populations
Asymptotic Normality of Support Vector Machine Variants and Other Regularized Kernel Methods
In nonparametric classification and regression problems, regularized kernel
methods, in particular support vector machines, attract much attention in
theoretical and in applied statistics. In an abstract sense, regularized kernel
methods (simply called SVMs here) can be seen as regularized M-estimators for a
parameter in a (typically infinite dimensional) reproducing kernel Hilbert
space. For smooth loss functions, it is shown that the difference between the
estimator, i.e.\ the empirical SVM, and the theoretical SVM is asymptotically
normal with rate . That is, the standardized difference converges
weakly to a Gaussian process in the reproducing kernel Hilbert space. As common
in real applications, the choice of the regularization parameter may depend on
the data. The proof is done by an application of the functional delta-method
and by showing that the SVM-functional is suitably Hadamard-differentiable
Optimal Bayes Classifiers for Functional Data and Density Ratios
Bayes classifiers for functional data pose a challenge. This is because
probability density functions do not exist for functional data. As a
consequence, the classical Bayes classifier using density quotients needs to be
modified. We propose to use density ratios of projections on a sequence of
eigenfunctions that are common to the groups to be classified. The density
ratios can then be factored into density ratios of individual functional
principal components whence the classification problem is reduced to a sequence
of nonparametric one-dimensional density estimates. This is an extension to
functional data of some of the very earliest nonparametric Bayes classifiers
that were based on simple density ratios in the one-dimensional case. By means
of the factorization of the density quotients the curse of dimensionality that
would otherwise severely affect Bayes classifiers for functional data can be
avoided. We demonstrate that in the case of Gaussian functional data, the
proposed functional Bayes classifier reduces to a functional version of the
classical quadratic discriminant. A study of the asymptotic behavior of the
proposed classifiers in the large sample limit shows that under certain
conditions the misclassification rate converges to zero, a phenomenon that has
been referred to as "perfect classification". The proposed classifiers also
perform favorably in finite sample applications, as we demonstrate in
comparisons with other functional classifiers in simulations and various data
applications, including wine spectral data, functional magnetic resonance
imaging (fMRI) data for attention deficit hyperactivity disorder (ADHD)
patients, and yeast gene expression data
Multivariate Analysis from a Statistical Point of View
Multivariate Analysis is an increasingly common tool in experimental high
energy physics; however, many of the common approaches were borrowed from other
fields. We clarify what the goal of a multivariate algorithm should be for the
search for a new particle and compare different approaches. We also translate
the Neyman-Pearson theory into the language of statistical learning theory.Comment: Talk from PhyStat2003, Stanford, Ca, USA, September 2003, 4 pages,
LaTeX, 1 eps figures. PSN WEJT00
Unexpected properties of bandwidth choice when smoothing discrete data for constructing a functional data classifier
The data functions that are studied in the course of functional data analysis
are assembled from discrete data, and the level of smoothing that is used is
generally that which is appropriate for accurate approximation of the
conceptually smooth functions that were not actually observed. Existing
literature shows that this approach is effective, and even optimal, when using
functional data methods for prediction or hypothesis testing. However, in the
present paper we show that this approach is not effective in classification
problems. There a useful rule of thumb is that undersmoothing is often
desirable, but there are several surprising qualifications to that approach.
First, the effect of smoothing the training data can be more significant than
that of smoothing the new data set to be classified; second, undersmoothing is
not always the right approach, and in fact in some cases using a relatively
large bandwidth can be more effective; and third, these perverse results are
the consequence of very unusual properties of error rates, expressed as
functions of smoothing parameters. For example, the orders of magnitude of
optimal smoothing parameter choices depend on the signs and sizes of terms in
an expansion of error rate, and those signs and sizes can vary dramatically
from one setting to another, even for the same classifier.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1158 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
An Alternative Approach to Functional Linear Partial Quantile Regression
We have previously proposed the partial quantile regression (PQR) prediction
procedure for functional linear model by using partial quantile covariance
techniques and developed the simple partial quantile regression (SIMPQR)
algorithm to efficiently extract PQR basis for estimating functional
coefficients. However, although the PQR approach is considered as an attractive
alternative to projections onto the principal component basis, there are
certain limitations to uncovering the corresponding asymptotic properties
mainly because of its iterative nature and the non-differentiability of the
quantile loss function. In this article, we propose and implement an
alternative formulation of partial quantile regression (APQR) for functional
linear model by using block relaxation method and finite smoothing techniques.
The proposed reformulation leads to insightful results and motivates new
theory, demonstrating consistency and establishing convergence rates by
applying advanced techniques from empirical process theory. Two simulations and
two real data from ADHD-200 sample and ADNI are investigated to show the
superiority of our proposed methods
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