16,184 research outputs found
The Degrees of Freedom of Partial Least Squares Regression
The derivation of statistical properties for Partial Least Squares regression
can be a challenging task. The reason is that the construction of latent
components from the predictor variables also depends on the response variable.
While this typically leads to good performance and interpretable models in
practice, it makes the statistical analysis more involved. In this work, we
study the intrinsic complexity of Partial Least Squares Regression. Our
contribution is an unbiased estimate of its Degrees of Freedom. It is defined
as the trace of the first derivative of the fitted values, seen as a function
of the response. We establish two equivalent representations that rely on the
close connection of Partial Least Squares to matrix decompositions and Krylov
subspace techniques. We show that the Degrees of Freedom depend on the
collinearity of the predictor variables: The lower the collinearity is, the
higher the Degrees of Freedom are. In particular, they are typically higher
than the naive approach that defines the Degrees of Freedom as the number of
components. Further, we illustrate how the Degrees of Freedom approach can be
used for the comparison of different regression methods. In the experimental
section, we show that our Degrees of Freedom estimate in combination with
information criteria is useful for model selection.Comment: to appear in the Journal of the American Statistical Associatio
Fitting Prediction Rule Ensembles with R Package pre
Prediction rule ensembles (PREs) are sparse collections of rules, offering
highly interpretable regression and classification models. This paper presents
the R package pre, which derives PREs through the methodology of Friedman and
Popescu (2008). The implementation and functionality of package pre is
described and illustrated through application on a dataset on the prediction of
depression. Furthermore, accuracy and sparsity of PREs is compared with that of
single trees, random forest and lasso regression in four benchmark datasets.
Results indicate that pre derives ensembles with predictive accuracy comparable
to that of random forests, while using a smaller number of variables for
prediction
Neural network ensembles: Evaluation of aggregation algorithms
Ensembles of artificial neural networks show improved generalization
capabilities that outperform those of single networks. However, for aggregation
to be effective, the individual networks must be as accurate and diverse as
possible. An important problem is, then, how to tune the aggregate members in
order to have an optimal compromise between these two conflicting conditions.
We present here an extensive evaluation of several algorithms for ensemble
construction, including new proposals and comparing them with standard methods
in the literature. We also discuss a potential problem with sequential
aggregation algorithms: the non-frequent but damaging selection through their
heuristics of particularly bad ensemble members. We introduce modified
algorithms that cope with this problem by allowing individual weighting of
aggregate members. Our algorithms and their weighted modifications are
favorably tested against other methods in the literature, producing a sensible
improvement in performance on most of the standard statistical databases used
as benchmarks.Comment: 35 pages, 2 figures, In press AI Journa
Random Forests: some methodological insights
This paper examines from an experimental perspective random forests, the
increasingly used statistical method for classification and regression problems
introduced by Leo Breiman in 2001. It first aims at confirming, known but
sparse, advice for using random forests and at proposing some complementary
remarks for both standard problems as well as high dimensional ones for which
the number of variables hugely exceeds the sample size. But the main
contribution of this paper is twofold: to provide some insights about the
behavior of the variable importance index based on random forests and in
addition, to propose to investigate two classical issues of variable selection.
The first one is to find important variables for interpretation and the second
one is more restrictive and try to design a good prediction model. The strategy
involves a ranking of explanatory variables using the random forests score of
importance and a stepwise ascending variable introduction strategy
Variable Selection for Nonparametric Gaussian Process Priors: Models and Computational Strategies
This paper presents a unified treatment of Gaussian process models that
extends to data from the exponential dispersion family and to survival data.
Our specific interest is in the analysis of data sets with predictors that have
an a priori unknown form of possibly nonlinear associations to the response.
The modeling approach we describe incorporates Gaussian processes in a
generalized linear model framework to obtain a class of nonparametric
regression models where the covariance matrix depends on the predictors. We
consider, in particular, continuous, categorical and count responses. We also
look into models that account for survival outcomes. We explore alternative
covariance formulations for the Gaussian process prior and demonstrate the
flexibility of the construction. Next, we focus on the important problem of
selecting variables from the set of possible predictors and describe a general
framework that employs mixture priors. We compare alternative MCMC strategies
for posterior inference and achieve a computationally efficient and practical
approach. We demonstrate performances on simulated and benchmark data sets.Comment: Published in at http://dx.doi.org/10.1214/11-STS354 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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