561 research outputs found
Functional Linear Mixed Models for Irregularly or Sparsely Sampled Data
We propose an estimation approach to analyse correlated functional data which
are observed on unequal grids or even sparsely. The model we use is a
functional linear mixed model, a functional analogue of the linear mixed model.
Estimation is based on dimension reduction via functional principal component
analysis and on mixed model methodology. Our procedure allows the decomposition
of the variability in the data as well as the estimation of mean effects of
interest and borrows strength across curves. Confidence bands for mean effects
can be constructed conditional on estimated principal components. We provide
R-code implementing our approach. The method is motivated by and applied to
data from speech production research
Functional Regression
Functional data analysis (FDA) involves the analysis of data whose ideal
units of observation are functions defined on some continuous domain, and the
observed data consist of a sample of functions taken from some population,
sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the
development of this field, which has accelerated in the past 10 years to become
one of the fastest growing areas of statistics, fueled by the growing number of
applications yielding this type of data. One unique characteristic of FDA is
the need to combine information both across and within functions, which Ramsay
and Silverman called replication and regularization, respectively. This article
will focus on functional regression, the area of FDA that has received the most
attention in applications and methodological development. First will be an
introduction to basis functions, key building blocks for regularization in
functional regression methods, followed by an overview of functional regression
methods, split into three types: [1] functional predictor regression
(scalar-on-function), [2] functional response regression (function-on-scalar)
and [3] function-on-function regression. For each, the role of replication and
regularization will be discussed and the methodological development described
in a roughly chronological manner, at times deviating from the historical
timeline to group together similar methods. The primary focus is on modeling
and methodology, highlighting the modeling structures that have been developed
and the various regularization approaches employed. At the end is a brief
discussion describing potential areas of future development in this field
Assessing, valuing and protecting our environment- is there a statistical challenge to be answered?
This short article describes some of the evolution in environmental regulation, management and monitoring and the information needs, closely aligned to the statistical challenges to deliver the evidence base for change and effect
Wavelet-based estimation with multiple sampling rates
We suggest an adaptive sampling rule for obtaining information from noisy
signals using wavelet methods. The technique involves increasing the sampling
rate when relatively high-frequency terms are incorporated into the wavelet
estimator, and decreasing it when, again using thresholded terms as an
empirical guide, signal complexity is judged to have decreased. Through
sampling in this way the algorithm is able to accurately recover relatively
complex signals without increasing the long-run average expense of sampling. It
achieves this level of performance by exploiting the opportunities for
near-real time sampling that are available if one uses a relatively high
primary resolution level when constructing the basic wavelet estimator. In the
practical problems that motivate the work, where signal to noise ratio is
particularly high and the long-run average sampling rate may be several hundred
thousand operations per second, high primary resolution levels are quite
feasible.Comment: Published at http://dx.doi.org/10.1214/009053604000000751 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonparametric regression for locally stationary random fields under stochastic sampling design
In this study, we develop an asymptotic theory of nonparametric regression
for locally stationary random fields (LSRFs) in observed at irregularly spaced locations
in . We first derive the uniform
convergence rate of general kernel estimators, followed by the asymptotic
normality of an estimator for the mean function of the model. Moreover, we
consider additive models to avoid the curse of dimensionality arising from the
dependence of the convergence rate of estimators on the number of covariates.
Subsequently, we derive the uniform convergence rate and joint asymptotic
normality of the estimators for additive functions. We also introduce
approximately -dependent RFs to provide examples of LSRFs. We find that
these RFs include a wide class of L\'evy-driven moving average RFs.Comment: 50 page
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