36,275 research outputs found
Regularized system identification using orthonormal basis functions
Most of existing results on regularized system identification focus on
regularized impulse response estimation. Since the impulse response model is a
special case of orthonormal basis functions, it is interesting to consider if
it is possible to tackle the regularized system identification using more
compact orthonormal basis functions. In this paper, we explore two
possibilities. First, we construct reproducing kernel Hilbert space of impulse
responses by orthonormal basis functions and then use the induced reproducing
kernel for the regularized impulse response estimation. Second, we extend the
regularization method from impulse response estimation to the more general
orthonormal basis functions estimation. For both cases, the poles of the basis
functions are treated as hyperparameters and estimated by empirical Bayes
method. Then we further show that the former is a special case of the latter,
and more specifically, the former is equivalent to ridge regression of the
coefficients of the orthonormal basis functions.Comment: 6 pages, final submission of an contribution for European Control
Conference 2015, uploaded on March 20, 201
A generalized Gaussian process model for computer experiments with binary time series
Non-Gaussian observations such as binary responses are common in some
computer experiments. Motivated by the analysis of a class of cell adhesion
experiments, we introduce a generalized Gaussian process model for binary
responses, which shares some common features with standard GP models. In
addition, the proposed model incorporates a flexible mean function that can
capture different types of time series structures. Asymptotic properties of the
estimators are derived, and an optimal predictor as well as its predictive
distribution are constructed. Their performance is examined via two simulation
studies. The methodology is applied to study computer simulations for cell
adhesion experiments. The fitted model reveals important biological information
in repeated cell bindings, which is not directly observable in lab experiments.Comment: 49 pages, 4 figure
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
The Harmonic Analysis of Kernel Functions
Kernel-based methods have been recently introduced for linear system
identification as an alternative to parametric prediction error methods.
Adopting the Bayesian perspective, the impulse response is modeled as a
non-stationary Gaussian process with zero mean and with a certain kernel (i.e.
covariance) function. Choosing the kernel is one of the most challenging and
important issues. In the present paper we introduce the harmonic analysis of
this non-stationary process, and argue that this is an important tool which
helps in designing such kernel. Furthermore, this analysis suggests also an
effective way to approximate the kernel, which allows to reduce the
computational burden of the identification procedure
Nonparametric inference in generalized functional linear models
We propose a roughness regularization approach in making nonparametric
inference for generalized functional linear models. In a reproducing kernel
Hilbert space framework, we construct asymptotically valid confidence intervals
for regression mean, prediction intervals for future response and various
statistical procedures for hypothesis testing. In particular, one procedure for
testing global behaviors of the slope function is adaptive to the smoothness of
the slope function and to the structure of the predictors. As a by-product, a
new type of Wilks phenomenon [Ann. Math. Stat. 9 (1938) 60-62; Ann. Statist. 29
(2001) 153-193] is discovered when testing the functional linear models.
Despite the generality, our inference procedures are easy to implement.
Numerical examples are provided to demonstrate the empirical advantages over
the competing methods. A collection of technical tools such as
integro-differential equation techniques [Trans. Amer. Math. Soc. (1927) 29
755-800; Trans. Amer. Math. Soc. (1928) 30 453-471; Trans. Amer. Math. Soc.
(1930) 32 860-868], Stein's method [Ann. Statist. 41 (2013) 2786-2819] [Stein,
Approximate Computation of Expectations (1986) IMS] and functional Bahadur
representation [Ann. Statist. 41 (2013) 2608-2638] are employed in this paper.Comment: Published at http://dx.doi.org/10.1214/15-AOS1322 in the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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