Robertas Gabrys
Department of Mathematics and Statistics
Utah
State University
Detecting Change-Point in the Mean Function of
Functional Observations
Functional
data analysis (FDA) has been enjoying increased popularity over the last decade
due to its applicability to problems which are difficult to cast into a
framework of scalar or vector observations. Even if such standard approaches
are available, the functional approach often leads to a more natural and
parsimonious description of the data, and to more accurate inference and
prediction results. Principal component analysis (PCA) has become a fundamental
tool of functional data analysis. It represents the functional data as Xi(t)
= µ(t) + Σ1≤ ℓ < ∞ ηi,ℓυℓ(t), where
µ(t) is the common mean, υℓ(t)
are the eigenfunctions of the covariance operator,
and the ηi,,ℓ are the scores.
Inferential procedures assume that the mean function µ(t)
is the same for all values of i. If, in fact, the observations do not come from
one population, but rather their mean changes at some point(s), the results of
PCA are confounded by the change(s). It
is therefore important to develop a methodology to test the assumption of a
common functional mean. We develop such a test using quantities which can be
readily computed in the R package fda. The null
distribution of the test statistic is asymptotically pivotal with a well-known
asymptotic distribution. The asymptotic test has excellent finite sample
performance. Its application is illustrated on temperature data from England.
Keywords: Change point
detection, Functional data analysis, Mean of functional data, Significance
test.
Joint
work with:
István Berkes, Graz University of Technology, Graz, Austria
Lajos Horváth University of Utah, Salt Lake City, USA
Piotr Kokoszka, Utah State University, Logan, USA