43 research outputs found
A note on quadratic forms of stationary functional time series under mild conditions
We study distributional properties of a quadratic form of a stationary
functional time series under mild moment conditions. As an important
application, we obtain consistency rates of estimators of spectral density
operators and prove joint weak convergence to a vector of complex Gaussian
random operators. Weak convergence is established based on an approximation of
the form via transforms of Hilbert-valued martingale difference sequences. As a
side-result, the distributional properties of the long-run covariance operator
are established
Locally Stationary Functional Time Series
The literature on time series of functional data has focused on processes of
which the probabilistic law is either constant over time or constant up to its
second-order structure. Especially for long stretches of data it is desirable
to be able to weaken this assumption. This paper introduces a framework that
will enable meaningful statistical inference of functional data of which the
dynamics change over time. We put forward the concept of local stationarity in
the functional setting and establish a class of processes that have a
functional time-varying spectral representation. Subsequently, we derive
conditions that allow for fundamental results from nonstationary multivariate
time series to carry over to the function space. In particular, time-varying
functional ARMA processes are investigated and shown to be functional locally
stationary according to the proposed definition. As a side-result, we establish
a Cram\'er representation for an important class of weakly stationary
functional processes. Important in our context is the notion of a time-varying
spectral density operator of which the properties are studied and uniqueness is
derived. Finally, we provide a consistent nonparametric estimator of this
operator and show it is asymptotically Gaussian using a weaker tightness
criterion than what is usually deemed necessary
A note on Herglotz's theorem for time series on function spaces
In this article, we prove Herglotz's theorem for Hilbert-valued time series.
This requires the notion of an operator-valued measure, which we shall make
precise for our setting. Herglotz's theorem for functional time series allows
to generalize existing results that are central to frequency domain analysis on
the function space. In particular, we use this result to prove the existence of
a functional Cram{\'e}r representation of a large class of processes, including
those with jumps in the spectral distribution and long-memory processes. We
furthermore obtain an optimal finite dimensional reduction of the time series
under weaker assumptions than available in the literature. The results of this
paper therefore enable Fourier analysis for processes of which the spectral
density operator does not necessarily exist
Optimizing the use of blood based biomarkers to improve the clinical management of advanced non-small cell lung cancer
A note on Herglotzâs theorem for time series on function spaces
In this article, we prove Herglotzâs theorem for Hilbert-valued time series. This requires the notion of an operator-valued measure, which we shall make precise for our setting. Herglotzâs theorem for functional time series allows to generalize existing results that are central to frequency domain analysis on the function space. In particular, we use this result to prove the existence of a functional CramĂ©r representation of a large class of processes, including those with jumps in the spectral distribution and long-memory processes. We furthermore obtain an optimal ïŹnite dimensional reduction of the time series under weaker assumptions than available in the literature. The results of this paper therefore enable Fourier analysis for processes of which the spectral density operator does not necessarily exist
Pivotal tests for relevant differences in the second order dynamics of functional time series
Motivated by the need to statistically quantify differences between modern
(complex) data-sets which commonly result as high-resolution measurements of
stochastic processes varying over a continuum, we propose novel testing
procedures to detect relevant differences between the second order dynamics of
two functional time series. In order to take the between-function dynamics into
account that characterize this type of functional data, a frequency domain
approach is taken. Test statistics are developed to compare differences in the
spectral density operators and in the primary modes of variation as encoded in
the associated eigenelements. Under mild moment conditions, we show convergence
of the underlying statistics to Brownian motions and construct pivotal test
statistics. The latter is essential because the nuisance parameters can be
unwieldy and their robust estimation infeasible, especially if the two
functional time series are dependent. Besides from these novel features, the
properties of the tests are robust to any choice of frequency band enabling
also to compare energy contents at a single frequency. The finite sample
performance of the tests are verified through a simulation study and are
illustrated with an application to fMRI data
Pivotal tests for relevant differences in the second order dynamics of functional time series
Motivated by the need to statistically quantify differences between modern (complex) datasets which commonly result as high-resolution measurements of stochastic processes varying over a continuum, we propose novel testing procedures to detect relevant differences between the second order dynamics of two functional time series. In order to take the between-function dynamics into account that characterize this type of functional data, a frequency domain approach is taken. Test statistics are developed to compare differences in the spectral density operators and in the primary modes of variation as encoded in the associated eigenelements. Under mild moment conditions, we show convergence of the underlying statistics to Brownian motions and obtain pivotal test statistics via a self-normalization approach. The latter is essential because the nuisance parameters can be unwieldly and their robust estimation infeasible, especially if the two functional time series are dependent. Besides from these novel features, the properties of the tests are robust to any choice of frequency band enabling also to compare energy contents at a single frequency. The finite sample performance of the tests are verified through a simulation study and are illustrated with an application to fMRI data
A general framework to quantify deviations from structural assumptions in the analysis of nonstationary function-valued processes
We present a general theory to quantify the uncertainty from imposing
structural assumptions on the second-order structure of nonstationary Hilbert
space-valued processes, which can be measured via functionals of time-dependent
spectral density operators. The second-order dynamics are well-known to be
elements of the space of trace-class operators, the latter is a Banach space of
type 1 and of cotype 2, which makes the development of statistical inference
tools more challenging. A part of our contribution is to obtain a weak
invariance principle as well as concentration inequalities for (functionals of)
the sequential time-varying spectral density operator. In addition, we
introduce deviation measures in the nonstationary context, and derive
estimators that are asymptotically pivotal. We then apply this framework and
propose statistical methodology to investigate the validity of structural
assumptions for nonstationary response surface data, such as low-rank
assumptions in the context of time-varying dynamic fPCA and principle separable
component analysis, deviations from stationarity with respect to the square
root distance, and deviations from zero functional canonical coherency
A statistical framework for analyzing shape in a time series of random geometric objects
We introduce a new framework to analyze shape descriptors that capture the
geometric features of an ensemble of point clouds. At the core of our approach
is the point of view that the data arises as sampled recordings from a metric
space-valued stochastic process, possibly of nonstationary nature, thereby
integrating geometric data analysis into the realm of functional time series
analysis. We focus on the descriptors coming from topological data analysis.
Our framework allows for natural incorporation of spatial-temporal dynamics,
heterogeneous sampling, and the study of convergence rates. Further, we derive
complete invariants for classes of metric space-valued stochastic processes in
the spirit of Gromov, and relate these invariants to so-called ball volume
processes. Under mild dependence conditions, a weak invariance principle in
is established for sequential empirical
versions of the latter, assuming the probabilistic structure possibly changes
over time. Finally, we use this result to introduce novel test statistics for
topological change, which are distribution free in the limit under the
hypothesis of stationarity.Comment: Submission versio