4,900 research outputs found
The spectral analysis of nonstationary categorical time series using local spectral envelope
Most classical methods for the spectral analysis are based on the assumption that the time
series is stationary. However, many time series in practical problems shows nonstationary
behaviors. The data from some fields are huge and have variance and spectrum which changes
over time. Sometimes,we are interested in the cyclic behavior of the categorical-valued time
series such as EEG sleep state data or DNA sequence, the general method is to scale the
data, that is, assign numerical values to the categories and then use the periodogram to find
the cyclic behavior. But there exists numerous possible scaling. If we arbitrarily assign the
numerical values to the categories and proceed with a spectral analysis, then the results will
depend on the particular assignment. We would like to find the all possible scaling that
bring out all of the interesting features in the data. To overcome these problems, there have
been many approaches in the spectral analysis.
Our goal is to develop a statistical methodology for analyzing nonstationary categorical
time series in the frequency domain. In this dissertation, the spectral envelope methodology
is introduced for spectral analysis of categorical time series. This provides the general
framework for the spectral analysis of the categorical time series and summarizes information
from the spectrum matrix. To apply this method to nonstationary process, I used the
TBAS(Tree-Based Adaptive Segmentation) and local spectral envelope based on the piecewise
stationary process. In this dissertation,the TBAS(Tree-Based Adpative Segmentation)
using distance function based on the Kullback-Leibler divergence was proposed to find the
best segmentation
On Locally Dyadic Stationary Processes
We introduce the concept of local dyadic stationarity, to account for
non-stationary time series, within the framework of Walsh-Fourier analysis. We
define and study the time varying dyadic ARMA models (tvDARMA). It is proven
that the general tvDARMA process can be approximated locally by either a tvDMA
and a tvDAR process.Comment: 27 pages, 2 figure
Time-frequency analysis of locally stationary Hawkes processes
Locally stationary Hawkes processes have been introduced in order to
generalise classical Hawkes processes away from stationarity by allowing for a
time-varying second-order structure. This class of self-exciting point
processes has recently attracted a lot of interest in applications in the life
sciences (seismology, genomics, neuro-science,...), but also in the modelling
of high-frequency financial data. In this contribution we provide a fully
developed nonparametric estimation theory of both local mean density and local
Bartlett spectra of a locally stationary Hawkes process. In particular we apply
our kernel estimation of the spectrum localised both in time and frequency to
two data sets of transaction times revealing pertinent features in the data
that had not been made visible by classical non-localised approaches based on
models with constant fertility functions over time.Comment: Bernoulli journal, A Para{\^i}tr
Covariance matrix estimation for stationary time series
We obtain a sharp convergence rate for banded covariance matrix estimates of
stationary processes. A precise order of magnitude is derived for spectral
radius of sample covariance matrices. We also consider a thresholded covariance
matrix estimator that can better characterize sparsity if the true covariance
matrix is sparse. As our main tool, we implement Toeplitz [Math. Ann. 70 (1911)
351-376] idea and relate eigenvalues of covariance matrices to the spectral
densities or Fourier transforms of the covariances. We develop a large
deviation result for quadratic forms of stationary processes using m-dependence
approximation, under the framework of causal representation and physical
dependence measures.Comment: Published in at http://dx.doi.org/10.1214/11-AOS967 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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