Anna E. Dudek

Generalized Seasonal Block Bootstrap in frequency analysis of cyclostationary signals

In this paper the Generalized Seasonal Block Bootstrap (GSBB) method is applied to the first and the second order characteristics of the cyclostationary (CS) signal. The new method for detection of the significant frequencies in the Fourier expansions of the mean and the autocovariance functions of CS processes is provided. The consistency of GSBB is shown. Finally, the bootstrap pointwise and simultaneous confidence intervals are constructed and hence hypothesis tests for the presence of the first and the second order cyclostationarity can be performed. The real data example is presented to show the possible applications of our results. Moreover, the simulation study was performed to check the efficiency and robustness to the noise of the considered method.

Block Bootstrap for Poisson‐Sampled Almost Periodic Processes

Let be an almost periodically correlated process and {N(t),t≥0} be a homogeneous Poisson process and {T,k≥1} be its jump moments. We assume that and {N(t),t≥0} are independent. Moreover, the process is not observed continuously but only in the time moments {T,k≥1}; In this paper, we focus on the estimation of the cyclic means of . The asymptotic normality of the rescaled error of the estimator is shown. Additionally, the bootstrap method based on the circular block bootstrap is proposed. The consistency of the bootstrap technique is proved, and the bootstrap pointwise and simultaneous confidence intervals for the cyclic means are constructed. The results are illustrated by a simulated data example.

A Bootstrap Algorithm for Data from a Periodic Multiplicative Intensity Function

The periodic multiplicative intensity model is considered. A new bootstrap method for non stationary counting processes which intensity function has some periodicity properties is presented. Its main advantage is that it does not destroy the temporal order and the original periodicity of the underlying counting process. The proposed algorithm is used to construct a bootstrap version of the maximum likelihood hazard function estimator. The consistency of the bootstrap method is shown. A possible modification of the proposed bootstrap method is discussed. The bootstrap simultaneous confidence intervals for the hazard function are presented. The telecommunication network traffic real data example is discussed.

PARMA Models with Applications in R

Periodic autoregressive–moving-average models (periodic ARMA models, PARMA models) are used to model non-stationary time series with periodic structure. They are similar to ARMA except the coefficients that are periodic in time with a common period. They are widely applied in climatology, hydrology, meteorology and economics data. In this paper we want to familiarize the readers with all the essential steps of PARMA model fitting. We present in detail the non-parametric spectral analysis, model identification, parameter estimation, diagnostic checking (model verification) and prediction on the real data example. Our aim is to provide appropriate tool for the complete analysis of periodic time series using PARMA modelling and to popularize this approach among non-specialists.

Bootstrap for the second-order analysis of Poisson-sampled almost periodic processes

In this paper we consider a continuous almost periodically correlated process {X(t), t ∈ R} that is observed at the jump moments of a stationary Poisson point process {N (t), t ≥ 0}. The processes {X(t), t ∈ R} and {N (t), t ≥ 0} are assumed to be independent. We define the kernel estimators of the Fourier coefficients of the autocovariance function of X(t) and investigate their asymptotic properties. Moreover, we propose a bootstrap method that provides consistent pointwise and simultaneous confidence intervals for the considered coefficients. Finally, to illustrate our results we provide a simulated data example.

Block Bootstrap for the Autocovariance Coefficients of Periodically Correlated Time Series

In this paper we propose a new technique of significant frequencies detection for periodically correlated time series. New method is based on bootstrap technique called Generalized Seasonal Block Bootstrap. Bootstrap procedure is applied in the time domain and then Fourier representation of autocovariance function for bootstrap samples is used. Finally, the simultaneous confidence intervals for the absolute values of the Fourier coefficients are calculated. The results are compared in the small simulation study with similar tools based on subsampling methodology and moving block bootstrap for almost periodic processes.

Deterministic/cyclostationary Signal Separation Using Bootstrap

Abstract Cyclostationarity (CS) is relatively a new technique that offers diagnostic advantages for analysis of faults related to a studied system. The aim of this paper is to address the issue of separating the second-order cyclostationary (CS2) component from the first-order cyclostationary component (CS1) of a signal when low speed fluctuations exist. A bootstrap-based method called Generalized Seasonal Block Bootstrap (GSBB) is applied on walking signals coming from elderly in order to characterize walking in this population and possible age-related walking disturbance, in one hand, and to explore and analyze the influence of low speed fluctuations on the first and second orders cyclostationary properties of signals, on the other hand. Two GSBB-based indicators have also been proposed to characterize the quality of the CS1 and CS2 estimates.

Bootstrap for almost cyclostationary processes with jitter effect

In this paper we consider almost cyclostationary processes with jitter effect. We propose a boot-strap approach based on the Moving Block Bootstrap method to construct pointwise and simultaneous confidence intervals for the Fourier coefficients of the autocovariance function of such processes. In the simulation study we show how our results can be used for second-order frequency detection. We compare behavior of our approach for jitter effects caused by perturbations from two distributions, namely uniform and truncated normal. Finally, we present a real data application of our methodology.

First and second order analysis for periodic random arrays using block bootstrap methods

In the paper row-wise periodically correlated triangular arrays are considered. The period length is assumed to grow in time. The Fourier decomposition of the mean and autocovariance functions for each row of the matrix is presented. To construct bootstrap estimators of the Fourier coefficients two block bootstrap techniques are used. These are the circular version of the Generalized Seasonal Block Bootstrap and the Circular Block Bootstrap. Consistency results for both methods are presented. Bootstrap-t equal-tailed confidence intervals for parameters of interest are constructed. Results are illustrated by an example based on simulated data.

Simulation Study of Performance of MBB in Overall Mean Estimation Problem for APC Time Series

In this paper the problem of the overall mean estimation of an almost periodically correlated (APC) time series is considered. The Moving Block Bootstrap method, its consistency result in the considered problem and the construction of the percentile equal-tailed bootstrap confidence intervals are recalled. The simulation study is performed to calculate the actual coverage probabilities for the various examples of the APC time series. The different lengths of the sample size and the wide range of possible block length choices are considered. The optimal block lengths for each time series and sample size are pointed out.