Autoregressive Sieve Bootstrap Research Articles

Sea clutter radar target detector based on autoregressive sieve bootstrap

The autoregressive sieve (ARS) bootstrap method can capture the correlation of time series and expand the original data set to the required volume. In this study, we introduce the ARS bootstrap into the matrix constant false alarm rate (CFAR) method to counter the inability of the Monte Carlo methods to extract the detection threshold due to the limited number of pulses under a certain false alarm probability. We assume the echo data of each range cell in a coherent processing interval (CPI) to be a stationary time series, and regard target detection as a two-sample hypothesis testing problem. We propose a statistic to test the equivalence of the autocovariance corresponding to two potential time series by using the correlation between pulses in a short pulse dataset, and analyze the validity of the ARS bootstrap for this statistic. The statistic distribution is obtained according to the bootstrap procedure and the threshold is obtained under fixed false alarm probabilities. Numerical experiments on the simulated and real sea clutter data were conducted to compare the detection results of the partial matrix CFAR detectors for the short and long pulse cases. With the introduction of the bootstrap method, we do not need to model the clutter amplitude distribution and the related statistical characteristics during target detection. To improve the detection probability with low signal-to-clutter ratio (SCR), we propose the two-sample autocovariance vector (ACV) detector based on the proposed statistic and use the correlation properties between pulses. We observed a 28% increase in detection probability when the SCR was 0 dB compared to other matrix CFAR methods, such as detectors based on Logarithmic Euclidean (LE) distance, Kernel Function-based Symmetric Kullback-Leibler (KSKL) divergence, and Total Bregman divergence (TBD) for the short pulse case.

A testing approach to clustering scalar time series

This article considers clustering stationary scalar time series using their marginal properties and a hierarchical method. Two major issues involved are to detect the existence of clusters and to determine their number. We propose a new test statistic for detecting whether a data set consists of multiple clusters and a new procedure to determine the number of clusters. The proposed method is based on the jumps, that is, the increments, in the heights of the dendrogram when a hierarchical clustering is applied to the data. We use autoregressive sieve bootstrap to obtain a reference distribution of the test statistics and propose an iterative procedure to find the number of clusters. The clusters found are internally homogeneous according to the test statistics used in the analysis. The performance of the proposed procedure in finite samples is investigated by Monte Carlo simulations and illustrated by some empirical examples. Comparisons with some existing methods for selecting the number of clusters are also investigated.

Simultaneous inference for autocovariances based on autoregressive sieve bootstrap

In this article, maximum deviations of sample autocovariances and autocorrelations from their theoretical counterparts over an increasing set of lags are considered. The asymptotic distribution of such statistics for physically dependent stationary time series, which is of Gumbel type, only depends on second‐order properties of the underlying time series. Since the autoregressive sieve bootstrap is able to mimic the second‐order structure asymptotically correctly it is an obvious problem whether the autoregressive (AR) sieve bootstrap, which has been shown to work for a number of relevant statistics in time series analysis, asymptotically works for maximum deviations of autocovariances and autocorrelations as well. This article shows that the question can be answered positively. Moreover, potential applications including spectral density estimation and an investigation of finite sample properties of the AR‐sieve bootstrap proposal by simulation are given.

Closed-form expression for finite predictor coefficients of multivariate ARMA processes

We derive a closed-form expression for the finite predictor coefficients of multivariate ARMA (autoregressive moving-average) processes. The expression is given in terms of several explicit matrices that are of fixed sizes independent of the number of observations. The significance of the expression is that it provides us with a linear-time algorithm to compute the finite predictor coefficients. In the proof of the expression, a correspondence result between two relevant matrix-valued outer functions plays a key role. We apply the expression to determine the asymptotic behavior of a sum that appears in the autoregressive model fitting and the autoregressive sieve bootstrap. The results are new even for univariate ARMA processes.

Nonlinear autoregressive sieve bootstrap based on extreme learning machines.

The aim of the paper is to propose and discuss a sieve bootstrap scheme based on Extreme Learning Machines for non linear time series. The procedure is fully nonparametric in its spirit and retains the conceptual simplicity of the residual bootstrap. Using Extreme Learning Machines in the resampling scheme can dramatically reduce the computational burden of the bootstrap procedure, with performances comparable to the NN-Sieve bootstrap and computing time similar to the ARSieve bootstrap. A Monte Carlo simulation experiment has been implemented, in order to evaluate the performance of the proposed procedure and to compare it with the NN-Sieve bootstrap. The distributions of the bootstrap variance estimators appear to be consistent, delivering good results both in terms of accuracy and bias, for either linear and nonlinear statistics (such as the mean and the median) and smooth functions of means (such as the variance and the covariance).

Bootstrap-assisted tests of symmetry for dependent data

ABSTRACTThe paper considers the problem of testing for symmetry (about an unknown centre) of the marginal distribution of a strictly stationary and weakly dependent stochastic process. The possibility of using the autoregressive sieve bootstrap and stationary bootstrap procedures to obtain critical values and P-values for symmetry tests is explored. Bootstrap-assisted tests for symmetry are straightforward to implement and require no prior estimation of asymptotic variances. The small-sample properties of a wide variety of tests are investigated using Monte Carlo experiments. A bootstrap-assisted version of the triples test is found to have the best overall performance.

Estimated Wold Representation and Spectral-Density-Driven Bootstrap for Time Series

Summary The second-order dependence structure of purely non-deterministic stationary processes is described by the coefficients of the famous Wold representation. These coefficients can be obtained by factorizing the spectral density of the process. This relationship together with some spectral density estimator is used to obtain consistent estimators of these coefficients. A spectral-density-driven bootstrap for time series is then developed which uses the entire sequence of estimated moving average coefficients together with appropriately generated pseudoinnovations to obtain a bootstrap pseudo-time-series. It is shown that if the underlying process is linear and if the pseudoinnovations are generated by means of an independent and identically distributed wild bootstrap which mimics, to the extent necessary, the moment structure of the true innovations, this bootstrap proposal asymptotically works for a wide range of statistics. The relationships of the proposed bootstrap procedure to some other bootstrap procedures, including the auto-regressive sieve bootstrap, are discussed. It is shown that the latter is a special case of the spectral-density-driven bootstrap, if a parametric auto-regressive spectral density estimator is used. Simulations investigate the performance of the new bootstrap procedure in finite sample situations. Furthermore, a real life data example is presented.

Baxter’s inequality and sieve bootstrap for random fields

The concept of the autoregressive (AR) sieve bootstrap is investigated for the case of spatial processes in Z2. This procedure fits AR models of increasing order to the given data and, via resampling of the residuals, generates bootstrap replicates of the sample. The paper explores the range of validity of this resampling procedure and provides a general check criterion which allows to decide whether the AR sieve bootstrap asymptotically works for a specific statistic of interest or not. The criterion may be applied to a large class of stationary spatial processes. As another major contribution of this paper, a weighted Baxter-inequality for spatial processes is provided. This result yields a rate of convergence for the finite predictor coefficients, i.e. the coefficients of finite-order AR model fits, towards the autoregressive coefficients which are inherent to the underlying process under mild conditions. The developed check criterion is applied to some particularly interesting statistics like sample autocorrelations and standardized sample variograms. A simulation study shows that the procedure performs very well compared to normal approximations as well as block bootstrap methods in finite samples.

Bootstrap methods for long-memory processes: a review

This manuscript summarized advances in bootstrap methods for long-range dependent time series data. The stationary linear long-memory process is briefly described, which is a target process for bootstrap methodologies on time-domain and frequency-domain in this review. We illustrate time-domain bootstrap under long-range dependence, moving or non-overlapping block bootstraps, and the autoregressive-sieve bootstrap. In particular, block bootstrap methodologies need an adjustment factor for the distribution estimation of the sample mean in contrast to applications to weak dependent time processes. However, the autoregressive-sieve bootstrap does not need any other modification for application to long-memory. The frequency domain bootstrap for Whittle estimation is provided using parametric spectral density estimates because there is no current nonparametric spectral density estimation method using a kernel function for the linear long-range dependent time process.

A distance test of normality for a wide class of stationary processes

A distance test for normality of the one-dimensional marginal distribution of stationary fractionally integrated processes is considered. The test is implemented by using an autoregressive sieve bootstrap approximation to the null sampling distribution of the test statistic. The bootstrap-based test does not require knowledge of either the dependence parameter of the data or of the appropriate norming factor for the test statistic. The small-sample properties of the test are examined by means of Monte Carlo experiments. An application to real-world data is also presented.

Using the Bootstrap to Test for Symmetry Under Unknown Dependence

This article considers tests for symmetry of the one-dimensional marginal distribution of fractionally integrated processes. The tests are implemented by using an autoregressive sieve bootstrap approximation to the null sampling distribution of the relevant test statistics. The sieve bootstrap allows inference on symmetry to be carried out without knowledge of either the memory parameter of the data or of the appropriate norming factor for the test statistic and its asymptotic distribution. The small-sample properties of the proposed method are examined by means of Monte Carlo experiments, and applications to real-world data are also presented.

On the Vector Autoregressive Sieve Bootstrap

The concept of autoregressive sieve bootstrap is investigated for the case of vector autoregressive (VAR) time series. This procedure fits a finite‐order VAR model to the given data and generates residual‐based bootstrap replicates of the time series. The paper explores the range of validity of this resampling procedure and provides a general check criterion, which allows to decide whether the VAR sieve bootstrap asymptotically works for a specific statistic or not. In the latter case, we will point out the exact reason that causes the bootstrap to fail.The developed check criterion is then applied to some particularly interesting statistics.

Valid Resampling of Higher-Order Statistics Using the Linear Process Bootstrap and Autoregressive Sieve Bootstrap

We show that the linear process bootstrap (LPB) and the autoregressive sieve bootstrap (AR sieve) are, in general, not valid for statistics whose large-sample distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or not. Inspired by the block-of-blocks bootstrap, we circumvent this non-validity by applying the LPB and AR sieve to suitably blocked data and not to the original data itself. In a simulation study, we compare the LPB, AR sieve, and moving block bootstrap applied directly and to blocked data.

On the Applicability of the Sieve Bootstrap in Time Series Panels*

In this article, we investigate the validity of the univariate autoregressive sieve bootstrap applied to time series panels characterized by general forms of cross-sectional dependence, including but not restricted to cointegration. Using the final equations approach we show that while it is possible to write such a panel as a collection of infinite order autoregressive equations, the innovations of these equations are not vector white noise. This causes the univariate autoregressive sieve bootstrap to be invalid in such panels. We illustrate this result with a small numerical example using a simple DGP for which the sieve bootstrap is invalid, and show that the extent of the invalidity depends on the value of specific parameters. We also show that Monte Carlo simulations in small samples can be misleading about the validity of the univariate autoregressive sieve bootstrap. The results in this article serve as a warning about the practical use of the autoregressive sieve bootstrap in panels where cross-sectional dependence of general form may be present.