A wavelet lifting approach to long-memory estimation

Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing Hurst parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors even in regular data settings. Armed with this evidence our method sheds new light on long-memory intensity results in environmental and climate science applications, sometimes suggesting that different scientific conclusions may need to be drawn.     

Exact simulation of Gaussian Time Series from Nonparametric Spectral Estimates with Application to Bootstrapping

The circulant embedding method for generating statistically exact simulations of time series from certain Gaussian distributed stationary processes is attractive because of its advantage in computational speed over a competitive method based upon the modified Cholesky decomposition. We demonstrate that the circulant embedding method can be used to generate simulations from stationary processes whose spectral density functions are dictated by a number of popular nonparametric estimators, including all direct spectral estimators (a special case being the periodogram), certain lag window spectral estimators, all forms of Welch's overlapped segment averaging spectral estimator and all basic multitaper spectral estimators. One application for this technique is to generate time series for bootstrapping various statistics. When used with bootstrapping, our proposed technique avoids some – but not all – of the pitfalls of previously proposed frequency domain methods for simulating time series.     

Adaptive lifting for nonparametric regression

Many wavelet shrinkage methods assume that the data are observed on an equally spaced grid of length of the form 2^J for some J. These methods require serious modification or preprocessed data to cope with irregularly spaced data. The lifting scheme is a recent mathematical innovation that obtains a multiscale analysis for irregularly spaced data. A key lifting component is the “predict” step where a prediction of a data point is made. The residual from the prediction is stored and can be thought of as a wavelet coefficient. This article exploits the flexibility of lifting by adaptively choosing the kind of prediction according to a criterion. In this way the smoothness of the underlying ‘wavelet’ can be adapted to the local properties of the function. Multiple observations at a point can readily be handled by lifting through a suitable choice of prediction. We adapt existing shrinkage rules to work with our adaptive lifting methods. We use simulation to demonstrate the improved sparsity of our techniques and improved regression performance when compared to both wavelet and non-wavelet methods suitable for irregular data. We also exhibit the benefits of our adaptive lifting on the real inductance plethysmography and motorcycle data.     

Test of hypotheses in panel data models when the regressor and disturbances are possibly non-stationary

This paper considers the problem of hypothesis testing in a simple panel data regression model with random individual effects and serially correlated disturbances. Following Baltagi et al. (Econom. J. 11:554–572, 2008), we allow for the possibility of non-stationarity in the regressor and/or the disturbance term. While Baltagi et al. (Econom. J. 11:554–572, 2008) focus on the asymptotic properties and distributions of the standard panel data estimators, this paper focuses on testing of hypotheses in this setting. One important finding is that unlike the time-series case, one does not necessarily need to rely on the “super-efficient” type AR estimator by Perron and Yabu (J. Econom. 151:56–69, 2009) to make an inference in the panel data. In fact, we show that the simple t-ratio always converges to the standard normal distribution, regardless of whether the disturbances and/or the regressor are stationary.     

Wavelet processes and adaptive estimation of the evolutionary wavelet spectrum

This paper defines and studies a new class of non-stationary random processes constructed from discrete non-decimated wavelets which generalizes the Cramér (Fourier) representation of stationary time series. We define an evolutionary wavelet spectrum (EWS) which quantifies how process power varies locally over time and scale. We show how the EWS may be rigorously estimated by a smoothed wavelet periodogram and how both these quantities may be inverted to provide an estimable time-localized autocovariance. We illustrate our theory with a pedagogical example based on discrete non-decimated Haar wavelets and also a real medical time series example.     

Estimation of 2D jump location curve and 3D jump location surface in nonparametric regression

A new procedure is proposed to estimate the jump location curve and surface in the two-dimensional (2D) and three-dimensional (3D) nonparametric jump regression models, respectively. In each of the 2D and 3D cases, our estimation procedure is motivated by the fact that, under some regularity conditions, the ridge location of the rotational difference kernel estimate (RDKE; Qiu in Sankhyā Ser. A 59, 268–294, 1997, and J. Comput. Graph. Stat. 11, 799–822, 2002; Garlipp and Müller in Sankhyā Ser. A 69, 55–86, 2007) obtained from the noisy image is asymptotically close to the jump location of the true image. Accordingly, a computational procedure based on the kernel smoothing method is designed to find the ridge location of RDKE, and the result is taken as the jump location estimate. The sequence relationship among the points comprising our jump location estimate is obtained. Our jump location estimate is produced without the knowledge of the range or shape of jump region. Simulation results demonstrate that the proposed estimation procedure can detect the jump location very well, and thus it is a useful alternative for estimating the jump location in each of the 2D and 3D cases.     

A Non‐Parametric Estimator of the Spectral Density of a Continuous‐Time Gaussian Process Observed at Random Times

Abstract. In numerous applications data are observed at random times and an estimated graph of the spectral density may be relevant for characterizing and explaining phenomena. By using a wavelet analysis, one derives a non‐parametric estimator of the spectral density of a Gaussian process with stationary increments (or a stationary Gaussian process) from the observation of one path at random discrete times. For every positive frequency, this estimator is proved to satisfy a central limit theorem with a convergence rate depending on the roughness of the process and the moment of random durations between successive observations. In the case of stationary Gaussian processes, one can compare this estimator with estimators based on the empirical periodogram. Both estimators reach the same optimal rate of convergence, but the estimator based on wavelet analysis converges for a different class of random times. Simulation examples and an application to biological data are also provided.     

Variance decompositions of nonlinear time series using stochastic simulation and sensitivity analysis

In this paper, A variance decomposition approach to quantify the effects of endogenous and exogenous variables for nonlinear time series models is developed. This decomposition is taken temporally with respect to the source of variation. The methodology uses Monte Carlo methods to affect the variance decomposition using the ANOVA-like procedures proposed in Archer et al. (J. Stat. Comput. Simul. 58:99–120, 1997), Sobol’ (Math. Model. 2:112–118, 1990). The results of this paper can be used in investment problems, biomathematics and control theory, where nonlinear time series with multiple inputs are encountered.     

Using recursive algorithms for the efficient identification of smoothing spline ANOVA models

In this paper we present a unified discussion of different approaches to the identification of smoothing spline analysis of variance (ANOVA) models: (i) the “classical” approach (in the line of Wahba in Spline Models for Observational Data, 1990; Gu in Smoothing Spline ANOVA Models, 2002; Storlie et al. in Stat. Sin., 2011) and (ii) the State-Dependent Regression (SDR) approach of Young in Nonlinear Dynamics and Statistics (2001). The latter is a nonparametric approach which is very similar to smoothing splines and kernel regression methods, but based on recursive filtering and smoothing estimation (the Kalman filter combined with fixed interval smoothing). We will show that SDR can be effectively combined with the “classical” approach to obtain a more accurate and efficient estimation of smoothing spline ANOVA models to be applied for emulation purposes. We will also show that such an approach can compare favorably with kriging.     

A note on “Data depths satisfying the projection property”

This note is on two theorems in a paper by Rainer Dyckerhoff (Allg. Stat. Arch. 88:163–190, 2004). We state a missing condition in Theorem 3. On the other hand, Theorem 2 can be weakened.     

Different estimators of the spectral matrix: an empirical comparison testing a new shrinkage estimator

ABSTRACT

In this paper we propose a new non parametric estimator of the spectral matrix of a multivariate stationary stochastic process, with the main goal to locally improve the deficiencies of the smoothed periodogram in terms of mean square error of the estimates. Our estimator is based on a convex linear combination of the frequency averaged periodogram and an estimate of the true mean spectral matrix across frequencies. In a wide simulation study we show that our estimator turns out to be able to markedly improve the frequency averaged periodogram especially at central frequencies.     

A ‘nondecimated’ lifting transform

Classical nondecimated wavelet transforms are attractive for many applications. When the data comes from complex or irregular designs, the use of second generation wavelets in nonparametric regression has proved superior to that of classical wavelets. However, the construction of a nondecimated second generation wavelet transform is not obvious. In this paper we propose a new ‘nondecimated’ lifting transform, based on the lifting algorithm which removes one coefficient at a time, and explore its behavior. Our approach also allows for embedding adaptivity in the transform, i.e. wavelet functions can be constructed such that their smoothness adjusts to the local properties of the signal. We address the problem of nonparametric regression and propose an (averaged) estimator obtained by using our nondecimated lifting technique teamed with empirical Bayes shrinkage. Simulations show that our proposed method has higher performance than competing techniques able to work on irregular data. Our construction also opens avenues for generating a ‘best’ representation, which we shall explore.     

Frequency domain bootstrap for ratio statistics under long-range dependence

A frequency domain bootstrap (FDB) is a common technique to apply Efron’s independent and identically distributed resampling technique (Efron, 1979) to periodogram ordinates – especially normalized periodogram ordinates – by using spectral density estimates. The FDB method is applicable to several classes of statistics, such as estimators of the normalized spectral mean, the autocorrelation (but not autocovariance), the normalized spectral density function, and Whittle parameters. While this FDB method has been extensively studied with respect to short-range dependent time processes, there is a dearth of research on its use with long-range dependent time processes. Therefore, we propose an FDB methodology for ratio statistics under long-range dependence, using semi- and nonparametric spectral density estimates as a normalizing factor. It is shown that the FDB approximation allows for valid distribution estimation for a broad class of stationary, long-range (or short-range) dependent linear processes, without any stringent assumptions on the distribution of the underlying process. The results of a large simulation study show that the FDB approximation using a semi- or nonparametric spectral density estimator is often robust for various values of a long-memory parameter reflecting magnitude of dependence. We apply the proposed procedure to two data examples.     

Non-parametric Curve Estimation by Wavelet Thresholding with Locally Stationary Errors

An important aspect in the modelling of biological phenomena in living organisms, whether the measurements are of blood pressure, enzyme levels, biomechanical movements or heartbeats, etc., is time variation in the data. Thus, the recovery of a 'smooth' regression or trend function from noisy time-varying sampled data becomes a problem of particular interest. Here we use non-linear wavelet thresholding to estimate a regression or a trend function in the presence of additive noise which, in contrast to most existing models, does not need to be stationary. (Here, non-stationarity means that the spectral behaviour of the noise is allowed to change slowly over time). We develop a procedure to adapt existing threshold rules to such situations, e.g. that of a time-varying variance in the errors. Moreover, in the model of curve estimation for functions belonging to a Besov class with locally stationary errors, we derive a near-optimal rate for the -risk between the unknown function and our soft or hard threshold estimator, which holds in the general case of an error distribution with bounded cumulants. In the case of Gaussian errors, a lower bound on the asymptotic minimax rate in the wavelet coefficient domain is also obtained. Also it is argued that a stronger adaptivity result is possible by the use of a particular location and level dependent threshold obtained by minimizing Stein's unbiased estimate of the risk. In this respect, our work generalizes previous results, which cover the situation of correlated, but stationary errors. A natural application of our approach is the estimation of the trend function of non-stationary time series under the model of local stationarity. The method is illustrated on both an interesting simulated example and a biostatistical data-set, measurements of sheep luteinizing hormone, which exhibits a clear non-stationarity in its variance.     

A stable estimator of the information matrix under EM for dependent data

This article develops a new and stable estimator for information matrix when the EM algorithm is used in maximum likelihood estimation. This estimator is constructed using the smoothed individual complete-data scores that are readily available from running the EM algorithm. The method works for dependent data sets and when the expectation step is an irregular function of the conditioning parameters. In comparison to the approach of Louis (J. R. Stat. Soc., Ser. B 44:226–233, 1982), this new estimator is more stable and easier to implement. Both real and simulated data are used to demonstrate the use of this new estimator.     

Nonstationary Data Analysis by Time Deformation

ABSTRACT

In this article we discuss methodology for analyzing nonstationary time series whose periodic nature changes approximately linearly with time. We make use of the M-stationary process to describe such data sets, and in particular we use the discrete Euler(p) model to obtain forecasts and estimate the spectral characteristics. We discuss the use of the M-spectrum for displaying linear time-varying periodic content in a time series realization in much the same way that the spectrum shows periodic content within a realization of a stationary series. We also introduce the instantaneous frequency and spectrum of an M-stationary process for purposes of describing how frequency changes with time. To illustrate our techniques we use one simulated data set and two bat echolocation signals that show time varying frequency behavior. Our results indicate that for data whose periodic content is changing approximately linearly in time, the Euler model serves as a very good model for spectral analysis, filtering, and forecasting. Additionally, the instantaneous spectrum is shown to provide better representation of the time-varying frequency content in the data than window-based techniques such as the Gabor and wavelet transforms. Finally, it is noted that the results of this article can be extended to processes whose frequencies change like at^α, a > 0, ?∞ < α < ? ∞.     

Estimation of a nonparametric regression spectrum for multivariate time series

Estimation of a nonparametric regression spectrum based on the periodogram is considered. Neither trend estimation nor smoothing of the periodogram is required. Alternatively, for cases where spectral estimation of phase shifts fails and the shift does not depend on frequency, a time domain estimator of the lag-shift is defined. Asymptotic properties of the frequency and time domain estimators are derived. Simulations and a data example illustrate the methods.     

Forecasting data revisions of GDP: a mixed frequency approach

Releases of GDP data undergo a series of revisions over time. These revisions have an impact on the results of macroeconometric models documented by the growing literature on real-time data applications. Revisions of U.S. GDP data can be explained and are partly predictable according to Faust et al. (J. Money Credit Bank. 37(3):403–419, 2005) or Fixler and Grimm (J. Product. Anal. 25:213–229, 2006). This analysis proposes the inclusion of mixed frequency data for forecasting GDP revisions. Thereby, the information set available around the first data vintage can be better exploited than the pure quarterly data. In-sample and out-of-sample results suggest that forecasts of GDP revisions can be improved by using mixed frequency data.     

VARIANCE ESTIMATION FOR SAMPLE AUTOCOVARIANCES: DIRECT AND RESAMPLING APPROACHES

The usual covariance estimates for data _n-1 from a stationary zero-mean stochastic process {X_t} are the sample covariances Both direct and resampling approaches are used to estimate the variance of the sample covariances. This paper compares the performance of these variance estimates. Using a direct approach, we show that a consistent windowed periodogram estimate for the spectrum is more effective than using the periodogram itself. A frequency domain bootstrap for time series is proposed and analyzed, and we introduce a frequency domain version of the jackknife that is shown to be asymptotically unbiased and consistent for Gaussian processes. Monte Carlo techniques show that the time domain jackknife and subseries method cannot be recommended. For a Gaussian underlying series a direct approach using a smoothed periodogram is best; for a non-Gaussian series the frequency domain bootstrap appears preferable. For small samples, the bootstraps are dangerous: both the direct approach and frequency domain jackknife are better.