Wavelet-based Estimation Research Articles

Wavelet Estimation Performance of Fractional Integrated Processes with Heavy-Tails

In this paper, we investigate the performance of four semi-parametric estimators in the wavelet domain in order to estimate the parameter of stationary long-memory models. The goal is to consider a wavelet estimate for the fractional differencing parameter d where the time series exhibit heavy tails. We show by Monte Carlo experiments that the wavelet Exact Local Whittle-type estimator improves considerably the other suggested wavelet-based estimators in terms of smaller bias, Root Mean Squared Error and variance. Furthermore, the simulation results show that the wavelet periodogram estimators perform better in most cases than wavelet ordinary least square estimate methods when the sample size is increased.

Estimation of long-range dependence in gappy Gaussian time series.

Knowledge of the long range dependence (LRD) parameter is critical to studies of self-similar behavior. However, statistical estimation of the LRD parameter becomes difficult when the observed data are masked by short range dependence and other noise, or are gappy in nature (i.e., some values are missing in an otherwise regular sampling). Currently there is a lack of theory for spectral- and wavelet-based estimators of the LRD parameter for gappy data. To address this, we estimate the LRD parameter for gappy Gaussian semiparametric time series based upon undecimated wavelet variances. We develop estimation methods by using novel estimators of the wavelet variances, providing asymptotic theory for the joint distribution of the wavelet variances and our estimator of the LRD parameter. We introduce sandwich estimators to compute standard errors for our estimates. We demonstrate the efficacy of our methods using Monte Carlo simulations, and provide guidance on practical issues such as how to select the range of wavelet scales. We demonstrate the methodology using two applications: one for gappy Arctic sea-ice draft data, and another for gap free and gappy daily average temperature data collected at 17 locations in south central Sweden.

Wavelet-Based and Fourier-Based Multivariate Whittle Estimation: multiwave

Multivariate time series with long-dependence are observed in many applications such as finance, geophysics or neuroscience. Many packages provide estimation tools for univariate settings but few are addressing the problem of long-dependence estimation for multivariate settings. The package multiwave is providing efficient estimation procedures for multivariate time series. Two semi-parametric estimation methods of the long-memory exponents and long-run covariance matrix of time series are implemented. The first one is the Fourier-based estimation proposed by Shimotsu (2007) and the second one is a wavelet-based estimation described in Achard and Gannaz (2016). The objective of this paper is to provide an overview of the R package multiwave with its practical application perspectives.

Directed wavelet covariance

A causal wavelet decomposition of the covariance structure for bivariate locally stationary processes, named directed wavelet covariance, is introduced and discussed. Theoretically, when compared to Fourier-based quantities, wavelet-based estimators are more appropriate to non-stationary processes and processes with local patterns, outliers and rapid regime changes. Results of directed coherence (DC), wavelet coherence (WTC) and directed wavelet covariance (DWC) with simulated data are also presented. All three quantities could identify the simulated covariances structures. Finally, an illustration of the proposed directed wavelet covariance in a task-based EEG experiment is given.

Two-step wavelet-based estimation for Gaussian mixed fractional processes

A Gaussian mixed fractional process $$\{Y(t)\}_{t \in {\mathbb {R}}} = \{PX(t)\}_{t \in {\mathbb {R}}}$$ is a multivariate stochastic process obtained by pre-multiplying a vector of independent, Gaussian fractional process entries X by a nonsingular matrix P. It is interpreted that Y is observable, while X is a hidden process occurring in an (unknown) system of coordinates P. Mixed processes naturally arise as approximations to solutions of physically relevant classes of multivariate fractional stochastic differential equations under aggregation. We propose a semiparametric two-step wavelet-based method for estimating both the demixing matrix $$P^{-1}$$ and the memory parameters of X. The asymptotic normality of the estimator is established both in continuous and discrete time. Monte Carlo experiments show that the estimator is accurate over finite samples, while being very computationally efficient. As an application, we model a bivariate time series of annual tree ring width measurements.

Wavelet-based estimators of multivariable mean regression function with long-memory data

ABSTRACTWavelet analysis has been proved to be a powerful statistical technique in the non parametric regression. In this paper, we propose non linear wavelet-based estimators for multivariable mean regression function with long-memory data. We also provide an asymptotic expansion for the mean integrated squared error (MISE) of the function estimators. This MISE expansion still works even when the underlying mean regression function is only piecewise smooth. This paper extends the corresponding results in the literature for single variable to multivariable case.

Do co-jumps impact correlations in currency markets?

We quantify how co-jumps impact correlations in currency markets. To disentangle the continuous part of quadratic covariation from co-jumps, and study the influence of co-jumps on correlations, we propose a new wavelet-based estimator. The proposed estimation framework is able to localize the co-jumps very precisely through wavelet coefficients and identify statistically significant co-jumps. Empirical findings reveal the different behaviors of co-jumps during Asian, European, and U.S. trading sessions. Importantly, we document that co-jumps significantly influence correlation in currency markets.

Multivariate Hadamard self-similarity: Testing fractal connectivity

While scale invariance is commonly observed in each component of real world multivariate signals, it is also often the case that the inter-component correlation structure is not fractally connected, i.e., its scaling behavior is not determined by that of the individual components. To model this situation in a versatile manner, we introduce a class of multivariate Gaussian stochastic processes called Hadamard fractional Brownian motion (HfBm). Its theoretical study sheds light on the issues raised by the joint requirement of entry-wise scaling and departures from fractal connectivity. An asymptotically normal wavelet-based estimator for its scaling parameter, called the Hurst matrix, is proposed, as well as asymptotically valid confidence intervals. The latter are accompanied by original finite sample procedures for computing confidence intervals and testing fractal connectivity from one single and finite size observation. Monte Carlo simulation studies are used to assess the estimation performance as a function of the (finite) sample size, and to quantify the impact of omitting wavelet cross-correlation terms. The simulation studies are shown to validate the use of approximate confidence intervals, together with the significance level and power of the fractal connectivity test. The test performance and properties are further studied as functions of the HfBm parameters.

Wavelet estimation of density for censored data with censoring indicator missing at random

ABSTRACTIn this paper, we define the nonlinear wavelet estimator of density for the right censoring model with the censoring indicator missing at random (MAR), and develop its asymptotic expression for mean integrated squared error (MISE). Unlike for kernel estimator, the MISE expression of the estimator is not affected by the presence of discontinuities in the curve. Meanwhile, asymptotic normality of the estimator is established. The proposed estimator can reduce to the estimator defined by Li [Non-linear wavelet-based density estimators under random censorship. J Statist Plann Inference. 2003;117(1):35–58] when the censoring indicator MAR does not occur and a bandwidth in non-parametric estimation is close to zero. Also, we define another two nonlinear wavelet estimators of the density. A simulation is done to show the performance of the three proposed estimators.

Estimation of long memory in volatility using wavelets

AbstractThis work studies wavelet-based Whittle estimator of the fractionally integrated exponential generalized autoregressive conditional heteroscedasticity (FIEGARCH) model often used for modeling long memory in volatility of financial assets. The newly proposed estimator approximates the spectral density using wavelet transform, which makes it more robust to certain types of irregularities in data. Based on an extensive Monte Carlo study, both behavior of the proposed estimator and its relative performance with respect to traditional estimators are assessed. In addition, we study properties of the estimators in presence of jumps, which brings interesting discussion. We find that wavelet-based estimator may become an attractive robust and fast alternative to the traditional methods of estimation. In particular, a localized version of our estimator becomes attractive in small samples.

Wavelet-based estimation of regression function with strong mixing errors under fixed design

ABSTRACTWe consider wavelet-based non linear estimators, which are constructed by using the thresholding of the empirical wavelet coefficients, for the mean regression functions with strong mixing errors and investigate their asymptotic rates of convergence. We show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes Bsp, q. The theory is illustrated with some numerical examples.A new ingredient in our development is a Bernstein-type exponential inequality, for a sequence of random variables with certain mixing structure and are not necessarily bounded or sub-Gaussian. This moderate deviation inequality may be of independent interest.

Do Co-Jumps Impact Correlations in Currency Markets?

We study how co-jumps influence covariance and correlation in currency markets. We propose a new wavelet-based estimator of quadratic covariation that is able to disentangle the continuous part of quadratic covariation from co-jumps. The proposed estimator is able to identify the statistically significant co-jumps that impact covariance structures by using bootstrapped test statistics. Empirical findings reveal the behavior of co-jumps during Asian, European and U.S. trading sessions. Our results show that the impact of co-jumps on correlations increased during the years 2012 -- 2015. Hence appropriately estimating co-jumps is becoming a crucial step in understanding dependence in currency markets.

Moment and Bayesian wavelet regression from spatially correlated functional data

A new wavelet-based estimation methodology, in the context of spatial functional regression, is proposed to discriminate between small-scale and large scale variability of spatially correlated functional data, defined by depth-dependent curves. Specifically, the discrete wavelet transform of the data is computed in space and depth to reduce dimensionality. Moment-based regression estimation is applied for the approximation of the scaling coefficients of the functional response. While its wavelet coefficients are estimated in a Bayesian regression framework. Both regression approaches are implemented from the empirical versions of the scaling and wavelet auto-covariance and cross-covariance operators, characterizing the correlation structure of the spatial functional response. Weather stations in ocean islands display high spatial concentration. The proposed estimation methodology overcomes the difficulties arising in the estimation of ocean temperature field at different depths, from long records of ocean temperature measurements in these stations. Data are collected from The World-Wide Ocean Optics Database. The performance of the presented approach is tested in terms of 10-fold cross-validation, and residual spatial and depth correlation analysis. Additionally, an application to soil sciences, for prediction of electrical conductivity profiles is also considered to compare this approach with previous related ones, in the statistical analysis of spatially correlated curves in depth.

Multivariate wavelet density and regression estimators for stationary and ergodic continuous time processes: Asymptotic results

In the present paper, we are mainly concerned with the nonparametric estimation of the density as well as the regression function, related to stationary and ergodic continuous time processes, by using orthonormal wavelet bases. We provide the strong uniform consistency properties with rates of these estimators, over compact subsets of ℝd, under a general ergodic condition on the underlying processes. We characterize the asymptotic normality of considered wavelet-based estimators under easily verifiable conditions. The asymptotic properties of these estimators are obtained by means of the martingale approach.

Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: Asymptotic results

ABSTRACTIn the present paper, we are mainly concerned with the non parametric estimation of the density as well as the regression function by using orthonormal wavelet bases. We provide the strong uniform consistency properties with rates of these estimators, over compact subsets of , under a general ergodic condition on the underlying processes. We characterize the asymptotic normality of considered wavelet-based estimators, under easily verifiable conditions. The asymptotic properties of these estimators are obtained, by means of the martingale approach.

Realized wavelet-based estimation of integrated variance and jumps in the presence of noise

We introduce wavelet-based methodology for estimation of realized variance allowing its measurement in the time-frequency domain. Using smooth wavelets and maximum overlap discrete wavelet transform, we allow for the decomposition of the realized variance into several investment horizons and jumps. Basing our estimator in the two-scale realized variance framework, we are able to utilize all available data and obtain a feasible estimator in the presence of microstructure noise as well. The estimator is tested in a large numerical study of the finite sample performance and is compared to other popular realized variation estimators. We use different simulation settings with changing noise as well as jump level in different price processes including the long memory fractional stochastic volatility model. The results reveal that our wavelet-based estimator is able to estimate and forecast the realized measures with the greatest precision. Our time-frequency estimators not only produce feasible estimates, but also decompose the realized variation into arbitrarily chosen investment horizons. We apply them to study the volatility of forex futures during the recent crisis at several investment horizons and obtain results which provide us with better understanding of the volatility dynamics.

Realized Wavelet-Based Estimation of Integrated Variance and Jumps in the Presence of Noise

We introduce wavelet-based methodology for estimation of realized variance allowing its measurement in the time-frequency domain. Using smooth wavelets and Maximum Overlap Discrete Wavelet Transform, we allow for the decomposition of the realized variance into several investment horizons and jumps. Basing our estimator in the two-scale realized variance framework, we are able to utilize all available data and get feasible estimator in the presence of microstructure noise as well. The estimator is tested in a large numerical study of the finite sample performance and is compared to other popular realized variation estimators. We use different simulation settings with changing noise as well as jump level in different price processes including long memory fractional stochastic volatility model. The results reveal that our wavelet-based estimator is able to estimate and forecast the realized measures with the greatest precision. Our time-frequency estimators not only produce feasible estimates, but also decompose the realized variation into arbitrarily chosen investment horizons. We apply it to study the volatility of forex futures during the recent crisis at several investment horizons and obtain the results which provide us with better understanding of the volatility dynamics.

A wavelet-based estimator of the degrees of freedom in denoised fMRI time series for probabilistic testing of functional connectivity and brain graphs

Connectome mapping using techniques such as functional magnetic resonance imaging (fMRI) has become a focus of systems neuroscience. There remain many statistical challenges in analysis of functional connectivity and network architecture from BOLD fMRI multivariate time series. One key statistic for any time series is its (effective) degrees of freedom, df, which will generally be less than the number of time points (or nominal degrees of freedom, N). If we know the df, then probabilistic inference on other fMRI statistics, such as the correlation between two voxel or regional time series, is feasible. However, we currently lack good estimators of df in fMRI time series, especially after the degrees of freedom of the “raw” data have been modified substantially by denoising algorithms for head movement. Here, we used a wavelet-based method both to denoise fMRI data and to estimate the (effective) df of the denoised process. We show that seed voxel correlations corrected for locally variable df could be tested for false positive connectivity with better control over Type I error and greater specificity of anatomical mapping than probabilistic connectivity maps using the nominal degrees of freedom. We also show that wavelet despiked statistics can be used to estimate all pairwise correlations between a set of regional nodes, assign a P value to each edge, and then iteratively add edges to the graph in order of increasing P. These probabilistically thresholded graphs are likely more robust to regional variation in head movement effects than comparable graphs constructed by thresholding correlations. Finally, we show that time-windowed estimates of df can be used for probabilistic connectivity testing or dynamic network analysis so that apparent changes in the functional connectome are appropriately corrected for the effects of transient noise bursts. Wavelet despiking is both an algorithm for fMRI time series denoising and an estimator of the (effective) df of denoised fMRI time series. Accurate estimation of df offers many potential advantages for probabilistically thresholding functional connectivity and network statistics tested in the context of spatially variant and non-stationary noise. Code for wavelet despiking, seed correlational testing and probabilistic graph construction is freely available to download as part of the BrainWavelet Toolbox at www.brainwavelet.org.

Commodity futures returns: more memory than you might think!

This paper investigates long-range dependence in 14 commodity and 3 other financial futures returns series from 1993 to 2009 and shows that long memory is a pervasive phenomenon in contrast to the extant evidence. Utilizing a semi-parametric wavelet-based estimator with time windows, the results provide overwhelming evidence of time-varying long-range dependence in all futures returns series. Structural break tests indicate multiple regimes of dependence, in the majority of which the persistence parameter is statistically significant. The results also provide evidence of predominantly negative parameter values which are known as anti-persistence. The latter is consistent with investor overreaction to shocks and suggests temporary departures from market efficiency.

Nonparametric regression on random fields with random design using wavelet method

We consider non-linear wavelet-based estimators of spatial regression functions with (known) random design on strictly stationary random fields, which are indexed by the integer lattice points in the \(N\)-dimensional Euclidean space and are assumed to satisfy some mixing conditions. We investigate their asymptotic rates of convergence based on thresholding of empirical wavelet coefficients and show that these estimators achieve nearly optimal convergence rates within a logarithmic term over a large range of Besov function classes \(B^{s}_{p,q}\). Therefore, wavelet estimators still achieve nearly optimal convergence rates for random fields and provide explicitly the extraordinary local adaptability.