On wavelet-based testing for serial correlation of unknown form using Fan’s adaptive Neyman method

Test procedures for serial correlation of unknown form with wavelet methods are investigated. A new test statistic is motivated using a canonical multivariate normal hypothesis testing model. It relies on empirical wavelet coefficients of a wavelet-based spectral density estimator. The choice of the Haar wavelet function is advocated, since evidence demonstrates that the choice of the wavelet function is not critical. Under the null hypothesis of no serial correlation, the asymptotic distribution of a vector of empirical wavelet coefficients is derived, which is asymptotically a multivariate normal distribution. A test statistic is proposed based on that asymptotic result, which presents the serious advantage to be completely data-driven or adaptive, avoiding the selection of any smoothing parameters. Furthermore, under a suitable class of fixed alternatives, the wavelet-based method is consistent against serial correlation of unknown form. The test statistic is expected to exhibit good power properties when the true spectral density displays significant spatial inhomogeneity, such as seasonal or business cycle periodicities. However, the convergence of the test statistic towards its asymptotic distribution is relatively slow. Thus, Monte Carlo methods based on random samples are suggested to determine the corresponding critical values. In a simulation study, the new methodology is compared with several test statistics, with respect to their exact levels and powers. The robustness properties of the spectral methods based on Monte Carlo critical values are also investigated empirically, when the error terms are weak white noises.     

On the power transformation of kernel-based tests for serial correlation in vector time series: Some finite sample results and a comparison with the bootstrap

Portmanteau test statistics represent useful diagnostic tools for checking the adequacy of multivariate time series models. For stationary and partially non-stationary vector time series models, Duchesne and Roy [Duchesne, P., Roy, R., 2004. On consistent testing for serial correlation of unknown form in vector time series models. Journal of Multivariate Analysis 89, 148-180] and Duchesne [Duchesne, P., 2005a. Testing for serial correlation of unknown form in cointegrated time series models. Annals of the Institute of Statistical Mathematics 57, 575-595] have proposed kernel-based test statistics, obtained by comparing the spectral density of the errors under the null hypothesis of non-correlation with a kernel-based spectral density estimator; these test statistics are asymptotically standard normal under the null hypothesis of non-correlation in the error term of the model. Following the method of Chen and Deo [Chen, W.W., Deo, R.S., 2004a. Power transformations to induce normality and their applications. Journal of the Royal Statistical Society, Ser. B 66, 117-130], we determine an appropriate power transformation to improve the normal approximation in small samples. Additional corrections for the mean and variance of the distance measures intervening in these test statistics are obtained. An alternative procedure to estimate the finite distribution of the test statistics is to use the bootstrap method; we introduce bootstrap-based versions of the original spectral test statistics. In a Monte Carlo study, comparisons are made under various alternatives between: the original spectral test statistics, the new corrected test statistics, the bootstrap-based versions, and finally the classical Hosking portmanteau test statistic.     

On testing for serial correlation of unknown form using wavelet thresholding

Omnibus procedures for testing serial correlation are developed, using spectral density estimation and wavelet shrinkage. The asymptotic distributions of the wavelet coefficients under the null hypothesis of no serial correlation are derived. Under some general conditions on the wavelet basis, the wavelet coefficients asymptotically follow a normal distribution. Furthermore, they are asymptotically uncorrelated. Adopting a spectral approach and using results on wavelet shrinkage, new one-sided test statistics are proposed. As a spatially adaptive estimation method, wavelets can effectively detect fine features in the spectral density, such as sharp peaks and high frequency alternations. Using an appropriate thresholding parameter, shrinkage rules are applied to the empirical wavelet coefficients, resulting in a non-linear wavelet-based spectral density estimator. Consequently, the advocated approach avoids the need to select the finest scale J, since the noise in the wavelet coefficients is naturally suppressed. Simple data-dependent threshold parameters are also considered. In general, the convergence of the spectral test statistics toward their respective asymptotic distributions appears to be relatively slow. In view of that, Monte Carlo methods are investigated. In a small simulation study, several spectral test statistics are compared, with respect to level and power, including versions of these test statistics using Monte Carlo simulations.     

Nonparametric Testing for Long-Run Neutrality with Applications to US Money and Output Data

We consider a nonparametric testing procedure for long-run monetary neutrality using spectral approaches. Long-run effects between bivariate integrated series are represented as the spectral density matrix of their first-differences evaluated at the zero frequency. The long-run neutrality, the core issue in this work, reduces to zero power of the cross spectral density function near the origin. We propose a statistic based on a kernel-based cross spectral density estimator. As designed to be consistent against cross correlations of unknown forms, the test differentiates it from tests based on parametric regression models. In implementing the tests, some feasible bandwidth selection procedures are detailed in terms of mean squared error criteria and of type I and type II errors criteria. Our testing procedures can be a complementary approach for neutrality testing. Simulation studies are shown to support theoretical results. Our methods are applied to testing long-run neutrality in the US nominal money and real output quarterly data from the first quarter of 1959 to the third quarter of 2009. Our tests unanimously reject the long-run neutrality for M2 regardless of the choice of bandwidths and of kernels.     

Testing the martingale difference hypothesis using integrated regression functions

An omnibus test for testing a generalized version of the martingale difference hypothesis (MDH) is proposed. This generalized hypothesis includes the usual MDH, testing for conditional moments constancy such as conditional homoscedasticity (ARCH effects) or testing for directional predictability. A unified approach for dealing with all of these testing problems is proposed. These hypotheses are long standing problems in econometric time series analysis, and typically have been tested using the sample autocorrelations or in the spectral domain using the periodogram. Since these hypotheses cover also nonlinear predictability, tests based on those second order statistics are inconsistent against uncorrelated processes in the alternative hypothesis. In order to circumvent this problem pairwise integrated regression functions are introduced as measures of linear and nonlinear dependence. The proposed test does not require to chose a lag order depending on sample size, to smooth the data or to formulate a parametric alternative model. Moreover, the test is robust to higher order dependence, in particular to conditional heteroskedasticity. Under general dependence the asymptotic null distribution depends on the data generating process, so a bootstrap procedure is considered and a Monte Carlo study examines its finite sample performance. Then, the martingale and conditional heteroskedasticity properties of the Pound/Dollar exchange rate are investigated.     

A robust forward weighted Lagrange multiplier test for conditional heteroscedasticity

Statistical tests routinely adopted for detecting nonlinear components in time series rely on the auxiliary regression of ARMA lagged residuals, and the Lagrange multiplier test to detect ARCH components is an example. The size distortion of such test suggests adopting a weighted test, where the weights are computed through a forward search algorithm. Simulations show that the forward weighted robust test is preferable to the classical Lagrange test and to existing robust tests, which are based on backward weighted regression or on estimated autocorrelation function. The forward weighted robust test is applied to daily financial and quarterly macroeconomic time series, showing its usefulness in detecting ARCH effects, even when outliers are present.     

On pseudo maximum likelihood estimation for multivariate time series models with conditional heteroskedasticity

We consider a general multivariate conditional heteroskedastic model under a conditional distribution that is not necessarily normal. This model contains autoregressive conditional heteroskedastic (ARCH) models as a special class. We use the pseudo maximum likelihood estimation method and derive a new estimator of the asymptotic variance matrix for the pseudo maximum likelihood estimator. We also study four special cases in this class, which are conditional heteroskedastic autoregressive moving-average models, regression models with ARCH errors, models with constant conditional correlations, and ARCH in mean models.     

Diagnostics for conditional heteroscedasticity models: some simulation results

In this paper, we study the size and power of various diagnostic statistics for univariate conditional heteroscedasticity models. These test statistics include the residual-based tests recently derived by Tse, Li and Mak, and Wooldridge, respectively. Monte-Carlo experiments with 1000 replications are conducted to generate conditional variances which follow the autoregressive conditional heteroscedasticity (ARCH)/GARCH processes. We use quasi-maximum likelihood estimation (MLE) method to obtain estimates of parameters under different ARCH/ generalized ARCH (GARCH) models. It is found that the Tse and Li–Mak diagnostics are more powerful.     

The application of a general time series model to floodplain fisheries in the Amazon

Time series analysis is a common tool in environmental and ecological studies to construct models to explain and forecast serially correlated data. There are several statistical techniques that are used to deal with univariate and multivariate (more than one series) chronological patterns of fisheries data. In this paper, an additive stochastic model is proposed with explicative and predictive features to capture the main seasonal patterns and trends of a fisheries system in the Amazon. The model is constructed on the assumption that the multivariate response variable – vector containing fishery yield of eight periodic species and the total fishery yield – can be decomposed into three terms: an autoregression of the response vector, an exogenous environmental variable (river level), and a seasonal component (significant frequencies obtained by using spectral analysis and the periodogram indicating the regularity of periodic cycles in the natural and fisheries system). The estimation procedure is carried out via maximum likelihood estimation. The model explained, on average, 78% of the variability in yield of the study species. The model represents the optimal solution (minimum mean square mean error) among the class of all multivariate autoregressive processes with exogenous and seasonal variables. Predictions for one period ahead are provided to illustrate how the model works in practice.     

Diagnostic checking integer-valued ARCH models using conditional residual autocorrelations

Time series of counts are commonly observed in real-world applications. The integer-valued ARCH(p) models are able to describe integer-valued processes and offer the potential to be widely applied in practice in future. This paper develops an asymptotic theory for (partial) autocorrelations of the conditional residuals from the integer-valued ARCH(p) model. Based on the above results, we propose five portmanteau test statistics, which are very useful in checking the adequacy of a fitted integer-valued ARCH specification. The asymptotic distributions of the statistics are derived and their finite sample properties are studied in detail through Monte Carlo simulations. Finally, we illustrate the results analyzing two empirical examples.     

几种小波融合方法在遥感影像融合中的应用与比较

遥感影像融合是在提高空间分辨率的同时尽量保持光谱信息的完整性。小波融合方法已经被证明在保持光谱信息方面具有一定的优势。本文利用SPOT5的全色影像和多光谱影像,对常用的几种小波融合方法进行了试验,使用多种评价方法,将他们与常用的IHS方法和PCA方法进行了比较,并研究了小波分解层数对融合结果的影响。     

A novel robust kernel for visual learning problems

A major challenge to appearance-based learning techniques is the robustness against data corruption and irrelevant within-class data variation. This paper presents a robust kernel for kernel-based approach to achieving better robustness on several visual learning problems. Incorporating a robust error function used in robust statistics together with a deformation invariant distance measure, the proposed kernel is shown to be insensitive to noise and robust to intra-class variations. We prove that this robust kernel satisfies the requirements for a valid kernel, so it has good properties when used with kernel-based learning machines. In the experiments, we validate the superior robustness of the proposed kernel over the state-of-the-art algorithms on several applications, including hand-written digit classification, face recognition and data visualization.     

Distributed Multivariate Regression Using Wavelet-Based Collective Data Mining

This paper presents a method for distributed multivariate regression using wavelet-based collective data mining (CDM). The method seamlessly blends machine learning and the theory of communication with the statistical methods employed in parametric multivariate regression to provide an effective data mining technique for use in a distributed data and computation environment. The technique is applied to two benchmark data sets, producing results that are consistent with those obtained by applying standard parametric regression techniques to centralized data sets. Evaluation of the method in terms of mode accuracy as a function of appropriateness of the selected wavelet function, relative number of nonlinear cross-terms, and sample size demonstrates that accurate parametric multivariate regression models can be generated from distributed, heterogeneous, data sets with minimal data communication overhead compared to that required to aggregate a distributed data set. Application of this method to linear discriminant analysis, which is related to parametric multivariate regression, produced classification results on the Iris data set that are comparable to those obtained with centralized data analysis.     

Difference-based target detection using Mahalanobis distance and spectral angle

Two difference-based target detection methods are proposed in this work. In contrast to many target detectors which only calculate the distance between the testing pixel to the target spectrum, the proposed methods calculate the distance of the testing pixel to both of target and of background spectra. In other words, they utilize the difference between target and background computed distances. The first proposed method uses the Mahalanobis distance and benefits the valuable information contained in the statistics of targets and background. The second proposed method uses the kernel-based spectral angle mapper to benefit the advantages of spectral angle and kernel trick to separate targets from background, especially in non-linear cases. The experiments done on three real hyperspectral images indicate the high detection probability of the proposed methods compared to several target detectors.     

稳定分布噪声下非线性PAR系统辨识新方法

描述了稳定分布的谱表示,提出了共变谱密度的概念,得到一种基于自共变序列与共变谱的稳定分布白噪声与有色噪声的概念及其判断标准,对传统意义上的白噪声进行了广义化,依据多项式自回归（PAR）系统模型,对基于稳定白噪声输入的系统输出非线性稳定有色噪声建立其非线性PAR模型,提出基于最小P范数的EIRLP算法对非线性PAR系统进行辨识。模拟和分析表明,这种算法是一种在高斯和分数低阶 稳定分布噪声条件下具有良好韧性的非线性系统辨识方法,是对传统的二阶统计量基础上的系统辨识方法的改造与推广。     

Response dimension reduction for the conditional mean in multivariate regression

Sufficient dimension reduction methodologies in regression have been developed in the past decade, focusing mostly on predictors. Here, we propose a methodology to reduce the dimension of the response vector in multivariate regression, without loss of information about the conditional mean. The asymptotic distributions of dimension test statistics are chi-squared distributions, and an estimate of the dimension reduction subspace is asymptotically efficient. Moreover, the proposed methodology enables us to test response effects for the conditional mean. Properties of the proposed method are studied via simulation.     

Response dimension reduction for the conditional mean in multivariate regression

Sufficient dimension reduction methodologies in regression have been developed in the past decade, focusing mostly on predictors. Here, we propose a methodology to reduce the dimension of the response vector in multivariate regression, without loss of information about the conditional mean. The asymptotic distributions of dimension test statistics are chi-squared distributions, and an estimate of the dimension reduction subspace is asymptotically efficient. Moreover, the proposed methodology enables us to test response effects for the conditional mean. Properties of the proposed method are studied via simulation.     

A kernel-based parametric method for conditional density estimation

A conditional density function, which describes the relationship between response and explanatory variables, plays an important role in many analysis problems. In this paper, we propose a new kernel-based parametric method to estimate conditional density. An exponential function is employed to approximate the unknown density, and its parameters are computed from the given explanatory variable via a nonlinear mapping using kernel principal component analysis (KPCA). We develop a new kernel function, which is a variant to polynomial kernels, to be used in KPCA. The proposed method is compared with the Nadaraya-Watson estimator through numerical simulation and practical data. Experimental results show that the proposed method outperforms the Nadaraya-Watson estimator in terms of revised mean integrated squared error (RMISE). Therefore, the proposed method is an effective method for estimating the conditional densities.     

Application of cepstral preprocessing for the improvement of spatial contrast of ocular fundus images

A method based on the cepstral analysis is proposed to preprocess fundus images. The method effectively reveals spectral peaks in a narrow band of spatial frequencies thus improving the spatial contrast of the image. The resulting images can be readily used for an automated structural analysis of eye pathologies. The method is applied to stitch several small-aperture fragments together. The article was translated by the authors.     

Size and power properties of some tests in the Birnbaum-Saunders regression model

The Birnbaum-Saunders distribution has been used quite effectively to model times to failure for materials subject to fatigue and for modeling lifetime data. In this paper we obtain asymptotic expansions, up to order n^−1/2 and under a sequence of Pitman alternatives, for the non-null distribution functions of the likelihood ratio, Wald, score and gradient test statistics in the Birnbaum-Saunders regression model. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the shape parameter. Monte Carlo simulation is presented in order to compare the finite-sample performance of these tests. We also present two empirical applications.