DETECTING SINUSOIDS IN NON-GAUSSIAN NOISE

Abstract. Spectral analysis is a well-established procedure for detecting harmonic signals in a noisy environment. Much research has been done on methods that use second-order statistics (i.e. the autocovariance function and power spectrum) such as Whittle's test, Bartlett's test, Hannan's test and the Priestley P ( Λ ) test. When the noise is non-Gaussian, statistics of order greater than two can provide more information to detect the periodicities in noisy data. We direct our main attention to the third (fourth) order cumulant and bispectral (trispectral) methods. New test statistics are derived and are shown to be more powerful than other methods based on second-order statistics under a mixed spectrum condition. The asymptotic power functions of the new test statistics and other tests are studied. Some Monte Carlo simulations are used to evaluate the performance of the new methods with moderate sample sizes.     

基层环境统计年报工作中存在的问题及对策

环境统计是环境保护的基础工作,随着经济社会的发展,越来越受到重视。在基层环境统计工作经历中,分析总结了当前环境统计年报工作中存在的一些问题,并提出了相应的对策和建议,以期为提高环境统计工作效率和数据质量提供一些参考。     

Diagnostics for Time Series Analysis

Test statistics are proposed to determine the goodness of fit of a time series model. The test statistics are based on a sequence of random variables that are independent and standard normal if the model is correct. The paper shows how to compute this sequence of random variables efficiently using a combination of Markov chain Monte Carlo and importance sampling. The power of the statistics to detect outliers and level shifts is studied for an autoregressive model. The methodology is illustrated using both simulated and real data.     

Tests for change—points based on recursive u—statistics

Tests for change—points for the location as well as regression models are often based on cumulative sums of recursive residuals. These recursive residuals are also employed in the sequential detection problem. In the context of general estimable parameters (funtionals of the underlying distribution functions), such recursive residuals may be defined in terms of recursive U—statistics. A class of tests for the change—points based on recursive U—statistics has been considered. Along with some invariance principles for recursive U—statistics, asymptotic properties of the proposed tests are studied.     

Characterization of the void size distribution in fluidized beds using statistics of pressure fluctuations

Experiments carried out in fluidized beds of Geldart A and B powders revealed that in-bed pressure fluctuations typically have power-law statistics, which are well described by the Student's distribution. We show that the pressure statistics encode information about the bubble size distribution, and propose a novel method that allows an easy determination of the approximate shape of the bubble size distribution from the statistics of pressure fluctuations.     

Rank Based Dickey–Fuller Test Statistics

This article provides various comprehensive comparisons between Breitung–Gouriéroux and Granger–Hallman rank statistics for the unit root test. New analytical asymptotic properties for the Granger–Hallman rank statistic are demonstrated. The statistic is of a Dickey–Fuller type, where the observations are replaced with their rank counterparts. Weak convergence results are given for the nonstationary random walk process when the errors are assumed to have higher than two moments. Empirical percentiles of both Breitung–Gouriéroux and Granger–Hallman rank statistics are presented for different sample sizes. In addition, empirical powers and sizes for these rank statistics and for the Dickey–Fuller test statistic are shown for different distributions of the error terms.     

Tests for the Equality of Two Processes' Spectral Densities with Unequal Lengths Using Wavelet Methods

Testing procedures for assessing whether two stationary and independent linear processes with unequal lengths have the same spectral densities or same auto‐covariance functions are investigated. New test statistics are proposed based on the difference of the two wavelet‐based estimates of the two spectral densities. The asymptotic normal distributions of the empirical wavelet coefficients are derived based on Bartlett type approximation of a quadratic form with dependent variables by the corresponding quadratic form with independent and identically distributed (i.i.d.) random variables. The limit distributions of the proposed test statistics are derived from those asymptotic results, and they asymptotically follow known chi‐square distributions. The advantage of those new procedures is that those test statistics are constructed very simply and can be used for two time series with arbitrary lengths. The performance of those new tests is compared with some recent test statistics, with respect to their exact levels and powers. Simulation studies show that our proposed tests are very comparable to the current tests.     

An optimal filter based MPC for systems with arbitrary disturbances

In this study, a linear model predictive control (MPC) approach with optimal filters is proposed for handling unmeasured disturbances with arbitrary statistics. Two types of optimal filters are introduced into the framework of MPC to relax the assumption of integrated white noise model in existing approaches. The introduced filters are globally optimal for linear systems with unmeasured disturbances that have unknown statistics. This enables the proposed MPC to better handle disturbances without access to disturbance statistics. As a result, the effort required for disturbance modeling can be alleviated. The proposed MPC can achieve offset-free control in the presence of asymptotically constant unmeasured disturbances. Simulation results demonstrate that the proposed approach can provide an improved disturbance õrejection performance over conventional approaches when applied to the control of systems with unmeasured disturbances that have arbitrary statistics.     

Testing for Change in Long‐Memory Stochastic Volatility Time Series

In this article, change‐point problems for long‐memory stochastic volatility (LMSV) models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM‐ and Wilcoxon‐type test statistics is derived. In this context, a limit theorem for the two‐parameter empirical process of LMSV time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long‐range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self‐normalized test statistics is proposed. The theoretical results are accompanied by an analysis of Standard & Poor's 500 daily closing indices with respect to structural changes and by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean and in variance.     

Scan B-statistic for kernel change-point detection

Abstract

Detecting the emergence of an abrupt change-point is a classic problem in statistics and machine learning. Kernel-based nonparametric statistics have been used for this task, which enjoys fewer assumptions on the distributions than the parametric approach and can handle high-dimensional data. In this article, we focus on the scenario when the amount of background data is large and propose a computationally efficient kernel-based statistics for change-point detection, inspired by the recently developed B-statistics. A novel theoretical result of the article is the characterization of the tail probability of these statistics using the change-of-measure technique, which focuses on characterizing the tail of the detection statistics rather than obtaining its asymptotic distribution under the null distribution. Such approximations are crucial to controlling the false alarm rate, which corresponds to the average run length in online change-point detection. Our approximations are shown to be highly accurate. Thus, they provide a convenient way to find detection thresholds for online cases without the need to resort to the more expensive simulations. We show that our methods perform well on both synthetic data and real data.     

JOINT HYPOTHESIS TESTS FOR A RANDOM WALK BASED ON INSTRUMENTAL VARIABLE ESTIMATORS

Abstract. In this paper we propose test statistics for the null hypothesis of a random walk or a random walk with drift for the case in which the innovations to the series are a moving-average process. The statistics are based on the instrumental variable estimators proposed by Hall and by Pantula and Hall and are shown to have the limiting distributions tabulated by Dickey and Fuller.     

基于SPA相似系数的故障识别方法

传统的主元分析（PCA）相似系数法没有充分利用数据的高阶统计量等有用的过程信息,导致故障识别效果较差。针对此问题,提出一种统计量模式分析（SPA）相似系数法。该方法首先使用SPA将原始数据转换到统计量空间中,然后在统计量空间中利用PCA获取主元方向,计算主元之间的相似性识别故障。在连续搅拌反应器（CSTR）过程上的仿真结果说明提出的SPA相似系数法比传统的PCA相似系数法能更有效地识别故障。     

Modeling the Microstructure of Energetic Materials with Realistic Constituent Morphology

In this article, we present a strategy for packing realistic crystals, critical for mesoscale simulations, and predictions. The current packing code uses a dynamic algorithm, with crystal shapes represented by level sets, to create appropriate packs of the microstructure for an energetic material. Crystal shapes include the nitramines HMX, RDX, PETN, and CL20. Two series of packs are considered: a bidisperse pack with size ratio 1 : 0.3 and a polydisperse pack. We also construct equivalent packs of spheres for comparison purposes. Higher‐order statistics are computed and compared. We show that the second‐order statistics are essentially independent of particle shape when the packing fraction is held constant. The second‐order statistics do, however, depend on the polydispersity.     

Testing for a Unit Root in Autoregressive Moving-average Models with Missing Data

Testing for a single autoregressive unit root in an autoregressive moving-average (ARMA) model is considered in the case when data contain missing values. The proposed test statistics are based on an ordinary least squares type estimator of the unit root parameter which is a simple approximation of the one-step Newton–Raphson estimator. The limiting distributions of the test statistics are the same as those of the regression statistics in AR(1) models tabulated by Dickey and Fuller (Distribution of the estimators for autoregressive time series with a unit root. J. Am. Stat. Assoc . 74 (1979), 427–31) for the complete data situation. The tests accommodate models with a fitted intercept and a fitted time trend.     

Simple Regressions with Linear Time Trends

Simple regressions of two trend stationary time series are considered. As the linear trend dominates the stochastic components the rates of convergence and the limiting distributions of ordinary least squares statistics are exactly the same as in the case of cointegrated regressions with drifts. In particular, asymptotic standard normal t statistics are readily available. Hence, asymptotic inference requires no distinction between simple regressions of trend stationary series and of cointegrated variables with drifts.     

基于最小充分统计量模式分析的故障检测方法

统计量模式分析（SPA）最近在故障检测领域取得了广泛应用,其实质是用数据的统计量矩阵来代替原始数据矩阵进行故障检测,然而其统计量的选取存在盲目性且各统计量之间存在复杂的非线性关联关系,难以满足后续应用主成分分析（PCA）完成故障检测所需的基本条件。为了解决这个问题,提出了基于最小充分统计量模式分析的故障检测方法（MSSPA）。该方法首先将原始数据矩阵进行正交变换以消除变量之间的关联性,然后估计出每个变量的概率密度函数或者多个变量的联合概率密度函数,进而求出原始数据的最小充分统计量,并用最小充分统计量来构造统计量矩阵。最小充分统计量的引入还能够有效应对数据的非高斯分布问题。最后,通过在TE过程上的仿真测试验证了该方法用于故障检测的可行性和有效性。     

A local and global statistics pattern analysis method and its application to process fault identification☆

Traditional principal component analysis (PCA) is a second-order method and lacks the ability to provide higher-order representations for data variables. Recently, a statistics pattern analysis (SPA) framework has been incor-porated into PCA model to make full use of various statistics of data variables effectively. However, these methods omit the local information, which is also important for process monitoring and fault diagnosis. In this paper, a local and global statistics pattern analysis (LGSPA) method, which integrates SPA framework and locality pre-serving projections within the PCA, is proposed to utilize various statistics and preserve both local and global in-formation in the observed data. For the purpose of fault detection, two monitoring indices are constructed based on the LGSPA model. In order to identify fault variables, an improved reconstruction based contribution (IRBC) plot based on LGSPA model is proposed to locate fault variables. The RBC of various statistics of original process variables to the monitoring indices is calculated with the proposed RBC method. Based on the calculated RBC of process variables' statistics, a new contribution of process variables is built to locate fault variables. The simula-tion results on a simple six-variable system and a continuous stirred tank reactor system demonstrate that the proposed fault diagnosis method can effectively detect fault and distinguish the fault variables from normal variables.     

Sequential signed rank statistics

In order to adapt rank methods to sequential analysis, we consider one-sample rank statistics which are built upon sequential instead of ordinary ranks. As both types of rank statistics are asymptotically equivalent within local models, we have to study their relative performance under a fixed distribution. This is exemplified by means of repeated significance tests (thereby making use of an idea borrowed from renewal theory). A function¬al limit theorem and a SLLN are derived as the main tools for any such study. In order to get the first one, we employ a refinement of the projection method and a decomposition by which a rank statistic is presented as a sum of a martingale, an inverse martingale and a linear function of order statistics. This decom¬position also yields a SLLN. We prefer a direct argument, however, which can do with much weaker regularity conditions.     

Specification of Exponential-Family Random Graph Models: Terms and Computational Aspects

Exponential-family random graph models (ERGMs) represent the processes that govern the formation of links in networks through the terms selected by the user. The terms specify network statistics that are sufficient to represent the probability distribution over the space of networks of that size. Many classes of statistics can be used. In this article we describe the classes of statistics that are currently available in the ergm package. We also describe means for controlling the Markov chain Monte Carlo (MCMC) algorithm that the package uses for estimation. These controls a ect either the proposal distribution on the sample space used by the underlying Metropolis-Hastings algorithm or the constraints on the sample space itself. Finally, we describe various other arguments to core functions of the ergm package.     

An overview of robust methods in medical research

Robust statistics is an extension of classical parametric statistics that specifically takes into account the fact that the assumed parametric models used by the researchers are only approximate. In this article, we review and outline how robust inferential procedures may routinely be applied in practice in the biomedical research. Numerical illustrations are given for the t-test, regression models, logistic regression, survival analysis and ROC curves, showing that robust methods are often more appropriate than standard procedures.