Decomposition of Scatter Ratios Used in Monitoring Multivariate Process Variability

Wilks’ ratio statistic can be defined in terms of the ratio of the sample generalized variances of two non-independent estimators of the same covariance matrix. Recently this statistic has been proposed as a control statistic for monitoring changes in the covariance matrix of a multivariate normal process in a Phase II situation, particularly when the dimension is larger than the sample size. In this article we derive a technique for decomposing Wilks’ ratio statistic into the product of independent factors that can be associated with the components of the covariance matrix. With these results, we demonstrate that, when a signal is detected in a control procedure for the Phase II monitoring of process variability using the ratio statistic, the signaling value can be decomposed and the process variables contributing to the signal can be specifically identified.     

Multivariate Quality Control Chart for Autocorrelated Processes

Traditional multivariate statistical process control (SPC) techniques are based on the assumption that the successive observation vectors are independent. In recent years, due to automation of measurement and data collection systems, a process can be sampled at higher rates, which ultimately leads to autocorrelation. Consequently, when the autocorrelation is present in the data, it can have a serious impact on the performance of classical control charts. This paper considers the problem of monitoring the mean vector of a process in which observations can be modelled as a first-order vector autoregressive VAR (1) process. We propose a control chart called Z-chart which is based on the single step finite intersection test (Timm, 1996). An important feature of the proposed method is that it not only detects an out of control status but also helps in identifying variable(s) responsible for the out of control situation. The proposed method is illustrated with the help of suitable illustrations.     

Chi-Square quantile-based multivariate variance monitoring for individual observations

Under the normality assumption, some statistics for monitoring a multivariate process variance for individual observations can be used to detect a variance shift, but the distribution of their in-control run length has a high variance as well as the median that is extremely smaller than the mean, which leads to many false alarms in the in-control process. In this paper, we propose a chi-square quantile-based monitoring statistic which is free of the problems. The numerical experiments show that the proposed monitoring statistics outperform the existing monitoring statistics in terms of the detection of a shift for the variance.     

A Comparison Between Two Multivariate EWMA Schemes

The signal issued by a control chart triggers the process professionals to investigate the special cause. Change point methods simplify the efforts to search for and identify the special cause. In this study, using maximum likelihood estimation, a multivariate joint change point estimation procedure for monitoring both location and dispersion simultaneously is proposed. After a signal is generated by the simultaneously used Hotelling's T ² and/or generalized variance control charts, the procedure starts detecting the time of the change. The performance of the proposed method for several structural changes for the mean vector and covariance matrix is discussed.     

对经过重建和调整后的我国婴儿死亡率的验证

本文根据婴儿死亡率随人均GDP的动态变化规律筛选最佳验证模型,验证了时间序列模型重新构建的1952－1980年婴儿死亡率和调整校正的1981－1990年婴儿死亡率。结果表明幂函数形式为相对较好的验证模型,拟合精度稍逊于时间序列预测模型,验证模型与时间序列模型的预测结果与历史婴儿死亡率比较,变异程度无显著差异,且预测结果与建立国家儿童死亡监测网络后的国家监测地区婴儿死亡率形成了平稳性过渡。文章认为以时间序列模型重新构建和调整校正的婴儿死亡率比较可靠,更加接近当时的实际水平。     

Addressing the effect of parameter estimation on phase II monitoring of multivariate multiple linear profiles via a new cluster-based approach

Abstract

Profile monitoring is applied when the quality of a product or a process can be determined by the relationship between a response variable and one or more independent variables. In most Phase II monitoring approaches, it is assumed that the process parameters are known. However, it is obvious that this assumption is not valid in many real-world applications. In fact, the process parameters should be estimated based on the in-control Phase I samples. In this study, the effect of parameter estimation on the performance of four Phase II control charts for monitoring multivariate multiple linear profiles is evaluated. In addition, since the accuracy of the parameter estimation has a significant impact on the performance of Phase II control charts, a new cluster-based approach is developed to address this effect. Moreover, we evaluate and compare the performance of the proposed approach with a previous approach in terms of two metrics, average of average run length and its standard deviation, which are used for considering practitioner-to-practitioner variability. In this approach, it is not necessary to know the distribution of the chart statistic. Therefore, in addition to ease of use, the proposed approach can be applied to other type of profiles. The superior performance of the proposed method compared to the competing one is shown in terms of all metrics. Based on the results obtained, our method yields less bias with small-variance Phase I estimates compared to the competing approach.     

Multivariate Control Chart Based on Multivariate Smirnov Test

Robust control charts are useful in statistical process control (SPC) when there is limited knowledge about the underlying process distribution, especially for multivariate observations. This article develops a new robust and self-starting multivariate procedure based on multivariate Smirnov test (MST), which integrates a multivariate two-sample goodness-of-fit (GOF) test based on multivariate empirical distribution function (MEDF) and the change-point model. As expected, simulation results show that our proposed control chart is robust to nonnormally distributed data, and moreover, it is efficient in detecting process shifts, especially large shifts, which is one of the main drawbacks of most robust control charts in the literature. As it avoids the need for a lengthy data-gathering step, the proposed chart is particularly useful in start-up or short-run situations. Comparison results and a real data example show that our proposed chart has great potential for application.     

Multivariate extreme-value distributions with applications to environmental data

Some parametric families of multivariate extreme-value distributions have been proposed in recent years; several additional parametric families are derived here. The parametric models are fitted, using numerical maximum likelihood, to some environmental multivariate extreme data sets consisting of extreme concentrations of a pollutant at several monitoring stations in a region. Some multivariate nonnormal data analysis techniques are proposed to aid in the likelihood analysis. The new models, together with previous models, appear to be adequate for inferences in that they cover a wide range of possible dependence patterns.     

Monitoring Multivariate Simple Linear Profiles in the Presence of between Profile Autocorrelation

In recent years, statistical profile monitoring has emerged as a relatively new and potentially useful subarea of statistical process control and has attracted attention of many researchers and practitioners. A profile, waveform, or signature is a function that relates a dependent or a response variable to one or more independent variables. Different statistical methods have been proposed by researchers to monitor profiles where each method requires its own assumptions. One of the common and implicit assumptions in most of the proposed procedures is the assumption of independent residuals. Violation of this assumption can affect the performance of control procedures and ultimately leading to misleading results. In this article, we study phase II analysis of monitoring multivariate simple linear profiles when the independency assumption is violated. Three time series based methods are proposed to eliminate the effect of correlation that exists between multivariate profiles. Performances of the proposed methods are evaluated using average run length (ARL) criterion. Numerical results indicate satisfactory performance for the proposed methods. A simulated example is also used to show the application of the proposed methods.     

Modelling mercury deposition through latent space–time processes

Summary.  The paper provides a space–time process model for total wet mercury deposition. Key methodological features that are introduced include direct modelling of deposition rather than of expected deposition, the utilization of precipitation information (there is no deposition without precipitation) without having to construct a precipitation model and the handling of point masses at 0 in the distributions of both precipitation and deposition. The result is a specification that enables spatial interpolation and temporal prediction of deposition as well as aggregation in space or time to see patterns and trends in deposition. We use weekly deposition monitoring data from the National Atmospheric Deposition Program–Mercury Deposition Network for 2003 restricted to the eastern USA and Canada. Our spatiotemporal hierarchical model allows us to interpolate to arbitrary locations and, hence, to an arbitrary grid, enabling weekly deposition surfaces (with associated uncertainties) for this region. It also allows us to aggregate weekly depositions at coarser, quarterly and annual, temporal levels.     

A Comparison of Spatiotemporal Surveillance Methods for Nonhomogeneous Change Size

Spatiotemporal surveillance, especially in detection of emerging outbreaks is of particular importance. When an outbreak spreads across some areas, the incidence rate at the center of the outbreak area might be expected to be much higher than the rate at its edge. However, to the best of our knowledge, all existing methods assume a uniformly increasing rate across the entire area of the outbreak. The purpose of this study is to compare the performance of the spatiotemporal surveillance methods such as multivariate cumulative sum (MCUSUM) or multivariate exponentially weighted moving average (MEWMA) when the changes in size are nonhomogeneous. Monte Carlo simulations were conducted to examine the properties of these spatiotemporal surveillance methods and compared them in terms of the detection speed and the identification rate under various scenarios. The results showed that when nonhomogeneous change sizes are involved, the MCUSUM method taking into account spatial nonhomogeneity of increase rates yields a better identification than the method ignoring such change size pattern although the detection speeds are similar. Further, a case study for the detection of male thyroid cancer data in New Mexico in the United States was performed to demonstrate the applicability of these methods.     

Sufficient reduction methods for multivariate time-dependent health surveillance data

ABSTRACT

Early detection with a low false alarm rate (FAR) is the main aim of outbreak detection as used in public health surveillance or in regard to bioterrorism. Multivariate surveillance is preferable to univariate surveillance since correlation between series (CBS) is recognized and incorporated. Sufficient reduction has proved a promising method for handling CBS, but has not previously been used when correlation within series (CWS) is present. Here we develop sufficient reduction methods for reducing a p-dimensional multivariate series to a univariate series of statistics shown to be sufficient to monitor a sudden, but persistent, shift in the multivariate series mean. Correlation both within and between series is taken into account, as public health data typically exhibit both forms of association. Simultaneous and lagged changes and different shift sizes are investigated. A one-sided exponentially weighted moving average chart is used as a tool for detection of a change. The performance of the proposed method is compared with existing sufficient reduction methods, the parallel univariate method and both VarR and Z charts. A simulation study using bivariate normal autoregressive data shows that the new method gives shorter delays and a lower FAR than other methods, which have high FARs when CWS is clearly present.     

Statistical Monitoring of Nominal Logistic Profiles in Phase II

Statistical analysis of profile monitoring, a relatively new sub-area of statistical process control due to its applications in different industries, have urged researchers and practitioners to contribute to the developments of new monitoring methods. A statistical profile is a relationship between a quality characteristic (a response) and one or more independent variables to characterize quality of a process or a product. In this article, statistical profiles based on nominal responses are studied, where logistic regression is used to model the responses. Three approaches including likelihood ratio test (LRT), multivariate exponentially weighted moving average (MEWMA), and support vector machines (SVM) approaches are proposed to monitor quality of a process or product in Phase II. Performances of the proposed approaches are evaluated and compared using a case study. Moreover, the effect of two important factors on average run length (ARL) performance, number of levels and number of covariates, has been considered. Results indicate that performance of all approaches depends on the number of covariates and levels. As the number of these factors increases, SVM performance improves while performance of the other approaches deteriorates.     

Phase-II monitoring and diagnosing of multivariate categorical processes using generalized linear test-based control charts

In this paper, two control charts based on the generalized linear test (GLT) and contingency table are proposed for Phase-II monitoring of multivariate categorical processes. The performances of the proposed methods are compared with the exponentially weighted moving average-generalized likelihood ratio test (EWMA-GLRT) control chart proposed in the literature. The results show the better performance of the proposed control charts under moderate and large shifts. Moreover, a new scheme is proposed to identify the parameter responsible for an out-of-control signal. The performance of the proposed diagnosing procedure is evaluated through some simulation experiments.     

Bayesian updating in reference centile charts

Reference centile charts, used for monitoring the health of an individual over time (e.g. the weight gain of pregnant women over successive periods in their pregnancy) do not take into account the longitudinal nature of the individual profiles. There is also generally more than one variable which affects the outcome of interest, and information regarding the path of a group of variables over time may prove advantageous in terms of sensitivity. We propose a Bayesian approach to this problem in which the successive deviations of the individual's observations from the mean of the corresponding reference distribution are used to compute updated reference centiles for future measurements on the individual. The univariate and multivariate situations are discussed, and consideration is given to non-normal cross-sectional distributions. The theory is illustrated on data obtained from the records of weight gain and fundal height of pregnant women visiting a clinic at intervals during pregnancy.     

Effect of Dependency in Systems for Multivariate Surveillance

Systems for multivariate on-line surveillance (e.g., outbreak detection) are investigated. Optimal systems for statistical surveillance are based on likelihood ratios. Three systems are compared: based on each marginal density, based on the joint density, and based on the Hotelling's T2. The effect of dependency between the monitored processes is investigated, and the effect of correlation between the change times. When the first change occurs immediately, the three methods give similar delay of an alarm, in the situation with independency. For late changes, T2 has the longest delay, both for independent processes and for processes with a positive covariance.     

A Monte Carlo comparison of Jarque–Bera type tests and Henze–Zirkler test of multivariate normality

In the paper, tests for multivariate normality (MVN) of Jarque-Bera type, based on skewness and kurtosis, have been considered. Tests proposed by Mardia and Srivastava, and the combined tests based on skewness and kurtosis defined by Jarque and Bera have been taken into account. In the Monte Carlo simulations, for each combination of p = 2, 3, 4, 5 number of traits and n = 10(5)50(10)100 sample sizes 10,000 runs have been done to calculate empirical Type I errors of tests under consideration, and empirical power against different alternative distributions. Simulation results have been compared to the Henze–Zirkler’s test. It should be stressed that no test yet proposed is uniformly better than all the others in every combination of conditions examined.     

Estimates of regression coefficients based on the sign covariance matrix

Summary. A new estimator of the regression parameters is introduced in a multivariate multiple-regression model in which both the vector of explanatory variables and the vector of response variables are assumed to be random. The affine equivariant estimate matrix is constructed using the sign covariance matrix (SCM) where the sign concept is based on Oja's criterion function. The influence function and asymptotic theory are developed to consider robustness and limiting efficiencies of the SCM regression estimate. The estimate is shown to be consistent with a limiting multinormal distribution. The influence function, as a function of the length of the contamination vector, is shown to be linear in elliptic cases; for the least squares (LS) estimate it is quadratic. The asymptotic relative efficiencies with respect to the LS estimate are given in the multivariate normal as well as the t -distribution cases. The SCM regression estimate is highly efficient in the multivariate normal case and, for heavy-tailed distributions, it performs better than the LS estimate. Simulations are used to consider finite sample efficiencies with similar results. The theory is illustrated with an example.     

An Algorithm to Simulate VMA Processes Having a Spectrum with Fixed Condition Number

This article proposes an algorithm to generate vector moving average (VMA) processes with a variable spectrum having a fixed condition number across frequencies. This method is based on the theory of multivariate linear spectrum for VMA processes, and is developed in a two-step procedure. Specific examples are provided, and the precision of generated time series is discussed. Such an algorithm is a useful tool to assess the performance of selected multivariate spectral estimators, and it turns out to be particularly appropriated in the Kolmogorov asymptotic estimation framework.     

Phase I monitoring of social network with baseline periods using poisson regression

Social network analysis is an important analytic tool to forecast social trends by modeling and monitoring the interactions between network members. This paper proposes an extension of a statistical process control method to monitor social networks by determining the baseline periods when the reference network set is collected. We consider probability density profile (PDP) to identify baseline periods using Poisson regression to model the communications between members. Also, Hotelling T² and likelihood ratio test (LRT) statistics are developed to monitor the network in Phase I. The results based on signal probability indicate a satisfactory performance for the proposed method.