Edgard M. Maboudou-Tchao

Multivariate exponentially weighted moving covariance matrix

Multivariate exponentially weighted moving average (MEWMA) charts are among the best control charts for detecting small changes in any direction. The well-known MEWMA is directed at changes in the mean vector. But changes can occur in either the location or the variability of the correlated multivariate quality characteristics, calling for parallel methodologies for detecting changes in the covariance matrix. This article discusses an exponentially weighted moving covariance matrix for monitoring the stability of the covariance matrix of a process. Used together with the location MEWMA, this chart provides a way to satisfy Shewharts dictum that proper process control monitor both mean and variability. We show that the chart is competitive, generally outperforming current control charts for the covariance matrix.

Self-Starting Multivariate Exponentially Weighted Moving Average Control Charting

Multivariate control charts are valuable tools for industrial quality control. The conventional discussion of them rests on the presumption that the in-control process parameters are known a priori. The more common reality is that practitioners plug in parameter estimates gathered from a special phase I sample to establish parameter values for the charts. But no sample will establish the exact process parameters, and quite small random errors translate into serious distortions of the run behavior, particularly of sensitive charts, and can affect chart performance. So-called “self-starting” methods can begin the control of the process right after startup without the preliminary step of a large phase I sample. Univariate self-starting methods for converting the unknown-parameter stream of process readings into a known-parameter sequence have been available for some time now. This article develops a multivariate equivalent by providing a way to transform the process readings into a stream of vectors following an exact known-parameter distribution. Although our approach is far from being the first proposal for self-starting charting of multivariate data, we believe it is the first that does so by transforming the unknown-parameter process vectors into known-parameter vectors of the same dimensionality. This stream of vectors has many potential uses. In particular, it may be used to construct any multivariate control chart, such as HotellingsT2, or any of the multivariate cusum methods. We illustrate using the transformed stream to set up a multivariate exponentially weighted moving average chart. With the self-starting front end, this (or any other) chart will have the same in-control properties as if the true process mean and covariance matrix were known exactly, thereby allowing multivariate control charting to proceed without a large and costly phase I data-gathering exercise.

Self-Starting Multivariate Control Charts for Location and Scale

Multivariate control charts are advisable when monitoring several correlated characteristics. The multivariate exponentially weighted moving average (MEWMA) is ideal for monitoring the mean vector, and the multivariate exponentially weighted moving covariance matrix (MEWMC) detects changes in the covariance matrix. Both charts were established under the assumption that the parameters are known a priori. This is seldom the case, and Phase I data sets are commonly used to estimate the charts in-control parameter values. Plugging in parameter estimates, however, fundamentally changes the run-length distribution from those assumed in the known-parameter theory and diminishes chart performance, even for large calibration samples. Self-starting methods, which correctly studentize the incoming stream of process readings, provide exact control right from start up. We extend the existing multivariate self-starting methodology to a combination chart for both the mean vector and the covariance matrix. This approach is shown to have good performance.

Detection of multiple change-points in multivariate data

The statistical analysis of change-point detection and estimation has received much attention recently. A time point such that observations follow a certain statistical distribution up to that point and a different distribution – commonly of the same functional form but different parameters after that point – is called a change-point. Multiple change-point problems arise when we have more than one change-point. This paper develops a method for multivariate normally distributed data to detect change-points and estimate within-segment parameters using maximum likelihood estimation.

A LASSO Chart for Monitoring the Covariance Matrix

Abstract Multivariate control charts are essential tools in multivariate statistical process control. In real applications, when a multivariate process shifts, it occurs in either location or scale. Several methods have been proposed recently to monitor the covariance matrix. Most of these methods use rational subgroups and are used to detect large shifts. In this paper, we propose a new accumulative method, based on penalized likelihood estimators, that uses individual observations and is useful to detect small and persistent shifts in a process when sparsity is present.

Monitoring the covariance matrix with fewer observations than variables

Multivariate control charts are essential tools in multivariate statistical process control. In real applications, when a multivariate process shifts, it occurs in either location or scale. Several methods have been proposed recently to monitor the covariance matrix. Most of these methods deal with a full rank covariance matrix, i.e., in a situation where the number of rational subgroups is larger than the number of variables. When the number of features is nearly as large as, or larger than, the number of observations, existing Shewhart-type charts do not provide a satisfactory solution because the estimated covariance matrix is singular. A new Shewhart-type chart for monitoring changes in the covariance matrix of a multivariate process when the number of observations available is less than the number of variables is proposed. This chart can be used to monitor the covariance matrix with only one observation. The new control chart is based on using the graphical LASSO estimator of the covariance matrix instead of the traditional sample covariance matrix. The LASSO estimator is used here because of desirable properties such as being non-singular and positive definite even when the number of observations is less than the number of variables. The performance of this new chart is compared to that of several Shewhart control charts for monitoring the covariance matrix.

Smoothed Linear Modeling for Smooth Spectral Data

Classification and prediction problems using spectral data lead to high-dimensional data sets. Spectral data are, however, different from most other high-dimensional data sets in that information usually varies smoothly with wavelength, suggesting that fitted models should also vary smoothly with wavelength. Functional data analysis, widely used in the analysis of spectral data, meets this objective by changing perspective from the raw spectra to approximations using smooth basis functions. This paper explores linear regression and linear discriminant analysis fitted directly to the spectral data, imposing penalties on the values and roughness of the fitted coefficients, and shows by example that this can lead to better fits than existing standard methodologies.

Significantly Lower Estimates of Volatility Arise from the Use of Open-High-Low-Close Price Data

Abstract This research provides an indication of the possible reduction in insurance liability valuations arising from the reduced volatility estimate of the Yang-Zhang refinement of volatility, when the liabilities are based on historic prices estimates arising from end-of-day prices in a jump-diffusion model. The paper also demonstrates the usefulness of change points. This research compares the standard measure of volatility (standard deviation of the log of close prices) for the total return of the S&P 500 to a recently developed volatility measure by Yang and Zhang that capitalizes on open-high-low-close prices. The latter volatility was developed to be the measure providing the narrowest confidence interval of all estimates satisfying certain desirable features and as such is the most desirable measure from a decision theory standpoint. This research shows that the Yang-Zhang volatility generally provides significantly lower estimates of volatility. This lower volatility estimate should lead to lower valuation levels for insurance products with guarantees, and this paper provides indicative reductions in liability valuations. Both volatility measures assume constant volatility and drift over a period. To accommodate this assumption, change points are employed to divide historical data into regimes of constant drift and volatility. To this end, the theory of change points is briefly introduced. The research shows that standard measure of volatility generally overestimates volatility, and the error increases with the absolute value of the underlying drift. There are several potential technical reasons why the lower volatility could be invalid, but this paper considers and rejects each, to conclude that the lower volatility estimate of Yang-Zhang is in fact the better estimate, not a result of a technical degeneracy. One conclusion is that valuations employing regime-switching generators, especially insurance liability valuations, should use the Yang-Zhang measure of volatility, otherwise any analysis embedded (or free-standing) options could overvalue prices or volatility. The simplicity of the Yang-Zhang calculation and its potentially large impact on valuations should justify its adoption for most companies.

A Predictive Model for Use of an Assistive Robotic Manipulator: Human Factors Versus Performance in Pick-and-Place/Retrieval Tasks

The goal of this study was to model the important individual differences to predict a users performance when operating an assistive robotic manipulator for a general population. Prior research done led to the identification of ten potential human factors to be observed including dexterity (gross and fine), spatial abilities (orientation and visualization), visual acuity in each eye, visual perception, depth perception, reaction time, and working memory. Eighty-nine individuals completed a test battery of potential human factors and, then, completed several tasks using a robotic manipulator designed to simulate find-and-fetch/pick-and-place tasks. During interaction with the robot, time on task, number of moves, and number of moves per minute were recorded. We successfully developed statistical models predicting performance that revealed several important human factors. Speed of information processing, spatial ability, dexterity, and working memory were all seen to be significant predictors of task performance. For time on task, linear and polynomial models showed roughly similar predictive performance on unseen test data achieving root-mean-square percentage error of about 7.3%; for number of moves per minute, a polynomial model was best with 9.1% error; and for number of moves, a linear model was best with 12.8% error.

Tests for mean vectors in high dimension

Traditional multivariate tests, Hotellings or Wilks , are designed for a test of the mean vector under the condition that the number of observations is larger than the number of variables. For high-dimensional data, where the number of features is nearly as large as or larger than the number of observations, the existing tests do not provide a satisfactory solution because of the singularity of the estimated covariance matrix. In this article, we consider a test for the mean vector of independent and identically distributed multivariate normal random vectors where the dimension is larger than or equal to the number of observations. To solve this problem, we propose a modified Hotelling statistic. Simulation results show that the proposed test is superior to other tests available in the literature. However, because we do not know the theoretical distribution of this modified statistic, Monte Carlo methods were used to reach this conclusion. Instead of using conventional Monte Carlo methods, which perform a fixed-number of simulations, we suggest using the sequential Monte Carlo test in order to decrease the number of simulations needed to reach a decision. Simulation results show that the sequential Monte Carlo test is preferable to a fixed-sample test, especially when using computationally intensive statistical methods.