Multivariate Monitoring with GPS Observations and Auxiliary Multi-Sensor Data

In a multi-sensor measurement or monitoring environment, p variables are measured simultaneously. The measured data are correlated and can be monitored to identify special causes of variation in order to establish control and to obtain reference samples to use as a basis in determining the control limits for future observations. One common method of constructing multivariate control charts is based on Hotelling’s 2 T statistic. When the monitoring process is at the start -up stage, F and chisquare distributions are used to construct the necessary multivariate control limits. An example from a seven-day GPS data measured concurrently with the accelerometer response, wind velocity and temperature illustrates that this technique can improve the interpretation of GPS results. Moreover, the computational complexity is reduced through a reduction in the data dimensionality. INTRODUCTION GPS technology has expanded in its areas of applications. Recently many authors have discussed projects and issues related to quality control and automatic deformation monitoring with GPS (e.g., Ka& & lber et al, 2000; Mertikas & Rizos 1996; Mertikas, 1998, 2000; Ogaja, 2001; Ogaja et al, 2000; Ogaja et al 2001a; 2001b; 2001c; Rizos, 1998; Rizos et al., 1999; Roberts & Rizos, 1997; Wang et al., 1997). Ogaja et al (2001b), for example, refers to a building monitoring project in which GPS is used to augment the existing monitoring instrumentation comprising accelerometers and wind anemometers. In many such situations, separate univariate control charts for each of the measured variables can be utilised to detect changes in the inherent variability of the process. When these variables are correlated, however, the univariate charts are not as sensitive as multivariate methods that capitalise on this correlation (see, for example, Barbara & Linda, 1989; Barnett, 1994; Joe, 1997; Morrison, 1990). The values plotted on multivariate control charts are usually statistics based on the 2 T distribution. This distribution is the multivariate counterpart to the Student’s t distribution. It is particularly appropriate when the variables of interest are correlated. There are two distinct phases in constructing control charts. The first phase involves testing whether the process was in control when the initial individual or subgroup data were collected. A subgroup represents a sample of observations taken at some point in the process, during a specific time interval. This phase is often termed the "start-up" stage of the process as the purpose is to obtain a set of data (reference samples) to establish the control limits for monitoring purposes. The goal of the first phase is to establish statistical control (i.e, a "clean" process) and find accurate control limits for phase two. The second phase consists of using the control chart to monitor the behaviour of the process by detecting any departure from the process settings as future samples are drawn. In many cases the multivariate 2 T statistic is often utilised as the charting statistic for both phases of control chart construction. In the first phase, with subgroup sizes greater than one, and in the second phase, where concern is in monitoring the process, the control limits are determined using the fact that the 2 T statistic (times a constant) follows an exact F distribution. Thus the control limits at the "start up" stage of the process can be computed based on the F or chi-square distribution. The purpose of this paper is to investigate a simple case of constructing multivariate control charts for use when GPS and complementary sensor data are collected simultaneously. The results are illustrated using an actual data example taken from a project experiment. ESTABLISHING THE CONTROL LIMITS Consider the case where p correlated variables are being measured simultaneously and are in need of control. Assume that these variables follow a p-dimensional multivariate normal distribution with mean vector ) ( p μ μ μ μ , . . . , , 2 1 = ' and covariance matrix ∑ , where i μ is the mean for the i th variable and ∑ is a p × p matrix consisting of the variances and covariances of the p variables. The multivariate normal distribution is the p-dimensional analogue to the univariate normal distribution assumed for each variable. Technically if the process is not in statistical control then there is no stable distribution for the data. At the "start up" stage of a monitoring process, the multivariate normality is assumed solely for the purpose of deriving the control limits. After the control limits are established, the data are assumed to be reasonably normally distributed. The results depend on the validity of the assumption, just as the validity of the usual control limits for a univariate control chart for individual variables requires the normality assumption. This assumption can be checked using appropriate multivariate normal goodness-of-fit tests. Assume that control of the process is in the "start up" stage and a sample of m subgroups of past data are available to estimate the parameters μ and ∑ . For notational purposes, represent the i epoch observation of the p variables from the reference samples with the vector: