Multivariate copulas on the MCUSUM control chart

The copula approach is a popular method for multivariate modeling applied in several fields; it defines non-parametric measures of dependence between random variables. In this paper, three families are proposed from elliptical and Archimedean copulas on the multivariate cumulative sum (MCUSUM) control chart when observations are draw from an exponential distribution. The performance of the control chart is based on the average run length (ARL)—via Monte Carlo simulations. A copula function for specifying the dependence between random variables is measured by Kendall’s tau. The numerical results indicate that the observations can be fitted and that the copula can be used on the MCUSUM for cases of small and large dependencies. Subjects: Science; Mathematics & Statistics; Statistics & Probability; Probability; Statistics; Multivariate Statistics; Technology; Engineering & Technology; Industrial Engineering & Manufacturing; Quality Control & Reliability


Introduction
Quality control charts are applicable for processes that have one variable or more, which are referred to as univariate or multivariate control charts, respectively. Multivariate control charts are widely used to simultaneously monitor several quality characteristics for detecting the mean changes in manufacturing industries. They are a powerful tool in statistical process control (SPC) for identifying an out-of-control process.
Generally, multivariate detection procedures are based on a multi-normality assumption and independence; however, many processes exhibit non-normality and correlation. Most of these multivariate control charts are generalizations of their univariate counterparts (Mahmoud & Maravelakis, 2013), such as the Hotelling's T 2 control chart, the multivariate exponentially weighted moving average (MEWMA) control chart proposed by Lowry, Woodall, Champ, and Rigdon (1992), and the multivariate cumulative sum (MCUSUM) control chart (Bersimis, Panaretos, & Psarakis, 2005;Bersimis, Psarakis, & Panaretos, 2007). Many practitioners have addressed the problem of correlated data in SPC (see Lowry & Montgomery, 1995), and multivariate control charts with the lack of related joint distribution and copula can reveal this characteristic.
In manufacturing processes, the time is used to represent some attributes or variable measures. One type of time distribution for an event is known as an exponential distribution, which is a continuous distribution of monitoring for successive event occurrences. This article presents a study of the MCUSUM control chart when observations are generated by an exponential distribution (Khoo, Atta, & Phua, 2009;Mason, Champ, Tracy, Wierda, & Young, 1997) and a copula describe the dependence between exponentially distributed components of a trivariate observation with mean shifts.

Properties of the MCUSUM control chart
In the univariate case, the cumulative sum (CUSUM) procedure is often designed for monitoring and detecting small changes. CUSUM was first introduced by Page (1954) to detect changes in the mean of an independent and identically distributed (i.i.d.) observed sequence of random variables. The MCUSUM control chart is the multivariate extension of a univariate CUSUM chart (see Busaba, Sukparungsee, Areepong, & Mititelu, 2012). Crosier (1988) proposed the MCUSUM control chart; which is the extension of a univariate CUSUM control chart. The MCUSUM control chart may be expressed as follows: t = 1, 2, …, where covariance matrix and t are the cumulative sums as determined by: where the reference value and is the aim point or target value for the mean vector (see Khoo et al., 2009). The control chart statistics for MCUSUM chart are The signal gives an out-of-control indication if Y t > h, where h is the control limit (see Alves, Samohyl, & Henning, 2010;Fricker, Knitt, & Hu, 2008).

Copula function
The copula function is a multivariate distribution with all univariate margins being U(0, 1). The multivariate copula function is used for capturing the dependence between two or more random variables. Suppose that a random vector (X 1 , …, X d ) has a joint distribution function H( then, there exists a unique d-dimensional copula C . In this section, the theory that is the central foundation of the copula is described and shown in Table 1 (see Genest & McKay, 1986;Joe, 2015;Sklar, 1959;Trivedi & Zimmer, 2005).

Sklar's theorem
This theorem is central to the theory of copulas and is applied using statistical theory. Assuming R denotes the ordinary real time (−∞, ∞), and the unit square 2 is the product × , where = [0, 1], Sklar's theorem can be written as Theorem 1.
If F is a d-variate distribution function of discrete random variables, then the copula is unique only on the set

Copula families
For the purposes of the statistical method, it is desirable to parameterize the copula function. Let θ denote the association parameter of the multivariate distribution; then, there exists a copula C (see Trivedi & Zimmer, 2005). This paper focuses on the normal copula and two families of Archimedean copulas, the Clayton and Frank copulas, because these copulas are well-known.

Normal copula
The normal copula is an elliptical copula. From the multivariate normal distribution with zero means, unit variances and d × d correlation matrix , the trivariate normal copula is determined by where (⋅; ) is the trivariate normal cdf, Φ is the univariate normal cdf and Φ −1 is the univariate normal inverse cdf or quantile function (see Ganguli & Reddy, 2013;Joe, 2015).

Measuring dependence
Spearman's rho is widely used to describe the association between random variables with linear dependence and it can be used for any dependence of a monotonic type. In the case of linear dependence Pearson's rho may be used, but it works well only for normal marginals. For association between random variables that are not linear, Kendall's tau is a measurement of the concordance. Kendall's tau is a non-parametric measurement of associations which is considered a copula-based dependence measurement (see Quessy, Said, & Favre, 2013). Kendall's tau is straightforward to calculate, and this measure is used for observation dependencies.

Performance characteristics and numerical results
In this section, the performance characteristics for the SPC chart is the average run length (ARL). It is classified into ARL 0 and ARL 1 , where ARL 0 is the average run length when the process is in control and ARL 1 is the average run length when the process is out-of-control (see . Generally, an acceptable ARL 0 should be sufficiently large when the process is in control, and ARL 1 should be small when the process is out-of-control. Empirical studies are implemented in the R statistical software and all packages used are available from CRAN at https:// CRAN.R-project.org/ (see Hofert, Mächler, & McNeil, 2012;Yan, 2007) with 50,000 simulation runs and a sample size of 1,000. Observations were drawn from the exponential distribution with parameter (α) equal to 1 for an in-control process (μ 0 = 1) and the shifts of the process level (δ) by μ = μ 0 + δ. The process means are 1.1, 1.2, 1.3, …, 2 for an out-of-control process. Copula estimations are restricted to the cases of positive dependence. For all copula models, the setting θ corresponds to Kendall's tau and the level of dependence is measured by Kendall's tau values (−1 ≤ τ ≤ 1). The authors define 0.2 and 0.8 as small and large dependencies, respectively.
Tables 2-4 present the numerical results for empirical observations. The different values of exponential parameters are (μ 1 , μ 2 , and μ 3 ) for the variables (X 1 , X 2 , X 3 ). For in-control processes, the MCUSUM control chart was chosen by setting the desired ARL 0 = 370, and μ 1 = μ 2 = μ 3 = 1 is fixed for each copula. Table 2 shows small and large positive dependencies in the case of all parameters shifts. Tables 3 and 4 show parameters shifts in the case of two parameters shifts. The results in Table 2 show the mean shifts of τ = 0.2 and 0.8. The ARL 1 values of the Clayton copula are less than the other copulas for nearly all shifts except τ = 0.8 for large shifts (1.8 ≤ μ 1 = μ 2 = μ 3 ≤ 2). Table 3 shows the case of small dependence; the ARL 1 values of the Clayton copula are less than the other (3.8)