The optimized CUSUM and EWMA multi-charts for jointly detecting a range of mean and variance change

ABSTRACT

This article considers the problem of jointly monitoring the mean and variance of a process by multi-chart schemes. Multi-chart is a combination of several single charts which detects changes in a process quickly. Asymptotic analyses and simulation studies show that the optimized CUSUM multi-chart has optimal performance than optimized EWMA multi-chart in jointly detecting mean and variance shifts in an $i.i.d.$ normal observation. A real example that monitors the changes in IBM's stock returns (mean) and risks (variance) is used to demonstrate the performance of the above two multi-charts. The proposed method has been compared to a benchmark and it performed better.

1. Introduction

A wealth of research is ongoing in qualimetry (also known as statistical process control (SPC)), an area of science bringing together various methods for the quantitative evaluation of product quality. Many of the existing works focused on methods for detecting mean shifts while others considered methods for detecting changes in variability (variance or standard deviation) or jointly ( mean–variance) change in the variables one wish to monitor. The application areas of these SPC schemes are varied such as industrial and chemical engineering, information and communication systems, biostatistics and public health, econometrics and financial surveillance, network science and graph data, and so on; see, for example, Frisén [13], Fricker [12], Woodall and Montgomery [43], Montgomery [35], Woodall et al. [44], Bersimis et al. [3] and Hosseini and Noorossana [22].

It is important to note that when special causes exist which triggers both the mean and variance to shift simultaneously, then it is more reasonable to combine the mean and variance information on one scheme and look at their behavior jointly. For instance, a wrongly fixed stencil in a circuit manufacturing may result in a concurrent change in the mean and variance of the thickness of the solder paste printed on circuit boards [14].

Several schemes have been proposed in open literatures for joint monitoring of both mean and variance or standard deviation. For example, Reynolds and Glosh [38] proposed the L chart for joint monitoring of mean and variance shifts. Domangue and Patch [9] proposed the omnibus exponentially weighted moving average (EWMA) chart for joint monitoring of mean and variance shifts. Costa [6] developed the so-called non-central chi-square statistic (also known as NCS chart) to monitor the mean and variance simultaneously. Grabov and Ingman [17] also presented the B-chart while Chen et al. [5] considered the MaxEWMA chart for mean–variance monitoring. Costa and Rahim [7] proposed EWMA non-central chi-square statistic to detect changes in mean and increase in variance, their chart does not detect decrease in variance. Wu and Tian [45] considered the weighted loss function (WLC) chart while Capizzi and Masarotto [4] developed generalized likelihood ratio (GLR) charts for joint monitoring of the mean and variance shifts of an autoregressive moving average process. Guh [18] used artificial neural network to monitor process mean and variance shifts. Guh's results indicated that the proposed model can effectively recognized single mean and variance control chart patterns and also mixed control chart patterns of mean and variance. Li et al. [26] considered a self-starting control chart for monitoring the mean and variance shifts simultaneously. McCracken et al. [33] considered a Shewhart-type control scheme for joint monitoring of mean and variance of a normal process. Li et al. [25] also presented the so-called Case-U type scheme which is the CUmulative SUM (CUSUM) modification of the scheme by McCracken et al. [33].

In recent years, Lu and Chang [31] proposed a fuzzy classification maximum likelihood change-point (FCML-CP) algorithm to detect shifts in the mean and variance, Wang and Cheng [42] presented an EWMA chart based on likelihood ratio test for monitoring Weibull mean and variance with subgroups, Quininoa et al. [37] presented novel control charts that are based on the inspection of attributes and the use of traditional control limits for the joint monitoring of mean and variance, Gao et al. [15] proposed a penalized weighted least-squares approach with an iterative estimation procedure that integrates variance change point detection and smooth mean function estimation, Messer [34] proposed a method for bivariate change point detection of changes in expectation and variance, and Kim et al. [23] proposed the so-called copula Markov statistical process control and conditional distribution method to monitor the mean and variance jointly.

It can be seen that all the control charts above mainly focus on monitoring whether there are changes in mean and variance, and rarely consider the magnitude of such changes at the same time and optimize the proposed control charts. Besides, some of the charts, for example FCML-CP algorithm has computational complexities and hence it is time-consuming in detecting shifts in mean and variance much faster due to the slow rate of convergence. The main purpose of this present paper is to deal with these problems by optimizing CUSUM and EWMA multi-charts for jointly detecting a range of mean and variance change.

Multi-chart consists of several single charts with different reference values that are used simultaneously to detect and monitor process changes. Multi-chart schemes have elegant properties in the sense that they are faster in detecting changes in a process and computationally less expensive than similar charts like Generalized EWMA by Han and Tsung [19] and the CUSUM-like control chart by Siegmund and Venkatraman [40]. Multi-chart schemes are different from multivariate CUSUM (MCUSUM) schemes (see Crosier [8], Apley and Fsung [1] and Golosnoy [16]), multivariate EWMA (MEWMA) schemes (see Lowry et al. [30]) and multi-hypothesis testing (see Baum and Veeravalli [2] and Lai [24]) in terms of methodology. For example, multi-chart schemes can tell which of the charts triggered some change, a property that falls short of multivariate charts (MCUSUM and MEWMA).

Generally, we rarely know the exact shift of a process before it is detected, hence it is imperative we consider a range of known or unknown mean and variance reference values which are the magnitude of shifts to be detected. Indeed, the possible change in many practical problems often forms a continuous region of two-dimensional plane. In fact, Lorden [28] considered and studied a model which consisted of several control charts. Since then, Lorden and Eisenberger [29], Lucas [32], Dragalin [10,11], Sparks [41], Han et al. [21] and Liu et al. [27] have further investigated and studied a combination of several CUSUM charts and a combined Shewhart-CUSUM to detect mean shifts in a range. However, most of the works centered on mean or variance change detection. Of course, the joint problem we faced here is how to design the best control chart (faster detection) which can detect shifts in the mean and variance simultaneously given some reference values. In the paper, we will give an optimal design of CUSUM and EWMA multi-charts procedures for monitoring shifts in the mean and variance simultaneously and seek to find optimal parameters for determining the multi-charts.

The reminder of the article is organized as follows; Section 2 presents the optimized CUSUM and EWMA multi-chart schemes. Section 3 gives the asymptotic analyses of the optimized multi-chart schemes. Section 4 presents the numerical simulation analyses that compare the CPIs of the charts. Section 5 gives a real example that illustrates the proposed method for financial surveillance using the IBM stock returns. Discussion, suggestion for further research and conclusion are presented in Section 6, with proof of preposition given in Appendix.

2. Optimized CUSUM and EWMA multi-charts

Assume that $X_i\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}}{\sim }N({\mu }_0,{\sigma }_0^2)$ where $i=1,2,\dots ,n$. Usually, at some time period τ, the probability distribution of $X_i$ changes from $N({\mu }_0,{\sigma }_0^2)$ to $N(\mu ,{\sigma }^2)$. We generally refer to τ as a change point where $\tau =1,2,3,\dots ,\mathrm{\infty }$ but here we assume $\tau =1$, which means the first time there is change in distribution. Intuitively, the mean and variance of $X_i$ undergo a shift of size $\mu -{\mu }_0$ and ${\sigma }^2$/( ${\sigma }_0^2$) respectively where ${\mu }_0$ and ${\sigma }_0^2$ are assumed known and are taken to be ${\mu }_0=0$ and ${\sigma }_0^2=1$.

Again let assume the common probability density (pdf of normal distribution) of $X_1,\dots ,X_{\tau -1}$ before change time be $f_{{\theta }_0}(X_n)$ and after the change time, the post-change probability density of $X_k,k\ge \tau$ be $f_{{\theta }_k}(X_n)$, where the parameter θ belongs to a bounded set $R^2$ with $f_{\theta }\ne f_{{\theta }^{\prime }}$ if and only if $\theta \ne {\theta }^{\prime }\in \mathrm{\Theta }$ and ${\theta }_0=(0,1)\notin \mathrm{\Theta }$, where $R^2$ denotes two-dimensional real number space. Again let denote ${\mathbf{P}}_{\theta }$ ( ${\mathbf{E}}_{\theta }$ ) as the probability distribution (expectation) with the post-change density function $f_{\theta }$ after the change time. When $\tau =\mathrm{\infty }$, i.e. the change never occurs, we denote by ${\mathbf{P}}_{{\theta }_0}({\mathbf{E}}_{{\theta }_0})$ the probability distribution (expectation) with the density function $f_{{\theta }_0}$ for all time $k\ge 0$.

Usually, the possible change region Θ forms a continuous region of two-dimensional plane and may be determined by engineering knowledge, practical experience or by statistical data. We may have to divide region Θ into several disjointed subsets $({\mathrm{\Theta }}_k$ for $1\le k\le m$) such that $\sum _{k=1}^m{\mathrm{\Theta }}_k=\mathrm{\Theta }$. The region Θ is a closed boundary set of the parameters and the boundary $(\mathrm{\partial })$ of Θ is known. For example, let $f_{\theta }$ be the normal density function with parameter $\theta =(\mu ,\sigma )$, we can take the set $\mathrm{\Theta }=\{\theta =(\mu ,\sigma ):{\mu }_1\le \mu \le {\mu }_2;{\sigma }_1\le \sigma \le {\sigma }_2\}\subset R\times R^{+}$, where $R=(-\mathrm{\infty },+\mathrm{\infty })$ and $R^{+}=(0,+\mathrm{\infty })$ are the possible change regions, given that μ and σ denote the mean and standard deviations respectively and ${\mu }_1,{\mu }_2,{\sigma }_1$ and ${\sigma }_2$ are four known numbers.

Here, we will use the charting performance index (CPI) proposed by Han and Tsung [20] to judge which chart $(T)$ performs better and is defined by

$$CPI(T)=\exp \{-{\int }_{\mathrm{\Theta }}w(\theta )[\frac{{ARL}_{\theta }(T)}{{\mathbf{A}\mathbf{R}\mathbf{L}}_{\theta }^{\ast }}-1]\mathrm{d}\theta \},$$ (1)

where $w(\theta )$ is a positive weight function for detection satisfying ${\int }_{\mathrm{\Theta }}w(\theta )d\theta =1$, $AR{L}_{\theta }(T)$ is the average run length (ARL) of the chart $(T)$ to be evaluated and ${\mathbf{A}\mathbf{R}\mathbf{L}}_{\theta }^{\ast }$ is the chart with the lowest ARL value. The average run length (ARL) is the average number of samples taken before a chart signal. That is, $AR{L}_{\theta }(T):=E_{\theta }(T)$. Usually, $AR{L}_{{\theta }_0}(T)$ denotes in-control average run length and $AR{L}_{{\theta }_1}(T)({\theta }_1\ne {\theta }_0)$ denotes out-of-control average run length. Moustakides [36] and subsequently Ritov [39] showed that among all control charts, the ARL of the chart with reference value as parameter is optimal. The chart with the greatest CPI performs best in jointly detecting the mean–variance change over a range. We have $0<CPI(T)\le 1$.

In general, we define multi-chart as $T_M$, given by

$$T_M=\underset{1\le k\le m}{min}\{T_k\}$$ (2)

where $T_k$ is a chart, m is the number of charts and k is a particular value of m.

We then define the CUSUM multi-chart as $T_{CM}$, where

$$T_{CM}=\underset{1\le k\le m}{min}\{T_C({\theta }_k)\}$$ (3)

$T_C({\theta }_k)$ is a one-sided CUSUM charts which we will define shortly and ${\theta }_k=({\mu }_k,{\sigma }_k^2)\in {\mathrm{\Theta }}_k$ are some reference parameters for $1\le k\le m$. The reference parameters are the sizes of shifts in the mean and variance one anticipates to detect quickly. Generally, smaller values of the reference values are effective for monitoring smaller shifts and larger values of the reference values are effective for monitoring larger shifts. The charting performance index (CPI) which is used to measure which chart (T) performs better, depends on the position of the reference value ${\theta }_k$ in the region ${\mathrm{\Theta }}_k$ for $1\le k\le m$. $T_C({\theta }_k)$ can be written as

$$T_C({\theta }_k)=T_C({\theta }_k)(c_k)=min\{n:Y_{k,n}\ge c_k\},Y_{k,n}=max\{Y_{k,n-1},0\}+\log \frac{f_{{\theta }_k}(X_n)}{f_{{\theta }_0}(X_n)}$$ (4)

for $1\le k\le m$, where n is the sample number, $Y_{k,n}$ is the CUSUM charting statistic and the control limits $c_1,\dots ,c_m$ are taken such that $T_C({\theta }_1),\dots ,T_C({\theta }_m)$ have a common in-control ARL ${}_0$, that is, ${\mathbf{E}}_{{\theta }_0}(T_C({\theta }_k))=\gamma$ for $1\le k\le m$.

We define the optimized CUSUM multi-chart as $T_{CM}^{\ast }$, where $T_{CM}^{\ast }$ is constructed with the optimal reference numbers; ${\theta }_k^{\ast }=({\mu }_k^{\ast },{\sigma }_k^{2\ast })\in {\mathrm{\Theta }}_k$ for $1\le k\le m$. Preposition 1 will enable us construct the optimized CUSUM multi-chart.

The EWMA scheme for joint monitoring of the mean and variance is defined as $T_E(\alpha ,\beta )$ where

$$T_E(\alpha ,\beta )=inf\{n:Y_n=Y_n^{(1)}(\alpha )+Y_n^{(2)}(\beta )\ge c\}$$ (5)

where

$Y_n^{(1)}(\alpha )=(1-\alpha )Y_{n-1}^{(1)}(\alpha )+\alpha X_n$,

$Y_n^{(2)}(\beta )=(1-\beta )Y_{n-1}^{(2)}(\beta )+\beta [(X_n-Y_n^{(1)}(\alpha )/{\alpha }_n{)}^2-1]$,

and ${\alpha }_n=1-(1-\alpha {)}^n$.

for $0<\alpha \le 1$ and $0<\beta \le 1$.

The smoothing parameters α and β are the mean charting statistic $(Y_n^{(1)})$ and variance charting statistic $(Y_n^{(2)})$ respectively. The parameter c is the width of the control limit.

We also define the EWMA multi-chart for joint monitoring of the mean and variance as $T_{EM}$ where

$$T_{EM}=\underset{1\le k\le m}{min}\{T_E({\alpha }_k,{\beta }_k)\}$$ (6)

The optimized EWMA multi-chart is defined as $T_{EM}^{\ast }$ where $T_{EM}^{\ast }$ is the EWMA multi-chart that has the greatest charting performance index (CPI).

3. Asymptotic analysis of optimized multi-chart scheme

The consequence of the following preposition will enable us construct the optimized CUSUM multi-chart.

Proposition 3.1

Let the real measure $\tilde{\theta }(i,j)=({\tilde{\mu }}_i,{\widetilde{{\sigma }^2}}_j)$ for $1\le i\le l$ and $1\le j\le q$, where $\tilde{\theta }(i,j)\in {\mathrm{\Theta }}_k$ then for large $\gamma =AR{L}_0(T_{CM})$ we have

$$\begin{array}{rl}CPI(T_{CM})& =(1+o(1))\exp \{1-\frac{1}{l\times q}\sum _{k=1}^m\sum _{\tilde{\theta }(i,j)\in {\mathrm{\Theta }}_k}^{}\\ & \frac{{\widetilde{{\sigma }^2}}_j-1-\log {\widetilde{{\sigma }^2}}_j+{\widetilde{{\mu }^2}}_i}{({\widetilde{{\sigma }^2}}_j+{\widetilde{{\mu }^2}}_i)(1-1/{\sigma }_k^2)-\log {\sigma }_k^2+\frac{{\mu }_k(2{\tilde{\mu }}_i-{\mu }_k)}{{\sigma }_k^2}}\}\end{array}$$ (7)

Proposition 3.1 implies that the charting performance index (CPI) which is used to measure which chart $(T)$ performs better, depends on the position of the real measure (real shifts) $\tilde{\theta }(i,j)$ = $({\tilde{\mu }}_i,{\widetilde{{\sigma }^2}}_j)$ for $1\le i\le l$ and $1\le j\le q$ in the region ${\mathrm{\Theta }}_k$ for $1\le k\le m$, where the number of real mean shifts is l and q is the number of real variance shifts. In general, we may consider l and q fewer than 10 to reduce computational burden. The proof of proposition 3.1 is in Appendix.

4. Numerical results and comparison

In this section, we shall consider the application of Proposition 3.1 in Section 1 and present results for the optimized CUSUM and EWMA multi-charts in Sections 2 and 3 respectively and round it up with comparison of the schemes in Section 4.

4.1. Application of Proposition 3.1

For the sake of convenience let's consider m = 2 to construct the optimized CUSUM multi-chart $(T_{CM})$ using the proposed method. The Kullback–Leibler information distance $I(\theta ,{\theta }_k)$ is defined as

$$I(\theta ,{\theta }_k)={\mathbf{E}}_{\theta }\{\log [\frac{f_{\theta }(X_1)}{f_{{\theta }_k}(X_1)}]\}=\frac{1}{2}\log (\frac{{\sigma }_k^2}{{\sigma }^2})+\frac{(\mu -{\mu }_k{)}^2}{2{\sigma }_k^2}+\frac{{\sigma }^2}{2}(\frac{1}{{\sigma }_k^2}-\frac{1}{{\sigma }^2})$$ (8)

where ${\mathbf{E}}_{\theta }$ is the mathematical expectation with respect to the parameter θ, $f_{\theta }(X_1)$ is the pdf of the normal distribution, ${\sigma }^2$ is the variance axis (coordinate) and μ is the mean axis (coordinate). For example, if we consider two parameters; ${\theta }_1=({\mu }_1,{\sigma }_1^2)$ and ${\theta }_2=({\mu }_2,{\sigma }_2^2)$, we can set the boundary (∂) equation between the two points on a plane using the Kullback–Leibler information distance as

$$\mathrm{\partial }{\mathrm{\Theta }}_1=\mathrm{\partial }{\mathrm{\Theta }}_2=\{\theta :I(\theta ,{\theta }_1)=I(\theta ,{\theta }_2)\}$$ (9)

where

$$I(\theta ,{\theta }_1)=\frac{1}{2}log(\frac{{\sigma }_1^2}{{\sigma }^2})+\frac{(\mu -{\mu }_1{)}^2}{2{\sigma }_1^2}+\frac{{\sigma }^2}{2}(\frac{1}{{\sigma }_1^2}-\frac{1}{{\sigma }^2})$$ (10)

and

$$I(\theta ,{\theta }_2)=\frac{1}{2}log(\frac{{\sigma }_2^2}{{\sigma }^2})+\frac{(\mu -{\mu }_2{)}^2}{2{\sigma }_2^2}+\frac{{\sigma }^2}{2}(\frac{1}{{\sigma }_2^2}-\frac{1}{{\sigma }^2})$$ (11)

The boundary $\theta =(\mu ,{\sigma }^2)$ is found such that $I(\theta ,{\theta }_1)=I(\theta ,{\theta }_2)$ with little algebraic simplification gives

$$\{\theta (\mu ,{\sigma }^2):(C-D){\mu }^2-2(D{\mu }_2-C{\mu }_1)\mu +B{\sigma }^2+C{\mu }_1^2-D{\mu }_2^2+A=0\}$$ (12)

where

$A=\frac{1}{2}\log (\frac{{\sigma }_1^2}{{\sigma }_2^2})$, $B=\frac{1}{2}(\frac{1}{{\sigma }_1^2}-\frac{1}{{\sigma }_2^2})$, $C=\frac{1}{2{\sigma }_1^2}$ and $D=\frac{1}{2{\sigma }_2^2}$.

We computed the CPI for each of the pairs as follows.

Let's consider that the reference parameter ${\theta }_1$ is from region ${\mathrm{\Theta }}_1$ and ${\theta }_2$ is from region ${\mathrm{\Theta }}_2$ separated by the boundary (12), that is to say ${\mathrm{\Theta }}_1\cap {\mathrm{\Theta }}_2=\varnothing$. We computed the boundary equation between each pairs and found the position and region where the real shifts belong.

We considered five real mean shifts $(l)$ and three real variance shifts $(q)$ hence, $l\times q=5\times 3=15$, we can therefore write Equation (7) as

$$\begin{array}{rl}CPI(T_{CM})& =(1+o(1))exp\{1-\frac{1}{15}\sum _{\tilde{\theta }(i,j)\in {\mathrm{\Theta }}_1}^{}\left[\frac{AR{L}_{\tilde{\theta }(i,j)}[T_C({\theta }_1)]}{{\mathbf{A}\mathbf{R}\mathbf{L}}_{\tilde{\theta }(i,j)}^{\ast }}\right]\\ & -\frac{1}{15}\sum _{\tilde{\theta }(i,j)\in {\mathrm{\Theta }}_2}\left[\frac{AR{L}_{\tilde{\theta }(i,j)}[T_C({\theta }_2)]}{{\mathbf{A}\mathbf{R}\mathbf{L}}_{\tilde{\theta }(i,j)}^{\ast }}\right]\}\end{array}$$ (13)

where

$AR{L}_{\tilde{\theta }(i,j)}[T_C({\theta }_1)]$ and $AR{L}_{\tilde{\theta }(i,j)}[T_C({\theta }_2)]$ are the ARL's of CUSUM chart with parameters ${\theta }_1=({\mu }_1,{\sigma }_1^2)$ and ${\theta }_2=({\mu }_2,{\sigma }_2^2)$ respectively and

$${\mathbf{A}\mathbf{R}\mathbf{L}}_{\tilde{\theta }(i,j)}^{\ast }=\frac{1}{I(\tilde{\theta }(i,j),{\theta }_0)}=\frac{1}{{\widetilde{{\sigma }^2}}_j-1-\log {\widetilde{{\sigma }^2}}_j+{\widetilde{{\mu }^2}}_i}$$ (14)

We consequently computed the CPIs for each of the pairs by (13) and the results presented in Table 1.

Table 1.

CPI results of CUSUM multi-charts with $AR{L}_0$ $\approx 500$.

Chart	CPI	Chart	CPI
$T_{CM}[{\theta }_1(0.1,1.13),{\theta }_2(1.00,1.38)]$	0.8690	$T_{CM}[{\theta }_1(1.5,1.63),{\theta }_2(2.5,1.88)]$	0.8215
$T_{CM}[{\theta }_1(0.1,1.13),{\theta }_2(1.00,1.38)]$	0.8690	$T_{CM}[{\theta }_1(1.5,1.75),{\theta }_2(3.0,2.00)]$	0.8208
$T_{CM}[{\theta }_1(0.1,1.13),{\theta }_2(1.25,1.50)]$	0.8880	$T_{CM}[{\theta }_1(2.0,1.63),{\theta }_2(2.5,1.88)]$	0.7580
$T_{CM}[{\theta }_1(0.1,1.25),{\theta }_2(1.00,1.50)]$	0.8628	$T_{CM}[{\theta }_1(2.0,1.63),{\theta }_2(3.0,2.00)]$	0.7586
$T_{CM}[{\theta }_1(0.1,1.25),{\theta }_2(1.25,1.38)]$	0.8646	$T_{CM}[{\theta }_1(2.0,1.75),{\theta }_2(2.5,2.00)]$	0.7579
$T_{CM}[{\theta }_1(0.1,1.25),{\theta }_2(1.25,1.50)]$	0.8695	$T_{CM}[{\theta }_1(1.5,1.75),{\theta }_2(3.0,1.88)]$	0.8211
$T_{CM}[{\theta }_1(0.5,1.13),{\theta }_2(1.25,1.38)]$	0.8587	$T_{CM}[{\theta }_1(1.5,1.63),{\theta }_2(3.0,2.00)]$	0.8215
$T_{CM}[{\theta }_1(0.5,1.13),{\theta }_2(1.25,1.50)]$	0.8540	$T_{CM}[{\theta }_1(2.0,1.75),{\theta }_2(3.0,1.88)]$	0.7586
$T_{CM}[{\theta }_1(0.5,1.13),{\theta }_2(1.00,1.50)]$	0.8481	$T_{CM}[{\theta }_1(2.0,1.63),{\theta }_2(3.0,2.00)]$	0.7586
$T_{CM}[{\theta }_1(0.5,1.25),{\theta }_2(1.25,1.38)]$	0.8847	$T_{CM}[{\theta }_1(1.5,1.63),{\theta }_2(2.5,2.00)]$	0.8217
$T_{CM}[{\theta }_1(0.5,1.25),{\theta }_2(1.00,1.50)]$	0.8786	$T_{CM}[{\theta }_1(1.5,1.75),{\theta }_2(2.5,1.88)]$	0.8203
$T_{CM}[{\theta }_1(0.5,1.25),{\theta }_2(1.00,1.38)]$	0.8705	$T_{CM}[{\theta }_1(2.0,1.63),{\theta }_2(3.0,1.88)]$	0.7589

4.2. Results for optimized CUSUM multi-chart

To construct the optimized CUSUM multi-chart $(T_{CM})$, we seek to find ${\theta }_1^{\ast }$ and ${\theta }_2^{\ast }$ that will optimize Equation (13), where ${\theta }_1^{\ast }=({\mu }_1^{\ast },{\sigma }_1^{2_{\ast }})$ and ${\theta }_2^{\ast }=({\mu }_2^{\ast },{\sigma }_2^{2_{\ast }})$. In other words, the value of mean (μ) and variance ( ${\sigma }^2$) for which Equation (13) has the greatest CPI.

Usually, the possible change region Θ forms a continuous region of two-dimensional plane, we consider the region $\mathrm{\Theta }=\{\theta (\mu ,{\sigma }^2):0<\mu \le 3;1<{\sigma }^2\le 2\}$. We hereby considered parameters of the mean $(\mu )$ from the range $(0,3]$ and parameters of the variance $({\sigma }^2)$ from the range $(1,2]$.

We choose values of μ as

$$\begin{array}{rl}& {\mu }_{11}=0.10,{\mu }_{12}=1.00,{\mu }_{13}=1.50,{\mu }_{14}=2.50,\\ & {\mu }_{21}=0.50,{\mu }_{22}=1.25,{\mu }_{23}=2.00,{\mu }_{24}=\mathrm{3.00.}\end{array}$$

We also choose values of ${\sigma }^2$ as

$$\begin{array}{rl}& {\sigma }_{11}^2=1.13,{\sigma }_{12}^2=1.38,{\sigma }_{13}^2=1.63,{\sigma }_{14}^2=1.88,\\ & {\sigma }_{21}^2=1.25,{\sigma }_{22}^2=1.50,{\sigma }_{23}^2=1.75,{\sigma }_{24}^2=\mathrm{2.00.}\end{array}$$

Instead of constructing schemes of all possible combinations of the mean (μ) and variance ( ${\sigma }^2$), we use pairs of the combinations of the mean (μ) and variance ( ${\sigma }^2$) in an efficient way utilizing orthogonal experimental design ( $[L_{12}(2{)}^8]$) (see Table A1, Appendix). The design reduces the computations from $8\times 8=64$ to 24 computations.

We selected values of the parameters; μ and ${\sigma }^2$ according to the orthogonal experimental design as follows:

$$\begin{array}{rl}& T_C({\theta }_1(0.10,1.13))\wedge T_C({\theta }_2(1.00,1.38)),T_C({\theta }_1(1.50,1.63))\wedge T_C({\theta }_2(2.50,1.88)),\\ & T_C({\theta }_1(0.10,1.13))\wedge T_C({\theta }_2(1.00,1.38)),T_C({\theta }_1(1.50,1.75))\wedge T_C({\theta }_2(3.00,2.00)),\\ & T_C({\theta }_1(0.10,1.13))\wedge T_C({\theta }_2(1.25,1.50)),T_C({\theta }_1(2.00,1.63))\wedge T_C({\theta }_2(2.50,1.88)),\\ & T_C({\theta }_1(0.10,1.25))\wedge T_C({\theta }_2(1.00,1.50)),T_C({\theta }_1(2.00,1.63))\wedge T_C({\theta }_2(3.00,2.00)),\\ & T_C({\theta }_1(0.10,1.25))\wedge T_C({\theta }_2(1.25,1.38)),T_C({\theta }_1(2.00,1.75))\wedge T_C({\theta }_2(2.50,2.00)),\\ & T_C({\theta }_1(0.10,1.25))\wedge T_C({\theta }_2(1.25,1.50)),T_C({\theta }_1(1.50,1.75))\wedge T_C({\theta }_2(3.00,1.88)),\\ & T_C({\theta }_1(0.50,1.13))\wedge T_C({\theta }_2(1.25,1.50)),T_C({\theta }_1(1.50,1.63))\wedge T_C({\theta }_2(3.00,2.00)),\\ & T_C({\theta }_1(0.50,1.13))\wedge T_C({\theta }_2(1.25,1.38)),T_C({\theta }_1(2.00,1.75))\wedge T_C({\theta }_2(3.00,1.88)),\\ & T_C({\theta }_1(0.50,1.13))\wedge T_C({\theta }_2(1.00,1.50)),T_C({\theta }_1(2.00,1.63))\wedge T_C({\theta }_2(3.00,2.00)),\\ & T_C({\theta }_1(0.50,1.25))\wedge T_C({\theta }_2(1.25,1.38)),T_C({\theta }_1(1.50,1.63))\wedge T_C({\theta }_2(2.50,2.00)),\\ & T_C({\theta }_1(0.50,1.25))\wedge T_C({\theta }_2(1.00,1.50)),T_C({\theta }_1(1.50,1.75))\wedge T_C({\theta }_2(2.50,1.88)),\end{array}$$

and

$$T_C({\theta }_1(0.50,1.25))\wedge T_C({\theta }_2(1.00,1.38)),T_C({\theta }_1(2.00,1.63))\wedge T_C({\theta }_2(3.00,1.88)),$$

where for example

$T_C({\theta }_1(\mu ,{\sigma }^2))\wedge T_C({\theta }_2({\mu }^{\mathrm{\prime }},{\sigma }^{2^{\prime }}))$ is $T_{CM}=min\{T_C({\theta }_1(\mu ,{\sigma }^2)),T_C({\theta }_2({\mu }^{\mathrm{\prime }},{\sigma }^{2^{\prime }}))\}.$

We take ${\theta }_0=(0,1)$, that is, ${\mu }_0=0$ and ${\sigma }_0^2$=1 when there is no change. We assume that after the change-point, the possible change region for constructing the CUSUM multi-chart is $\mathrm{\Theta }=\{\theta (\mu ,{\sigma }^2):0<\mu \le 3;1<{\sigma }^2\le 2\}$. If we consider two parameters; ${\theta }_1=({\mu }_1,{\sigma }_1^2)$ and ${\theta }_2=({\mu }_2,{\sigma }_2^2)$, we can get the curve from Equation (12) which is the red line as shown in Figure 1. The curve divides the region Θ into two parts, ${\mathrm{\Theta }}_1$ and ${\mathrm{\Theta }}_2$ such that ${\theta }_1=({\mu }_1,{\sigma }_1^2)$ ∈ ${\mathrm{\Theta }}_1$ and ${\theta }_2=({\mu }_2,{\sigma }_2^2)$ ∈ ${\mathrm{\Theta }}_2$. For each of the 24 pairs of parameters, we computed the boundary (12) between ${\theta }_1$ and ${\theta }_2$ and found the position and region where the real shifts belong. We computed the ARL's of the charts and used (13) to compute the CPIs of each of the 24 pairs. The chart with the highest CPI is the optimized CUSUM multi-chart.

Figure 1.

Parameter boundary of CUSUM multi-chart.

Table 1 shows the CPIs for different combinations of the mean and variance. The pairs $T_C({\theta }_1(0.1,1.13))$ and $T_C({\theta }_2(1.25,1.50))$ produced the greatest CPI of 0.8880. We therefore conclude that the optimized CUSUM multi-chart can be constructed with the parameters; ${\theta }_1^{\ast }(0.1,1.13)$ and ${\theta }_2^{\ast }(1.25,1.50)$ according to our design.

4.3. Results for optimized EWMA multi-chart

We presented the results for the EWMA multi-chart $(T_{EM})$ for joint monitoring of the mean and variance in Table 2. The smoothing parameters α and β were selected from the range $(0,1)$.

Table 2.

CPI of EWMA multi-charts with $AR{L}_0$ $\approx 500$.

Chart	CPI	Chart	CPI
$T_{EM}[(0.1,0.1),(0.3,0.3)]$	0.2895	$T_{EM}[(0.5,0.5),(0.7,0.7)]$	0.3579
$T_{EM}[(0.1,0.1),(0.3,0.3)]$	0.2895	$T_{EM}[(0.5,0.6),(0.8,0.8)]$	0.3873
$T_{EM}[(0.1,0.1),(0.4,0.4)]$	0.3027	$T_{EM}[(0.6,0.5),(0.7,0.7)]$	0.3882
$T_{EM}[(0.1,0.2),(0.3,0.4)]$	0.1157	$T_{EM}[(0.6,0.5),(0.8,0.8)]$	0.4330
$T_{EM}[(0.1,0.2),(0.4,0.3)]$	0.3334	$T_{EM}[(0.6,0.6),(0.7,0.8)]$	0.3268
$T_{EM}[(0.1,0.2),(0.4,0.4)]$	0.2030	$T_{EM}[(0.5,0.6),(0.8,0.7)]$	0.4162
$T_{EM}[(0.2,0.1),(0.4,0.4)]$	0.4895	$T_{EM}[(0.5,0.5),(0.8,0.8)]$	0.4115
$T_{EM}[(0.2,0.1),(0.4,0.3)]$	0.5234	$T_{EM}[(0.6,0.6),(0.8,0.7)]$	0.4391
$T_{EM}[(0.2,0.1),(0.3,0.4)]$	0.4856	$T_{EM}[(0.6,0.6),(0.7,0.8)]$	0.3268
$T_{EM}[(0.2,0.2),(0.4,0.3)]$	0.3545	$T_{EM}[(0.5,0.5),(0.7,0.8)]$	0.3179
$T_{EM}[(0.2,0.2),(0.3,0.4)]$	0.2060	$T_{EM}[(0.5,0.6),(0.7,0.7)]$	0.3415
$T_{EM}[(0.2,0.2),(0.3,0.3)]$	0.2431	$T_{EM}[(0.6,0.5),(0.8,0.7)]$	0.4629

We choose values of the smoothing parameter α as

$$\begin{array}{r}{\alpha }_{11}=0.1,{\alpha }_{12}=0.3,{\alpha }_{13}=0.5,{\alpha }_{14}=0.7,\\ {\alpha }_{21}=0.2,{\alpha }_{22}=0.4,{\alpha }_{23}=0.6,{\alpha }_{24}=\mathrm{0.8.}\end{array}$$

We also choose values of the smoothing parameter β as

$$\begin{array}{r}{\beta }_{11}=0.1,{\beta }_{12}=0.3,{\beta }_{13}=0.5,{\beta }_{14}=0.7,\\ {\beta }_{21}=0.2,{\beta }_{22}=0.4,{\beta }_{23}=0.6,{\beta }_{24}=\mathrm{0.8.}\end{array}$$

Instead of carrying out $64\times 64-64=4032$ computations for $T_{EM}$, we use orthogonal experimental design $[L_{12}(2{)}^8]$ (see Table A1, Appendix) to choose possible combinations of α and β which reduces the computations to 24. We select values of the smoothing parameter α and β according to the orthogonal experimental design ( $[L_{12}(2{)}^8]$) to construct EWMA multi-chart as follows:

$$\begin{array}{rl}& T_E(0.1,0.1)\wedge T_E(0.3,0.3),T_E(0.5,0.5)\wedge T_E(0.7,0.7),\\ & T_E(0.1,0.1)\wedge T_E(0.3,0.3),T_E(0.5,0.6)\wedge T_E(0.8,0.8),\\ & T_E(0.1,0.1)\wedge T_E(0.4,0.4),T_E(0.6,0.5)\wedge T_E(0.7,0.7),\\ & T_E(0.1,0.2)\wedge T_E(0.3,0.4),T_E(0.6,0.5)\wedge T_E(0.8,0.8),\\ & T_E(0.1,0.2)\wedge T_E(0.4,0.3),T_E(0.6,0.6)\wedge T_E(0.7,0.8),\\ & T_E(0.1,0.2)\wedge T_E(0.4,0.4),T_E(0.5,0.6)\wedge T_E(0.8,0.7),\\ & T_E(0.2,0.1)\wedge T_E(0.4,0.4),T_E(0.5,0.5)\wedge T_E(0.8,0.8),\\ & T_E(0.2,0.1)\wedge T_E(0.4,0.3),T_E(0.6,0.6)\wedge T_E(0.8,0.7),\\ & T_E(0.2,0.1)\wedge T_E(0.3,0.4),T_E(0.6,0.6)\wedge T_E(0.7,0.8),\\ & T_E(0.2,0.2)\wedge T_E(0.4,0.3),T_E(0.5,0.5)\wedge T_E(0.7,0.8),\\ & T_E(0.2,0.2)\wedge T_E(0.3,0.4),T_E(0.5,0.6)\wedge T_E(0.7,0.7),\end{array}$$

and

$$T_E(0.2,0.2)\wedge T_E(0.3,0.3),T_E(0.6,0.5)\wedge T_E(0.8,0.7),$$

where

$T_E(\alpha ,\beta )\wedge T_E({\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }})$ is $T_{EM}=min\{T_E(\alpha ,\beta ),T_E({\alpha }^{\mathrm{\prime }},{\beta }^{\mathrm{\prime }})\}.$

The EWMA multi-chart parameter space spans between $0<\alpha \le 1$ and $0<\beta \le 1$ as shown in Figure 2.

Figure 2.

Parameter boundary of EWMA multi-chart.

Considering the real measure $\tilde{\theta }(i,j)=({\tilde{\mu }}_i,{\widetilde{{\sigma }^2}}_j)$ for $1\le i\le 5$ and $1\le j\le 3$, we compute the CPIs of EWMA multi-charts $(T_{EM})$ by

$$CPI(T_{EM})=\exp \{-\frac{1}{15}\sum _{i=1}^5\sum _{j=1}^3[\frac{{ARL}_{\tilde{\theta }(i,j)}(T_{EM})}{{\mathbf{A}\mathbf{R}\mathbf{L}}_{\tilde{\theta }(i,j)}^{\ast }}-1]\},$$ (15)

where

${ARL}_{\tilde{\theta }(i,j)}(T_{EM})$ is the ARL of EWMA multi-chart to be evaluated and

$${\mathbf{A}\mathbf{R}\mathbf{L}}_{\tilde{\theta }(i,j)}^{\ast }=\frac{1}{I(\tilde{\theta }(i,j),{\theta }_0)}=\frac{1}{{\widetilde{{\sigma }^2}}_j-1-\log {\widetilde{{\sigma }^2}}_j+{\widetilde{{\mu }^2}}_i}$$ (16)

The EWMA multi-chart that has the greatest CPI is the optimized EWMA multi-chart.

Table 2 presents the CPIs results of EWMA multi-charts ( $T_{EM}$) for jointly detecting mean–variance shifts. The EWMA multi-charts were constructed with ${10}^5$ repetition experiments and the $AR{L}_0$ of the single charts were made to be approximately equal using Monte Carlo simulations. We considered five real mean shifts ( ${\mu }_1=0.2$, ${\mu }_2=0.5$, ${\mu }_3=1.0$, ${\mu }_4=2.0$ and ${\mu }_5=3.0$) and three real variance changes ( ${\sigma }_1^2=1.05$, ${\sigma }_2^2=1.50$ and ${\sigma }_3^2=2.00$). The change point $\tau =1$, that is the first time there is change. The smoothing parameter α and β were chosen according to orthogonal experimental design $[L_{12}(2{)}^8]$. We aim to find the EWMA multi-chart that will optimize $T_{EM}$ in respect to the CPI. In other words, which $T_{EM}$ has the greatest CPI.

Apparently, the EWMA multi-chart of $T_E(\alpha =0.2,\beta =0.1)$ and $T_E({\alpha }^{\mathrm{\prime }}=0.4,{\beta }^{\mathrm{\prime }}=0.3)$ produced the greatest CPI of 0.5234, hence the optimized EWMA multi-chart is $T_E(\mathbf{0.2}\mathbf{,}\mathbf{0.1}\mathbf{)}$ and $T_E(\mathbf{0.4}\mathbf{,}\mathbf{0.3}\mathbf{)}$.

4.4. Comparison of results

Here in this section, we shall compare the CPI results of the optimized CUSUM multi-chart with optimized EWMA multi-chart. The optimized CUSUM multi-chart $(T_{CM}^{\ast })$ has a CPI of 0.8880 whilst the optimized EWMA multi-chart $(T_{EM}^{\ast })$ has CPI of 0.5234. The optimized CUSUM multi-chart $(T_{CM}^{\ast })$ has higher CPI value than the optimized EWMA multi-chart $(T_{EM}^{\ast })$ hence the optimized CUSUM multi-chart scheme has better detection performance than the optimized EWMA multi-chart scheme.

We also compared our results to that of Han et al. [21]. The CPI's of Table 3 in page 1149 of [21] are: CUSUM multi-chart $=0.865$, optimized CUSUM multi-chart $=0.8797$ and EWMA multi-chart $=0.808$ for m = 5, where m is the number of charts.

Table 3.

CPIs of CUSUM and EWMA multi-charts for monitoring IBM stock returns with $AR{L}_0\approx 500$.

Chart	CPI
$T_{CM}[{\theta }_{01}(-0.0660,1.7640),{\theta }_{02}(-0.0924,1.5618)]$	0.5510
$T_{CM}^{\ast }[{\theta }_{01}(-0.0660,1.7640),{\theta }_{02}(-0.0656,1.2486)]$	0.6341
$T_{CM}[{\theta }_{01}(-0.0924,1.5618),{\theta }_{02}(-0.0656,1.2486)]$	0.5615
$T_{EM}^{\ast }[(0.2,0.1),(0.4,0.3)]$	0.4755

The proposed method gave a CPI of optimized CUSUM multi-chart with m = 2 to be 0.8880 which is higher than the CPI of optimized CUSUM multi-chart in Han et al. [21] for m = 5.

By inequality (4.8) and Remark 2 in [21], the CPIs will increase if we add more reference values (more charts). This means the CPI will increase if we add more charts since our work considered only two charts $(m=2)$.

5. A real example

We use the International Business Machines (IBM) stock market returns from 2013 to 2018 to illustrate the implementation of the optimized CUSUM and EWMA multi-charts for financial surveillance. We monitored the changes in IBM's stock returns (mean) and risks (variance) using the two multi-charts. Let the daily closing prices be $P_1,\dots ,P_n$ where $P_k$ is the closing price on trading day k. We denote the stock returns for the IBM as

$$X_k=100\log \left(\frac{P_{k+1}}{P_k}\right)k\ge 1.$$ (17)

Figure 3 shows the IBM stock market returns from 2013 to 2018. The mean of the returns seems to be dynamic while the variance (risks) seems to be increasing at some time period and decreasing at some time period. Many researchers have developed methods for monitoring financial variables. However, most of the works monitored the mean or variance but rarely has work been done concerning the joint mean–variance monitoring of financial variables by multi-chart schemes. Financial surveillance is important as investors are advised with a range of investment options based on statistical/ econometric models. It is in this light that we monitored the mean of the IBM returns and the variance (risk) jointly using the optimized CUSUM and optimized EWMA multi-chart schemes. We tested whether the stock returns follow the normal distribution using Shapiro Wilk test with a 0.05 level of significance. The hypothesis of interest is $H_0:$ the stock returns are normally distributed versus $H_1:$ the stock returns are not normally distributed. The p-value is $2.2\times {10}^{16}$. Since the p-value is small, we reject $H_0$ and claim that the stock returns do not follow normal distribution.

Figure 3.

IBM stock returns from 2013 to 2018.

The histogram of IBM stock returns with the normal density having same mean and variance as returns superimposed on it shows some heaviness at the tails as shown in Figure 4. The t-distribution is known to fit data that is heavy tailed. We fitted several t-distributions with different degrees of freedom to the stock returns data as shown in Figure 5.

Figure 4.

Histogram of the IBM stock returns with the normal density having the same mean and variance superimposed on it.

Figure 5.

Plots of quantiles of IBM stock returns with quantiles of several t-distributions having different degrees of freedom.

The quantile of t-distribution with five degrees of freedom fitted well with the quantiles of IBM stock returns as shown in Figure 5. The quantiles are very close to the straight line (qqline). We consequently used t-distribution with five degrees freedom to generate parameters for monitoring.

We estimate the change point $(\stackrel{\wedge }{\tau })$ time by

$$\stackrel{\wedge }{\tau }=arg\underset{1\le k\le n-1}{max}|\frac{1}{k}\sum _{i=1}^k{X}_i-\frac{1}{n-k+1}\sum _{j=k+1}^n{X}_j|$$ (18)

Generally, the in-control values are unknown, hence, we use data before change point as phase 1 data to estimate the in-control parameter. We denote the in-control reference value as ${\theta }_{00}({\mu }_{00},{\sigma }_{00}^2)$, where ${\mu }_{00}$ is the mean of returns and ${\sigma }_{00}^2$ is the risk before change point. After change point we partitioned the data into three sections and found the mean of returns and risks for each section which represents the reference values. For example, the reference values are parameterized as ${\theta }_{01}({\mu }_{01},{\sigma }_{01}^2)$, ${\theta }_{02}({\mu }_{02},{\sigma }_{02}^2)$ and ${\theta }_{03}({\mu }_{03},{\sigma }_{03}^2).$

Table 3 presents the numerical results of the optimized CUSUM and EWMA multi-chart schemes for monitoring IBM stock returns. The change point was estimated at the 226th trading day (21st November 2013) probably because the festive season (Christmas) was approaching so there was surge in the purchase of stocks. We consequently estimated the in-control reference parameter as ${\theta }_{00}=(-0.0828,1.2065)$. The post-change parameters were also estimated as ${\theta }_{01}=(-0.0660,1.7640)$, ${\theta }_{02}=(-0.0924,1.5618)$ and ${\theta }_{03}=(-0.0656,1.2486)$.

We formulate the parameters of the CUSUM multi-charts $(T_{CM})$ for joint monitoring of the mean and risks of the IBM stock returns as

$$\begin{array}{rl}& T_C[{\theta }_{01}(-0.0660,1.7640)]\wedge T_C[{\theta }_{02}(-0.0924,1.5618)],\\ & T_C[{\theta }_{01}(-0.0660,1.7640)]\wedge T_C[{\theta }_{02}(-0.0656,1.2486)],\end{array}$$

and

$$T_C[{\theta }_{01}(-0.0924,1.5618)]\wedge T_C[{\theta }_{02}(-0.0656,1.2486)].$$

We then seek to determine the detection capability of the optimized CUSUM and EWMA multi-charts to detect changes in the mean and risk of returns starting from the 226th trading day (21st November 2013). We constructed the CUSUM charts $T_C({\theta }_{01})$, $T_C({\theta }_{02})$ and $T_C({\theta }_{03})$ with an in-control ARL of approximately 500 by choosing $c_{01}=2.592639$, $c_{02}=2.125684$ and $c_{03}=0.5356052$, where $c_{01},$ $c_{02}$ and $c_{03}$ are the control limits. We constructed the CUSUM multi-chart $(T_{CM})$ using Equation (13) and the results presented in Table 3.

We also used parameters of the optimized EWMA multi-chart $(T_{EM}^{\ast })$ that is $T_E(\mathbf{0.2}\mathbf{,}\mathbf{0.1}\mathbf{)}$ and $T_E(\mathbf{0.4}\mathbf{,}\mathbf{0.3}\mathbf{)}$ to jointly monitor the mean and risks of the IBM stock returns. We constructed the optimized EWMA multi-chart $(T_{EM}^{\ast })$ with an in-control ARL of approximately 500 by choosing $AR{L}_0[T_E(0.2,0.1)]=689.93$ with $c_{01}^{\mathrm{\prime }}$ = 1.26648 and $AR{L}_0[T_E(0.4,0.3)]=689.26$ with $c_{02}^{\mathrm{\prime }}$ = 1.882597, where $c_{01}^{\mathrm{\prime }}$, $c_{02}^{\mathrm{\prime }}$ and $c_{03}^{\mathrm{\prime }}$ are the control limits. That is, we chose the $AR{L}_0$ of the single charts to be approximately equal.

We considered five real mean shifts ( ${\tilde{\mu }}_{01}={\stackrel{\wedge }{\mu }}_{00}+0.2$, ${\tilde{\mu }}_{02}={\stackrel{\wedge }{\mu }}_{00}+0.5$, ${\tilde{\mu }}_{03}={\stackrel{\wedge }{\mu }}_{00}+1.0$, ${\tilde{\mu }}_{04}={\stackrel{\wedge }{\mu }}_{00}+2.0$ and ${\tilde{\mu }}_{05}={\stackrel{\wedge }{\mu }}_{00}+3.0$) and three real variance changes ( ${\widetilde{{\sigma }^2}}_{01}={\stackrel{\wedge }{{\sigma }^2}}_{00}(1.05)$, ${\widetilde{{\sigma }^2}}_{02}={\stackrel{\wedge }{{\sigma }^2}}_{00}(1.50)$ and ${\widetilde{{\sigma }^2}}_{03}={\stackrel{\wedge }{{\sigma }^2}}_{00}(2.00))$.

The optimized CUSUM multi-chart $(T_{CM}^{\ast })$ has CPI of 0.6341 whilst the optimized EWMA multi-chart $(T_{EM}^{\ast })$ has CPI of 0.4755 as shown in Table 3, hence the optimized CUSUM multi-chart $(T_{CM}^{\ast })$ has better detection performance than the optimized EWMA multi-chart $(T_{EM}^{\ast })$ for detecting mean and variance (risks) change in the IBM stock returns.

6. Conclusion and discussion

Basically, this study seeks to jointly monitor the mean and variance of an i.i.d process by using the optimized CUSUM and EWMA multi-charts such that they can detect changes in mean–variance much faster and also evaluates the efficiency of these schemes by the charting performance index (CPI). To achieve this purpose, we constructed the optimized CUSUM multi-charts as follows; we derived the theoretical relation between the CPI, the post-change reference value of mean and variance, and the position of the real mean–variance shifts given some two reference parameters ${\theta }_1=({\mu }_1,{\sigma }_1^2)$ and ${\theta }_2=({\mu }_2,{\sigma }_2^2)$.

Usually, the possible change region Θ forms a continuous region of two-dimensional plane, we hereby selected parameters of the mean $(\mu )$ from the range $(0,3]$ and variance $({\sigma }^2)$ from the range $(1,2]$. Instead of constructing schemes of all possible combinations of the mean (μ) and variance ( ${\sigma }^2$), we use pairs of the combinations of the mean (μ) and variance ( ${\sigma }^2$) in an efficient way utilizing orthogonal experimental design $[L_{12}(2{)}^8]$. We computed the CPIs for each pair of parameters using Equation (13). The pair ${\theta }^{\ast }(\mathbf{0.10}\mathbf{,}\mathbf{1.13}\mathbf{)}$ and ${\theta }^{\ast }(\mathbf{1.25}\mathbf{,}\mathbf{1.50}\mathbf{)}$ produced the greatest CPI of 0.8880. We therefore conclude that the optimized CUSUM multi-charts can be constructed with the parameters ${\theta }^{\ast }(\mathbf{0.10}\mathbf{,}\mathbf{1.13}\mathbf{)}$ and ${\theta }^{\ast }(\mathbf{1.25}\mathbf{,}\mathbf{1.50}\mathbf{)}$ according to our design.

We also constructed the optimized EWMA multi-charts for joint monitoring of the mean and variance as follows: we selected the smoothing parameters; α and β such that α, β ∈ $(0,1)$. Again we use $[L_{12}(2{)}^8]$ design to choose possible combinations of α and β to construct EWMA multi-charts. The EWMA multi-chart scheme that has the greatest CPI is the optimized EWMA multi-chart schemes. The EWMA multi-chart [ $T_E(\mathbf{0.2}\mathbf{,}\mathbf{0.1}\mathbf{)}$ and $T_E(\mathbf{0.4}\mathbf{,}\mathbf{0.3}\mathbf{)}$] produced the greatest CPI of 0.5234, hence the optimized EWMA multi-chart scheme is [ $T_E(\mathbf{0.2}\mathbf{,}\mathbf{0.1}\mathbf{)}$ and $T_E(\mathbf{0.4}\mathbf{,}\mathbf{0.3}\mathbf{)}$].

The simulation results show that the optimized CUSUM multi-chart has a higher charting performance index (CPI) than the optimized EWMA multi-chart hence the optimized CUSUM multi-chart scheme has better detection performance. We also compared our results to that of Han et al. [21], where they considered five charts and our results performed better with just two charts. Also, the asymptotic analyses were consistent with the simulation results.

It is known that the stock returns have elegant statistical properties hence we used the proposed methods to jointly monitor the mean and risks of IBM stock returns from the period of 2013 to 2018. The stock returns seems fairly not normal as there is evidence of heaviness at the tails of the returns distribution hence we fitted several t-distributions with different degrees of freedom to the stock returns data as the t-distribution is known to fit approximately data that is heavy tailed. The quantile of t-distribution with five degrees of freedom fitted the data well. We generated the parameters for monitoring IBM stock returns from the t-distribution with five degrees of freedom. We estimated the change point at the 226th trading day (21st November 2013) and consequently used the data before 21st November 2013 as phase 1 historical data to find the in-control reference values which was estimated as ${\theta }_{00}=(-0.0828,1.2065)$. After change point we partitioned the data into three sections and estimated post-change parameters as ${\theta }_{01}=(-0.0660,1.7640)$, ${\theta }_{02}=(-0.0924,1.5618)$ and ${\theta }_{03}=(-0.0656,1.2486)$. We used these parameters to construct the CUSUM multi-charts for jointly monitoring changes in the mean and risks of IBM stock returns. We also used parameters of the optimized EWMA multi-chart $(T_{EM}^{\ast })$ that is $T_E(\mathbf{0.2}\mathbf{,}\mathbf{0.1}\mathbf{)}$ and $T_E(\mathbf{0.4}\mathbf{,}\mathbf{0.3}\mathbf{)}$ to jointly monitor changes in the mean and risks of the IBM stock returns. Monitoring results have it that the optimized CUSUM multi-chart $(T_{CM}^{\ast })$ has higher CPI, hence better detection performance in the changes of IBM stock returns than the optimized EWMA multi-chart $(T_{EM}^{\ast })$.

Early detection of abrupt changes in the mean and risks of IBM returns could be an important information for financial and economic analysts to advise clients and investors appropriately. Further research can focus on how the procedures considered in the article may be adapted using nonparametric monitoring methods. Also, some volatility models could be fitted and the errors monitored for abrupt changes using the method discussed in this paper.

Appendix 1. Design of Experiment.

Table A1.

$L_{12}(2{)}^8$ design.

1	2	3	4	5	6	7	8
1	1	1	1	1	1	1	1
1	1	1	1	1	2	2	2
1	1	2	2	2	1	1	1
1	2	1	2	2	1	2	2
1	2	2	1	2	2	1	2
1	2	2	2	1	2	2	1
2	1	2	2	1	1	2	2
2	1	2	1	2	2	2	1
2	1	1	2	2	2	1	2
2	2	2	1	1	1	1	2
2	2	1	2	1	2	1	1
2	2	1	1	2	1	2	1

Appendix 2. Proof of Preposition 3.1.

Assuming that $I(\tilde{\theta }(i,j),{\theta }_0)$ is the Kullback–Leibler information (number) defined as

$$I(\tilde{\theta }(i,j),{\theta }_0)={\mathbf{E}}_{\tilde{\theta }(i,j)}\{\log [\frac{f_{\tilde{\theta }(i,j)}(X_1)}{f_{{\theta }_0}(X_1)}]\}$$

For large $AR{L}_0(T_{CM})$, we have

$$AR{L}_{\tilde{\theta }(i,j)}(T_{CM})\to \frac{1}{I(\tilde{\theta }(i,j),{\theta }_0)-I(\tilde{\theta }(i,j),{\theta }_k)}$$

for $\tilde{\theta }(i,j)\in {\mathrm{\Theta }}_k$

where

$$I(\tilde{\theta }(i,j),{\theta }_0)-I(\tilde{\theta }(i,j),{\theta }_k)=I(\tilde{\theta }(i,j),\tilde{\theta }(i,j),{\theta }_0)={\mathbf{E}}_{\tilde{\theta }(i,j)}\{\log [\frac{f_{\tilde{\theta }(i,j)}(X_1)}{f_{{\theta }_0}(X_1)}]\}.$$

We also have

$$AR{L}_{\tilde{\theta }(i,j)}^{\ast }=\frac{1}{I(\tilde{\theta }(i,j),{\theta }_0)}$$

It follows from Theorem 3.3 in Han and Tsung [20] that for large γ and $1\le k\le m$, the charting performance index (CPI) in (1) can be written as

$$CPI(T_{CM})=(1+o(1))\exp \{1-\frac{1}{l\times q}\sum _{k=1}^m\sum _{\tilde{\theta }(i,j)\in {\mathrm{\Theta }}_k}^{}\frac{I(\tilde{\theta }(i,j),{\theta }_0)}{I(\tilde{\theta }(i,j),{\theta }_0)-I(\tilde{\theta }(i,j),{\theta }_k)}\}$$ (A1)

Specifically, under the normal distribution we can derive that

$$I(\tilde{\theta }(i,j),{\theta }_0)=\frac{1}{2}[{\widetilde{{\sigma }^2}}_j-1-\log {\widetilde{{\sigma }^2}}_j+{\widetilde{{\mu }^2}}_i]$$ (A2)

and

$$I(\tilde{\theta }(i,j),{\theta }_0)-I(\tilde{\theta }(i,j),{\theta }_k)=\frac{1}{2}[({\widetilde{{\sigma }^2}}_j+{\widetilde{{\mu }^2}}_i)(1-1/{\sigma }_k^2)-\log {\sigma }_k^2+\frac{{\mu }_k(2{\tilde{\mu }}_i-{\mu }_k)}{{\sigma }_k^2}]$$ (A3)

We can substitute (A2) and (A3) into (A1) to get (7).

Funding Statement

This research was supported by National Basic Research Program of China (973 Program, 2015CB856004) and the National Natural Science Foundation of China (11531001).

Disclosure statement

No potential conflict of interest was reported by the author(s).