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Open AccessArticle

Cumulative Sum Chart Modeled under the Presence of Outliers

1
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
2
Preparatory Year Mathematics Program, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3
Department of Systems Engineering and Engineering Management, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(2), 269; https://doi.org/10.3390/math8020269
Received: 26 December 2019 / Revised: 9 February 2020 / Accepted: 11 February 2020 / Published: 18 February 2020
(This article belongs to the Special Issue Statistics and Modeling in Reliability Engineering)

Abstract

Cumulative sum control charts that are based on the estimated control limits are extensively used in practice. Such control limits are often characterized by a Phase I estimation error. The presence of these errors can cause a change in the location and/or width of control limits resulting in a deprived performance of the control chart. In this study, we introduce a non-parametric Tukey’s outlier detection model in the design structure of a two-sided cumulative sum (CUSUM) chart with estimated parameters for process monitoring. Using Monte Carlo simulations, we studied the estimation effect on the performance of the CUSUM chart in terms of the average run length and the standard deviation of the run length. We found the new design structure is more stable in the presence of outliers and requires fewer amounts of Phase I observations to stabilize the run-length performance. Finally, a numerical example and practical application of the proposed scheme are demonstrated using a dataset from healthcare surveillance where received signal strength of individuals’ movement is the variable of interest. The implementation of classical CUSUM shows that a shift detection in Phase II that received signal strength data is indeed masked/delayed if there are outliers in Phase I data. On the contrary, the proposed chart omits the Phase I outliers and gives a timely signal in Phase II.
Keywords: average run length; control chart; cumulative sum; outlier; health care; statistical process control average run length; control chart; cumulative sum; outlier; health care; statistical process control

1. Introduction

The cumulative sum (CUSUM) control chart is an effective monitoring tool widely used in industries and medical processes for quality improvement [1]. The scheme was introduced by [2] as the substitution of the traditional Shewhart control chart. The CUSUM chart statistic accumulates the past and current information of the process, which provides more sensitivity to detect small and moderate shifts as compared to the traditional Shewhart control chart. Designing a CUSUM control chart requires setting up of the control limit, where the known in-control parameters are often assumed. However, this assumption is not realistic, and hence the CUSUM chart is implemented in a two-phase method. In Phase I, random observations are collected from a stable process and used to estimate the unknown parameters. In Phase II, the estimates from the earlier observations are used for the construction of the CUSUM chart to monitor and detect changes in a process [3].
The performance of a CUSUM chart to effectively handle changes in the process in Phase II largely depends on the accuracy of the estimated parameters in Phase I. Furthermore, higher chances of estimation error may occur when there exist some extreme values or outliers in the Phase I observations [4]. Outliers may occur by chance in the process data or could be due to some incorrect specifications of instruments or as a result of human reporting error. The presence of outliers in a process data can adversely affect parametric computations. Of course, dropping the outliers from the sampled observations is the simplest remedy often used to avoid such a problem. However, this may not be appropriate for small sample data. Thus, outlier detection is key to adequate monitoring of process parameters. Recently, some non-parametric and robust outlier detection procedures have been suggested to enhance the performance of control charts in the presence of outliers. For example, see Schoonhoven, Nazir [5], Nazir, Riaz [6], Amdouni, Castagliola [7], Abid, Nazir [8], Zhang, Li [9] and Mahmood, Nazir [10], and the references therein.
Hawkins [11], Beckman and Cook [12] and Barnett and Lewis [13] have studied several outlier detectors. The common parametric outlier detectors are the Student-type and Grubbs-type detectors mostly used in the regression residuals and when the data is normally distributed (cf. Grubbs [14] and Tietjen and Moore [15]). For non-normal data, the Tukey’s outlier detection model is more robust since its independence of the sample mean and standard deviation [16]. Teoh, Khoo [17] suggested the local outlier factor, a non-parametric outlier detector for detecting the outliers in the multivariate setup. Knorr, Ng [18] designed a detector based on classification methodology while a detector based on order statistics was studied by Tse and Balasooriya [19]. Hubert, Dierckx [20] proposed a procedure based on Hill’s estimator for detecting the influential point in Pareto-type distributions. Recently, Castagliola, Amdouni [21] introduced a new non-parametric outlier detector for all types of univariate distributions.
In this article, we study the effect of outliers on the performance of a two-sided CUSUM control chart for monitoring process location with the estimated parameters using the run length (RL) properties. Furthermore, the study proposed a non-parametric outlier detector, the robust Tukey outlier detection model in the design structure of a CUSUM control chart for efficient monitoring of the process location parameters in the presence of the extremes. These measures are evaluated in three cases. The first case is when the in-control mean is known, and the standard deviation is estimated. Second is when the in-control standard deviation is known, and the mean is estimated, and the third case is when both the mean and the standard deviation are unknown. A synthesis table about the research on a two-sided CUSUM chart is given in Table 1.
The rest of the article is organized as follows. In the next section, we gave overview information on the two-sided CUSUM chart with estimated parameters followed by the performance measure metrics in terms of the RL properties. Section 3 presents the practitioner-to-practitioner variation on the performance of the CUSUM chart. The section also discusses the effect of error estimation on CUSUM control limits. In Section 4, we gave the design structure of the CUSUM chart in the presence of outliers and analyzed the effect of extremes on its in-control performance. The introduction of the Tukey outlier detection model in the CUSUM chart is presented in Section 5. An application example to illustrate the practical use of the scheme is given in Section 6. Finally, we provide some concluding remark in Section 7.

2. Overview of CUSUM Charts with Estimated Parameters

Let X i 1 , X i 2 , X i 3 , , X i n . for i = 1 , 2 , 3 , be independent random observations of size n from a normal process, with a known in-control mean μ 0 and standard deviation σ 0 . The upper and lower sided CUSUM chart statistics for monitoring the upward and downward changes in the process location parameters are respectively, given by
C U S U M i + = max [ 0 , C U S U M i 1 + + n ( X ¯ i μ 0 ) / σ 0 k ] , C U S U M i = min [ 0 , C U S U M i 1 + n ( X ¯ i μ 0 ) / σ 0 + k ]
where max   [ a , b ] and min   [ a , b ] are the maximum and minimum of a and b , respectively. The statistic, X ¯ i = ( 1 / n ) j = 1 n X i j is the mean of i t h sample, and k is the reference value. The initial values, C U S U M 0 + and C U S U M 0 , are usually set equal to zero. The chart gives an out-of-control signal when either C U S U M i + or C U S U M i exceeds the predetermined control limit, h . The h is usually chosen to satisfy the desired in-control RL property.
However, if the process parameters are unknown, then μ 0 and σ 0 are replaced by their corresponding Phase I estimates. Let X i j , i = 1 ,   2 ,   3 , , m and j = 1 ,   2 ,   3 , , n denote m random samples each of size n of Phase I observations from a stable process. Then the unbiased estimator for μ 0 , is the overall sample mean given by
μ ^ 0 = 1 m i = 1 m ( j = 1 n X i j n ) = 1 m i = 1 m X ¯ i
and for the unbiased estimator of σ 0 when subgroup size n > 1 , we used the pooled standard deviation,
S p = 1 m i = 1 m S i 2
recommended by some researchers like Chen [22], Mahmoud, Henderson [23] and Nazir, Abbas [24]. Here, S i 2 = 1 / ( n 1 ) j = 1 n ( X i j X ¯ i ) is the variance the of i t h Phase I sample. The unbiased estimator is defined by
σ ^ 0 =   S p c 4 [ m ( n 1 ) + 1 ]
where the constant, c 4 ( w ) = 2 / ( w 1 )   Γ ( w / 2 ) / Γ [ ( w 1 ) / 2 ] is the bias correction constant that depends on the m and n . Thus, the corresponding two-sided CUSUM chart statistics based on the estimated parameters are defined as
C U S U M i + = max [ 0 , C U S U M i 1 + + n ( X ¯ i μ ^ 0 ) / σ ^ 0 k ] , C U S U M i = min [ 0 , C U S U M i 1 + n ( X ¯ i μ ^ 0 ) / σ ^ 0 + k ] .
The statistical performance of a CUSUM chart is often evaluated in terms of its RL distribution [25]. For a two-sided CUSUM chart with initial value of C U S U M 0 = z , where z ( h , h ) , the probability mass function [26] is given by
p r ( r l | z ) = P ( R L = r l   |   C U S U M 0 = z )
For a single case, r l = 1 , we have
p r ( 1 | z ) = P ( R L = 1   |   C U S U M 0 = z ) = P ( C U S U M 1 < h   |   C U S U M 0 = z ) + P ( C U S U M 1 > h   |   C U S U M 0 = z ) = P ( z + n ( X ¯ 1 μ ^ 0 ) / σ ^ 0 + k < h ) + P ( z + n ( X ¯ 1 μ ^ 0 ) / σ ^ 0 k > h ) = P ( z + Y 1 + k < h ) + P ( z + Y 1 k > h ) = P ( Y 1 < h z k ) + P ( Y 1 > h z + k ) = 1 Φ ( v λ ( h z k ) + δ λ u λ m ) + Φ ( v λ ( h z + k ) + δ λ u λ m )
where v = σ ^ 0 / σ 0 , λ = σ / σ 0 , δ = n ( μ μ 0 ) / σ 0 , u = m n ( μ ^ 0 μ 0 ) / σ 0 and Φ ( . ) denotes the standard normal distribution function. For the case when r l > 1 , we have
p r ( r l | z ) = P ( R L = r l   |   C U S U M 0 = z ) = P ( R L 1 = r l 1 ,   C U S U M 1 = 0   |   C U S U M 0 = z ) +   P ( R L 1 = r l 1 ,   h < C U S U M 1 < 0   |   C U S U M 0 = z ) +   P ( R L 1 = r l 1 ,   0 < C U S U M 1 < h   |   C U S U M 0 = z ) = P ( R L 1 = r l 1   |   C U S U M 0 = z ,   C U S U M 1 = 0 )   P ( C U S U M 1 = 0   |   C U S U M 0 = z   ) +   P ( R L 1 = r l 1   |   C U S U M 0 = z , h < C U S U M 1 < 0 )   P ( h < C U S U M 1 0 |   C U S U M 0 = z ) +   P ( R L 1 = r l 1   |   C U S U M 0 = z , 0 < C U S U M 1 < h )   P ( 0 < C U S U M 1 < h   |   C U S U M 0 = z ) = p r ( r l 1   |   0 )   [ Φ ( v λ ( 0 z k ) + δ λ u λ m ) Φ ( v λ ( h z + k ) + δ λ u λ m ) ] +   h 0 p r ( r l 1   |   C U S U M 1 = y )   v λ   ϕ ( v λ ( y z k ) + δ λ u λ m )   d y +   0 h   p r ( r l 1   |   C U S U M 1 = y )   v λ   ϕ ( v λ ( y z + k ) + δ λ u λ m )   d y
where ϕ ( . ) is the standard normal density function. The most common used RL property to evaluate the performance of a control chart is the average run length (ARL), which represents the average number of samples plotted on a control chart before a process issues a signal. The ARL measures how quickly a control chart responds to changes in a process. If Equation (7) is denoted by g ( u , v ) , for simplicity, then the ARL can be defined by the integral equation [26,27].
A R L = E ( RL ) = 0 1 g ( u , v )   1 2 π exp ( u 2 2 ) f ( v ) d u d v
where f ( v ) is the scaled chi ( χ ) distribution with m   ( n 1 ) degrees of freedom from c χ / m   ( n 1 ) , and c is a scaled factor. There is also the standard deviation of run length (SDRL) that sometimes is used as a supplementary measure. The SDRL is the standard deviation of samples until the chart gives an out-of-control signal, that is,
S D R L = E ( R L 2 ) [ E ( R L ) ] 2
where E ( R L 2 ) = 0 2 g ( u , v ) g 2 ( u , v )   1 2 π exp ( u 2 2 ) f ( v )   d u d v . For an in-control process, denote the ARL by ARL 0 , which in practice, should be sufficiently large to avoid unnecessary false signals. Furthermore, denote the out-of-control ARL by ARL 1 , which should be small enough to enable early detection of changes in a process. The above RL properties of a two-sided CUSUM chart may be obtained by evaluating f ( v ) , but unfortunately, it cannot be computed exactly. Hence, the need for approximation using either Gaussian quadrature, Markov chain approximation or Monte Carlo simulation. With the technological advancements in computing software, we followed the simulation approach as recommended by several authors of the quality control chart.

3. Variability in the CUSUM Chart Performance

For the location control chart, the process is assumed to be initially stable with an in-control mean μ 0 and standard deviation σ 0 . After a certain point in time, it changes from the target value μ 0 to an out-of-control value μ 1 = μ 0 + δ σ 0 thus, requiring immediate and quick detection of such changes. Without loss of generality, we assumed that the in-control process is normally distributed. To study the so-called practitioner-to-practitioner variation on the performance of the CUSUM chart, 100,000 seeded iterations, each sample size n = 5 , were generated from the standard normal distribution N ( 0 , 1 ) . We then set up the charts with k = 0.25 ,   0.50 ,   0.75 and 1.00 , using the combinations of the control limit h = 6.8516 ,   4.1713 ,   2.9332 and 1.0894 that corresponds to the in-control ARL 0 of 200 . We used the simulation approach based on an algorithm developed in R, to compute the distributional properties of the CUSUM chart in terms of the ARL and SDRL for different shift values δ when the control chart parameters μ 0 and σ 0 are known and the results obtained are presented in Table 2. These results are in agreement with the theoretical values of a classical two-sided CUSUM chart [2].
The unknown in-control process parameters, on the other hand, are estimated from m = 10 ,   50 ,   100 ,   500 and 1000 in-control Phase I samples each of subgroup size n = 5 . Substituting the unknown parameters with their corresponding estimates, the Phase II two-sided CUSUM control charts were developed. For each fixed value of k = 0.25 ,   0.50 ,   0.75 and 1.00 , the control limit h was determined through simulations to obtain the desired in-control ARL 0 of 200 . Here, all the observations are from N ( μ ^ 0 , σ ^ 0 2 ) . The ARL and SDRL values are computed using 100,000 simulation iterations. For a clear consequence on the effect of each estimated process parameter on the performance of a CUSUM chart, we considered the cases when either the sample mean or sample standard deviation or both were estimated. Results obtained are given in Table 3, Table 4 and Table 5.

3.1. Effect of Estimation on the Two-Sided CUSUM Chart Performance

Results in Table 2 and Table 3 shows that a small number of Phase I samples, produced out-of-control ARL and SDRL values (cf. Table 3) that were higher than the known standard values in Table 2, for a fixed ARL 0 = 200 . This is an indication that the use of small Phase I samples to estimate the process mean had direct consequences on the performance of a two-sided CUSUM chart. It follows from Table 4 that the out-of-control ARL was relatively smaller than the desired. Hence, the effect of estimating the standard deviation from Phase I samples had less impact on the ARL performance of the CUSUM chart. However, the very large values of the accompanying SDRLs when was small required the availability of a large amount of Phase I samples. This was also the case when both the parameters were estimated (cf. Table 5). In all the three cases, Table 3, Table 4 and Table 5, the ARL and SDRL values were closer to the desired values in Table 2 as the number of Phase I observations, m increased. Furthermore, parameter estimation had a more adverse impact on the performance of a two-sided CUSUM chart based on smaller reference value k and designed for quick detection of very small changes in the process mean.

3.2. Effect of Estimation on Two-Sided CUSUM Control Limits

To study the effect of estimation error on the two-sided CUSUM control limits, we used a sample size of n = 5 and set the in-control ARL 0 to 200 . For each value of k = 0.25 ,   0.50 ,   0.75 ,   1.00 ,   1.25 ,   1.50 ,   1.75 and 2.00 , the corresponding value of the control limits h were computed based on 100,000 iterations. Table 6 presents the two-sided CUSUM control limits using values of m ranging from 10 to 1000. Once again, the use of a small number of Phase I observations m to estimate the unknown in-control chart parameters give the control limit that is higher or lower than the desired value when the mean or the standard deviation is estimated, respectively. Similar to the ARL performance and the displayed percentage error curves in Figure 1, quite a larger number of Phase I samples was required to achieve the desired control limit. The problem, however, is the availability of such an amount of Phase I data in practical applications. Hence, the need to design a more robust scheme that can minimize the practitioner-to-practitioner variation, particularly when extreme values or outliers was involved.

4. The Outliers and CUSUM Chart with Estimated Parameters

The effect of estimation errors on the performance of a CUSUM chart may further be strained if there exist some extreme values in the Phase I samples. Both the in-control and the out-of-control ARL and SDRL values will be different from those of the theoretical CUSUM charts. In this section, we evaluated the effects of the outliers on the performance of a two-sided CUSUM control chart with estimated parameters. Using a simulation approach, outliers were generated from the mixture distribution, where ( 1 α )   100 % regular observations were from N ( μ ^ , σ ^ 2 ) and the remaining α 100 % observations came from a multiple of χ ( n ) 2 with n degrees of freedom, [28]. That is, each observation was generated from a mixture distribution
f ( x ) = ( 1 α )   N ( μ ,   σ 2 ) + α [ N ( μ ,   σ 2 ) + w χ ( 1 ) 2 ]
where α is the probability of having a multiple of χ ( 1 ) 2 added and w 1 is the outlier model multiplier. A value of α = 0 indicates no presence of an outlier in the sampled data. Without loss of generality, we set μ = 0 and σ 2 = 1 . The values of w is set equal to 1, 2 or 3 corresponding to the small, medium and large outlier, respectively.
The mean and the variance of mixture distribution in Equation (10) are derived in Equations (11) and (12) respectively.
E ( X ) = p = 1 2 ω p μ p = ( 1 α ) E [ N ( μ ,   σ 2 ) ] + α E [ N ( μ ,   σ 2 ) + w χ ( 1 ) 2 ] = ( 1 α ) μ + α ( μ + w ) = μ + α w
V a r ( X ) = p = 1 2 ω p ( μ p 2 + σ p 2 { E ( X ) } 2 ) = ( 1 α ) [ ( E ( N ( μ ,   σ 2 ) ) ) 2 + V ( N ( μ ,   σ 2 ) ) { E ( X ) } 2 ] +   α [ ( E ( N ( μ ,   σ 2 ) + w χ ( 1 ) 2 ) ) 2 + V ( N ( μ ,   σ 2 ) + w χ ( 1 ) 2 ) { E ( X ) } 2 ] = ( 1 α ) [ μ 2 + σ 2 { μ + α w } 2 ] + α [ { μ + w } 2 + σ 2 + 2 w 2 { μ + α w } 2 ] = ( 1 α ) [ μ 2 + σ 2 μ 2 α 2 w 2 2 α w μ ] + α [ μ 2 + w 2 + 2 w μ + σ 2 + 2 w 2 μ 2 α 2 w 2 2 α w μ ] = ( 1 α ) [ σ 2 α 2 w 2 2 α w μ ] + α [ w 2 + 2 w μ + σ 2 + 2 w 2 α 2 w 2 2 α w μ ] = σ 2 ( 1 α + α ) 2 α w μ ( 1 α + α ) α 2 w 2 ( 1 α + α ) + α w 2 + 2 α w μ + 2 α w 2 = σ 2 2 α w μ α 2 w 2 + α w 2 + 2 α w μ + 2 α w 2 = σ 2 + 3 α w 2 α 2 w 2 = σ 2 + α ( 3 α ) w 2
We set up a CUSUM chart using the same design parameters, n ,   k ,   h and m as in Section 3. he in-control ARL and SDRL values for the two-sided CUSUM chart based on this model with α = 0.00 ,   0.01 ,   0.02 ,   0.03 and 0.04 are presented in Table 7, Table 8 and Table 9. To save space, we restricted the study to in-control cases having seen the behavioral pattern for the out-of-control cases in Table 3, Table 4 and Table 5.
From Table 7, Table 8 and Table 9, it was observed that estimating μ , σ or both in the presence of outliers, α > 0 to set up a CUSUM chart had a significant effect on the ARL and SDRL performance of the chart. Particularly, when the number of Phase I samples, m was small. The in-control ARLs were approximately equal to the limiting value of 200 when α = 0 . As expected, the RL values were directly proportional to k and α . That is, the in-control ARL and SDRL deteriorated with the increasing number of the false alarm rate as k , α or both the design parameters increased. In fact, the deterioration level became more alarming with the increase in an outlier metric multiplier, w > 1 . Furthermore, as the number of Phase I samples increased, the ARL approached its theoretical value and much faster than its corresponding SDRL (cf. Table 7). However, this was not the case for Table 8 and Table 9, when m 500 . In general, increasing the number of Phase I data will reduce the occurrence of false alarm and bring the RL to be closer to the theoretical value. Unfortunately, this may not be visible in practice. Thus, we suggest a design structure based on the robust Tukey outlier detection model.

5. Performance of the Tukey CUSUM Control Chart

In this section, we studied the performance of the proposed Tukey model based CUSUM control chart with estimated parameters. Let X 1 ,   X 2 , ,   X n denote Phase I samples and X ˇ be the median samples. Then an observation X k from X 1 ,   X 2 , ,   X n is declared as an outlier if | X k X ˇ | > p × I Q R , where I Q R = Q 3 Q 1 is the interquartile range. Q 1 and Q 3 are the first and third quartile of X 1 ,   X 2 , ,   X n , corresponding to the 25th and 75th percentile, respectively. The constant, p is the confidence factor commonly chosen between 1.5 and 3.0. The confidence factor of Tukey’s detector is selected so that it is not too small leading to unnecessary screening of observations that are not outliers, and at the same time it should not be too large implying the inability of the detector to detect any outliers. For the said reason, p is chosen to be 2.2 for the current study (for more details on the Tukey’s outlier detector see, Tukey [28]).
Once an outlier is detected from the Phase I sample using Tukey’s model, it is screened and the remaining data points are used to estimate mean and variance of the process. After screening the suspected outliers, distribution of the remaining data points in Phase I is revised from a mixture distribution to a truncated mixture distribution. Here, the truncation limits are set to be L D L = X ˇ 2.2 × I Q R and U D L = X ˇ + 2.2 × I Q R where L D L and U D L are lower and upper detection limits, respectively. Finally, the truncated mean and variance for the Phase I data points are defined, respectively, as follows:
E ( X   |   L D L < X < U D L ) = L D L U D L x g ( x ) d x F X ( U D L ) F x ( L D L )
V a r ( X   |   L D L < X < U D L ) = L D L U D L x 2 g ( x ) d x F X ( U D L ) F x ( L D L ) [ E ( X   |   L D L < X < U D L ) ] 2
where g ( x ) = { f ( x )         L D L < x < U D L 0                   otherwise and F X ( . ) . is the cumulative distribution function of X . The truncated mean and variance in Equations (13) and (14) are evaluated for different values of α and w , and are given in Table 10.
Table 10 clearly indicates that mixing α ( 100 ) % outliers in the distribution disturbs the mean and variance, especially for the larger values of w . On contrary, when the distribution is truncated (i.e., Tukey’s outlier detector is applied) this disturbance in the mean and variance is negligible. In view of this discussion, the estimates of the process mean, and variance obtained from the truncated distribution (i.e., after screening the data using Tukey’s model) will have the minimal effect of outliers introduced in the Phase I samples.
Using the same design structure and parameters as in Section 3 and Section 4, we computed the in-control ARL and SDRL values for the two-sided CUSUM control chart based on the Tukey outlier detection model with the estimated parameters. Three cases were considered, when the mean, the standard deviation or both were estimated. To access the performance of the proposed charts, we present in Figure 2, Figure 3 and Figure 4, a graphical display of the in-control ARL values with m = 10 ,   100 ,   500 and 1000 when the magnitude of outlier multiplier w is small ( w = 1 ), medium ( w = 2 ) and large ( w = 3 ). We presented only the case when both the mean and the standard deviation were estimated, as the other two cases had similar conclusions. Furthermore, we also showed the in-control ARL values in the presence of outliers without screening in Figure 2, Figure 3 and Figure 4 for a quick comparison. With the two charts side-by-side, we outlined our findings under the following headings.

5.1. Performance Comparison with Respect to m

We saw earlier that the number of Phase I data, m , did have a significant effect on the performance of a CUSUM chart. From Figure 2, Figure 3 and Figure 4, we saw that there was a vast difference in the reported ARL 0 between non-screened data and when the robust Tukey outlier detection model was applied to construct a CUSUM chart, particularly when m was small. For example, in Figure 3, if m = 10 ,   k = 1.0 and α > 0.04 , the ARL 0 for non-screened data were in five figures while the corresponding Tukey screened data were relatively closed to the target value. Even an increase in the number of Phase I observations with no screening did not appear to have a significant impact on the chart’s performance as the outlier multiplier increased. The Tukey screened counterpart, however, was getting closer to the limiting value ARL 0 = 200 , as m increased.
In other words, the use of the Tukey outlier detector in the construction of a CUSUM chart would maintain the performance of the chart, even with the handful amount of Phase I data.

5.2. Performance Comparison with Respect to α

If α = 0 , the in-control ARL values of CUSUM charts were approximately equal to the theoretical value of 200 and indicates the absence of outliers in the Phase I sampled data. However, as the magnitude of α increased, the non-screened data blew out of proportion, particularly when m was small and k > 0.5 . For example, if m = 10 ,   k = 1.0 and α = 0.05 , in Figure 2, the ARL 0 for the non-screened observations was 770 as against to 280 when the Tukey outlier detection model was applied. Even with the large values of k and α , the Tukey screened data appeared to be getting closer to the nominal value as m increased. The same conclusion could not be made for non-screened data, as the in-control ARL values remained high when α was relatively large (cf. Figure 2, Figure 3 and Figure 4). This means that the Tukey’s model would not only keep the ARL 0 on target but also maintain the performance of the CUSUM control chart. In general, we observed that the effect of α was minimal when k was small.

5.3. Performance Comparison with Respect to w

The larger the magnitude of outlier multiplier w , the worst the in-control ARL value of a two-sided CUSUM chart. If the outliers in a Phase I data were not screened, the ARL 0 was so huge as w increased, that the capability of the CUSUM chart in process monitoring was seriously affected. Unlike the Tukey based chart that tried to maintain the ARL 0 at the target value. For example, if m = 10 ,   k = 0.25 and α = 0.05 , the in-control ARL values for the non-screened data were 302 ,   634 and 1025 when w = 1 ,   2 and 3 , respectively. Compared to the screened Phase I data by the Tukey’s model with ARL values of 217, 222 and 225. Thus, the Tukey CUSUM chart could relatively withstand the impact of outlier multiplier w as compared to the chart based on non-screened data.

6. Illustrative Example

For illustrating the application of Tukey’s outlier detectors with the CUSUM control chart, we used a dataset from [3]. The variable of interest was the flow width measurement (in microns) for the hard-brake process. The data consisted of twenty-five in-control Phase I samples and twenty out-of-control Phase II samples where the average width had increased due to an assignable cause(s). The process mean and standard deviation were estimated (cf. Equations (2)–(4)) from Phase I samples and were found to be 1.5056 and 0.14 , respectively. These estimates were used to set up a CUSUM control chart for Phase II samples.
It is clearly observed from the scatter plot given in Figure 5a that the observations were relocated in Phase II. Further, it might also be confirmed from the CUSUM chart plotted in Figure 5b, which indicates several out-of-control signals in Phase-II. These findings led to the evidence that the hard-brake process had a positive shift at subgroup number fifteen and onwards.
Now using the data perturbation technique (cf. Kargupta, Datta [29] and Liu, Kargupta [30]), we introduced random outliers in different subgroups. Further, the process mean and standard deviation were estimated and found out to be 1.554 and 0.21544 , respectively. Based on these estimates, we constructed the limits, which were further used to monitor the location of Phase II samples. In Figure 5c, the scatter plot depicts a slight upward change in Phase II, and control chart presented in Figure 5d shows that the out-of-control situation in Phase II was delayed (to subgroup number twenty) due to a small number of outliers present in Phase I. This happened because the limits widened due to the variation in Phase I estimates of process mean and standard deviation.
Finally, by using the above-mentioned contaminated Phase I data, we estimated the limit of the Tukey’s outlier detector, which was found to be p × I Q R = 0.44726 . Now for any value, the absolute deviation from the median (i.e., | X 1.5171 | ) greater than 0.44726 implies that the corresponding value is an outlier and needs to be screened from the data. Hence, by using the outlier detector, six observations were screened from the Phase I data. Further, the process mean and standard deviation were estimated and found out to be 1.5123 and 0.152 , respectively. These new estimates were similar to the estimates of the original data and the scatter plot of the data is given in Figure 5e, which also showed upward trend in Phase II. In Figure 5f, the control chart is presented, which revealed that the there was no change in the limits, but the chart had detected an increase in the process mean at subgroup number sixteen.

7. Practical Application

In recent years, activity recognition (AR) became an emerging research topic due to the advancement of electronic devices. AR is commonly used in pattern recognition, ubiquitous computing, human behavior modeling and human–machine interaction. In health care studies, different electronic devices are commonly used to recognize everyday life activities. In eldercare centers, these facilities provide assistance and care to the elders and help to ensure their safety and successful aging. Commonly, wearable devices and cameras are used to monitor everyday life activities, but these approaches suffer from several disadvantages such as intrusiveness, time-consuming processing and low resolution. Therefore, to overcome these challenges in real-time activity recognition, Hong, Kang [31] used an alternative method named as multisensor data fusion (assembly reliability evaluation method—AReM). For a more detailed introduction on the AReM see [32]. In the AReM system, information is gathered from an inertial sensor embedded in a smartphone and wireless sensor system, which is plugged between the user and environment. Further, in a wireless sensor network, the movement of an individual is measured in the received signal strength (RSS) between the user and environment. For the AR dataset [31], designed a competition. In which three IRIS motes are used and placed on the chest, the right and left ankle of an actor (cf. Figure 6).
From this wireless sensor network, data was recorded on the actor’s activities such as; bending, cycling, standing, sitting, laying and walking. Further, for the first task of heterogeneous AReM, they considered activities such as cycling and standing. For the application purpose, we were concerned to detect a change in the pattern of RSS generated through the heterogeneous AReM setup. The AR time series dataset contained 480 observations in total, and each observation was obtained after 250 milliseconds. The average of RSS against the three IRIS motes (i.e., rss12, rss13, and rss23) was available in 15 different sequences of each activity. In our application, we considered the first sequence of the rss13 IRIS mote (chest-left ankle). The average of RSS of cycling was considered as in-control Phase I samples and average RSS of standing was considered as the out-of-control Phase II sample points. To access the normality of Phase I data set, we plotted a probability plot at the 95% confidence interval (cf. Figure 7) and also applied the Anderson–Darling test ( AD = 0.517 and p - value = 0.189 ), which provided the evidence that the Phase I data set was normal.
The RSS values of the chest-left ankle mote belonging to the cycling activity were clubbed into Phase I subgroups, and only the first 50 subgroups were used for the plotting purpose. Moreover, the first 25 subgroups based on the RSS values of chest-left ankle mote belonging to the standing activity were used as Phase II samples. The dataset of 75 subgroups is reported in Table 11. The process mean and standard deviation were estimated from the Phase I samples and found to be 16.9734 and 3.4764, respectively. These estimates were then used to construct the CUSUM chart for the Phase II samples. Figure 8a,b presents the scatter plot for the original data and the control chart output, respectively.
It was evident from Figure 8a that there was a downward relocation in Phase II samples, a point equally supported by the corresponding CUSUM chart, which gave an out-of-control signal right from the start of the plots in Figure 8b. Now to access the effect of the outliers, we followed the same procedure as described in Section 6, by first contaminating the Phase I data and used the estimates obtained, μ ^ 0 = 18.9924 and σ ^ 0 = 7.4018 to setup a CUSUM control chart for the Phase II data (cf. Figure 8c,d). Secondly, we used the Tukey outlier detector to screen the Phase I samples, computed the control chart parameters and used the estimates, μ ^ 0 = 17.0593 and σ ^ 0 = 3.5543 to construct the CUSUM chart for the Phase II samples (cf. Figure 8e,f).
The introduction of outliers in the Phase I samples, Figure 8c gave rise to wider control limits, which in turn delayed the out-of-control signal in the Phase II control chart setup (cf. Figure 8d). However, the application of the outlier detector on the contaminated Phase I data resulted in the screening out of about ten data points (cf. Figure 8e). Subsequently, the corresponding CUSUM chart in Figure 8f shows a similar behavioral pattern as those of the original data in Figure 8b.

8. Conclusions

In this article, we evaluated the in-control performance of a two-sided CUSUM control chart when the parameters were estimated in the presence of outliers based on the robust Tukey detection model. Using a Monte Carlo simulation approach, the ARL and SDRL were computed for a different number of Phase I data.
The results show that a large number of Phase I data was required to minimize the practitioner-to-practitioner variability. In the presence of outliers, a larger amount of Phase I data was needed, which might not be realistic in practical applications. The results further revealed that the use of the Tukey outlier detector in the construction of a two-sided CUSUM control chart required fewer Phase I observations to stabilize the chart’s performance. Therefore, it was plausible to use the Tukey’s model in the design structure of a CUSUM chart when the parameters were estimated for efficient process monitoring, particularly when the observations were prone to outliers. The advantage of this proposal is its simplicity to design and it is easy to use. A point demonstrated by the illustrative and application examples of the new Tukey CUSUM control chart. The scope of this study might be extended to other control charts design strategies like the Shewhart and exponentially weighted moving average.

Author Contributions

Study planning, mathematical derivations, calculation of results and draft writing was done by authors N.A. and M.R.A. under the supervision of author M.R. Application of proposed model on real-life dataset was done and written by author T.M. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by Deanship of Scientific Research (DSR) at King Fahd University of Petroleum and Minerals, under grant number IN171010.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

CUSUMCumulative Sum
ARLAverage Run Length
SDRLStandard Deviation of Run Length
RSSReceived Signal Strength
RLRun Length
ARActivity Recognition
AReMAssembly Reliability Evaluation Method

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Figure 1. Control limits for the two-sided CUSUM chart when the in-control mean and standard deviation are either known or estimated ( n = 5 ,   ARL 0 = 200 ) .
Figure 1. Control limits for the two-sided CUSUM chart when the in-control mean and standard deviation are either known or estimated ( n = 5 ,   ARL 0 = 200 ) .
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Figure 2. In-control ARL values for the two-sided CUSUM control chart in the presence of an outlier, with and without screening, when the parameters are estimated ( w = 1 ,   n = 5 ,   ARL 0 = 200 ) .
Figure 2. In-control ARL values for the two-sided CUSUM control chart in the presence of an outlier, with and without screening, when the parameters are estimated ( w = 1 ,   n = 5 ,   ARL 0 = 200 ) .
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Figure 3. In-control ARL values for the two-sided CUSUM control chart in the presence of an outlier, with and without screening, when the parameters are estimated ( w = 2 ,   n = 5 ,   ARL 0 = 200 ) .
Figure 3. In-control ARL values for the two-sided CUSUM control chart in the presence of an outlier, with and without screening, when the parameters are estimated ( w = 2 ,   n = 5 ,   ARL 0 = 200 ) .
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Figure 4. In-control ARL values for the two-sided CUSUM control chart in the presence of an outlier, with and without screening, when the parameters are estimated ( w = 3 ,   n = 5 ,   ARL 0 = 200 ) .
Figure 4. In-control ARL values for the two-sided CUSUM control chart in the presence of an outlier, with and without screening, when the parameters are estimated ( w = 3 ,   n = 5 ,   ARL 0 = 200 ) .
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Figure 5. Scatter plots and the CUSUM control chart outputs for the dataset on the width of the hard-brake process.
Figure 5. Scatter plots and the CUSUM control chart outputs for the dataset on the width of the hard-brake process.
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Figure 6. Placement of IRIS nodes on the actor’s body (cf. (Palumbo, Gallicchio [33]).
Figure 6. Placement of IRIS nodes on the actor’s body (cf. (Palumbo, Gallicchio [33]).
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Figure 7. Probability plot of the received signal strength (RSS) values of the rss13 mote belonging to the cycling activity.
Figure 7. Probability plot of the received signal strength (RSS) values of the rss13 mote belonging to the cycling activity.
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Figure 8. Scatter plots and CUSUM control chart outputs for the dataset on the received signal strength process.
Figure 8. Scatter plots and CUSUM control chart outputs for the dataset on the received signal strength process.
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Table 1. A synthesis table for the past and current research on a two-sided cumulative sum (CUSUM) chart.
Table 1. A synthesis table for the past and current research on a two-sided cumulative sum (CUSUM) chart.
Shewhart ChartCUSUMRobust CUSUMCurrent Study
Year1931195420132020
Author(s)Shewhart W.A.Page E.S.Nazir et al.Abbas et al.
Parameter of InterestProcess MeanProcess MeanProcess MeanProcess Mean
Plotting StatisticSample MeanCumulative sum of sample meanCumulative sum of robust estimators like sample median etc.Cumulative sum of sample mean
Advantages
  • Simplicity.
  • Quick detection of large shifts
  • Has a sensitivity parameter that can be adjusted according to shift size.
  • Indirectly, assigns small weights to the outliers in Phase-II using robust estimators.
  • Detects and deletes the outliers from Phase-I samples.
  • Phase-II performance is not much affected by the presence of outliers in Phase-I
Disadvantages
  • Has no sensitivity parameter that can be adjusted according to shift size.
  • Takes too long for detecting small shifts.
  • Performance of the chart is highly affected by the presence of outliers in Phase-I samples
  • Does not take care of the outliers present in Phase-I samples.
Table 2. Run length (RL) properties for the two-sided CUSUM control chart when the in-control mean and standard deviation are known ( n = 5 ,   ARL 0 = 200 ) .
Table 2. Run length (RL) properties for the two-sided CUSUM control chart when the in-control mean and standard deviation are known ( n = 5 ,   ARL 0 = 200 ) .
khRL Propertiesδ
0.000.250.500.751.001.502.003.004.005.00
0.256.8516ARL200.00063.58824.22914.0769.8626.1954.5583.0602.3282.013
SDRL187.8251.5814.736.663.901.921.200.660.480.22
0.504.1713ARL199.99783.10028.43813.9218.7244.9183.4562.2591.7721.372
SDRL195.1377.6123.279.354.842.041.190.590.470.26
0.752.9332ARL200.008102.89737.45816.7929.4464.6413.0631.8981.3811.094
SDRL197.6899.6834.2913.746.682.431.290.640.50.29
1.002.2137ARL200.006119.96848.84121.70511.4064.8642.9561.6951.2191.037
SDRL198.41117.9246.8319.589.353.101.500.680.420.19
Table 3. RL properties for the two-sided CUSUM control chart when the in-control standard deviation is known, and mean is estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 3. RL properties for the two-sided CUSUM control chart when the in-control standard deviation is known, and mean is estimated ( n = 5 ,   ARL 0 = 200 ) .
mkhRL Propertiesδ
0.000.250.500.751.001.502.003.004.005.00
100.257.609ARL199.383104.34333.24116.70911.2296.8845.0193.3432.5512.093
SDRL215.236139.68237.55910.3845.0392.2231.3310.7000.5380.299
0.504.447ARL200.059117.11839.46116.6979.7595.2803.6682.3701.8651.479
SDRL211.211147.58051.00115.2056.4072.3261.2870.6220.4410.501
0.753.070ARL199.955129.52549.91220.36010.6644.9353.2051.9711.4381.122
SDRL206.777152.89163.94521.9989.0662.7941.3950.6540.5130.327
1.002.290ARL199.845139.73461.59226.11012.9725.1743.0731.7421.2471.045
SDRL204.832158.02375.77029.93212.7693.6011.6280.6990.4420.207
500.257.071ARL199.98972.68625.88014.69410.2146.3874.6893.1412.3892.033
SDRL191.10170.53017.2007.2284.1081.9851.2280.6690.5050.230
0.504.246ARL200.32891.51530.40414.4768.9585.0103.5102.2891.7991.401
SDRL196.72194.95027.13910.2185.0972.1011.2080.5950.4620.491
0.752.968ARL200.300109.73639.79017.4809.6874.7093.0971.9161.3961.100
SDRL198.429113.86639.05415.0377.0572.5041.3090.6430.5020.300
1.002.233ARL200.129125.35851.43322.54811.7224.9282.9831.7061.2261.039
SDRL199.137128.59452.20021.3149.9443.1841.5260.6830.4270.193
1000.256.970ARL199.94668.27325.05714.39510.0486.2974.6273.1042.3602.024
SDRL188.66460.97315.8906.9494.0051.9501.2120.6650.4960.223
0.504.212ARL200.43087.51229.50114.2008.8464.9683.4872.2751.7861.388
SDRL196.06186.55625.1519.7734.9732.0721.2000.5930.4670.488
0.752.952ARL200.415106.37138.64917.1349.5614.6773.0821.9071.3891.097
SDRL197.742107.16036.67014.3616.8352.4701.3000.6410.5000.297
1.002.224ARL200.613122.84950.14422.13411.5804.9022.9701.7011.2231.038
SDRL199.037123.51749.43620.4639.6763.1471.5170.6810.4250.192
5000.256.878ARL200.25764.52824.40014.1469.9026.2194.5753.0692.3342.015
SDRL187.97553.60214.9336.7153.9171.9271.2010.6620.4860.220
0.504.180ARL200.34883.93428.70013.9818.7484.9253.4642.2621.7751.374
SDRL195.11479.43423.5969.4404.8612.0471.1900.5890.4700.485
0.752.938ARL200.330103.98137.72816.8809.4804.6503.0671.9001.3831.095
SDRL197.702101.51834.76313.8836.7002.4451.2930.6400.4980.294
1.002.215ARL199.622120.37749.08421.81411.4274.8672.9571.6951.2211.037
SDRL197.769119.22847.30819.8179.3943.1051.5030.6780.4230.189
10000.256.870ARL200.43564.13724.32714.1149.8936.2144.5673.0652.3322.014
SDRL187.97752.72314.8126.6803.9201.9211.1970.6610.4850.220
0.504.174ARL199.92183.37328.48813.9398.7294.9233.4602.2601.7731.372
SDRL195.12878.42523.3329.3914.8372.0491.1890.5890.4710.484
0.752.936ARL200.261103.48137.55716.8329.4564.6443.0661.9001.3821.095
SDRL197.588100.85734.56013.8116.6702.4341.2920.6410.4980.293
1.002.215ARL200.364120.20848.94221.74611.4224.8702.9591.6951.2201.038
SDRL198.748118.62246.94519.7489.3943.1071.5050.6790.4230.190
Table 4. RL properties for the two-sided CUSUM control chart when the in-control mean is known, and the standard deviation is estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 4. RL properties for the two-sided CUSUM control chart when the in-control mean is known, and the standard deviation is estimated ( n = 5 ,   ARL 0 = 200 ) .
mkhRL Propertiesδ
0.000.250.500.751.001.502.003.004.005.00
100.256.363ARL200.40560.10222.76213.2099.2625.8214.2842.8812.2281.928
SDRL307.83758.83015.2426.8304.0011.9901.2580.7120.4730.347
0.503.769ARL200.18679.65426.64612.9158.0804.5543.2052.1031.6111.256
SDRL373.977116.71327.1559.9475.0042.0891.2100.6260.5210.437
0.752.610ARL200.504101.34435.91015.7878.7894.2892.8261.7411.2791.063
SDRL404.643185.34650.47516.5027.3232.5181.3210.6620.4580.244
1.001.947ARL199.994119.32047.93320.90910.7964.5282.7341.5681.1621.026
SDRL419.821241.97982.66827.81011.5473.3481.5570.6710.3770.159
500.256.750ARL200.03362.84423.94613.8909.7336.1164.5043.0212.3072.000
SDRL208.51853.15814.8556.7013.9191.9331.2120.6750.4780.245
0.504.087ARL200.59682.35728.04813.7318.5994.8413.4022.2281.7371.344
SDRL229.81884.12923.9699.4974.8762.0541.1910.5960.4860.476
0.752.863ARL200.545102.18837.04716.5749.3014.5603.0121.8631.3581.086
SDRL240.560113.08336.67114.2176.7712.4531.2970.6460.4910.281
1.002.155ARL200.400119.42848.46021.50411.2544.7862.9071.6661.2061.034
SDRL246.062139.06851.87420.9289.7113.1411.5140.6780.4130.182
1000.256.794ARL199.29463.15524.05213.9669.7866.1544.5263.0372.3162.006
SDRL197.07752.48414.7466.6753.9061.9271.2060.6680.4800.232
0.504.131ARL200.57882.76628.22513.8238.6524.8843.4312.2441.7551.359
SDRL212.75380.76323.5739.4174.8412.0511.1890.5930.4790.480
0.752.898ARL200.473102.69437.30016.6859.3754.6073.0371.8801.3701.090
SDRL218.562106.84535.48413.9836.7242.4501.2940.6430.4940.287
1.002.183ARL199.677119.56848.55921.54711.3254.8242.9291.6801.2121.035
SDRL220.710127.96549.29120.1899.5343.1261.5060.6780.4170.185
5000.256.848ARL200.57263.55224.22614.0809.8626.1934.5573.0592.3282.013
SDRL190.74951.95114.7466.6753.9041.9231.2000.6630.4830.222
0.504.165ARL200.22383.17728.41513.9138.7074.9143.4532.2571.7691.369
SDRL198.68178.45523.3019.3774.8312.0491.1890.5900.4730.483
0.752.928ARL200.675103.04137.40416.7839.4384.6353.0601.8941.3801.093
SDRL201.922101.27634.55713.7736.6722.4361.2930.6410.4970.291
1.002.209ARL200.376120.01948.87821.73711.3924.8602.9511.6931.2191.036
SDRL203.135120.33547.38919.7989.4033.1081.5000.6780.4210.187
10000.256.851ARL200.37763.64824.23914.0829.8626.1964.5583.0592.3282.013
SDRL189.27952.10714.7556.6723.8971.9201.2000.6620.4840.221
0.504.171ARL201.03583.25428.51513.9408.7264.9183.4552.2581.7721.372
SDRL197.39878.29423.3169.3844.8442.0451.1860.5890.4710.484
0.752.933ARL200.892103.32737.53116.8209.4354.6443.0631.8981.3811.094
SDRL200.377100.86034.49113.8056.6552.4401.2910.6400.4970.292
1.002.210ARL199.787120.04148.75621.68811.3874.8592.9561.6941.2181.037
SDRL199.806119.23147.03619.6689.3743.1041.5060.6780.4210.189
Table 5. RL properties for the two-sided CUSUM control chart when the in-control mean and standard deviation are estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 5. RL properties for the two-sided CUSUM control chart when the in-control mean and standard deviation are estimated ( n = 5 ,   ARL 0 = 200 ) .
mkhRL Propertiesδ
0.000.250.500.751.001.502.003.004.005.00
100.257.059ARL199.839102.36531.35815.66110.5226.4584.7163.1432.4082.036
SDRL343.854200.55942.96210.6565.1122.2941.3990.7620.5380.334
0.504.010ARL200.741115.38437.44015.5489.0384.8743.3922.2041.7001.333
SDRL403.561253.06972.92217.3116.6742.3601.3080.6490.5150.473
0.752.719ARL199.462128.27848.25019.2989.9064.5512.9451.8011.3191.080
SDRL428.983293.013104.85328.93610.2912.8911.4200.6810.4800.272
1.002.010ARL200.078139.63860.75125.38112.3704.8102.8371.6071.1811.031
SDRL444.178321.987137.96346.28317.3573.9321.6840.6940.3960.174
500.256.917ARL199.69967.77124.90614.3129.9896.2554.6013.0852.3482.019
SDRL198.38961.66615.9556.9574.0181.9621.2220.6710.4930.233
0.504.169ARL200.33687.24529.24014.0978.7744.9243.4622.2591.7691.373
SDRL212.79990.31025.5309.8434.9872.0781.2030.5950.4750.484
0.752.913ARL199.648105.68438.33316.9949.4934.6313.0531.8891.3761.093
SDRL217.876113.73237.85114.6026.9332.4781.3040.6460.4960.291
1.002.193ARL199.956122.28849.91922.00311.4794.8512.9461.6851.2151.037
SDRL221.744133.82552.08121.1109.8233.1631.5200.6800.4200.188
1000.256.917ARL199.69967.77124.90614.3129.9896.2554.6013.0852.3482.019
SDRL198.38961.66615.9556.9574.0181.9621.2220.6710.4930.233
0.504.169ARL200.33687.24529.24014.0978.7744.9243.4622.2591.7691.373
SDRL212.79990.31025.5309.8434.9872.0781.2030.5950.4750.484
0.752.913ARL199.648105.68438.33316.9949.4934.6313.0531.8891.3761.093
SDRL217.876113.73237.85114.6026.9332.4781.3040.6460.4960.291
1.002.193ARL199.956122.28849.91922.00311.4794.8512.9461.6851.2151.037
SDRL221.744133.82552.08121.1109.8233.1631.5200.6800.4200.188
5000.256.868ARL199.85764.35524.39814.1229.8856.2104.5663.0672.3322.014
SDRL189.93653.53115.0086.7153.9201.9261.2010.6630.4850.222
0.504.171ARL200.04483.86528.60713.9538.7344.9213.4572.2601.7701.372
SDRL198.63880.16823.6849.4324.8652.0491.1900.5890.4720.484
0.752.930ARL200.139103.40537.59516.8569.4624.6373.0631.8961.3801.094
SDRL201.232102.43234.95613.9426.7072.4431.2940.6410.4970.292
1.002.209ARL199.771120.23948.99521.79311.4314.8592.9531.6941.2181.037
SDRL202.617120.90047.73419.9479.4603.1051.5050.6790.4210.189
10000.256.860ARL199.88763.96324.30214.0979.8706.2054.5633.0632.3302.014
SDRL188.74952.60014.8426.6923.9081.9241.2010.6610.4840.221
0.504.172ARL200.17583.43228.54213.9438.7284.9173.4582.2601.7721.372
SDRL196.71278.73123.4829.4004.8492.0461.1900.5890.4720.484
0.752.933ARL200.433103.18737.60616.8389.4654.6423.0631.8981.3821.094
SDRL199.533101.22134.68813.8486.6972.4391.2920.6400.4980.292
1.002.213ARL200.351120.45549.06021.77611.4144.8662.9581.6951.2191.037
SDRL201.344120.01747.49619.8089.4073.1091.5050.6790.4220.189
Table 6. Control limits for the two-sided CUSUM chart when the in-control mean and standard deviation are either known or estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 6. Control limits for the two-sided CUSUM chart when the in-control mean and standard deviation are either known or estimated ( n = 5 ,   ARL 0 = 200 ) .
µσmk
0.250.500.751.001.251.501.752.00
knownknown-6.8524.1712.9332.2141.7411.3871.0890.819
estimatedknown107.6094.4473.0702.2901.7891.4231.1200.845
207.3064.3313.0102.2571.7681.4071.1060.832
307.1994.2902.9862.2441.7601.4001.1000.829
507.0714.2462.9682.2331.7531.3951.0960.826
1006.9704.2122.9522.2241.7471.3901.0920.822
5006.8784.1802.9382.2151.7421.3881.0900.820
10006.8704.1742.9362.2151.7411.3871.0900.819
knownestimated106.3633.7692.6101.9471.5181.1950.9160.650
206.5983.9572.7582.0701.6201.2810.9940.729
306.6894.0302.8162.1171.6601.3151.0250.758
506.7504.0872.8632.1551.6901.3441.0500.781
1006.7944.1312.8982.1831.7161.3651.0700.799
5006.8484.1652.9282.2091.7361.3821.0860.814
10006.8514.1712.9332.2101.7391.3861.0880.818
estimatedestimated107.0594.0102.7192.0101.5571.2230.9380.672
207.0514.1102.8272.1071.6451.2991.0090.743
307.0194.1402.8662.1441.6751.3281.0350.768
506.9734.1592.8962.1711.7011.3521.0570.788
1006.9174.1692.9132.1931.7211.3701.0740.804
5006.8684.1712.9302.2091.7371.3841.0860.815
10006.8604.1722.9332.2131.7401.3861.0880.818
Table 7. In-control average run length (ARL) and standard deviation run length (SDRL) values for the two-sided CUSUM control chart in the presence of outlier when the in-control standard deviation is known, and mean is estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 7. In-control average run length (ARL) and standard deviation run length (SDRL) values for the two-sided CUSUM control chart in the presence of outlier when the in-control standard deviation is known, and mean is estimated ( n = 5 ,   ARL 0 = 200 ) .
mkhRL Propertiesw = 1w = 2w = 3
ααα
0.000.010.020.030.040.000.010.020.030.040.000.010.020.030.04
100.257.609ARL199198196194192199195190182178199192183173163
SDRL215214213212212215213211206205215213208204198
0.504.447ARL200199198196195200195191187180200194185177167
SDRL211210211210209211209206206200211210204201196
0.753.070ARL200199199198196200198194189184200193187179171
SDRL207207207206207207206204203198207203202198193
1.002.290ARL200199198197196200197195189186200195189182176
SDRL205203204202202205202202198196205201199195191
500.257.071ARL200198197194192200197190181169200192179164147
SDRL191189190187186191189184177169191187177166154
0.504.246ARL200199198196194200197192186178200194186172158
SDRL197195195194192197195191186179197193185176165
0.752.968ARL200199199198195200198195189184200196188179167
SDRL198197198197195198198195190185198195189182172
1.002.233ARL200200199199197200198195192188200197191183174
SDRL199199198198196199197195191190199196192186177
1000.256.97ARL200199197195191200196189180168200193178161141
SDRL189188186185180189187179172163189183172159142
0.504.212ARL200200200196195200198193187177200196185173156
SDRL196196196192192196193189185176196192183173158
0.752.952ARL200201199197196200200194191185200196189178168
SDRL198199197195195198196192189184198195189178168
1.002.224ARL201199201199197201199196194188201199192185173
SDRL199198200197197199197195193188199198192185174
5000.256.878ARL200199197195189200196189177164200194179156134
SDRL188187187183178188184177166154188181168146126
0.504.18ARL200201199197193200199192186175200196185171153
SDRL195195194192189195193188181172195191181167151
0.752.938ARL200201199197195200200195190183200198191180166
SDRL198198196195193198197193187180198195190179164
1.002.215ARL200200199199196200198197192188200198193185174
SDRL198199197197193198196196190185198196191182173
10000.256.870ARL200201198193190200198190178163200194178156134
SDRL188189186181179188186178166153188182167144124
0.504.174ARL200200198197194200198193186176200196185171153
SDRL195195193191189195193188181171195190180167149
0.752.936ARL200199199197196200200197191183200198191179166
SDRL198195197194194198198194189180198195189175165
1.002.215ARL200199200198198200200198192188200198192185173
SDRL199199199196197199198195190186199196191183172
Table 8. In-control ARL and SDRL values for the two-sided CUSUM control chart in the presence of an outlier when the in-control mean is known, and the standard deviation is estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 8. In-control ARL and SDRL values for the two-sided CUSUM control chart in the presence of an outlier when the in-control mean is known, and the standard deviation is estimated ( n = 5 ,   ARL 0 = 200 ) .
mkhRL Propertiesw = 1w = 2w = 3
ααα
0.000.010.020.030.040.000.010.020.030.040.000.010.020.030.04
100.256.363ARL2002322713083692001252282751807632200576613,02922,02833,346
SDRL30872012711442220230825,58841,78159,47272,55030866,775101,057131,849163,026
0.503.769ARL20025332040249820023745025867713,606200857419,93832,75549,639
SDRL374121618392660338437439,57159,41679,131100,61237482,482127,093162,747200,013
0.752.610ARL2012663744715692012614611510,30615,47820110,44023,69839,03956,370
SDRL405139726653349389140541,68666,63186,598107,91540592,015139,038178,484213,382
1.001.947ARL2002893864845992003152675110,52716,60520010,32224,09240,72059,187
SDRL420185127453539409042046,73369,18587,750111,80942091,131140,118181,993218,900
500.256.750ARL20021322924526220024931339952220032059810202048
SDRL20922524927129820930445589213812099586844900819,563
0.504.087ARL201220240263287201274377550767201413104323325481
SDRL230259294328379230453109341444185230285411,61024,59745,072
0.752.863ARL201221245272299201288425614970201499128434897734
SDRL241274314379417241655150924437384241540614,26132,41756,026
1.002.155ARL2002232482753092002954537131113200517162239659219
SDRL246285340383450246628364650868738246529719,60335,46063,990
1000.256.794ARL199212226242257199240294360440199284416621984
SDRL19721022924626619725032942757319733463614392903
0.504.131ARL20121823725828120125833543758220133355510051883
SDRL2132352582843152133014286169532131159107950109240
0.752.898ARL20022124326529220026735849467120035164912902600
SDRL21924427630233821933049579912972195862578731914,579
1.002.183ARL20022124227029520026937651771820036070014173124
SDRL22124927732134922134055594813542215952335614118,020
5000.256.848ARL201213227242257201238284340411201272375523735
SDRL191203218232248191228278337412191267381550807
0.504.165ARL2002182372552762002513173995102002984547021098
SDRL1992162362562801992533264165431993064927961317
0.752.928ARL2012182402612862012573334355652013154998141350
SDRL2022212432652922022643474636132023305549631698
1.002.209ARL2002192412642912002613434555972003215278821509
SDRL20322224727229820326936148865220334159510521921
10000.256.851ARL200214227240256200237284339405200269368514717
SDRL189203216228244189227273330401189260362517740
0.504.171ARL2012192352542772012513133965012012954456811058
SDRL1972162332512771972493144025121972984577261138
0.752.933ARL2012202402622852012583324325582013114907841280
SDRL2002202402632882002593384425782003185168401420
1.002.210ARL2002192422652892002613404465862003175118441408
SDRL2002212442672902002643494606122003295429111580
Table 9. In-control ARL and SDRL values for the two-sided CUSUM control chart in the presence of an outlier when the in-control mean and standard deviation are estimated ( n = 5 ,   ARL 0 = 200 ) .
Table 9. In-control ARL and SDRL values for the two-sided CUSUM control chart in the presence of an outlier when the in-control mean and standard deviation are estimated ( n = 5 ,   ARL 0 = 200 ) .
mkhRL Propertiesw = 1w = 2w = 3
ααα
0.000.010.020.030.040.000.010.020.030.040.000.010.020.030.04
100.257.059ARL2002202382572642002894205065222004156678341018
SDRL34457980213258803442389840088367702344697012,59713,93715,538
0.504.010ARL20124431533339020169112212017255620124654835772210,719
SDRL404137552704092561440415,50721,36730,53734,15740438,10054,56470,64084,741
0.752.719ARL1992913404094971991402283942586234199485610,45218,23025,507
SDRL429509152905868820742927,82140,32750,68962,25442958,25887,142116,950139,084
1.002.010ARL2002803964686422001919374061038942200703815,60124,67335,470
SDRL44424717494708811,81244433,96048,18562,32077,49044472,692109,882138,233165,823
500.256.917ARL201212227233242201238268297319201265323366391
SDRL212228250259270212276331381429212355474560616
0.504.169ARL201218236253271201260326405482201333495715977
SDRL231259290312345231371528684894231678137122393977
0.752.913ARL20022224226528920027637249764220039371912812057
SDRL2402823133523922404547381374176924018854811955914,183
1.002.193ARL19922124727329719928240255477019943592518263613
SDRL244281333382419244488928170433502441775776114,79928,661
1000.256.917ARL200211223232241200234266291310200262313350368
SDRL198210225237248198242285318345198287356409440
0.504.169ARL200217234252268200252311375441200302434591767
SDRL2132312562772992132853784705702133886069061233
0.752.913ARL2002192382612802002603374305462003305268161212
SDRL21824326629832521831143458481121847289516822721
1.002.193ARL20022224226829020026935847461920034759610161662
SDRL222252278311339222336489703994222586133027104963
5000.256.868ARL200210222232240200233263286302200260311340352
SDRL190201212222231190224255280295190253306341352
0.504.171ARL200215232248264200247302359414200288402528662
SDRL199216231247265199248307369429199297420561712
0.752.93ARL200218237258277200256323403496200307459670947
SDRL2012202402642812012623364245302013204997471071
1.002.209ARL2002192402622852002603354305452003174957621164
SDRL2032242442692962032693504565902033355458651382
10000.256.860ARL200212223232239200234264287302200258310339346
SDRL189200211222228189224254275293189248302329336
0.504.172ARL200215232248263200247300357416200289399523653
SDRL197213230245261197244301359418197289406534675
0.752.933ARL200218238257277200256322398491200304453659927
SDRL200219239258278200257326409507200309466690986
1.002.213ARL2002182402622852002593334255422003134897521124
SDRL2012182402652892012643384365622013205118011219
Table 10. Non-truncated and truncated mean and variance of mixture distribution of N ( 0 , 1 ) and χ ( 1 ) 2 .
Table 10. Non-truncated and truncated mean and variance of mixture distribution of N ( 0 , 1 ) and χ ( 1 ) 2 .
α w
123
X X   |   L D L < X < U D L X X   |   L D L < X < U D L X X   |   L D L < X < U D L
0 E ( . ) 0−0.0000700.000100.00006
V ( . ) 10.9709710.9710110.971
0.01 E ( . ) 0.010.00510.020.005640.030.00539
V ( . ) 1.02990.977271.11960.978811.26910.97878
0.02 E ( . ) 0.020.010640.040.011270.060.01083
V ( . ) 1.05960.983731.23840.98661.53640.98683
0.03 E ( . ) 0.030.016240.060.017130.090.01664
V ( . ) 1.08910.990161.35640.994581.80190.99498
0.04 E ( . ) 0.040.021580.080.023230.120.0223
V ( . ) 1.11840.996661.47361.003112.06561.00392
0.05 E ( . ) 0.050.0270.10.029280.150.0284
V ( . ) 1.14751.003231.591.011562.32751.01339
0.06 E ( . ) 0.060.032740.120.035680.180.03429
V ( . ) 1.17641.009881.70561.020882.58761.02238
0.07 E ( . ) 0.070.038470.140.042030.210.04068
V ( . ) 1.20511.017091.82041.029942.84591.03251
0.08 E ( . ) 0.080.044240.160.048590.240.04752
V ( . ) 1.23361.023811.93441.039423.10241.04268
0.09 E ( . ) 0.090.050170.180.055360.270.0539
V ( . ) 1.26191.031072.04761.049283.35711.05381
0.1 E ( . ) 0.10.055960.20.062090.30.06083
V ( . ) 1.291.038112.161.059683.611.06487
Table 11. Phase-I subgroups of RSS values chest-left ankle mote. Phase-II subgroups of RSS values chest-left ankle mote.
Table 11. Phase-I subgroups of RSS values chest-left ankle mote. Phase-II subgroups of RSS values chest-left ankle mote.
Phase-I Subgroups of RSS Values Chest-Left Ankle Mote
NoRSS X ¯ i S i
117.5021.5017.7520.5014.2518.3002.847
211.8021.5023.3318.5018.0018.6264.399
318.0017.5015.3316.2518.7517.1661.372
412.5017.5016.0019.2513.0015.6502.892
522.0020.2516.2515.3317.7518.3162.776
616.5024.0022.7520.7520.7520.9502.847
724.5013.2518.2523.0019.0019.6004.418
813.0015.3317.3312.3322.5016.0984.089
920.5014.5016.7517.2513.3316.4662.769
1017.5017.2513.7517.6714.6716.1681.823
1114.0013.0019.0016.0011.2514.6502.977
1218.0017.0017.5024.0016.0018.5003.162
1314.7512.5017.7510.6720.2515.1843.874
1418.0011.7512.0019.7512.7514.8503.744
1518.0016.6717.6712.8019.2516.8782.459
1619.7524.0015.6721.3319.6720.0843.027
1726.7517.7522.7516.0011.7519.0005.858
1811.2519.6723.7515.7517.0017.4844.641
1919.2516.5015.2519.2516.2517.3001.841
206.0013.5015.0010.2518.0012.5504.604
2111.3313.0016.5019.5019.2515.9163.669
2210.0013.0019.5019.0018.5016.0004.257
2312.2519.0014.5015.0018.0015.7502.739
2419.7516.0012.5016.7517.5016.5002.640
2515.7517.7519.0019.0010.3316.3663.626
2619.0024.0025.7517.3319.7521.1663.552
2719.2522.5016.7511.5018.7517.7504.058
2810.6716.5017.3316.3313.2514.8162.788
2919.0020.7513.5018.2516.5017.6002.753
305.5017.5016.009.7511.5012.0504.843
3120.678.5014.2514.0018.7515.2344.737
3218.7516.2514.0021.0022.3318.4663.402
3313.5017.5018.3317.0015.7516.4161.879
3420.5016.7517.0021.0019.5018.9501.972
3518.3315.7515.0020.7515.6717.1002.404
3616.0020.2524.5015.0016.6718.4843.902
3718.7519.7512.3315.2521.6717.5503.735
3822.2513.7520.7521.0016.7518.9003.543
3918.7516.6713.7512.0018.5015.9342.971
4012.2514.7511.0017.258.3312.7163.431
4121.2520.2514.0017.5017.3318.0662.842
4221.7515.7515.0019.2516.3317.6162.817
4319.7516.6717.0016.5015.0016.9841.726
4418.6721.3318.2516.6717.7518.5341.732
4515.0016.3319.0017.0017.5016.9661.474
4614.0019.0017.7526.7515.0018.5005.034
4717.2520.5018.5019.0020.0019.0501.280
4818.0010.5011.6716.6718.0014.9683.610
498.7520.7521.008.0021.0015.9006.875
5016.0012.0018.2512.0014.7514.6002.684
Phase-II Subgroups of RSS Values Chest-Left Ankle Mote.
NoRSS X ¯ i C U S U M i + C U S U M i H H
111.5013.5012.0011.5012.0012.1000.000−4.0966.41−6.41
211.0012.0016.0011.6717.6013.6540.000−6.6386.41−6.41
312.7515.506.5010.3318.0012.6160.000−10.2186.41−6.41
49.502.0011.6012.0012.009.4200.000−16.9946.41−6.41
512.006.0012.5012.5013.3311.2660.000−21.9246.41−6.41
612.0013.2511.2512.0013.2512.3500.000−25.7706.41−6.41
716.3311.7518.009.7515.0014.1660.000−27.8006.41−6.41
84.5016.7513.678.2515.0011.6340.000−32.3626.41−6.41
912.254.0011.756.5012.009.3000.000−39.2586.41−6.41
108.5011.009.5012.2511.7510.6000.000−44.8546.41−6.41
1112.0012.2511.0012.0013.2512.1000.000−48.9506.41−6.41
1218.2512.0017.0011.0012.2514.1000.000−51.0466.41−6.41
1315.5014.0014.009.008.7512.2500.000−54.9926.41−6.41
1415.0014.0013.0011.2510.0012.6500.000−58.5386.41−6.41
154.3312.5014.0012.008.0010.1660.000−64.5686.41−6.41
1612.7512.0010.7512.0012.0011.9000.000−68.8646.41−6.41
1718.759.7513.0012.0016.5014.0000.000−71.0606.41−6.41
1816.5012.5012.2514.7510.5013.3000.000−73.9566.41−6.41
1913.7513.6712.005.003.509.5840.000−80.5686.41−6.41
2011.0012.0013.0010.758.0010.9500.000−85.8146.41−6.41
2113.2512.0011.0012.0011.0011.8500.000−90.1606.41−6.41
2218.0011.0011.6716.0016.2514.5840.000−91.7726.41−6.41
2313.7511.004.6717.5011.5011.6840.000−96.2846.41−6.41
246.6711.6712.7511.337.6710.0180.000−102.4626.41−6.41
2511.5012.336.6710.002.678.6340.000−110.0246.41−6.41
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