Time-adaptive support vector data description for nonstationary process monitoring

Engineering Applications of Artificial Intelligence 68 (2018) 18–31

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Engineering Applications of Artificial Intelligence journal homepage: www.elsevier.com/locate/engappai

Time-adaptive support vector data description for nonstationary process monitoring Seulki Lee, Seoung Bum Kim * Department of Industrial Management Engineering, Korea University, 145 Anam-Ro, Seoungbuk-Gu, Anam-dong, Seoul 136-713, South Korea

a r t i c l e

i n f o

Keywords: Multivariate control chart Support vector data description Time-varying process Process control Machine learning Nonstationary process

a b s t r a c t Statistical process control techniques are widely used for quality control to monitor the stability of a process over time. In modern manufacturing systems with complex and variable processes, appropriate control chart techniques that can efficiently address nonnormal processes are required. Furthermore, in real manufacturing environments, process changes occur frequently because of various factors such as product and setpoint changes, catalyst degradation, seasonal variations, and sensor drift. However, conventional control chart schemes cannot necessarily accommodate all possible future conditions of a process because they are formulated based on information recorded in the early stages of the process. Several attempts have been made to accommodate process changes over time. In the present paper, we propose a time-adaptive support vector data descriptionbased control chart that can address not only nonnormal in-control observations, but also time-varying processes. The effectiveness and applicability of the proposed chart was demonstrated through experiments with simulated data and real data from the metal frame process in mobile device manufacturing. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Statistical process control (SPC) methods are widely used in many industries such as manufacturing and service operations to monitor and improve process performance over time (Woodall et al., 2000). The goal of SPC methods is to reduce predictable quality variations and monitor the complete system to detect unexpected root causes of variation (Ferracuti et al., 2015; Vander Wiel et al., 1992). A control chart is a representative tool of SPC that distinguishes between the inherent variations within the process and variations from unwanted process disruptions (Gitlow, 2009; Oakland, 2008). Control charts are commonly used as graphical tools to quickly detect changes in manufacturing processes (Noorossana et al., 2015; Stoumbos and Sullivan, 2002). The Shewhart control chart is the most representative SPC chart for manufacturing processes. Shewhart (1939) developed a univariate control chart to monitor a single quality characteristic. However, monitoring these quality characteristics independently can be misleading because modern manufacturing systems involve many inter-correlated quality characteristics (Hwang and Lee, 2015; Lu, 1998). This limitation prompted the development of multivariate control charts that can simultaneously consider the correlations between multiple quality characteristics and effectively manage the overall probability

of Type I errors. Hotelling’s 𝑇 2 control chart has been widely used to monitor multivariate processes. The monitoring statistics for the 𝑇 2 chart are computed from the following equation: )𝑇 ( ) ( 𝑇 2 = 𝑥 − 𝑥 𝑆 −1 x − x ,

(1)

where 𝑥 and 𝑆 are, respectively, the sample mean vector and sample covariance matrix determined from the in-control data. The 𝑇 2 statistic can be considered the distance of an observation from the center of incontrol observations, while considering the correlation among variables. The control limit of a 𝑇 2 chart is proportional to the percentile of an Fdistribution assuming that in-control observations follow a multivariate normal distribution (Mason and Young, 2002). As modern manufacturing processes become more complex, incontrol observations of many industrial processes do not follow a normal distribution (Hu et al., 2015; Yang and Arnold, 2013; Gani et al., 2011). Thus, traditional control charts such as 𝑇 2 charts do not effectively reflect the quality characteristics of observations that follow the nonnormal distribution. To overcome the shortcomings of conventional control charts under nonnormal situations, a number of methods have been proposed to use one-class classification (OCC) algorithms to monitor nonnormal processes (Liu et al., 2015; Tuerhong et al., 2014; Sukchotrat et al., 2009).

* Corresponding author.

E-mail addresses: [email protected] (S. Lee), [email protected] (S.B. Kim). https://doi.org/10.1016/j.engappai.2017.10.016 Received 1 November 2016; Received in revised form 14 May 2017; Accepted 20 October 2017 0952-1976/© 2017 Elsevier Ltd. All rights reserved.

S. Lee, S.B. Kim

Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Both OCC algorithms and control charts assume that only the incontrol observations are available for measuring the degree of abnormality of new observations. The novelty scores of OCC algorithms are used as the monitoring statistics of OCC-based control charts. This stream of research began with the introduction of the K chart based on a support vector data description (SVDD) algorithm (Sun and Tsung, 2003). The K charts performed well for nonnormal or unknown distributed in-control processes. Kumar et al. (2006) constructed robust K charts through normalized monitoring statistics and demonstrated that these charts can efficiently handle autocorrelated process data. Ning and Tsung (2013) proposed a guideline to determine the K chart parameters in practice. Gani et al. (2011) provided an assessment of the K chart by applying it to a real industrial process and revealed that the K chart is more sensitive to small mean shifts than the 𝑇 2 chart. Khediri et al. (2012) proposed the kernel k-means-based SVDD control chart for multimodal processes. Sukchotrat et al. (2009) proposed a bootstrapping strategy to establish robust control limits. In addition to the SVDD algorithm, other OCC control charts include the hybrid novelty score-based control chart and the 𝐾 2 chart, based on the k-nearest neighbors data description algorithm (Tuerhong et al., 2014; Sukchotrat et al., 2009). In summary, OCC-based control charts have demonstrated their improved performance in many of the nonnormal and nonlinear situations frequently encountered in modern manufacturing systems (Tuerhong and Kim, 2015; Grasso et al., 2015; Kim et al., 2011). However, in addition to the nonnormal situations, the time-varying operation of process data is also common in modern industrial processes (Haimi et al., 2016; Soares and Araújo, 2015; Ge and Song, 2013). Such behavior can be caused by several factors such as setpoint and throughput changes, catalyst degradation, sensor drift, and the presence of unmeasured disturbances (Ketelaere et al., 2011; Choi et al., 2006). Time-varying process operation is considered a critical issue in many electronic and chemical engineering fields (Li et al., 2000; Qin, 1998). Because conventional SPC schemes are formulated based on the data recorded in the early stage of the process, it is difficult to describe all possible future conditions of a process. Consequently, traditional control charts may detect normal variations as faults in time-varying situations, leading to a high level of Type I error rates (i.e., false alarms). To enhance the monitoring performance and reduce false alarms in timevarying situations, some studies have proposed an adaptive technique in multivariate control charts (Chakour et al., 2015; Ge and Song, 2008; Lee et al., 2006). These approaches are based on updating the parameters of the control charts for time-varying processes. Time-varying process monitoring methods can be divided into two categories: recursive estimation with a weighting parameter and moving window. In the first category, a weighting parameter is added to the control chart to allow old data to be gradually forgotten. The adaptive principal component analysis (PCA)-based 𝑇 2 chart was proposed in which the sample mean and sample covariance of the in-control data can be updated in such a manner that more weight is assigned to current observations than past observations (Zhang et al., 2012; Choi et al., 2006; Li et al., 2000; Wold, 1994). As for an example of the second category, the moving window-based methods exclude the oldest data and include the newest data simultaneously with the data window. The moving window PCA-based 𝑇 2 chart was developed to control timevarying processes (Liu et al., 2009; Wang et al., 2005). Xie and Shi (2012) suggested an adaptive form of Gaussian mixture model (GMM) charts using a moving window scheme for time-varying processes. In the moving window GMM, having found the most suitable Gaussian components for the new data, the parameters of the components are updated by a moving window. Despite these efforts, the majority of time-adaptive control charts rely on the assumption that in-control observations follow a normal distribution. The above-mentioned adaptive PCA-based process monitoring methods used 𝑇 2 statistics in the projected space. Thus, a normality assumption of in-control observations in the projected space is necessary. The moving window GMM method also assumes that each group of the mixture follows a Gaussian distribution.

The present study focuses on developing a multivariate control chart for both nonnormal and time-varying processes. The proposed chart is an extension of the existing SVDD-based chart adding a weight factor to effectively address the time-varying situations. We define the updating region for the efficient model-updating structure of the control chart. The remainder of the paper is organized as follows. Section 2 reviews the existing SVDD-based control charts. Section 3 describes the proposed time-adaptive SVDD-based control chart by emphasizing its adaptive capability for time-varying processes. In Section 4, a simulation study is conducted to examine the performance of the proposed time-adaptive SVDD-based control chart under various scenarios. Section 5 presents the results of a case study using actual data from the metal frame process in mobile device manufacturing exhibiting nonnormal and time-varying characteristics. Section 6 provides concluding remarks. 2. SVDD-based control charts The SVDD algorithm is one of the representative OCC algorithms (Tax and Duin, 2004). The objective of the SVDD algorithm is to determine a sphere with minimal volume that can envelop all of the data points in the training set 𝐱𝒊 = [𝑥𝑖1 , 𝑥𝑖2 , … , 𝑥𝑖𝑝 ]𝑇 , for 𝑖 = 1, 2, … , 𝑛. This sphere is characterized by two factors, sphere center 𝒂 and radius 𝑅. That is, the problem is to: 𝑛 ∑ ( ) 𝑚𝑖𝑛𝐹 𝑅, 𝒂, 𝜀𝑖 = 𝑅2 + 𝐶 𝜀𝑖 ,

(2)

𝑖=1 2 ‖2 s.t. ‖ ‖𝐱𝒊 − 𝒂‖ ≤ 𝑅 + 𝜀𝑖 ,

𝜀𝑖 ≥ 0,

(3)

∀𝑖,

where 𝐶 is the trade-off parameter between the sphere’s volume and the misclassification error (also referred to as the regularization parameter), and 𝜀𝑖 is the slack variable that allows 𝐱𝒊 to be outside the sphere. Eqs. (2) and (3) can be solved by the following Lagrange dual formulation: 𝑚𝑎𝑥 s.t.

𝑛 ∑

𝑖=1 𝑛 ∑

𝑛 ( ) ( ) ∑ 𝛼𝑖 𝛼𝑗 𝐱𝒊 ⋅ 𝐱𝒋 , 𝛼𝑖 𝐱𝒊 ⋅ 𝐱𝒊 −

𝛼𝑖 = 1,

(4)

𝑖,𝑗=1

0 ≤ 𝛼𝑖 ≤ 𝐶,

(5)

∀𝑖,

𝑖=1

where 𝛼𝑖 is Lagrange multiplier. Having solved the above Lagrange formulation, the values of 𝛼𝑖 and data point 𝐱𝒊 with 𝛼𝑖 > 0 are obtained. The data points 𝐱𝒊 are called support vectors. SVDD algorithms can generate a flexible boundary by employing a kernel trick, which maps an input space into a higher dimensional feature space by replacing the inner product with kernel functions. Although many kernel functions are available, Tax and Duin (1999) demonstrated that the following Gaussian kernel is one of the most effective functions for SVDD. ( ) ( ) ‖ ‖2 𝐾 𝐱𝒊 ⋅ 𝐱𝒋 = exp − ‖𝐱𝒊 − 𝐱𝒋 ‖ ∕𝑠2 , (6) ‖ ‖ where 𝑠 ≠ 0 is the width parameter that controls the level of detail of the SVDD boundary. For new observation 𝐳, the kernel distance to the center 𝒂 can be calculated by ‖𝐳 − 𝒂‖2 = 𝐾 (𝐳 ⋅ 𝐳) − 2

𝑛 ∑

𝑛 ( ) ∑ 𝛼𝑖 𝐾 𝐳 ⋅ 𝐱𝒊 + 𝛼𝑖 𝛼𝑗 𝐾(𝐱𝒊 ⋅ 𝐱𝒋 ).

𝑖=1

𝑖,𝑗=1

(7)

For classification, testing observation 𝐳 is classified as the target when this distance is less than or equal to 𝑅2 . Several studies have implemented the SVDD algorithm to solve SPC problems. However, previous studies have not addressed time-varying situations. This motivates the focus of this paper on the development of an SVDD-based control chart to handle time-varying and nonnormal situations. 19

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 1. Comparison boundaries of (a) original SVDD algorithm and (b) time-adaptive SVDD algorithm when the weights of old data points are 0.1.

3. Proposed methodology: time-adaptive SVDD-based control chart

calculate the monitoring statistics. The monitoring statistics of a kernel version of the time-adaptive SVDD can be calculated as:

3.1. Time-adaptive SVDD formulation for monitoring statistics

𝑇 𝐴𝑆𝑉 𝐷𝐷 (𝐳) = 𝐾 (𝐳 ⋅ 𝐳) − 2

𝑛 ∑

𝑛 ( ) ( ) ∑ 𝛼𝑖 𝛼𝑗 𝐾 𝐱𝒊 ⋅ 𝐱𝒋 . 𝛼𝑖 𝐾 𝐳 ⋅ 𝐱𝒊 +

𝑖=1

Conventional control charts are constructed based on data recorded in the early stages of a process. However, the patterns of in-control observations may change because of various reasons such as catalyst degradation, seasonal variations, and sensor drift. In this case, past observations should be progressively forgotten when the model is updated to properly capture the pattern of the recent in-control observations. In the original SVDD algorithm, the trade-off parameter 𝐶 is assigned the same value for all data. However, the proposed method assigns different trade-off values to each of the training observations I according to their age such that older observations can be located outside the sphere. The proposed time-adaptive SVDD can be formulated as follows: 𝑛 ∑ ( ) 𝑊𝑖(𝑇 ) 𝜀𝑖 , 𝑚𝑖𝑛𝐹 𝑅, 𝒂, 𝜀𝑖 = 𝑅2 + 𝐶

Fig. 1 illustrates a two-dimensional scatter plot displaying how the time-adaptive SVDD boundary can be controlled by the weights. Given the same parameter value (𝐶 = 0.05) and kernel width parameter value (𝑠 = 3), the original and time-adaptive SVDD algorithms render different shapes of boundaries (Fig. 1). It can be observed that the time-adaptive SVDD algorithm can properly capture the patterns of recent data points, while the original SVDD algorithm cannot. To further demonstrate the time-adaptive ability of the proposed method, we generated the data from the bivariate normal distribution under the time-varying situation (Fig. 2). It can be clearly seen that each time a new observation comes in, the decision boundary moves to reflect the patterns of new observations.

(8) 3.2. Control limits and updating criteria

𝑖=1 2 ‖2 s.t. ‖ ‖𝐱𝒊 − 𝒂‖ ≤ 𝑅 + 𝜀𝑖 ,

𝜀𝑖 ≥ 0,

∀𝑖.

(9)

All the existing adaptive control charts described in Introduction are updated only when a new observation is determined to be in control. This updating scheme is inefficient for application to real manufacturing cases. The actual processes have not only a time-varying situation, but also a static pattern. In this case, the existing adaptive models are unnecessarily reconstructed despite the static processes. Thus, it is important to capture a time-varying situation and determine when the model is updated. When a time-varying situation occurs, patterns of incontrol observations are shifted slowly. In this situation, new data may have a tendency to display the edge of the in-control data distributions. Thus, it can logically and efficiently address time-varying situations to update the control chart only when a new observation is assigned to the edge of the in-control dataset. Therefore, we present an efficient modelupdating structure containing three regions divided by the control limit (CL) and the updating limit (UL) (Fig. 3). The region between zero and UL is the safe region, indicating that the process is stable. The region between UL and CL is the updating region, indicating that the process is exhibiting time-varying patterns. The region exceeding CL is the alarm region, indicating the out-of-control process. The proposed adaptive control chart is updated only when the monitoring statistics for the observations are in the updating region. CL and UL are derived from the bootstrap-estimated quantile of monitoring statistics. The 100(1 − 𝛼)th and 100(1 − (𝛼 + 𝑑))th bootstrap percentiles of the time-adaptive SVDD statistics are used as the CL and

𝑊𝑖(𝑇 ) ,

a weight for training observation I at time T can be calculated as follows: ( ) 𝑊𝑖(𝑇 ) = {𝜆 𝑡𝑙 − 𝑡𝑖 + 1}−1 , (10) where 0 ≤ 𝜆 < 1 is the weighting parameter, 𝑡𝑙 is the time of the latest observation, 𝑡𝑖 is the time of ith observation, and T indicates the time point at which the new observation belongs to the updating region. A detailed description of the updating region is described in Section 3.2. For example, if 𝜆 is large, more weight is given to the current observation. In general, the weighting parameter 𝜆 can be determined empirically. Because the value 𝑊𝑖(𝑇 ) that corresponds to an older observation is smaller than that of the newer observation, the older one tends to have larger values of 𝜀𝑖 . Consequently, older observations are likely to be located outside the boundary. The Lagrangian dual problem of Eqs. (8) and (9) becomes: 𝑚𝑎𝑥 s.t.

𝑛 ∑

𝑖=1 𝑛 ∑

𝑛 ( ) ∑ ( ) 𝛼𝑖 𝐱𝒊 ⋅ 𝐱𝒊 − 𝛼𝑖 𝛼𝑗 𝐱𝒊 ⋅ 𝐱𝒋 ,

(11)

𝑖,𝑗=1

𝛼𝑖 = 1,

0 ≤ 𝛼𝑖 ≤ 𝐶𝑊𝑖(𝑇 ) ,

∀𝑖.

(13)

𝑖,𝑗=1

(12)

𝑖=1

The support vectors are determined when the optimum value 𝛼𝑖 is in 0 < 𝛼𝑖 ≤ 𝐶𝑊𝑖(𝑇 ) . These 𝛼𝑖 and a set of support vectors can be used to 20

S. Lee, S.B. Kim

Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 2. The boundaries of time-adaptive SVDD under the time-varying situation.

process data, and banana-shaped distributions to represent nonlinearly distributed process data. We generated 400 in-control observations for the four simulation scenarios for Phase I. We also generated testing observations with a time-varying property for each scenario. For the time-varying situation, we generated 900 observations that were gradually shifted by 0.003 for every new incontrol observation. Moreover, an additional 100 out-of-control observations were generated, gradually shifted by 0.006. Fig. 5 shows how the time-varying in-control observations from each simulation case were distributed.

UL, respectively. The percentile values such as 𝛼 and 𝑑 can be specified by the user. The parameters 𝛼 and 𝑑, respectively, determine the Type I error rate and size of the updating region. The choice of parameter 𝑑 is selected empirically. A detailed description of the selection of the parameter is described in Section 4.2. Fig. 4 shows the whole process of the proposed time-adaptive SVDD chart. The proposed chart is updated each time a new observation is included in the updating region. The weights for training observations are recalculated and consequently the chart is newly constructed. 4. Simulation study 4.1. Simulation setup

4.2. Parameters selection

We conducted a simulation study to examine the properties of the proposed time-adaptive SVDD-based chart and to compare it with the original SVDD-based chart under various situations. Simulations were conducted for both normal and nonnormal situations. For nonnormal cases, we investigated lognormal distributions to represent skewed process data, t-distributions to represent symmetric and high kurtosis

For parameter selection of time-adaptive SVDD charts, four parameters are required: a kernel width parameter s, trade-off parameter C, weighting parameter 𝜆, and percentile value of the update region d. We first find optimal parameters C and s which determine the shape of the SVDD boundary and then find optimal parameters 𝜆 and d, which are associated with the model update. In this section, we show the 21

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Table 1 Actual Type II error rates from artificial out-of-control dataset and Actual Type I error rates over different choices of parameters C and s of time-adaptive SVDD-based chart under the normal distribution when expected Type I error rate is 0.05. The boldface indicates the best parameters. C

s 1

2

3

4

5

6

7

8

9

10

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.188 (0.41) 0.330 (0.19) 0.389 (0.25) 0.394 (0.31) 0.518 (0.16) 0.535 (0.12) 0.468 (0.32) 0.465 (0.34) 0.456 (0.36) 0.458 (0.35)

0.219 (0.27) 0.315 (0.12) 0.376 (0.08) 0.444 (0.07) 0.498 (0.07) 0.535 (0.05) 0.579 (0.05) 0.614 (0.04) 0.640 (0.04) 0.649 (0.04)

0.226 (0.24) 0.303 (0.12) 0.375 (0.07) 0.459 (0.05) 0.495 (0.04) 0.515 (0.03) 0.565 (0.03) 0.576 (0.03) 0.600 (0.02) 0.609 (0.02)

0.221 (0.24) 0.334 (0.11) 0.404 (0.07) 0.425 (0.05) 0.501 (0.04) 0.534 (0.03) 0.550 (0.03) 0.584 (0.02) 0.613 (0.02) 0.614 (0.02)

0.223 (0.25) 0.339 (0.11) 0.414 (0.06) 0.445 (0.06) 0.510 (0.03) 0.524 (0.03) 0.574 (0.03) 0.581 (0.03) 0.591 (0.02) 0.605 (0.02)

0.225 (0.25) 0.343 (0.11) 0.418 (0.06) 0.478 (0.05) 0.510 (0.04) 0.550 (0.03) 0.555 (0.03) 0.586 (0.03) 0.610 (0.02) 0.611 (0.02)

0.230 (0.25) 0.344 (0.11) 0.419 (0.07) 0.468 (0.06) 0.529 (0.04) 0.559 (0.03) 0.564 (0.03) 0.583 (0.02) 0.595 (0.03) 0.633 (0.02)

0.234 (0.25) 0.360 (0.11) 0.425 (0.07) 0.474 (0.06) 0.543 (0.04) 0.563 (0.03) 0.564 (0.03) 0.569 (0.03) 0.584 (0.02) 0.653 (0.01)

0.248 (0.22) 0.360 (0.11) 0.430 (0.07) 0.495 (0.05) 0.534 (0.04) 0.560 (0.03) 0.571 (0.03) 0.581 (0.03) 0.614 (0.02) 0.686 (0.01)

0.234 (0.25) 0.360 (0.12) 0.435 (0.07) 0.475 (0.06) 0.548 (0.04) 0.561 (0.03) 0.575 (0.03) 0.589 (0.03) 0.629 (0.02) 0.704 (0.01)

case, various pairs of parameter values C and s are obtained. However, some cannot properly capture the structure of in-control data. Therefore, we generated artificial out-of-control data from independent uniform distributions where their parameters are estimated by the maximum and minimum values of each variable. That is, we find the optimal parameters to minimize the Type II error rate from artificial data when the actual Type I error rate is same as the expected Type I error rate. As shown in Table 1, Type II error rates are listed along with Type I error rates (in parentheses). For instance, in the normal distribution case, parameters C = 0.04 and s = 4 were considered the best because the decision boundary of SVDD with these parameters produced the minimum Type II error rate when the actual Type I error rate is 0.05 (Table 1). Note that the actual Type I error rates with a value of 0.05 are underlined. Tables 1, 2, 3, and 4, respectively, represent the Type II error rates from artificial data and the actual Type I error rates for selection of parameters C and s under normal, lognormal, 𝑡, banana-shaped distributions. The corresponding decision boundaries for each of the four scenarios are illustrated in Fig. 6. After we obtained the optimal parameters C and s, we selected values for parameters 𝜆 and d that yielded an actual false alarm rate closest to the expected false alarm rate. Tables 5, 6, 7, and 8 list the actual Type I error rates of the time-adaptive SVDD chart under normal, lognormal, t, and banana-shaped distribution cases, respectively. We compare the performances by changing the parameters 𝜆 and d with the expected Type I error rate as 0.05. For example, in normal distribution case, the actual Type I error rate of time-adaptive SVDD chart is similar to 0.05 when we set parameters 𝜆 = 0.02 and d = 0.25.

Fig. 3. Structure of time-adaptive SVDD chart.

example of parameter selection when the actual Type I error rate is 0.05. It is preferable that the expected Type I error rate of a control chart is the same as or close to the actual Type I error rate. Therefore, we find the optimal parameters using greedy search to minimize the difference between the expected and actual Type I error rates. In this

Fig. 4. Overview of the proposed method.

22

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 5. In-control observations from (a) normal distribution, (b) lognormal distribution, (c) t-distribution, and (d) banana-shaped distribution with 0.003 shift size with every new in-control observation. Table 2 Actual Type II error rates from artificial out-of-control dataset and Actual Type I error rates over different choices of parameters C and s of time-adaptive SVDD-based chart under the lognormal distribution when expected Type I error rate is 0.05. The boldface indicates the best parameters. C

s 1

2

3

4

5

6

7

8

9

10

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.035 (0.26) 0.054 (0.13) 0.070 (0.07) 0.076 (0.06) 0.086 (0.05) 0.093 (0.05) 0.095 (0.05) 0.093 (0.05) 0.093 (0.05) 0.093 (0.05)

0.048 (0.24) 0.066 (0.11) 0.075 (0.07) 0.086 (0.05) 0.101 (0.06) 0.109 (0.05) 0.120 (0.04) 0.123 (0.03) 0.123 (0.03) 0.121 (0.03)

0.060 (0.25) 0.080 (0.12) 0.098 (0.09) 0.115 (0.07) 0.125 (0.05) 0.123 (0.05) 0.128 (0.05) 0.139 (0.04) 0.139 (0.04) 0.138 (0.04)

0.070 (0.25) 0.100 (0.12) 0.130 (0.08) 0.148 (0.05) 0.153 (0.04) 0.144 (0.05) 0.151 (0.04) 0.160 (0.04) 0.153 (0.04) 0.160 (0.03)

0.075 (0.24) 0.113 (0.12) 0.146 (0.08) 0.185 (0.05) 0.214 (0.04) 0.221 (0.03) 0.231 (0.02) 0.226 (0.02) 0.239 (0.02) 0.233 (0.02)

0.078 (0.25) 0.121 (0.12) 0.164 (0.08) 0.204 (0.05) 0.236 (0.03) 0.255 (0.03) 0.265 (0.02) 0.273 (0.02) 0.293 (0.02) 0.296 (0.02)

0.079 (0.24) 0.125 (0.12) 0.181 (0.08) 0.213 (0.05) 0.244 (0.04) 0.279 (0.03) 0.285 (0.03) 0.305 (0.02) 0.333 (0.02) 0.346 (0.02)

0.079 (0.25) 0.141 (0.11) 0.189 (0.08) 0.223 (0.05) 0.256 (0.04) 0.306 (0.03) 0.313 (0.03) 0.343 (0.02) 0.356 (0.02) 0.373 (0.03)

0.078 (0.25) 0.141 (0.11) 0.189 (0.08) 0.234 (0.05) 0.279 (0.04) 0.324 (0.03) 0.330 (0.03) 0.366 (0.02) 0.378 (0.02) 0.409 (0.02)

0.078 (0.25) 0.149 (0.11) 0.190 (0.08) 0.235 (0.05) 0.288 (0.04) 0.338 (0.03) 0.343 (0.03) 0.381 (0.02) 0.390 (0.02) 0.428 (0.02)

setup, we generated normal mean shift from t = 1 to t = 900 for time-varying situations in four simulation cases, normal, lognormal, 𝑡, and banana-shaped distributions. It can be observed that the original SVDD-based control charts could not appropriately accommodate the time-varying patterns on the normal change range. Hence, many alarms were generated. Conversely, the monitoring statistics for the proposed

4.3. Simulation results 4.3.1. Comparison results between the original SVDD-based and timeadaptive SVDD-based charts Fig. 7 shows the patterns of the monitoring statistics of the original SVDD-based and time-adaptive SVDD-based charts. In the simulation 23

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Table 3 Actual Type II error rates from artificial out-of-control dataset and Actual Type I error rates over different choices of parameters C and s of time-adaptive SVDD-based chart under the t-distribution when expected Type I error rate is 0.05. The boldface indicates the best parameters. C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

s 1

2

3

4

5

6

7

8

9

10

0.035 (0.52) 0.070 (0.46) 0.086 (0.27) 0.085 (0.44) 0.085 (0.46) 0.088 (0.38) 0.085 (0.41) 0.085 (0.45) 0.085 (0.4) 0.088 (0.37)

0.034 (0.26) 0.071 (0.16) 0.103 (0.23) 0.125 (0.13) 0.151 (0.16) 0.163 (0.32) 0.173 (0.12) 0.173 (0.11) 0.166 (0.31) 0.173 (0.12)

0.034 (0.25) 0.068 (0.12) 0.100 (0.09) 0.124 (0.10) 0.151 (0.09) 0.154 (0.15) 0.168 (0.21) 0.179 (0.08) 0.184 (0.12) 0.195 (0.13)

0.036 (0.25) 0.070 (0.12) 0.093 (0.09) 0.129 (0.07) 0.163 (0.05) 0.166 (0.04) 0.166 (0.04) 0.185 (0.03) 0.186 (0.04) 0.201 (0.06)

0.035 (0.25) 0.069 (0.12) 0.095 (0.08) 0.134 (0.06) 0.149 (0.05) 0.175 (0.04) 0.181 (0.04) 0.194 (0.03) 0.206 (0.03) 0.206 (0.03)

0.036 (0.25) 0.074 (0.12) 0.100 (0.08) 0.133 (0.06) 0.141 (0.05) 0.169 (0.04) 0.186 (0.04) 0.186 (0.04) 0.208 (0.03) 0.208 (0.03)

0.039 (0.25) 0.076 (0.12) 0.104 (0.08) 0.123 (0.06) 0.150 (0.05) 0.165 (0.04) 0.181 (0.04) 0.186 (0.03) 0.196 (0.03) 0.231 (0.02)

0.039 (0.25) 0.079 (0.12) 0.103 (0.08) 0.130 (0.06) 0.148 (0.05) 0.165 (0.04) 0.185 (0.04) 0.196 (0.03) 0.201 (0.03) 0.229 (0.02)

0.040 (0.25) 0.080 (0.13) 0.100 (0.08) 0.128 (0.06) 0.146 (0.05) 0.171 (0.04) 0.179 (0.04) 0.198 (0.03) 0.211 (0.03) 0.220 (0.03)

0.043 (0.24) 0.081 (0.12) 0.098 (0.09) 0.133 (0.06) 0.166 (0.04) 0.174 (0.04) 0.186 (0.04) 0.204 (0.03) 0.206 (0.03) 0.225 (0.02)

Table 4 Actual Type II error rates from artificial out-of-control dataset and Actual Type I error rates over different choices of parameters C and s of time-adaptive SVDD-based chart under the banana-shaped distribution case when expected Type I error rate is 0.05. The boldface indicates the best parameters. C

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

s 1

2

3

4

5

6

7

8

9

10

0.211 (0.47) 0.328 (0.34) 0.401 (0.22) 0.414 (0.2) 0.411 (0.24) 0.365 (0.4) 0.416 (0.2) 0.414 (0.16) 0.401 (0.27) 0.399 (0.3)

0.246 (0.25) 0.351 (0.14) 0.370 (0.16) 0.440 (0.08) 0.418 (0.17) 0.428 (0.18) 0.495 (0.09) 0.515 (0.06) 0.485 (0.11) 0.469 (0.13)

0.250 (0.25) 0.356 (0.12) 0.414 (0.09) 0.455 (0.06) 0.478 (0.06) 0.496 (0.04) 0.500 (0.04) 0.518 (0.03) 0.529 (0.03) 0.544 (0.02)

0.334 (0.25) 0.451 (0.11) 0.470 (0.09) 0.506 (0.06) 0.515 (0.05) 0.529 (0.04) 0.529 (0.04) 0.540 (0.03) 0.561 (0.02) 0.565 (0.02)

0.364 (0.25) 0.486 (0.12) 0.539 (0.08) 0.558 (0.06) 0.576 (0.05) 0.590 (0.04) 0.591 (0.04) 0.601 (0.03) 0.620 (0.03) 0.625 (0.03)

0.395 (0.25) 0.519 (0.13) 0.560 (0.08) 0.586 (0.06) 0.594 (0.05) 0.613 (0.04) 0.618 (0.04) 0.650 (0.03) 0.653 (0.03) 0.656 (0.03)

0.408 (0.25) 0.546 (0.1) 0.575 (0.08) 0.593 (0.06) 0.613 (0.05) 0.629 (0.04) 0.651 (0.04) 0.661 (0.03) 0.664 (0.03) 0.670 (0.03)

0.421 (0.25) 0.530 (0.12) 0.610 (0.06) 0.609 (0.06) 0.623 (0.05) 0.644 (0.04) 0.661 (0.04) 0.680 (0.03) 0.679 (0.03) 0.690 (0.03)

0.426 (0.25) 0.545 (0.12) 0.596 (0.08) 0.629 (0.05) 0.635 (0.05) 0.664 (0.04) 0.664 (0.04) 0.688 (0.03) 0.701 (0.03) 0.706 (0.03)

0.434 (0.25) 0.550 (0.13) 0.601 (0.08) 0.631 (0.06) 0.636 (0.05) 0.661 (0.04) 0.685 (0.03) 0.699 (0.03) 0.704 (0.02) 0.738 (0.02)

Table 5 Actual Type I error rates over different choices of parameters 𝝀 and d of time-adaptive SVDD-based chart under normal distribution when expected Type I error rate is 0.05. The boldface indicates the best parameters. 𝜆

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

d 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.122 0.093 0.132 0.129 0.129 0.127 0.807 0.841 0.941 0.951

0.109 0.072 0.077 0.076 0.080 0.697 0.807 0.939 0.942 0.951

0.107 0.070 0.076 0.078 0.084 0.701 0.807 0.939 0.944 0.951

0.107 0.067 0.069 0.074 0.078 0.701 0.817 0.939 0.944 0.946

0.103 0.065 0.072 0.077 0.078 0.701 0.817 0.939 0.947 0.951

0.102 0.068 0.071 0.076 0.081 0.702 0.817 0.939 0.947 0.951

0.102 0.067 0.071 0.077 0.087 0.703 0.821 0.939 0.947 0.951

0.102 0.067 0.071 0.077 0.081 0.704 0.821 0.939 0.948 0.951

0.102 0.068 0.071 0.078 0.081 0.704 0.822 0.940 0.950 0.953

0.102 0.068 0.071 0.079 0.088 0.704 0.822 0.940 0.950 0.953

Table 6 Actual Type I error rates over different choices of parameters 𝝀 and d of time-adaptive SVDD-based chart under lognormal distribution when expected Type I error rate is 0.05. The boldface indicates the best parameters. 𝜆

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

d 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.208 0.307 0.302 0.346 0.394 0.766 0.776 0.930 0.938 0.938

0.112 0.166 0.301 0.221 0.232 0.789 0.774 0.911 0.937 0.954

0.112 0.181 0.170 0.309 0.308 0.789 0.801 0.911 0.944 0.954

0.092 0.163 0.174 0.228 0.164 0.789 0.801 0.927 0.944 0.954

0.090 0.157 0.097 0.132 0.118 0.789 0.801 0.929 0.944 0.954

0.090 0.094 0.097 0.119 0.118 0.789 0.801 0.929 0.944 0.954

0.088 0.093 0.097 0.119 0.117 0.789 0.801 0.929 0.952 0.954

0.088 0.093 0.098 0.116 0.118 0.792 0.841 0.930 0.952 0.954

0.088 0.093 0.094 0.110 0.120 0.792 0.841 0.929 0.952 0.954

0.088 0.089 0.096 0.110 0.120 0.792 0.841 0.936 0.952 0.954

time-adaptive SVDD-based control charts were relatively stable, leading to fewer false alarms than the original SVDD-based chart. Further, both monitoring statistics of control charts from t = 901 to t = 1000 stayed over the control limit. That is, the time-adaptive SVDD-based chart can distinguish between a normal time-varying process and real faults. These results clearly demonstrated the advantage of the proposed chart in time-varying situations.

Moreover, we examined the average of the actual false alarm rate with different expected false alarm rates (0.01–0.1); the result is presented in Fig. 8. A chart with similar actual and expected false alarm rates can be considered superior. That is, control charts that are close to the 45degree line are preferred. Fig. 8 confirms that the time-adaptive SVDDbased charts are closer to the 45-degree line, while the traditional 24

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 6. SVDD boundaries from (a) normal distribution, (b) lognormal distribution, (c) t-distribution, and (d) banana-shaped distribution.

Table 7 Actual Type I error rates over different choices of parameters 𝝀 and d of time-adaptive SVDD-based chart under t-distribution when expected Type I error rate is 0.05. The boldface indicates the best parameters. 𝜆

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

d 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.067 0.069 0.211 0.570 0.729 0.848 0.934 0.947 0.947 0.947

0.066 0.069 0.214 0.570 0.760 0.848 0.934 0.947 0.947 0.947

0.066 0.069 0.072 0.570 0.760 0.852 0.934 0.947 0.950 0.954

0.066 0.069 0.074 0.570 0.760 0.853 0.934 0.947 0.950 0.954

0.066 0.069 0.074 0.570 0.760 0.853 0.934 0.947 0.950 0.954

0.066 0.069 0.242 0.570 0.760 0.853 0.934 0.947 0.950 0.954

0.067 0.069 0.074 0.572 0.760 0.847 0.934 0.947 0.950 0.954

0.067 0.069 0.074 0.572 0.760 0.853 0.934 0.947 0.950 0.954

0.067 0.069 0.258 0.572 0.760 0.853 0.937 0.947 0.951 0.954

0.067 0.069 0.072 0.572 0.760 0.853 0.937 0.947 0.951 0.954

cases, indicating that the proposed time-adaptive SVDD-based chart could significantly reduce the number of false alarms and correctly detect process faults compared with the original SVDD-based chart. In control charts, the average run length (ARL) is widely used in performance comparisons (Li et al., 2014). However, we felt that the use of ARL was not appropriate in our case because correlated and time-dependent process observations could cause the results of ARL

SVDD-based charts deviate significantly from the 45-degree line. In the t-distribution case, because of the long tails in this distribution, the control limit calculated by the bootstrapping method has a relatively large value. Hence, the original SVDD-based control chart in the tdistribution case demonstrates superior performance compared to the other simulation cases in terms of actual false alarm rates. In our simulation scenarios, Type II error rates were observed as zero in all 25

S. Lee, S.B. Kim

Engineering Applications of Artificial Intelligence 68 (2018) 18–31 Table 8 Actual Type I error rates over different choices of parameters 𝝀 and d of time-adaptive SVDD-based chart under banana-shaped distribution case when expected Type I error rate is 0.05. The boldface indicates the best parameters. 𝜆

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

d 0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.088 0.108 0.148 0.207 0.653 0.714 0.898 0.937 0.951 0.956

0.091 0.108 0.150 0.218 0.701 0.718 0.813 0.938 0.952 0.957

0.091 0.108 0.152 0.423 0.687 0.774 0.814 0.944 0.952 0.957

0.091 0.109 0.151 0.426 0.693 0.756 0.937 0.944 0.952 0.957

0.091 0.110 0.149 0.231 0.678 0.713 0.937 0.944 0.952 0.957

0.091 0.107 0.151 0.229 0.679 0.722 0.937 0.944 0.952 0.957

0.091 0.109 0.151 0.228 0.686 0.719 0.937 0.947 0.952 0.957

0.091 0.110 0.151 0.228 0.687 0.719 0.937 0.947 0.952 0.957

0.091 0.109 0.153 0.244 0.687 0.719 0.937 0.947 0.952 0.957

0.091 0.109 0.158 0.246 0.687 0.720 0.937 0.947 0.952 0.957

Fig. 7. Comparison of monitoring statistics between original SVDD-based and time-adaptive SVDD-based charts in four simulation scenarios (𝜶 = 0.05): (a) normal distribution, (b) lognormal distribution, (c) t-distribution, and (d) banana-shaped distribution.

to be unreliable (Montgomery, 2011). Therefore, we compared the performance of the control charts in terms of Type I and Type II error rates.

Thus, in this section, we compare the time-adaptive SVDD-based chart with the existing adaptive GMM-based chart described in Introduction. Fig. 9 shows the performance of the two control charts in terms of false alarm rates. Note that the time-adaptive SVDD-based charts are closer to the 45-degree line and more stable than the adaptive GMM-based charts. Especially, in the lognormal distribution and tdistribution cases, the false alarm rates of the adaptive GMM-based chart were significantly higher because a major limitation of GMM is its lack of robustness to outliers (Svensen and Bishop, 2005). Moreover, the number of false alarms with the adaptive GMM-based chart rapidly increased corresponding to an increase of the expected false alarm rate.

4.3.2. Comparison results between an adaptive GMM chart and timeadaptive SVDD-based chart The results of the previous simulations demonstrated that proposed time-adaptive SVDD-based chart reduced the number of false alarms compared with a traditional SVDD-based control chart. We did not consider the comparative results surprising because the proposed chart has an updating feature, whereas the original SVDD-based chart does not. 26

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 8. Changes in average actual Type I error rates corresponding to different expected Type I error rates for time-adaptive SVDD-based chart and original SVDD-based chart with four simulation cases: (a) normal distribution, (b) lognormal distribution, (c) t-distribution, and (d) banana-shaped distribution.

The main reason for the increasing false alarm in the GMM charts is the updating scheme. Because the adaptive GMM-based chart is updated when a new observation belongs to the in-control region, this control chart can be updated for the data as opposed to in the time-varying direction. Further, as the control limit is smaller, the probability that the model is updated becomes smaller, leading to many false alarms. Therefore, this scheme may be inappropriate to reflect time-varying patterns with a large expected false alarm rate. Compared with the adaptive GMM-based chart, the main advantage of the proposed timeadaptive SVDD-based chart is not only the correctness of its monitoring statistics, but also its rational false alarm rates, which are the result of its updating structure.

The CNC machining step accounts for half of all the steps and is the most important in the overall process. In the CNC process, all parts are cut from rolled aluminum using a machine tool. The number of steps for cutting and post-processing depends on the component being manufactured. These steps are performed simultaneously in thousands of facilities leading to complex patterns in the data. For this reason, in this study, we focused on the CNC step in the overall process. These quality characteristics are responsible for the quality of the final product. Therefore, an efficient and reliable monitoring tool is required to detect variations caused by various causes in the processes. 5.2. Experimental results of the case study For the experiment, we used 800 observations characterized by 12 quality characteristics from the CNC process to compare the performance of the SVDD-based and time-adaptive SVDD-based charts. We used 400 in-control observations recorded in the early stage of the process to construct the control charts. A total of 400 observations (12 out-of-control and 388 in-control) were used to evaluate the control charts in terms of false alarms and misdetection rates. Fig. 11 displays three examples of time-varying variables in the incontrol CNC process. In this process, the original SVDD-based chart can generate a large number of false alarms because it does not accommodate the time-varying patterns. Because of confidentiality agreements with the company that provided this dataset, we could not disclose the names of the variables and their explanations in this paper. The monitoring results of the original SVDD-based chart and the proposed time-adaptive SVDD-based chart when the actual Type I error rate is 0.05 are shown in Fig. 12(a) and (b), respectively. The majority

5. Case study: metal frame process in mobile device manufacturing 5.1. Description of metal frame manufacturing process We applied the proposed time-adaptive SVDD-based control chart to a real case of a metal frame process in mobile device manufacturing. Recently, the use of a metal frame of devices in the mobile industry has rapidly increased. However, the metal frame process is one of the difficult manufacturing processes because it not only has complex processes but also requires sophisticated control of equipment during the entire machining time. The manufacturing of a metal frame consists of more than 20 steps. The main processes are computerized numerical control (CNC) machining and anodizing, and each process requires several repetitions. Fig. 10 is an overview of the metal frame process in mobile device manufacturing. 27

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 9. Changes in average actual Type I error rates corresponding to different expected Type I error rates of a time-adaptive SVDD-based chart and adaptive GMM chart with four simulation cases: (a) normal distribution, (b) lognormal distribution, (c) t-distribution, and (d) banana-shaped distribution.

Fig. 10. Overview of metal frame process in mobile device manufacturing.

of the monitoring statistics of the SVDD-based chart are higher than those of the time-adaptive chart. In particular, after the 270th time point, the SVDD-based chart tended to generate more false alarms than the proposed chart. The actual fault is marked with a red asterisk. Both charts could detect the true process faults, except one out-ofcontrol observation at the 24th time point. This result confirms that the proposed control chart accommodates the normal gradual changes, yet detects real process faults effectively.

The performances of the time-adaptive SVDD-based and GMM-based charts were evaluated with the CNC process (Fig. 13). Fig. 13(a) indicates the behavior of the actual Type I error rates from the adaptive GMM-based chart and the proposed time-adaptive SVDD-based chart over a range of expected Type I error rates (0.01∼0.1). It is preferable that the control chart has similar expected and actual Type I error rates. The proposed time-adaptive SVDD-based chart produced more similar values between the actual and expected Type I error rates 28

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 11. Example of time-varying variables in the CNC process for metal frame manufacturing.

Fig. 12. Monitoring statistics from (a) original SVDD-based control chart and (b) time-adaptive SVDD-based control chart with CNC process data.

29

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Engineering Applications of Artificial Intelligence 68 (2018) 18–31

Fig. 13. Comparison between an adaptive GMM-based chart and time-adaptive SVDD-based chart with CNC process data: (a) actual Type I error rates corresponding to expected Type I error rates and (b) actual Type II error rates corresponding to actual Type I error rates.

than the adaptive GMM-based chart. This demonstrates that in a real manufacturing process with a time-varying pattern, the proposed timeadaptive SVDD chart yield less number of false alarms than the GMMbased chart. Moreover, Fig. 13(b) indicates the actual Type II error rates corresponding to the actual Type I error rates of the two charts. Lower Type II error rates, given the similar level of Type I error rates, are considered to represent superior performance and a better method. The proposed time-adaptive SVDD-based chart outperformed the GMMbased control chart in terms of Type II error rates. Note that because the two charts have different ranges of actual Type I error rates, the values of the x-axis in Fig. 13(b) between the two charts are not exactly matched. These results indicate that the proposed time-adaptive SVDD-based chart is superior to the existing adaptive GMM-based chart in more appropriately handling time-varying patterns in a real manufacturing process.

Foundation of Korea funded by the Ministry of Science, ICT and Future Planning (NRF-2016R1A2B1008994), and the Ministry of Trade, Industry & Energy under Industrial Technology Innovation Program (R1623371). References Chakour, C., Harkat, M.F., Djeghaba, M., 2015. Neuronal principal component analysis for nonlinear time-varying processes monitoring. IFAC-PapersOnLine 48 (21), 1408– 1413. Choi, S.W., Martin, E.B., Morris, A.J., Lee, I.B., 2006. Adaptive multivariate statistical process control for monitoring time-varying processes. Ind. Eng. Chem. Res 45 (9), 3108–3118. Ferracuti, F., Giantomassi, A., Iarlori, S., Ippoliti, G., Longhi, S., 2015. Electric motor defects diagnosis based on kernel density estimation and Kullback–Leibler divergence in quality control scenario. Eng. Appl. Artif. Intell. 44, 25–32. Gani, W., Taleb, H., Limam, M., 2011. An assessment of the kernel-distance-based multivariate control chart through an industrial application. Qual. Reliab. Eng. Int. 27 (4), 391–401. Ge, Z., Song, Z., 2008. Online monitoring of nonlinear multiple mode processes based on adaptive local model approach. Control. Eng. Pract. 16 (12), 1427–1437. Ge, Z., Song, Z., 2013. Multivariate Statistical Process Control: Process Monitoring Methods and Applications. Springer, London. Gitlow, H.S., 2009. A Guide To Lean Six Sigma Management Skills. CRC Press. Grasso, M., Colosimo, B.M., Semeraro, Q., Pacella, M., 2015. A comparison study of distribution-free multivariate SPC methods for multimode data. Qual. Reliab. Eng. Int. 31 (1), 75–96. Haimi, H., Mulas, M., Corona, F., Marsili-Libelli, S., Lindell, P., Heinonen, M., Vahala, R., 2016. Adaptive data-derived anomaly detection in the activated sludge process of a large-scale wastewater treatment plant. Eng. Appl. Artif. Intell. 52, 65–80. Hu, J., Zhang, L., Cai, Z., Wang, Y., 2015. An intelligent fault diagnosis system for process plant using a functional HAZOP and DBN integrated methodology. Eng. Appl. Artif. Intell. 45, 119–135. Hwang, W.Y., Lee, J.S., 2015. Shifting artificial data to detect system failures. Int. Trans. Oper. Res. 22 (2), 363–378. Ketelaere, B.D., Mertens, K., Mathijs, F., Diaz, D.S., Baerdemaeker, J.D., 2011. Nonstationarity in statistical process control - issues, cases, ideas. Appl. Stoch. Model. Bus. Ind. 27 (4), 367–376. Khediri, I.B., Weihs, C., Limam, M., 2012. Multimodal and complex process monitoring using kernel methods. In: 3th Meeting on Statistics and Data Mining. pp. 115-121. Kim, S.B., Sukchotrat, T., Park, S.K., 2011. A nonparametric fault isolation approach through one-class classification algorithms. IIE Trans. 43, 505–517. Kumar, S., Choudhary, A.K., Kumar, M., Shankar, R., Tiwari, M.K., 2006. Kernel distancebased robust support vector methods and its application in developing a robust Kchart. Int. J. Prod. Res. 44 (1), 77–96. Lee, Y.H., Jin, H.D., Han, C., 2006. On-line process state classification for adaptive monitoring. Ind. Eng. Chem. Res. 45 (9), 3095–3107. Li, W., Yue, H.H., Valle-Cervantes, S., Qin, S.J., 2000. Recursive PCA for adaptive process monitoring. J. Process. Control 10 (5), 471–486. Li, Z., Zou, C., Gong, Z., Wang, Z., 2014. The computation of average run length and average time to signal: an overview. J. Stat. Comput. Simul. 84 (8), 1779–1802. Liu, X., Kruger, U., Littler, T., Xie, L., Wang, S., 2009. Moving window kernel PCA for adaptive monitoring of nonlinear processes. Chem. Intell. Lab. Syst. 96 (2), 132–143. Liu, Y., Pan, Y., Wang, Q., Huang, D., 2015. Statistical process monitoring with integration of data projection and one-class classification. Chem. Intell. Lab. Syst. 149, 1–11.

6. Conclusions This paper proposed a time-adaptive SVDD-based control chart that can address time-varying situations. We modified an SVDD formulation by adding a time-weighting factor and suggested an updating region of the control chart for efficient model updating. Because the proposed control chart can be promptly adjusted to time-varying patterns, it is expected that unwanted false alarms can be reduced in process control. The simulations and case study from the metal frame of mobile device manufacturing process demonstrated the efficiency and applicability of the proposed chart. A comparison of the monitoring results confirmed that the proposed chart can appropriately address nonnormally distributed-time-varying processes. Conventional SPC techniques cannot benefit from available out-ofcontrol data. However, if a model can train it, a control chart will be sensitive to all types of out-of-control processes. Using the proposed weighting mechanism of the proposed method, available out-of-control observations can be located outside of the sphere. Future work will be focused on how to determine the weight of real fault observation in time-varying situations. Moreover, some more efforts are still needed to isolate faulty variables. We believe the proposed concept can be extended to research on other OCC algorithm-based monitoring methods with a time-varying approach. Acknowledgments The authors would like to thank the editor and reviewers for their useful comments and suggestions, which were greatly help in improving the quality of the paper. This research was supported by Brain Korea PLUS, Basic Science Research Program through the National Research 30

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