A Residual Control Chart Based on Convolutional Neural Network for Normal Interval-Censored Data

Abstract

To reduce reliability testing time, experiments are often terminated at a predetermined time, producing right-censored lifetime data. Alternatively, when test samples are inspected at fixed intervals, failures are only observed within these intervals, resulting in interval-censored lifetime data. Although quality control methods for right-censored data are well established, relatively little attention has been given to interval-censored observations. Motivated by the success of residual control charts based on convolutional neural network (CNN) for right-censored data, this study extends the chart for monitoring normally distributed interval-censored lifetime data. Simulation results based on average run length (ARL) indicate that the proposed method outperforms the traditional exponentially weighted moving average (EWMA) chart in detecting decreases in mean lifetime. The findings also highlight the practical benefits of employing high- or low-order autoregressive CNN models depending on the magnitude of process shifts.

1. Introduction

Quality control (QC) plays a critical role across industries by ensuring process stability, product consistency, and customer satisfaction. Effective quality control not only reduces production costs and enhances operational efficiency but also strengthens competitive advantage and safeguards the long-term survival of enterprises. Among various QC activities, monitoring product lifetime is particularly important, as it directly reflects product reliability and performance.

Product lifetime information is typically collected through reliability testing. To reduce testing time and cost, practitioners often terminate tests at a pre-specified end time, regardless of whether all test units have failed. As a result, the observed lifetime data are frequently subject to censoring. Right-censored data, where the exact failure times of unfailed units are unknown but exceed a certain threshold, are especially common in reliability tests.

To address right-censored data in statistical process control, Steiner and Mackay introduced the conditional expected value (CEV) to estimate the lifetimes of unfailed samples and incorporated it into the $\overline{X}$ control chart framework for monitoring both normal and nonnormal processes [1,2]. Subsequent studies integrated the CEV approach with EWMA control charts to improve sensitivity to small and moderate shifts. Raza et al. proposed a double EWMA CEV chart for detecting scale changes in gamma-distributed censored data, which was later extended in various directions [3]. Other researchers replaced CEV with the conditional median (CM) to construct control charts for Weibull and generalized exponential censored data [4,5,6,7]. Further developments include EWMA CEV charts for window-censored data, as well as cost-based design approaches [8,9].

Despite these advances, traditional control charts rely on reducing sample information to a single summary statistic, which limits their ability to capture complex temporal patterns and evolving process dynamics. With the rapid development of artificial intelligence, machine learning and deep learning methods have been increasingly adopted to enhance quality monitoring performance [10,11,12,13,14,15,16,17,18]. Unlike conventional statistical methods, deep learning models can automatically extract nonlinear features and temporal dependencies from sequential data.

Recent studies have applied neural networks to the design of control charts, particularly for censored data. Deep-learning-based residual control charts have been developed by modeling in-control EWMA CEV or CM statistics and monitoring the residuals. These convolutional neural network (CNN)-based approaches have demonstrated superior detection performance compared with traditional EWMA CEV or CM charts, especially for detecting mean decreases under high levels of right censoring [19,20].

Compared with right-censored data, interval-censored data are also widely encountered in reliability testing, yet they have received relatively little attention in quality control research. An observation is said to be interval-censored when its exact value is unknown but is known to lie within a specific interval. In reliability tests, interval-censored data typically arise when inspections are conducted at fixed time intervals. If a unit is found to have failed at an inspection, the failure is only known to have occurred between two inspection times rather than at an exact moment.

For example, consider a reliability test involving three test units, where functionality is inspected every hour. At the end of the first hour, one unit is found to have failed, indicating that the failure occurred between 0 and 1 h. No failures are observed during the second hour. During the third hour, the remaining two units fail. In this scenario, the recorded data consist of the number of failures within each inspection interval. Such interval-censored data differ fundamentally from right-censored data, which only indicate that certain units have survived beyond a given time point. Beyond reliability testing, interval-censored data also commonly appear in medical studies, social science surveys, animal experiments, and field observations.

Although right-censored data can be viewed as a special case of interval-censored data, most existing monitoring methods are tailored exclusively to right-censored observations. Monitoring techniques specifically designed for interval-censored data remain largely underdeveloped, despite their practical importance and rich information content. This limitation highlights a significant research gap in the quality control literature [21,22].

The normal distribution is widely used in quality control to model lifetime or quality characteristics when variability arises from the aggregation of multiple independent effects or when wear-out mechanisms dominate. Moreover, many traditional control charts, including Shewhart, EWMA, and CUSUM charts, are originally developed under the assumption of normality. In practical reliability testing, interval-censored normal data commonly arise when failure times are subject to periodic inspections, limited measurement resolution, or discretized recording schemes, while the underlying lifetime variability can still be reasonably approximated by a normal distribution. From a methodological perspective, focusing on normal interval-censored data allows the proposed control method to be developed within a well-established statistical framework, thereby providing a clear and interpretable setting for investigating the effects of interval censoring on monitoring performance.

Motivated by these considerations, this study extends the work of Lee and Liao to develop a novel control method specifically designed for normal interval-censored data [19]. The proposed approach aims to fully exploit the information contained in interval-censored observations while maintaining compatibility with classical quality control frameworks. Average run length (ARL) simulations are conducted to evaluate the monitoring performance of the proposed method under various process shift scenarios and censoring conditions. The results demonstrate that the proposed control scheme provides improved detection performance compared with existing methods, thereby offering an effective and practical solution for monitoring interval-censored lifetime data.

2. EWMA Control Chart

Assuming that practitioners conduct a sample inspection every $\gamma$ hours for a reliability test, the $\kappa$-th inspection interval is $\left(\left.{Hq}_{\kappa -1},{Hq}_{\kappa }\right]\right.$, where $\gamma ={Hq}_{\kappa }-{Hq}_{\kappa -1}$ and ${Hq}_0=0$. For a sample of size $n$, let the number of failures in the inspection interval $\kappa$ be $q_{\kappa }$, and then ${\sum }_{\kappa =1}^{\infty }{q}_{\kappa }=n$. Figure 1 illustrates an example of interval-censored data. Three samples ($n=3$) undergo a life test. The inspector checks the samples’ functionality every hour. In the first inspection, Sample #2 is found to have failed, and $q_1=1$ is recorded. The inspector only knows that this sample failed between 0 and 1 h, but not its actual failure time. The second inspection finds no failed samples, so $q_2=0$ is recorded. The remaining two samples are found to have failed in the third inspection, so $q_3=2$ is recorded. The inspector only knows that these two samples failed between ${Hq}_2$ and ${Hq}_3$, but not their actual failure time.

Let the lifetime of test samples follow a normal distribution with a mean ${\mu }_0$ and a variance ${\sigma }_0^2$, the inspection rate $R$ is $\gamma /{\mu }_0$, and Meeker and Escobar used the midpoint (MP) of the inspection interval to estimate the lifetime of this failed sample [23]. MP of $\kappa$-th inspection interval is

$${MP}_{\kappa }=\left({Hq}_{\kappa }+{Hq}_{\kappa -1}\right)/2$$

In the example shown in Figure 1, assume that the lifetime of these three samples follows a normal distribution, the midpoint of the first test interval is ${MP}_1=\left(1+5\right)/2=0.5$. Using the same method, we can obtain ${MP}_2=1.5$ and ${MP}_2=2.5$.

Truncated normal distribution is a probability distribution derived from that of a normal random variable by bounding the random variable from below (${Hq}_{\kappa -1}$) and above (${Hq}_{\kappa }$) [24]. The expected value of truncated normal distribution can be the CEV of a normal distribution with an inspection interval $\left(\left.{Hq}_{\kappa -1},{Hq}_{\kappa }\right]\right.$. CEV for $\kappa$-th inspection interval is

$${CEV}_{\kappa }={\mu }_0-{\sigma }_0{\displaystyle \frac{\phi \left(\frac{{Hq}_{\kappa }-{\mu }_0}{{\sigma }_0}\right)-\phi \left(\frac{{Hq}_{\kappa -1}-{\mu }_0}{{\sigma }_0}\right)}{\mathsf{\Phi }\left(\frac{{Hq}_{\kappa }-{\mu }_0}{{\sigma }_0}\right)-\mathsf{\Phi }\left(\frac{{Hq}_{\kappa -1}-{\mu }_0}{{\sigma }_0}\right)}}$$

The CM of a normal random variable is

$${CM}_{\kappa }={\mu }_0+{\sigma }_0{\mathsf{\Phi }}^{-1}\left\{\left[\mathsf{\Phi }\left({\displaystyle \frac{{Hq}_{\kappa }-{\mu }_0}{{\sigma }_0}}\right)+\mathsf{\Phi }\left({\displaystyle \frac{{Hq}_{\kappa -1}-{\mu }_0}{{\sigma }_0}}\right)\right]/2\right\}$$

where $\phi \left(·\right)$ and $\mathsf{\Phi }\left(·\right)$ are the probability density function and cumulative distribution function of standard normal distribution, respectively. ${\mathsf{\Phi }}^{-1}\left(·\right)$ is the inverse of cumulative distribution function of standard normal distribution. For the derivation of ${CM}_{\kappa }$, please refer to Appendix A.

Assuming $x_{\kappa }$ is ${MP}_{\kappa }$, then the sample mean $\overline{X}$ of size $n$ is

$$\overline{X}={\sum }_{\kappa =1}^n{q}_{\kappa }{x}_{\kappa }/n$$

Replacing $x_{\kappa }$ with ${CEV}_{\kappa }$ or ${CM}_{\kappa }$ can calculate the sample mean $\overline{X}$ of CEV or CM.

Zhang and Chen presented that the EWMA statistic at sampling period t for monitoring the mean decrease is:

$$E_t=min\left[{\mu }_0,\lambda {\overline{X}}_t+\left(1-\lambda \right)E_{t-1}\right]$$

(1)

where $\lambda$ is the smoothing coefficient of the EWMA statistic, ${\mu }_0$ is an in-control process mean, and $E_0={\mu }_0$ [25].

The process is running in a stable state, and its process mean is maintained at ${\mu }_0$, which is called an in-control state. When an assignable cause occurs to result in the decrease of lifetime mean that is reduced from ${\mu }_0$ to ${\mu }_1$, it can be indicated the out-of-control state. These two means are related mathematically by ${\mu }_1=\delta \times {\mu }_0$ where $\delta$ is a shift size coefficient. This study focuses on the monitoring of a product lifetime decrease, so a lower control limit (LCL) is set to detect the reduction of average lifetime.

ARL is always used as a measurement indicator to perform the monitoring efficiency of a control chart [12,16,18,25,26]. A larger in-control ARL value signifies a lower false-alarm rate. However, in an out-of-control state, a smaller ARL value indicates higher detection efficiency of mean reduction.

3. A Residual Control Chart Based on CNN

3.1. Set Up the Control Chart

CNN is a type of deep learning neural network that combines convolutional layers, activation layers, pooling layers, and a fully connected layer. The convolutional layer is mainly responsible for extracting features of the data. The activation layer usually considers the rectified linear unit (ReLU) function to perform non-linear processing on the output of the convolution layer. The pooling layer is usually used for decreasing the size of the feature map while retaining the most important feature information. Max pooling is a common pooling method. Each neuron in a fully connected layer is connected to all neurons in the previous layer and uses this information to make the final prediction.

Assuming that $w_{l-1}^{Con}$ be the filter kernel at layer $l-1$, the output of $l$-th convolutional layer ${\theta }_l^{Con}$ is:

$${\theta }_l^{Con}={\theta }_{l-1}^{Con}\times w_{l-1}^{Con}$$

The output of the $l$-th activation layer is

$${ReLU}_l=max\left(\begin{array}{cc}0,& {\theta }_l^{Con}\end{array}\right)$$

The max pooling at layer $l$ is given by:

$${Mp}_l=max\left({ReLU}_l\right)$$

Considering that the sample mean $\overline{X}$ is converted into EWMA statistics $E_t$, these EWMA sample sequence plotted on a chart may be not a linear time series data. This study proposes a CNN network architecture for a $k$-th order autoregressive model. The CNN network is composed of ten combination layers and one fully connected layer as shown in Figure 2. Each combination layer (Com. layer) consists of a one-dimensional convolution layer (Con1D), a ReLu layer and a max pooling layer. The input and output layers of this CNN are both sequence data. As a $k$-th order autoregressive model, the input layer considers three variables as $E_{t-1}$, $E_{t-2}$, …, $E_{t-k}$, and the variable of output layer is $E_t$. $E_t$ can be obtained by Equation (1).

Figure 3 presents a flow chart for the creation of a residual control chart based on CNN (CNN-based residual chart). For step 1, this study first generates the sample mean $\overline{X}s$ of $\mathrm{10,000}+k$ from an in-control state. Here, ${\mu }_0=5$ and ${\sigma }^2=1$ are considered to generate normal random data for the $\overline{X}$s of $\mathrm{10,000}+k$. In step 2, the $E_t$s of $\mathrm{10,000}+k$ can be calculated form the $\overline{X}$s of $\mathrm{10,000}+k$ using Equation (1). A dataset of 10,000 records is be created in step 3. Each record of the dataset includes three input variables as $E_{t-1}$, $E_{t-2}$, …, $E_{t-k}$ and an output variable $E_t$. For step 6, the residual is defined as:

$${\epsilon }_t=E_t-\widehat{E_t}$$

In step 7, practitioners need to first determine an appropriate in-control ARL value based on their own experience. In the literature, this in-control ARL value is usually set at 200 or 370.4. Then practitioners can randomly determine an LCL value and count the number of residual values that fall outside the LCL. The false alarm rate can be obtained by:

$$\mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e} \mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{m}=\left(\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r} \mathrm{o}\mathrm{f} \mathrm{r}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{u}\mathrm{a}\mathrm{l} \mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} \mathrm{f}\mathrm{a}\mathrm{l}\mathrm{l} \mathrm{o}\mathrm{u}\mathrm{t}\mathrm{s}\mathrm{i}\mathrm{d}\mathrm{e} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{L}\mathrm{C}\mathrm{L}\right)/\mathrm{10,000}$$

If the$\mathrm{in}\text{-}\mathrm{control} \mathrm{ARL}\ne 1/\mathrm{F}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{e} \mathrm{a}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{m}$, practitioners adjust the LCL value and then re-estimate the false alarm rate, otherwise the practitioners obtain the best LCL for the implementation of process monitoring.

3.2. Implementation of Control Chart

Based on the sample size and inspection interval for setting the control chart, the practitioner collects life test data and calculates the sample mean and EWMA statistic. EWMA statistics $E_{t-1}$, $E_{t-2}$, …, $E_{t-k}$ are inputted into the well-trained CNN network to output the predicted value of $E_t$ for the period t. Next, practitioners collect the next period (period t) of test data for calculating $E_t$ and then calculate the prediction error $e_t$, which is:

$$e_t=E_t-\widehat{E_t}$$

Finally, the prediction error $e_t$ is plotted on the CNN-based residual control chart. If the $e_t$ falls outside of LCL, the process is indicated as an out-of-control state, otherwise the process is an in-control state. The flow chart of implementing control chart is shown in Figure 4.

4. ARL Performance

4.1. ARL Simulation Procedure

ARL measurement can evaluate the monitoring performance of control charts to select the best control chart to monitor the process. The literature also presents simulation methods for EWMA control charts [4,5,6,7]. This section focuses on the simulation process description of the CNN-based residual chart.

In the simulation process, ${\mu }_1$, $R$, $n$ and ${\sigma }^2$ are first determined to generate the interval-censored data from normal distribution, and sample mean $\overline{X}$ can be obtained from the generated interval censored data. The next step is to generate $\overline{X}$s of $\mathrm{10,000}+k$ based on step 1 and then calculate the EWMA statistics for these $\overline{X}$s. Let $E_{t-1}$, $E_{t-2}$, …, $E_{t-k}$ be input variables and $E_t$ be an output variable for creating 10,000 records of the dataset. The dataset is inputted into a well-trained CNN network for outputting 10,000 predicted values of $E_t$. The next step is to calculate the prediction error for 10,000 predicted values of $E_t$ and plot these errors on the residual chart. From the residual chart, the first error that falls the outsize of LCL can be found out; let run length (RL) be the sample period of this error. The last step is to repeat the above process 10,000 times to obtain 10,000 RL values and calculate the ARL value form the 10,000 RL values. The simulation procedure is shown in Figure 5. This study used MATLAB R2025a to code the program for the simulation procedure of Figure 5. The deep learning toolbox of MATLAB R2025a can easily and quickly construct the CNN architecture in Figure 2, generate training data, and train the model. The kernel size and number of kernels for each network layer are set in reference [19]. The MATLAB code for the CNN architecture in this study is shown in Appendix B.

4.2. ARL Values for Comparison

Generally, the production process starts in an in-control state. After the process has been running for some sampling periods, it will be disturbed by occurrences of an assignable cause, and result in a shift in the mean. The measurement of out-of-control ARL value in this situation is the steady-state ARL (SSARL). If we assume that a variation occurs at the beginning of the process, the out-of-control ARL is measured immediately, which is the zero-state ARL (ZSARL). This section will exhibit SSARL and ZSARL to compare the EWMA chart and the CNN-based residual chart.

The initial mean ${\mu }_0$ of the process is 5 and the variance ${\sigma }^2$ is 1. The inspection rate $R$ set 3 levels as 0.25, 0.5 and 0.75 to indicates low, medium, and high rates, respectively. Let the shift size $\delta$ be 0.9, 0.8, 0.7, 0.6, 0.5 and 0.2, where $\delta$ = 0.9 to 0.7 are small shift sizes, $\delta$ = 0.6 and 0.5 are medium shift sizes, and $\delta$ = 0.2 is a large shift size. The EWMA smoothing coefficient $\lambda$ considers two values as 0.1 and 0.2 for both charts. The sample size for all charts is fixed at 5. Consider training 1st, 2nd, 3rd, and 4th order autoregressive CNN models ($k=1,2,3 \mathrm{a}\mathrm{n}\mathrm{d} 4$) to build residual charts and examine their out-of-control ARL values. The in-control ARL values of EWMA and CNN-based charts are fixed at 200.

Table 1. The LCLs of control charts for in-control ARL = 200.

$\mathit{\lambda }$		0.1			0.2
$\mathit{R}$		0.25	0.5	0.75	0.25	0.5	0.75
Zero-State
EWMA	CEV	4.7709	4.7986	4.8355	4.6439	4.6853	4.7312
	CM	4.7809	4.8231	4.8181	4.6612	4.7253	4.7241
	MP	4.7410	4.6838	4.8097	4.5985	4.5079	4.6459
CNN ($k=1)$	CEV	−0.1581	−0.1291	−0.1064	−0.2506	−0.1830	−0.1830
	CM	−0.1480	−0.1393	−0.1157	−0.2240	−0.1791	−0.1754
	MP	−0.1858	−0.2022	−0.1938	−0.2546	−0.3379	−0.3293
CNN ($k=2$)	CEV	−0.1454	−0.1250	−0.1269	−0.2245	−0.2037	−0.1908
	CM	−0.1695	−0.1024	−0.1149	−0.2344	−0.1901	−0.1729
	MP	−0.1715	−0.2066	−0.1809	−0.2814	−0.3445	−0.3025
CNN ($k=3$)	CEV	−0.1496	−0.1321	−0.1141	−0.2533	−0.2078	−0.1844
	CM	−0.1335	−0.1192	−0.1249	−0.2144	−0.1904	−0.1957
	MP	−0.1696	−0.1875	−0.1763	−0.2732	−0.3110	−0.3159
CNN ($k=4$)	CEV	−0.1401	−0.1396	−0.1109	−0.2628	−0.2130	−0.1931
	CM	−0.1367	−0.1073	−0.1281	−0.2411	−0.1772	−0.1842
	MP	−0.1582	−0.1883	−0.1765	−0.2641	−0.3314	−0.3088
Steady-State
EWMA	CEV	4.7689	4.7972	4.8345	4.6411	4.6857	4.7305
	CM	4.7788	4.8209	4.8156	4.6592	4.7233	4.7241
	MP	4.7385	4.6795	4.8075	4.5971	4.5065	4.6455
CNN ($k=1)$	CEV	−0.1395	−0.1401	−0.1258	−0.2554	−0.2176	−0.1833
	CM	−0.1502	−0.1336	−0.1069	−0.2574	−0.1899	−0.1673
	MP	−0.1721	−0.2236	−0.1795	−0.2774	−0.3232	−0.3000
CNN ($k=2$)	CEV	−0.1480	−0.1394	−0.1330	−0.2445	−0.2075	−0.1924
	CM	−0.1447	−0.1140	−0.1139	−0.2317	−0.1839	−0.1904
	MP	−0.1812	−0.2103	−0.1927	−0.2887	−0.3127	−0.3059
CNN ($k=3$)	CEV	−0.1406	−0.1324	−0.1185	−0.2459	−0.2120	−0.1881
	CM	−0.1483	−0.1126	−0.1052	−0.2381	−0.1928	−0.1807
	MP	−0.1773	−0.2004	−0.1853	−0.2741	−0.3178	−0.3011
CNN ($k=4$)	CEV	−0.1527	−0.1234	−0.1150	−0.2210	−0.1943	−0.1943
	CM	−0.1354	−0.1277	−0.1098	−0.2323	−0.1941	−0.1788
	MP	−0.1681	−0.1962	−0.1867	−0.2817	−0.3337	−0.3068

shows the LCLs of control charts for in-control ARL = 200.

Table 2. Zero-state ARL values for $\lambda =0.1.$

Control Charts	EWMA			CNN ($\mathit{k}=1$)			CNN ($\mathit{k}=2$)			CNN ($\mathit{k}=3$)			CNN ($\mathit{k}=4$)
	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP
$\delta$	$R=0.25$
0.9	7.24	7.22	7.25	7.50	7.28	7.84	6.82	8.07	6.47	5.72	5.73	5.82	4.77	4.77	4.73
0.8	3.29	3.30	3.28	2.68	2.64	2.77	1.87	2.15	1.80	1.27	1.26	1.35	1.08	1.11	1.10
0.7	2.23	2.20	2.22	1.48	1.46	1.51	1.06	1.10	1.04	1.00	1.00	1.00	1.00	1.00	1.00
0.6	1.79	1.78	1.79	1.06	1.04	1.07	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.41	1.41	1.41	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.83	2.82	2.83	2.45	2.40	2.53	2.13	2.39	2.05	1.83	1.83	1.86	1.64	1.65	1.64
$\delta$	$R=0.50$
0.9	8.54	8.57	8.56	8.25	10.09	9.48	8.51	8.04	8.24	7.13	8.27	6.87	7.09	6.44	6.08
0.8	3.93	3.88	3.96	3.07	3.64	3.57	2.49	2.45	2.64	1.75	1.95	1.73	1.37	1.27	1.30
0.7	2.65	2.51	2.71	1.71	2.02	2.08	1.21	1.21	1.32	1.02	1.06	1.03	1.01	1.00	1.00
0.6	2.03	1.89	2.09	1.15	1.30	1.37	1.01	1.01	1.02	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.65	1.45	1.75	1.01	1.05	1.05	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.01	1.00	1.02	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	3.30	3.22	3.35	2.70	3.18	3.09	2.54	2.45	2.54	2.15	2.38	2.10	2.08	1.95	1.90
$\delta$	$R=0.75$
0.9	10.03	10.49	10.29	11.29	12.51	11.97	11.66	11.83	11.16	9.46	12.44	10.72	10.00	12.33	9.83
0.8	3.78	4.17	3.19	3.21	3.79	2.68	2.56	2.95	1.92	1.88	2.49	1.74	1.60	2.02	1.46
0.7	2.31	2.59	1.81	1.60	1.96	1.31	1.18	1.30	1.04	1.03	1.11	1.03	1.01	1.03	1.00
0.6	1.72	2.00	1.31	1.13	1.28	1.03	1.00	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.39	1.75	1.08	1.01	1.06	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.01	1.25	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	3.37	3.71	3.11	3.21	3.60	3.17	3.07	3.18	2.85	2.56	3.17	2.75	2.60	3.06	2.55

,

Table 3. Zero-state ARL values for $\lambda =0.2.$

Control Charts	EWMA			CNN ($\mathit{k}=1$)			CNN ($\mathit{k}=2$)			CNN ($\mathit{k}=3$)			CNN ($\mathit{k}=4$)
	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP
$\delta$	$R=0.25$
0.9	6.86	6.79	6.92	6.77	6.61	6.47	5.34	6.56	5.99	6.18	5.05	5.52	5.92	4.88	4.96
0.8	2.79	2.79	2.78	2.04	2.08	2.02	1.40	1.52	1.43	1.23	1.15	1.18	1.10	1.08	1.09
0.7	1.88	1.86	1.87	1.14	1.17	1.14	1.01	1.01	1.01	1.00	1.00	1.00	1.00	1.00	1.00
0.6	1.42	1.42	1.41	1.00	1.00	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.11	1.11	1.11	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.51	2.49	2.52	2.16	2.14	2.11	1.79	2.02	1.90	1.90	1.70	1.78	1.84	1.66	1.67
$\delta$	$R=0.50$
0.9	8.43	8.43	8.41	9.51	8.92	8.02	7.70	7.88	8.86	8.02	7.22	7.62	7.18	7.31	6.55
0.8	3.39	3.31	3.42	2.44	2.76	2.65	1.84	1.86	2.12	1.51	1.52	1.51	1.32	1.30	1.23
0.7	2.18	2.10	2.29	1.27	1.40	1.40	1.06	1.09	1.12	1.02	1.01	1.01	1.00	1.00	1.00
0.6	1.56	1.51	1.81	1.03	1.05	1.05	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.20	1.16	1.43	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.96	2.92	3.06	2.71	2.69	2.52	2.27	2.31	2.52	2.26	2.12	2.19	2.08	2.10	1.96
$\delta$	$R=0.75$
0.9	10.14	10.37	11.04	9.51	9.81	12.23	10.72	8.86	11.35	9.49	9.73	10.38	9.58	9.07	9.99
0.8	3.15	3.56	3.17	2.44	2.64	2.51	2.07	2.01	2.01	1.61	1.83	1.71	1.48	1.46	1.38
0.7	1.81	2.12	1.80	1.27	1.34	1.20	1.09	1.06	1.07	1.01	1.03	1.02	1.01	1.01	1.01
0.6	1.30	1.59	1.31	1.03	1.04	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.07	1.28	1.08	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	3.08	3.32	3.23	2.71	2.80	3.16	2.81	2.49	2.90	2.52	2.60	2.68	2.51	2.42	2.56

,

Table 4. Steady-state ARL values for $\lambda =0.1.$

Control Charts	EWMA			CNN ($\mathit{k}=1$)			CNN ($\mathit{k}=2$)			CNN ($\mathit{k}=3$)			CNN ($\mathit{k}=4$)
	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP
$\delta$	$R=0.25$
0.9	6.24	6.24	6.22	6.22	7.22	6.58	5.27	5.64	6.32	4.51	4.58	5.03	4.84	5.03	4.29
0.8	2.83	2.80	2.80	2.24	2.46	2.32	1.53	1.58	1.67	1.18	1.17	1.25	1.09	1.13	1.08
0.7	1.91	1.91	1.92	1.33	1.38	1.32	1.02	1.02	1.03	1.00	1.00	1.00	1.00	1.00	1.00
0.6	1.51	1.52	1.52	1.04	1.03	1.03	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.24	1.23	1.24	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.46	2.45	2.45	2.14	2.35	2.21	1.80	1.87	2.00	1.61	1.63	1.71	1.66	1.69	1.56
$\delta$	$R=0.50$
0.9	7.44	7.57	7.61	8.23	9.69	8.90	8.13	7.46	7.98	5.97	6.76	6.15	5.60	7.87	6.11
0.8	3.31	3.29	3.35	2.96	3.22	3.15	2.32	2.12	2.18	1.46	1.63	1.61	1.21	1.55	1.30
0.7	2.22	2.18	2.30	1.68	1.83	1.88	1.18	1.13	1.17	1.02	1.03	1.03	1.00	1.00	1.00
0.6	1.70	1.65	1.78	1.20	1.20	1.26	1.01	1.02	1.01	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.40	1.30	1.50	1.02	1.03	1.03	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.01	1.00	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.85	2.83	2.92	2.68	2.99	2.87	2.44	2.29	2.39	1.91	2.07	1.97	1.80	2.24	1.90
$\delta$	$R=0.75$
0.9	9.04	9.07	9.89	11.66	10.12	10.88	11.40	10.49	11.62	10.16	9.46	10.64	8.62	9.72	11.11
0.8	3.27	3.42	3.02	3.22	3.04	2.34	2.46	2.36	2.06	1.78	1.81	1.71	1.38	1.59	1.50
0.7	1.98	2.10	1.74	1.52	1.54	1.19	1.16	1.18	1.07	1.03	1.03	1.02	1.01	1.01	1.01
0.6	1.51	1.63	1.26	1.11	1.11	1.01	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.25	1.43	1.07	1.01	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.12	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	3.01	3.13	3.00	3.25	2.97	2.90	3.01	2.84	2.96	2.66	2.55	2.73	2.33	2.55	2.77

and

Table 5. Steady-state ARL values for $\lambda =0.2.$

Control Charts	EWMA			CNN ($\mathit{k}=1$)			CNN ($\mathit{k}=2$)			CNN ($\mathit{k}=3$)			CNN ($\mathit{k}=4$)
	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP	CEV	CM	MP
$\delta$	$R=0.25$
0.9	6.47	6.30	6.28	6.39	6.37	6.03	5.15	5.60	5.63	4.80	5.39	5.47	4.21	4.64	4.83
0.8	2.54	2.53	2.50	1.93	1.94	1.82	1.29	1.39	1.33	1.16	1.16	1.17	1.06	1.06	1.07
0.7	1.69	1.66	1.67	1.11	1.13	1.10	1.01	1.01	1.01	1.00	1.00	1.00	1.00	1.00	1.00
0.6	1.29	1.29	1.29	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.07	1.07	1.07	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.34	2.31	2.30	2.07	2.07	1.99	1.74	1.83	1.83	1.66	1.76	1.77	1.54	1.62	1.65
$\delta$	$R=0.50$
0.9	7.83	7.81	7.77	8.41	7.66	7.52	7.33	7.42	7.64	7.45	7.68	6.40	8.19	6.60	6.55
0.8	3.01	3.00	3.07	2.51	2.31	2.30	1.78	1.77	1.78	1.49	1.47	1.40	1.38	1.30	1.20
0.7	1.97	1.93	2.05	1.32	1.27	1.27	1.06	1.06	1.07	1.01	1.02	1.01	1.00	1.01	1.00
0.6	1.44	1.41	1.59	1.04	1.03	1.03	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.14	1.13	1.28	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	2.73	2.71	2.79	2.55	2.38	2.35	2.20	2.21	2.25	2.16	2.19	1.97	2.26	1.99	1.96
$\delta$	$R=0.75$
0.9	9.75	9.36	10.71	10.58	9.97	10.85	9.94	10.54	10.68	9.43	8.81	9.76	8.19	7.78	9.99
0.8	3.03	3.10	3.05	2.57	2.41	2.36	1.90	2.08	1.77	1.55	1.55	1.51	1.38	1.41	1.40
0.7	1.82	1.86	1.72	1.25	1.32	1.18	1.06	1.07	1.04	1.03	1.02	1.01	1.00	1.00	1.00
0.6	1.40	1.41	1.26	1.03	1.02	1.01	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.5	1.19	1.18	1.07	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
0.2	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00	1.00
AARL	3.03	2.99	3.13	2.90	2.79	2.90	2.65	2.78	2.75	2.50	2.40	2.55	2.26	2.20	2.56

show the ZSARL and SSARL values for all charts. The AARL value is the average of ARL values for 6 shift sizes. Comparing AARL values, it can find out the CNN-based residual chart is more sensitive than EWMA chart to detect the reduction of mean lifetime. In most cases of small and medium shift sizes, the CNN-based residual chart is better than EWMA chart. There are only a few cases of small and medium shift sizes where EWMA chart is better than the CNN-based residual chart. For large shift size, EWAM and CNN-based residual charts have similar detection efficiency. As $R$ increases, the AARL values of EWMA and CNN-based residual charts slightly increase. For EWMA and CNN-based residual charts, the steady-state AARL values are slightly less than zero-state AARL values, but the difference is not significant. In most cases, increasing $\lambda$ will slightly decrease the AARL values of EWMA and CNN-based residual charts.

Comparing the AARL values of Table 2, Table 3, Table 4 and Table 5, it can be found there is little difference in detection efficiency when using MP, CEV, an-d CM to create control charts. MP is a reasonable choice to create EWMA or CNN-based residual charts because MP is easier to calculate than CEV and CM. Focusing on CNN-based residual chart with MP, the AARL value decreases slightly as $k$ increases. Comparing the ARL values of CNN-based residual chart with MP, it can see that at $\delta =0.9$, the ARL values decrease significantly with increasing $k$. The ARL values for the other shift sizes are not significantly different, or even remain at 1.

For monitoring normal interval-censored data, this study recommends that practitioners use MP to set up EWMA chart or CNN-based residual chart. If practitioners prefer CNN-based residual chart, unless there is a need for rapid detection of tiny shifts, it is recommended to train a low-order autoregressive CNN model to set up the residual chart, thereby reducing the computational load on the manufacturing execution system.

4.3. A Practical Case

In a continuous chemical reaction (such as free-radical reactions or oxidation reactions), an inhibitor is added to control the reaction rate and prevent undesired side reactions. The inhibitor is gradually consumed over time, and once it is depleted, the reaction rate increases markedly within a few hours, potentially causing the product to deviate from specifications. An inspector performs the analyses of the reaction rate or key component concentrations for the test samples every 2.5 h. If the reaction rate remains below the upper specification limit, the inhibitor is considered to be effective. If the reaction rate exceeds the upper specification limit for the first time, the inhibitor is considered to be depleted and the test sample is deemed to have failed. It is known that the lifetime follows a normal distribution with ${\mu }_0=5$ and ${\sigma }^2=1$. 5 samples were taken from each batch of production for testing, and the inspection rate $R$ is $\gamma /{\mu }_0=2.5/5=0.5$.

Table 6. The interval-censored lifetime data of inhibitor.

$\mathit{\kappa }$	$\mathit{\kappa }=1$	$\mathit{\kappa }=2$	$\mathit{\kappa }=3$	$\mathit{\kappa }=4$	$\overline{\mathit{X}}$	${\mathit{E}}_{\mathit{t}}$	$\widehat{{\mathit{E}}_{\mathit{t}}}$	${\mathit{e}}_{\mathit{t}}$
(${\mathit{H}\mathit{q}}_{\mathit{\kappa }-1},{\mathit{H}\mathit{q}}_{\mathit{\kappa }}$]	( $0,2.5$ ]	( $2.5,5$]	( $5,7.5$]	( $7.5,10$ ]
${\mathit{M}\mathit{P}}_{\mathit{\kappa }}$	${\mathit{M}\mathit{P}}_1=1.25$	${\mathit{M}\mathit{P}}_2=3.75$	${\mathit{M}\mathit{P}}_3=6.25$	${\mathit{M}\mathit{P}}_4=8.75$
Batches	${\mathit{q}}_1$	${\mathit{q}}_2$	${\mathit{q}}_3$	${\mathit{q}}_4$
#1	0	4	1	0	4.25	4.9250
#2	0	4	1	0	4.25	4.8575
#3	0	2	3	0	5.25	4.8968
#4	0	4	1	0	4.25	4.8321	5.0172	−0.1851
#5	0	2	3	0	5.25	4.8739	5.0561	−0.1822
#6	0	4	1	0	4.25	4.8115	4.9981	−0.1866
#7	0	2	3	0	5.25	4.8553	5.0418	−0.1864
#8	1	1	3	0	4.75	4.8448	5.0266	−0.1818
#9	0	1	4	0	5.75	4.9353	5.1188	−0.1835
#10	0	3	1	1	5.25	4.9668	5.1506	−0.1839
#11	0	3	2	0	4.75	4.9451	5.1308	−0.1857
#12	0	4	1	0	4.25	4.8756	5.0536	−0.1780
#13	0	2	3	0	5.25	4.9130	5.0953	−0.1823
#14	0	3	2	0	4.75	4.8967	5.0751	−0.1784
#15	0	2	3	0	5.25	4.9321	5.1146	−0.1825
#16	0	3	2	0	4.75	4.9139	5.0925	−0.1786
#17	0	2	3	0	5.25	4.9475	5.1304	−0.1830
#18	1	2	2	0	4.25	4.8777	5.0609	−0.1832
#19	0	1	4	0	5.75	4.9650	5.1484	−0.1834
#20	0	3	2	0	4.75	4.9435	5.1313	−0.1879
#21	1	4	0	0	3.25	4.7741	4.9624	−0.1883
#22	0	5	0	0	3.75	4.6717	4.8599	−0.1882
#23	0	5	0	0	3.75	4.5795	4.7683	−0.1887
#24	1	4	0	0	3.25	4.4466	4.6354	−0.1888
#25	0	5	0	0	3.75	4.3769	4.5652	−0.1883

shows the testing data for 25 batches. The inhibitor testing of each batch is conducted four inspections ($\kappa =1,\dots ,4$), and all failed samples can be observed in the four inspections.

For the first inspection ($\kappa =1$), the interval is $\left(\left.\mathrm{0,2.5}\right]\right.$. ${MP}_1=\left(2.5+0\right)/2=1.25$. The ${MP}_{\kappa }$ values for the other three intervals can be obtained using the same method. $q_1$ to $q_4$ represent the number of failures in the four intervals, respectively. The batch #1 had 4 failed samples found in the second inspection and 1 failed sample found in the third inspection, so $q_2=4$ and $q_3=1$. The sample mean of the batch #1 is $\overline{X}=\left(0\times 1.25+4\times 1.75+1\times 6.25+0\times 8.75\right)/5=4.25$. Practitioner uses $\lambda$ = 0.1 to calculate the EWMA statistic, so $E_{t=1}=0.1\times 4.25+\left(1-0.1\right)\times 5=4.9250$. The same method can be used to obtain the sample means and EWMA statistics of the other 24 batches, as shown in Table 6.

Since this reduction in mean lifetime always occurs in the initial stages of the process, practitioner set up an EWMA chart of zero-state to monitor the inhibitor’s lifetime; therefore, the LCL will be 4.6838 according to Table 1. The left side of Figure 6 shows the EWMA chart for the $E_t$ values of 25 batches. If practitioner prefers a 3rd-order CNN-based residual chart ($k=3$) to monitor mean lifetime reduction, the $E_t$ values of batches #1, #2, and #3 are input into the CNN network to output an $\widehat{E_t}$ value of 5.0172 for batch #4. The inspector obtains the lifetime data for batch #4 and calculates its $E_t$ value to be 4.8321, resulting in an error $e_t$ of −0.1851. The same method can be used to obtain the $e_t$ values for batches #5 to #25. Referring to Table 1, the LCL of this 3rd-order CNN-based residual chart with zero-state is −0.1875. The $e_t$ values of batches #4 to #25 are plotted on the residual chart, as shown on the right side of Figure 6.

Comparing the two charts in Figure 6, the EWMA chart detected a mean shift in batch #22, while the CNN-based chart detected the same shift in batch #20. After investigating process variation, practitioner discovered that an assignable cause occurred after batch #18, causing the mean shift. The EWMA chart required 4 batches to detect the shift, while the CNN-based chart detected it in just 2 batches. Therefore, the CNN-based chart is more efficient at detecting mean reduction than the EWMA chart.

5. Conclusions

This study proposes a CNN-based residual control chart for monitoring normally distributed interval-censored data, addressing the limited availability of effective monitoring methods for this commonly encountered data type. Simulation results based on average run length (ARL) show that the proposed method outperforms the traditional EWMA control chart in detecting process shifts. The monitoring performance of EWMA statistics constructed using the MP, CEV, and CM is found to be comparable. Owing to its computational simplicity, MP is recommended for practical implementation. The results also indicate that high-order autoregressive CNN models enhance sensitivity to small shifts, whereas low-order models provide adequate performance for medium to large shifts with reduced training complexity, offering useful guidance for model selection in practice. Future research may extend the proposed framework to interval-censored data under non-normal distributions and to mixed interval- and right-censored data commonly observed in reliability testing.