Unit Interval Time and Magnitude Monitoring Using Beta and Unit Gamma Distributions

Quick detection of an assignable cause is necessary for process accuracy with respect to the specifications. +e aim of this study is to monitor the time and magnitude processes based on unit-interval data. To this end, maximum exponentially weighted moving average (Max-EWMA) control chart for simultaneous monitoring time andmagnitude of an event is proposed. To be precise, beta and unit gamma distributions are considered to develop theMax-EWMA chart.+e chart’s performance is accessed using average run length (ARL), the standard deviation of run length (SDRL), and different quantiles of the run length distribution through extensive Monte Carlo simulations. Besides a comprehensive simulation study, the proposed charting methodology is applied to a real data set. +e results show that the proposed chart is more efficient in detecting small to medium-sized shifts. +e results also indicate that simultaneous shifts are detected more quickly as compared to the pure shift.


Introduction
Statistical quality control (SQC) is the science of using statistical techniques in maintaining and monitoring the quality level of products and services. ere are two main components of SQC: acceptance sampling and statistical process control (SPC). Acceptance sampling is a technique of selecting a lot (of products, goods, and raw material) based on the result (or quality) of selected samples. On the other hand, SPC uses sampling along with other statistical methods to assess the quality of output of a manufacturing process. A graphical representation known as the control chart of the process helps to determine whether the process output achieves the acceptable quality level or not.
In SPC, monitoring of a process to check if it is incontrol (IC) or out-of-control (OOC) is done only after the stable process parameters, including mean, fraction nonconforming, or variance, are obtained. Control charts have long been and extensively used in the areas of manufacturing to monitor a process' stability, and recently, their applications have extended beyond the manufacturing processes.
For instance, control charts have been used to monitor the stability of stock price or the quality of a service. Furthermore, they are used in disease surveillance systems to assess whether a disease has reached to the pandemic level or not.
One of the techniques used in SPC to monitor a process is the Shewhart approach where the process is termed as IC or OOC based on the result of a current sample. is technique usually performs better when we have large size disturbances in a process. Since this technique completely ignores the past information (because of which this scheme is also called memoryless charting scheme), it cannot detect small shifts in the process. Contrary to Shewhart approach, the memory-type charting schemes (such as cumulative sum (CUSUM) and exponentially weighted moving average (EWMA)) consider not only the current but also the past information about the process. is feature enables these charts to detect small shifts in the process.
Many real-life settings involve data of rates or proportions, e.g., monitoring the ratio of ingredients of life-saving drugs, etc., and hence, there arises a need to develop the charting mechanism for such types of data. e data of rates and proportions may not be resulted from a Bernoulli process, and therefore, control charts other than the traditional attribute charts, such as p and np charts, are required. For instance, monitoring the rate of unemployment to ensure the better utilization of resources in public politics can be performed with these charts. While dealing with different processes, magnitude associated with an event occurrence plays an important role to define its quality. is study aims at presenting the simultaneous monitoring methodology of time and magnitude based on the distributions having support between 0 and 1. To this end, the exponentially weighted moving average (EWMA) charting mechanism with a slight modification is used in order to develop the proposed chart. Different shift sizes are introduced, and the average speed, known as the average run length (ARL) to measure the average of the out-of-control signals, is used. e average run length is the most widely and frequently used criterion for evaluating the performance of a control chart. Our proposed chart is flexible and can be applied to many real-life scenarios.
For Bernoulli data, attribute control charts such as p chart is applied to monitor the fraction and the number of nonconforming items, respectively [1]. An advantage of an attribute control chart over variable control chart is that it is economical and less time-consuming compared to the former because attribute data collection is much easier than variable data. Ho and Quinino [2] used an attribute control chart to monitor the variability of a process. ere are a number of practical situations where one might be interested in the monitoring of proportions or rates in a product; however, the available data set may not follow the Bernoulli distribution [3]. e data of proportions and rates can be described as collection of observations where the variable of interest is confined to an interval of (0,1). Such data can be found extensively in many fields of study and often analyzed using normal distribution because of simplicity associated with normal distribution, yet misleading conclusions [4].
Certain distributions are available in the literature which can be utilized to deal with proportion data. One of such distributions is known as the beta distribution. Sant'Anna and Ten Caten [5] considered this distribution along with Shewhart strategy to construct the beta control chart for monitoring fractional type variables of a manufacturing process. Later, Lima-Filho et al. [6] constructed an inflated beta control chart. Apart from beta distribution, the unit gamma (UG) can exhibit constant, unimodal, increasing, and decreasing shapes, whereas its hazard function has bathtub and decreasing shapes [7]. Grassia [8] gave a detailed overview of this distribution as well as of its variants. Ratnaparkhl and Mosimann [9] studied this distribution to derive some new distributions. e UG distribution is similar to the beta distribution in terms of some of the properties, and hence, some of its applications are the same as that of the beta distribution. is distribution can be used to estimate the density of virus or bacteria. Tadikamalla [10] showed that the UG distribution can be used as an alternative for Johnson SB and beta distributions. Recently, Mousa et al. [11] developed a regression model using this distribution.
Time-between events (TBE) charts have recently been the focus of many researchers as these charts are very effective in monitoring high quality processes such as high yield production lines with extremely small defect rates [12]. Several TBE control charts are available in literature [13]. ese charts also overcome the drawbacks of traditional Shewhart charting schemes, including high false alarm rate, the assumption of normality, and negative lower limit. Ali [14] proposed the TBE control chart based on the renewal process assuming a continuous exponentiated family of distributions. e authors concluded that the proposed chart is more efficient and flexible when compared to existing TBE charts based on exponential distribution. For some recent contributions in this direction, we refer to the works of Xie et al. [15], Zhang et al. [16], Shafae et al. [17], Shah et al. [18], Talib et al. [19], and references cited therein.
Wu et al. [20] developed a rate chart for monitoring time and magnitude simultaneously by using the ratio of magnitude and time as the plotting statistic. Practically, during a certain time period, the effect of presence of an event, i.e., the damage or disturbance it can cause, is determined by its frequency or by its magnitude. us, there arises a need to monitor both of these characteristics (frequency and magnitude) simultaneously. Taking this into account, Wu et al. [21] adopted a Shewhart control charting technique for monitoring the frequency and magnitude of an event simultaneously. In earlier studies, simultaneous monitoring was done using two separate charting statistics. is technique was later found to be less effective when compared to the charts based on single statistic except for some exceptional situations. Most of the work on simultaneous monitoring is based on Shewhart mechanism [22,23]. However, Shewhart charts are of memoryless nature, and the historical information of the process is ignored as the new points are plotted [24]. As a result, these charts cannot detect small shifts in process parameters satisfactorily. Consequently, memory-type control charts such as exponentially weighted moving average chart (EWMA) and cumulative sum (CUSUM) chart are required. e EWMA chart is more sensitive as compared to the Shewhart chart when it comes to reflecting critical past information. In addition, unlike Shewhart and CUSUM schemes, the EWMA chart is not sensitive to nonnormality of data for smaller values of the smoothing parameter [25]. Most recently, Sanusi et al. [26] introduced the maximum EWMA (Max-EWMA) control chart to simultaneously monitor the frequency and magnitude of events where the frequency was assumed to follow the exponential and magnitude followed a gamma distribution. e rest of the article is organized as follows. e probability density functions of the beta and unit gamma distributions with transformed parameters to have the same means are discussed in Section 2. Max-EWMA chart using the beta and unit gamma distributions is defined in Section 3. e charting procedure is also defined in the same section. Section 4 presents the in and out of control performance of the Max-EWMA chart. Real data examples are discussed in Section 5, while concluding remarks are given in Section 6.

Beta and Unit Gamma Distribution
For the proposed Max-EWMA control chart, we consider the beta and unit gamma distribution in this section to provide in-depth analysis of the unit data monitoring problem.

Beta Distribution.
A random variable T follows a beta distribution with shape parameters δ > 0 and c > 0, if its probability density function (PDF) is given by where B(δ, c) � (Γ(δ)Γ(c)/Γ(δ + c)) is called the beta function and Γ(δ) � ∞ 0 ω δ− 1 e − ω dt. e mean and variance of T are respectively. Suppose, we transform the parameters in equation (1) as Now, using equation (3), the PDF of the beta distribution in equation (1) takes the form Furthermore, the mean and the variance of T with transformed parameters are E(T) � μ and Var(T) � (μ(1 − μ)/ϕ + 1), respectively.

Unit Gamma Distribution.
A random variable X follows the unit gamma distribution with shape parameter τ > 0 and rate parameter θ > 0, i.e., X ∼ uG(τ, θ), if it has the following PDF: e mean and variance associated with equation (5) are respectively. Let us transform θ in the above PDF as θ � (μ 1/τ /(1 − μ 1/τ )). en, the PDF of X takes the form where τ > 0 and 0 < μ < 1. Under this parameterization, we have E(X) � μ and Var(X)

Max-EWMA Control Chart Using Beta and Unit Gamma Distributions
To jointly monitor unit interval time and magnitude, the Max-EWMA charting procedure is adopted. e plotting statistic of the Max-EWMA control chart consists of the maximum value of two independent EWMA statistics, both in absolute terms. For joint monitoring of the TBE (T) and magnitude (X) of the event, the maximum of absolute values of TBE and magnitude is plotted. When an OOC signal is observed, the chart has the capability to identify whether the shift has occurred in T or X.
In particular, this study considers that the TBE follows the beta distribution, T ∼ B(δ, c), and magnitude of the event follows the unit gamma distribution, X ∼ uG(τ, θ). It is also assumed that the time between the occurrences of two successive disturbances and magnitude is independent. Furthermore, the process is assumed to start at time t � 0, and the IC parameters are assumed to be (δ 0 , c 0 ) and (τ 0 , θ 0 ). Let the shifts in the parameters of T and X, respectively, are represented as follows, δ � Δ δ δ 0 , c � Δ c c 0 , τ � Δ τ τ 0 , and θ � Δ θ θ 0 , respectively, where Δ δ , Δ c , Δ τ , and Δ θ represent the shifts in the shape parameters of the beta distribution and shape and rate parameter of the unit gamma distribution, respectively. To this end, we have further considered two cases.

Case 1(a): Shifts in Shape Parameters of Beta and Unit
Gamma Distribution. In the first case, shifts in the first shape parameter δ of the beta distribution and shape parameter τ of unit gamma distribution are considered. e process is termed as IC when both Δ δ and Δ τ are equal to one. e following two independent statistics are defined in the construction of the Max-EWMA chart e EWMA statistics, based on U i and V i , are defined as where λ e ∈ (0, 1] is one of the parameters of the EWMA control chart called the smoothing constant (the other parameter is known as the constant multiplier L). e starting Journal of Mathematics 3 values of two EWMA statistics are Y 0 � E(U i ) � 0 and Z 0 � E(V i ) � 0, respectively. Traditionally, a small value of λ e is considered to be suitable for detecting small shifts, whereas a large value is deemed appropriate to detect large shifts. Practically, λ e ∈ [0.05, 0.25] is considered to detect small or medium-sized shifts in the process [27]. e monitoring statistic of the Max-EWMA control chart comprises the maximum of absolute of Y i and Z i , i.e., If there is shift in either T (shape1 parameter of the beta distribution) or X (shape parameter of the unit gamma distribution) or in both parameters, the M i statistic becomes larger as compared to when there is no shift in either of the parameters. Furthermore, the proposed Max-EWMA chart only has the upper control limit (UCL) due to the fact that the plotting statistic is real and nonnegative number. e UCL is computed as Here, E(M) and Var(M) are the IC values of mean and variance of the plotting statistic, and L is the constant multiplier that controls the width of control limit. When E(M) and Var(M) are unknown, one can replace them with sample mean and variance of M, calculated from the IC historical data. e Max-EWMA chart can detect the upward shift (UW) shift as well as the downward (DW) shift in the X and/or T simultaneously.

Case 1(b): Shifts in the Shape Parameter of Beta Distribution and Rate Parameter of the Unit Gamma Distribution.
In the second case, shifts in c parameter of the beta and θ parameter of the unit gamma distribution are considered. Similar to the first case, the process is IC when both Δ c and Δ θ are equal to one.

Charting Procedure for the Max-EWMA Control Chart.
It is of utmost importance to identify the variables (T and/or X) being the cause of the OOC signal once it is triggered by the chart. As mentioned previously, follow-up approach, once an OOC state has been established, can be labelled easily using the Max-EWMA chart. e summary of the construction of the Max-EWMA control chart for simultaneous monitoring of frequency and magnitude is given as follows: (1) Initialize Y 0 � Z 0 � 0 and specify the desired ARL 0 value. Also, select the suitable value of constant multiplier L and smoothing parameter λ e to achieve the desired ARL 0 . Different values of L, for different combinations of τ, μ, and ϕ are given in Table 1

Performance of the Proposed Chart
e average run length (ARL) is the most common measure to access the performance of a control chart. It is defined as the average of run length, i.e., average number of subgroups (samples) plotted until an OOC signal is observed. Two types of ARL are defined in the literature: the in-control ARL (ARL 0 ) and the out-of-control ARL (ARL 1 ). Under ideal conditions, ARL 0 should be as large as possible so that unnecessary pauses in the production line can be avoided. On the other hand, the smaller the value of ARL 1 , the better the performance of the control chart. In fact, in comparative analysis, the chart with a smaller value of ARL 1 is considered the best. Since the ARL distribution is skewed, we need to study its standard deviation (SDRL) as well as the percentiles (q10, q25, q50, q75, and q90) of the run length. erefore, a detailed analysis of the ARL along with its standard deviation (SDRL) and different percentiles of run length distribution are given in Tables 3 and 4 and Table S1 (Supplementary Materials) corresponding to Case 1(a), and in Tables S2-S4 corresponding to Case 1(b). Even though our interest lies in the simultaneous monitoring of the magnitude and TBE, shifts in a single parameter of each distribution, keeping IC the corresponding parameter of other distribution, is also considered. For example, shift in shape 1 parameter of the beta distribution is considered in Table 3 keeping the shape parameter of the unit gamma distribution in-control. Furthermore, shift in shape parameter of unit gamma, keeping IC the shape1 parameter of the beta distribution, is considered in Table 4. Finally, Journal of Mathematics simultaneous shifts in shape1 parameter of the beta and shape parameter of unit gamma distribution is considered in Table S1. Similarly, for Case 1(b), a shift in shape 2 parameter of the beta distribution, keeping IC the rate parameter of the unit gamma distribution is considered in Table S2. Shift in the rate parameter of the unit gamma distribution, keeping IC the shape 2 parameter of the beta distribution, is considered in Table S3. Finally, simultaneous shifts in both the parameters are considered in Table S4.
To be specific, we considered the shift size Δ ∈ (1, 1.10, 1.20, 1.30, 1.40, 1.90, 0.80, 0.70, 0.60, 0.50). e shifts are considered in the percentage form, that is, the value of Δ � 0.90 means that there is a 10% decrease in the parameter value, whereas the value Δ � 1.10 means that the value of the parameter is increased by 10%. e value Δ � 1 corresponds to no shift in the parameter value. e IC values of mean and variance of plotting statistic M, i.e., EM and VM are also given in the tables. e analysis of the proposed scheme is conducted under the following general conditions. For all of the cases given below, we have μ � 0.20 ARL 0 is fixed at 370 and 500 to obtain L. e results are obtained using R software (R version 4.0.0, 2020-04-24).

Results and Discussion.
e analysis listed in Tables 3, 4  and Table S1 is conducted to assess the performance of the proposed control chart under Case 1(a). More specifically, we have considered pure shifts in shape 1 parameter of T in Table 3. For the set of parameters given in Case 1, when λ e � 0.05 and ARL 0 � 370, the ARL is observed to reduce by 94.46%, 97.62%, 98.45%, and 98.82% with respect to the nominal ARL value when the shape 1 parameter of T is increased by 10%, 20%, 30%, and 40%, respectively, and reduced by 94.73%, 97.83%, 98.64%, 99.00%, and 99.20% when the shape 1 parameter of T is increased by 10%, 20%, 30%, 40%, and 50%, respectively. For the value of λ e � 0. 10 .73% for the DW shifts in the shape parameter of X. e interpretation of the remaining cases can be done similarly. According to the ARL criteria, the chart based on pure shift in the shape parameter of X performs better than the one based on pure shift in shape 1 parameter of T.
For simultaneous shifts in the two parameters of X and T, Table S1, the similar degenerating pattern of ARL is observed for increasing UW and DW shifts. Specifically, when Δ τ � 1.10 and Δ δ ∈ (1.10, 1.20, 1.30, 1.40) in Case 1, i.e., simultaneous 10% UW shift in shape parameter of X and 10%, 20%, 30%, and 40% UW shift in the shape 1 parameter of T, the ARL is reduced by 97.35%, 97.90%, 98.47%, and 98.83%, respectively. On the other hand, when Table 2: Symbols representing the source of the out-of-control signal.

Time
Magnitude Table  3: Max-EWMA results considering shift in the shape1 parameter of beta distribution (Δ δ ∈ (1.10,       e percentile analysis shows that, in most cases, the ARL is greater than the median (q50) but less than the 75 th percentile (q75) suggesting that the run length distribution is positively skewed. For example, when Δ δ � 1.10, λ e � 0.05, and ARL 0 � 370 in Case 1 of Table 3, the ARL is 20.497, the median is 18, and q75 is 25. Interestingly, in some cases the ARL is slightly less than the median which means that the distribution is slightly negatively skewed. For example, in case 1 of Table 3 assuming Δ δ � 1.30 and λ e � 0.05, the ARL is 5.741 and the median is 6; however, in Table 4, when Δ τ � 1.30 and λ e � 0.05 in case 1, the ARL is 3.675 and the median is 4. us, the proposed chart is more sensitive to small shifts (Δ δ , Δ τ ∈ (1.10, 0.90)) for pure or simultaneous shifts with λ e � 0.05. It is to be noted that the sensitivity to moderate shifts increased when λ e increased. For example, in case 1 of  Table S1 when λ e ∈ (0.05, 0.15) the ARL 1 is (9.83,8.28) for 10% simultaneous UW shift in both parameters. It is worth noting that the behavior of the ARL does not conform to the traditional fashion, that is, small smoothing parameters are suitable for small shifts and vice versa. As this discrepancy exists only for a few ARL 1 values, it is safe to say that smaller values of λ e are suitable for detecting smaller shifts and larger values are suitable to detect larger shifts.
In Table S2, we have considered pure shift in the shape 2  parameter of T, whereas in Table S3, pure shift in the rate parameter of X is considered. Finally, a simultaneous shift in shape 2 parameter of T and rate parameter of X is considered in Table S4. From the tables, we observed a very similar degenerating pattern in the ARL as well as in the SDRL as noted earlier. More specifically, in the case of pure shift, we can see that an UW shift in the shape 1 parameter of T is detected more quickly as compared to the UW shift in the shape 2 parameter of T. On the other hand, a DW shift in shape 2 is detected more quickly. For example, when λ e � 0.05, ARL 0 � 370, and Δ δ ∈ (1.30, 1.40) in Case 1 of Table 3, ARL 1 � (5.741, 4.354), and when Δ c � (1.30, 1.40) under similar settings in Table 4, ARL 1 � (6.735, 5.278). On the other hand, when Δ δ ∈ (0.90, 0.80) in Case 1, ARL 1 � (19.524, 8.024) and ARL 1 � (17.912, 6.971) when Δ c � (0.90, 0.80) in Case 1. While comparing the efficiency of the proposed chart for detecting pure shifts in shape and rate parameter of X, we can see that an UW shift in the rate parameter is detected more quickly, whereas a DW shift in the shape parameter of X is detected more quickly. e performance of the chart for simultaneous shifts in Table S1 is more or less the same as given in Table S4.

Real Data Application
In this section, the proposed methodology is applied to a real data set taken from a disaster management department that controls fire [20]. Controlling fire in an efficient manner leaves a positive impact than an uncontrolled occurrence of fire that may cause severe destruction. A disaster management department always tries to monitor the damages inflicted by the fire outbreaks to minimize the future losses.
ere are 25 observations in the data set including two variables: the amount of losses (in 1000$) and the time interval (in days) between the successive fire outbreaks. Both of these variables are transformed into proportion. e magnitude (X) represents the proportion of losses, whereas the TBE (T) represents the proportion of days between the successive fire outbreaks. Our interest here is to detect an increase in the amount of losses and/or decrease in the time between the fire outbreaks. e variable T is fitted to beta distribution, and variable X is fitted to unit gamma distribution. To estimate the parameters, we have used the maximum likelihood approach, and all the calculations are done using the statistical software R. Apart from the first 25 observations which are used as the phase I sample, we have further simulated 25 values as the phase II observations by introducing shifts of Δ δ � 0.80 and Δ τ � 0.80 to exhibit the application of the proposed charting scheme. Empirical mean and variance of the plotting statistic are 0.254 5 and 0.021 4, respectively. Using λ e � 0.15 and L � 1.90, we obtained UCL � 0.5325 with ARL 0 � 25. e maximum likelihood estimates of the parameters along with their standard errors (in parentheses), the Akaike information criterion (AIC) and Bayesian information criterion (BIC), for the 25 IC observations are presented in Table 5. Figure 1 presents the Max-EWMA control chart constructed for the data set  (phase I and phase II). ere appears no signal in phase I data as the chart gives no OOC signal; however, the first OOC signal is observed at the 28 th sample. When a shift in the fire outbreaks, such as this one, is observed, the disaster management department can look into the reasons behind the phenomenon and try to eliminate them. Furthermore, we have constructed the Max-EWMA chart for 80% pure shift in the parameters. Specifically, when there is a pure shift in the shape 1 parameter of T, Figure 2 shows that the first OOC signal is triggered at the 34 th sample. On the other hand, if there is a pure shift in the shape parameter of the unit gamma distribution, Figure 3, the first OOC signal is triggered at the 30 th sample. ese charts indicate that the simultaneous shift is detected more quickly as compared to the pure shift. e OOC points are labelled with respective labels to identify the signals.

Conclusion
With the advancement in technology, many human centric jobs have been taken over by machines and it is imperative to monitor the output of these machines to assess whether the performance level is up to the expectations or not. e SPC provides tools and techniques which enabled us to distinguish between assignable and natural causes in a process. A number of techniques have already been developed to monitor processes with different assumptions, distributional forms, and range of data involved; however, there are very few studies related to unit interval data. erefore, motivated by the importance of unit interval data, we introduced a Max-EWMA chart for simultaneous monitoring of the magnitude and TBE of an event under the assumption that TBE follows the beta distribution and magnitude of the event follows unit gamma distribution. In order to evaluate the performance of the proposed chart, the most frequently used criterion, i.e., the ARL has been used. e results indicate that the overall performance of the proposed chart is substantially good for small to medium-sized shifts. e current study also emphasizes on the fact that simultaneous monitoring of magnitude and frequency is more efficient as compared to individual monitoring of both these characteristics. Finally, the charting procedure is applied to a real life data set. In the future, the current work can be extended by considering some new distributions dealing with unit interval data. Furthermore, the effect of parameter estimation on phase I and phase II may be studied in detail. Also, a recent idea of neutrosophic statistics [28] can be utilized to extend the present work.
Data Availability e dataset used in the analysis is available upon request from the first author.