Shewhart-Type TBEA Charts for Monitoring Frequency and Amplitude with Symmetry Structure Under Generalized Weibull and Generalized Log-Logistic Distributions

Abstract

Control charts for monitoring time between events (T) and amplitude (X) have been developed in recent years. Many TBEA charts depend on limited models such as exponential, normal, and gamma distributions and mainly rely on the ratio statistic (${\displaystyle \frac{X}{T}}$). This representation ignores the symmetric relationship between event occurrence and event magnitude. This paper proposes Shewhart-type TBEA charts constructed from three statistics $\left(Z_1\right)$, $\left(Z_2\right)$, and $\left(Z_3\right)$ based on $\left(X\right)$ and $\left(T\right)$. The approach models symmetry between frequency and amplitude using generalized Weibull and generalized log-logistic distributions. The statistics maintain proportional invariance when both variables shift together, which enables balanced monitoring of the process. Several scenarios are examined for detecting upward shifts. Performance is assessed using numerical measures of detection efficiency and average run length. The results show improved detection compared with classical ratio-based TBEA charts. A real data example from a French forest fire database illustrates the ability of the proposed charts to detect simultaneous changes in occurrence rate and burn intensity.

1. Introduction

Control charts are widely utilized in various industrial and non-industrial sectors for statistical process monitoring approaches. Time between event and amplitude charts (TBEA) were employed in these procedures to track the duration of two successive events, T of event E and ampliude X of the event. Control charts must often be watched for scenarios where time reduces and amplitude increases since they might have dangerous and detrimental effects. For instance, in the observation of natural disasters, a rise in magnitude could be a sign of increased hazard. Similar to this, more serious and frequent patient occurrences may necessitate immediate attention in hospital settings.

Time between events (TBE) charts are necessary to enhance typical control charts, which are inadequate for monitoring high-quality processes. To enhance the conventional control charts for high-quality operations, Calvin [1] tracked the total number of conforming items between two nonconforming ones. Lucas [2], Vardeman and Ray [3], and numerous other scholars expanded on this idea of TBE. Radaelli [4] introduced the one- and two-sided Shewhart TBE charts and used an example where the count process was taken to be homogeneous to demonstrate their use. Gan [5] then presented an exponentially weighted moving average (EWMA) based TBE chart for tracking the rate of recurrence of unusual events using the inter-arrival intervals between occurrences. Subsequently, TBE monitoring techniques based on the exponential distribution were developed by Xie et al. [6]. An exponential TBE cumulative sum control chart (CUSUM) was created and examined for the Weibull and lognormal distributions by Borror et al. [7]. A gamma-based TBE chart with a novel approach to random shift modeling was presented by Zhang et al. [8].

Monitoring the frequency and amplitude of events is important, according to Cheng and Mukherjee [9].According to Cheng et al. [10], the time interval between two consecutive events provides a concept of frequency, while amplitude X provides an idea of how much it influences. In TBEA charts, an event E is affected by both its amplitude X and the time interval T between subsequent occurrences. Wu et al. [11] proposed tracking the time interval and the amplitude using an X-chart and a T chart (TBE chart). This work was enhanced by Wu et al. [12], who developed a TC-CUSUM chart that tracks TBEA using two CUSUM charts. Furthermore, Wu et al. [13] created a single rate chart based on a single statistic R to monitor simultaneous changes in both T and X, where T and X have Gamma and exponential distributions. Qu et al. [14] provided a single GCUSUM chart with exponential and normal distributions for amplitude X and time T, respectively. For more details see, [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29]

Three statistics ($Z_1$, $Z_2$, and $Z_3$) were introduced by Rahali et al. [30]) utilizing Shewhart type controal charts with T and X as different distributions. To improve responsiveness, real time and bivariate TBEA monitoring techniques have also been developed Xie et al. [31]. For increased dependability, measurement error modeling has also been incorporated into chart design [32]. Advanced distributional families have also been investigated recently for tracking time-between-events data. To enhance shift detection in skewed lifetime data a TBE chart based on the transmuted weighted exponential distribution was introduced by Bhatti et al. [33]. Chen [34] proposed to monitor correlated time-between-events variables while taking parameter estimation effects into account, control charts based on Gumbel’s bivariate exponential distribution. EWMA-based TBE charts have been developed by Xie [35] for gamma-distributed time-between-events data. Recent research has concentrated on employing control charts like Max-EWMA and rate-based EWMA under distributions like Weibull, gamma, exponential and beta to jointly monitor magnitude and time-between-events [36,37,38,39]. Furthermore, renewal reward process-based methods expand this framework by adding random failure thresholds and adaptable stochastic structures for more accurate modeling [40]. The previous work of TBEA chart primarily focused on standard distributions like exponential for T and X and also sloely focused on only one statistics ${\displaystyle \frac{X}{T}}$. Nevertheless, these distributions might not be enough for modeling skewed and heavy-tailed data. Pandey and Gupta [41] used these three statistics for generalized distributions as generalized exponential, generalized Rayleigh and three parameter Weibull distribution. Therefore, we use T and X as generalized Weibull and generalized log logistics distributions using these three statistics in this study. The Rayleigh and exponential distributions, in particular, emerge as special cases of the Weibull distribution corresponding to particular values of the shape parameter. As a result, under suitable parameter settings, their generalized forms are frequently included within generalized Weibull frameworks. On the other hand, a broad variety of hazard shapes and tail behaviors can be accommodated by the generalized Weibull and generalized log-logistic distributions, which offer far more flexibility. This nesting trait improves their appropriateness for sophisticated statistical process monitoring applications by allowing more accurate modeling of complicated data while guaranteeing that simpler models are kept as special cases.

The chart design and control limitations are explained in Section 2. In Section 3, we first describe the employed distributions before offering the performance evaluation approach and numerical comparisons based on simulated data. Section 4 illustrates the application of the suggested charts using actual forest fire data. The paper’s conclusion is found in Section 5.

2. TBEA Chart Design

Let T and X denote the inter-arrival time and amplitude of event E, respectively. Assume that T and X are independent continuous random variables on $[0,\infty )$. For $U\in \{T,X\}$, let $f_U$ and $F_U$ denote the PDF and CDF with parameter vector ${\theta }_U$.

The process parameters are given by ${\theta }_{U_0}$ (in-control) and ${\theta }_{U_1}$ (out-of-control). The corresponding means and standard deviations are $({\mu }_{U_0},{\sigma }_{U_0})$ and $({\mu }_{U_1},{\sigma }_{U_1})$, respectively.

Since the scales of T and X can differ greatly, we define their standardized form in in-control as follows to ensure that they are treated fairly:

$$T^{\prime }={\displaystyle \frac{T}{{\mu }_{T_0}}},X^{\prime }={\displaystyle \frac{X}{{\mu }_{X_0}}}.$$

These standardizations lead to $\mathbb{E}\left(T^{\prime }\right)=\mathbb{E}\left(X^{\prime }\right)=1$ under in-control conditions. We look at statistics, $Z\in \{Z_1,Z_2,Z_3\}$, which are functions of T and X, in order to monitor T and X simultaneously. Z increases when time declines and amplitude increases, and Z lowers when time rises and amplitude decreases, according to the characteristics of these statistics.

2.1. Statistic $Z_1$

The first statistic is defined as

$$Z_1=X^{\prime }-T^{\prime },$$

(1)

This statistic shows a possible process shift as $X^{\prime }$ increases and/or $T^{\prime }$ lowers. $Z_1$ has limits of $(-\infty ,+\infty )$, and its CDF and PDF are provided by

$$F_{Z_1}(z\mid {\theta }_Z)=1-{\mu }_{X_0}{\int }_0^{\infty }{F}_T((x-z){\mu }_{T_0}\mid {\theta }_T)f_X(x{\mu }_{X_0}\mid {\theta }_X)dx,$$

(2)

$$f_{Z_1}(z\mid {\theta }_Z)={\mu }_{T_0}{\mu }_{X_0}{\int }_0^{\infty }{f}_T((x-z){\mu }_{T_0}\mid {\theta }_T)f_X(x{\mu }_{X_0}\mid {\theta }_X)dx,$$

(3)

where ${\theta }_Z=({\theta }_T,{\theta }_X)$ represents the vector of parameters for T and X.

2.2. Statistic $Z_2$

The second statistic is defined as

$$Z_2={\displaystyle \frac{X^{\prime }}{T^{\prime }}},$$

(4)

and behaves similarly to $Z_1$. $Z_2$ has limits as $[0,+\infty )$, and its CDF and PDF are expressed as

$$F_{Z_2}(z\mid {\theta }_Z)=1-{\mu }_{X_0}{\int }_0^{\infty }{F}_T\left({\displaystyle \frac{x{\mu }_{T_0}}{z}}\mid {\theta }_T\right)f_X(x{\mu }_{X_0}\mid {\theta }_X)dx,$$

(5)

$$f_{Z_2}(z\mid {\theta }_Z)={\displaystyle \frac{{\mu }_{T_0}{\mu }_{X_0}}{z^2}}{\int }_0^{\infty }x{f}_T\left({\displaystyle \frac{x{\mu }_{T_0}}{z}}\mid {\theta }_T\right)f_X(x{\mu }_{X_0}\mid {\theta }_X)dx.$$

(6)

2.3. Statistic $Z_3$

The third statistic, considered a hybrid of $Z_1$ and $Z_2$, is defined as

$$Z_3=X^{\prime }+{\displaystyle \frac{1}{T^{\prime }}}.$$

(7)

This statistic also increases when $X^{\prime }$ increases and/or $T^{\prime }$ decreases. The random variable $Z_3$ is defined on $[0,+\infty )$, and its CDF and PDF are given by

$$F_{Z_3}(z\mid {\theta }_Z)=F_X(z{\mu }_{X_0}\mid {\theta }_X)-{\mu }_{X_0}{\int }_0^z{F}_T\left({\displaystyle \frac{{\mu }_{T_0}}{z-x}}\mid {\theta }_T\right)f_X(x{\mu }_{X_0}\mid {\theta }_X)dx,$$

(8)

$$f_{Z_3}(z\mid {\theta }_Z)={\mu }_{T_0}{\mu }_{X_0}{\int }_0^z{\displaystyle \frac{1}{{(z-x)}^2}}{f}_T\left({\displaystyle \frac{{\mu }_{T_0}}{z-x}}\mid {\theta }_T\right)f_X(x{\mu }_{X_0}\mid {\theta }_X)dx.$$

(9)

For the statistics $Z_1$, $Z_2$, and $Z_3$ as defined above, closed-form expressions for the Equations (2), (3), (5), (6), (8) and (9) are generally not available. However, these functions can be evaluated using numerical integration techniques.

2.4. Control Limits

The related CDF and PDF equations are substituted by ${\theta }_Z=({\theta }_T,{\theta }_X)$ to find the distributions of the statistics $Z\in \{Z_1,Z_2,Z_3\}$ for in-control and out-of-control cases. In particular, these parameters are substituted with ${\theta }_{Z_0}=({\theta }_{T_0},{\theta }_{X_0})$ for the in-control case and ${\theta }_{Z_1}=({\theta }_{T_1},{\theta }_{X_1})$ for the out of control case.

Based on these statistics, the control limits for TBEA charts are as follows,

$${\mathrm{LCL}}_Z=F_Z^{-1}({\alpha }_L\mid {\theta }_{Z_0}),$$

$${\mathrm{UCL}}_Z=F_Z^{-1}(1-{\alpha }_U\mid {\theta }_{Z_0}),$$

where the inverse CDF of Z under control is represented by $F_Z^{-1}(·\mid {\theta }_{Z0})$. In this case, $\alpha$ represents a type-I error, with ${\alpha }_L$ representing the lower-sided and ${\alpha }_U$ representing the upper-sided type I error components:

$${\alpha }_L=m\alpha ,{\alpha }_U=(1-m)\alpha ,$$

Here, n represent the detection power allocation factor

$$m={\displaystyle \frac{{\alpha }_L}{{\alpha }_L+{\alpha }_U}}.$$

The situation $n=0$ is taken into consideration because it is frequently more crucial to identify increases in Z. As a result, the control limits are reduced to

$${\alpha }_L=0,{\alpha }_U=1-\alpha ,$$

resulting in a single upper sided control limit

$${\mathrm{LCL}}_Z=-\infty ,{\mathrm{UCL}}_Z=F_Z^{-1}(1-\alpha \mid {\theta }_{Z_0}).$$

It is noteworthy that for the statistics $Z_1$, $Z_2$, and $Z_3$, the inverse CDF $F_Z^{-1}(·\mid {\theta }_{Z_0})$ cannot be stated in closed form. Consequently, the equation must be solved in order to calculate the control limits numerically,

$$F_Z(z\mid {\theta }_{Z_0})=\alpha ,$$

using a one dimensional root finding algorithm.

2.5. Algorithm

The inverse cumulative distribution function for the statistics $Z_1$, $Z_2$, and $Z_3$ does not have a closed-form expression; instead, a numerical root-finding method is used to determine the upper control limit. The following is a description of the algorithm.

• Identify the vector of in-control parameters ${\theta }_{Z_0}=({\theta }_{T_0},{\theta }_{X_0})$ for the time-between-events variable T and the amplitude variable X.
• Determine the type-I error probability $\alpha$, and choose the desired in-control average time to signal (ATS).
• Since the chart is intended to be an upper-sided monitoring system, set ${\alpha }_L=0$, ${\alpha }_U=\alpha$.
• Define the cumulative distribution function of z.
• Apply one dimensional root function to solve

  $$F_Z(z\mid {\theta }_{Z_0})=1-\alpha ,$$
• The root value obtained fron step 6 is taken as UCL.
• This procedure is repeat for all the statistics and combination of parameters.

2.6. Time to Signal Properties

Based on TBEA charts, the statistic $Z\in \{Z_1,Z_2,Z_3\}$ yields the valur of $\beta$ (type-II error) as

$$\beta =F_Z({\mathrm{UCL}}_Z\mid {\theta }_Z)-F_Z({\mathrm{LCL}}_Z\mid {\theta }_Z),$$

The upper and lower control limits for the statistic Z are denoted by ${\mathrm{UCL}}_Z$ and ${\mathrm{LCL}}_Z$, respectively.

The Shewhart type TBEA control control chart run length (RL) is followed by a geometric distribution with parameter $1-\beta$. Consequently, the average run length (ARL) and standard deviation of the run length (SDRL) are provided by

$$\mathrm{ARL}=E\left(RL\right)={\displaystyle \frac{1}{1-\beta }},\mathrm{SDRL}=\sigma \left(RL\right)={\displaystyle \frac{\sqrt{\beta }}{1-\beta }}.$$

Let $T_1,T_2,T_3,\dots$ be the intervals between two successive events, respectively. The total of the run length times is the time to signal.

$$\mathrm{TS}=\sum _{l=1}^{RL}{T}_l.$$

The average time to signal (ATS) and the standard deviation of time to signal (SDTS) given as follows:

$$\mathrm{ATS}=\mathbb{E}\left(\mathrm{TS}\right)=\mathbb{E}\left(T\right)·\mathbb{E}\left(RL\right)={\mu }_T·\mathrm{ARL}={\mu }_T·{\displaystyle \frac{1}{1-\beta }},$$

and

$$\mathrm{SDTS}=\sigma \left(\mathrm{TS}\right)=\sqrt{\mathbb{V}\left(T\right)·\mathbb{E}\left(RL\right)+{\mathbb{E}}^2\left(T\right)·\mathbb{V}\left(RL\right)}.$$

$$\mathrm{SDTS}=\sqrt{{\sigma }_T^2·{\displaystyle \frac{1}{1-\beta }}+{\mu }_T^2·{\displaystyle \frac{\beta }{{(1-\beta )}^2}}}.$$

When the process is in control, $1-\beta =\alpha$, and the ATS becomes

$${\mathrm{ATS}}_0={\mu }_{T_0}·{\displaystyle \frac{1}{\alpha }},$$

So,

$$\alpha ={\displaystyle \frac{{\mu }_{T_0}}{{\mathrm{ATS}}_0}}.$$

${\mathrm{ATS}}_1$ and ${\mathrm{SDTS}}_1$ are out-of-control values that can be derived by replacing the expressions for ATS and SDTS with ${\theta }_Z$ = ${\theta }_{Z1}$.

3. Comparative Studies

For the random variables $U\in \{T,X\}$, we examine two types of distributions, each with three parameters a, b and c. As a result, the parameter vector is determined by ${\theta }_{U_0}=(a_0,b_0,c_0)$ under the in-control state and ${\theta }_{U_1}=(a_1,b_1,c_1)$ under the out-of-control state. The following describes the distributions taken into consideration in this study.

3.1. Generalized Weibull Distribution

The generalized Weibull distribution is a popular method for model the life time data and also use in the survival analysis data. The generalized Weibull have properties as

$$f_T(t;a,b,c)=\left\{\begin{array}{cc}abc{t}^{c-1}{e}^{-b{t}^c}{(1-e^{-b{t}^c})}^{a-1},\hfill & \mathrm{if}t>0,a>0,b>0,c>0,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

$$F_T(t;a,b)=\left\{\begin{array}{cc}{(1-e^{-b{t}^c})}^k,\hfill & \mathrm{if}t>0,a>0,b>0,c>0,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

3.2. Generalized Log-Logistic Distribution

We used another distribution as the generalized log logistic distribution. In many fields, we have wide range of applications for the log logistic distribution. Here, we use the generalization of the log-logistic distribution. It has some progressive modifications instead of log-logistic distribution.

$$f_T(t;a,b,c)=\left\{\begin{array}{cc}abc{t}^{c-1}{(1+b{t}^c)}^{-1-a},\hfill & \mathrm{if}t>0,a>0,b>0,c>0\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

$$F_T(t;a,b,c)=\left\{\begin{array}{cc}1-{(1+{\left(bt\right)}^c)}^{-a},\hfill & \mathrm{if}t>0,a>0,b>0,c>0\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

In order to represent the amplitude variable X and event time T, we take into consideration the generalized Weibull and generalized log logistic distributions. By changing the underlying distributions utilized to simulate the time between occurrences T and the amplitude X, different scenarios were taken into consideration in the simulation study. In this case, the standard deviation ${\sigma }_0$ varied between the values $\{1,2,5\}$ but the in-control mean ${\mu }_0=10$ was fixed.

The in-control distribution parameters $a_0$, $b_0$ and $c_0$ as well as the related skewness coefficients ${\gamma }_0$ based on their in-control mean and variance are given in

Table 1. Values of the parameters for the fixed mean and varying variance of the distributions under consideration.

Distribution	${\mathit{a}}_0$	${\mathit{b}}_0$	${\mathit{c}}_0$	${\mathit{\mu }}_0$	${\mathit{\sigma }}_0$	${\mathit{\gamma }}_0$
Generalized Weibull	11.6188	0.0003	3.9341	10	1	0.0076
	6.1979	0.0008	3.4342	10	2	−0.4810
	1.3287	0.0001	4.3329	10	5	−0.2639
Generalized log logistics	1.6344	1.9742 × 10−16	15.4229	10	1	0.3001
	11.3513	1.1336 × 10−10	8.6995	10	2	0.2154
	160.2011	3.0159 × 10−8	5.1323	10	5	−0.1076

. In order to guarantee that the theoretical mean and standard deviation nearly matched the intended values, the parameter values were calculated numerically by solving systems of moment matching equations, as closed form for the mean and variance for these distributions are typically not available. A numerical moment-matching method was used to determine the parameter values. In particular, a system of equations that equated the theoretical moments of the assumed distribution to the corresponding sample moments was solved in order to retrieve the unknown values. Numerical optimization techniques were used to obtain stable and trustworthy approximations because these equations do not accept closed-form solutions. This method guarantees that the fitted distributions accurately capture the salient features of the data. Density charts for each of the three distributions are shown in Figure 1. The upper control limits for each of the three statistics are shown in

Table 2. The upper control limits $UC{L}_Z$ for the TBEA charts are determined by statistics $Z_1$, $Z_2$, and $Z_3$.

T	X	GW			GLL
	${\mathbf{\sigma }}_{\mathbf{0}}$	1	2	5	1	2	5
Statistic $Z_1$
GW	1	0.0050	0.0637	0.2136	0.2572	0.3442	0.4889
	2	0.0572	0.0001	0.1499	0.3000	0.3703	0.5126
	5	0.1540	0.0968	0.0003	0.4176	0.485	0.5751
GLL	1	0.2432	0.287	0.3668	0.2738	0.3547	0.4916
	2	0.2842	0.3232	0.4152	0.3139	0.389	0.5164
	5	0.4229	0.4548	0.5198	0.4438	0.5014	0.6016
Statistic $Z_2$
GW	1	1.0100	1.0809	1.3356	1.3068	1.4704	1.8984
	2	1.0500	1.0003	1.2356	1.3476	1.5037	1.9225
	5	1.1347	1.0806	1.0012	1.4690	1.6288	1.945
GLL	1	1.2653	1.3471	1.5207	1.3183	1.4837	1.8721
	2	1.3085	1.3832	1.5627	1.3639	1.5191	1.9134
	5	1.4489	1.5237	1.6792	1.5014	1.6503	1.9942
Statistic $Z_3$
GW	1	2.0334	2.1216	2.4211	2.3009	2.4671	2.8866
	2	2.0753	2.0723	2.3574	2.3339	2.4867	2.9020
	5	2.1721	2.1305	2.2111	2.4522	2.6095	2.8954
GLL	1	2.2541	2.3311	2.5055	2.3027	2.4598	2.8334
	2	2.2951	2.3579	2.5287	2.3439	2.4885	2.8578
	5	2.4290	2.4847	2.6045	2.4686	2.5917	2.9222

. The table contains 54 potential combinations of the nine chosen distributions for T and X. The control ATS value is ${\mathrm{ATS}}_0=370.4$.

Changes in the mean of the underlying variables could cause the process to shift upward. In particular, the mean amplitude X may vary from ${\mu }_{X_0}$ to ${\mu }_{X_1}={\delta }_X{\mu }_{X_0}$, where ${\delta }_X\ge 1$ denotes the amplitude change. As an alternative, the mean inter-arrival time T could vary from ${\mu }_{T_0}$ to ${\mu }_{T_1}={\delta }_T{\mu }_{T_0}$, where the amount of the change in time is indicated by ${\delta }_T\ge 1$. Both variables might change at the same time, so that ${\mu }_{X_1}={\delta }_X{\mu }_{X_0}$ and ${\mu }_{T_1}={\delta }_T{\mu }_{T_0}$.

We do not evaluate the three TBEA charts using the ATS criterion for the statistics $Z\in \{Z_1,Z_2,Z_3\}$ directly since the actual values of ${\delta }_X$ and ${\delta }_T$, which measure the magnitude of mean changes in amplitude and time, respectively, are typically unknown. Instead, we evaluate them using the following expected average times to signal(EATS). The estimated time needed for a control chart to generate a signal is represented by the expected average time to signal. A higher EATS is preferred under in-control circumstances since it shows fewer false alarms. Smaller EATS values are favored in out-of-control situations because they indicate quicker process change identification.

• EATS_X: We consider no change in time T when we calculate EATS for shifts in X (i.e., ${\delta }_T=1$):

  $${\mathrm{EATS}}_X=\sum _{{\delta }_X\in {\mathcal{X}}_X}{f}_{{\delta }_X}\left({\delta }_X\right)·\mathrm{ATS}({\delta }_X,1)$$
• EATS_T: We consider no change in amplitude X when we calculate EATS for shifts in T (i.e., ${\delta }_X=1$):

  $${\mathrm{EATS}}_T=\sum _{{\delta }_T\in {\mathcal{X}}_T}{f}_{{\delta }_T}\left({\delta }_T\right)·\mathrm{ATS}(1,{\delta }_T)$$
• EATS_XT: The expected ATS for simultaneous shifts in both X and T:

  $${\mathrm{EATS}}_{XT}=\sum _{{\delta }_X\in {\mathcal{X}}_X}\sum _{{\delta }_T\in {\mathcal{X}}_T}{f}_{{\delta }_X}\left({\delta }_X\right)f_{{\delta }_T}\left({\delta }_T\right)·\mathrm{ATS}({\delta }_X,{\delta }_T)$$

In this case, the range of potential increases in the mean of X (up to double the nominal value ${\mu }_{X_0}$)) is represented by ${\mathcal{X}}_X=\{1.1,1.2,\dots ,1.9,2.0\}$ and the range of potential decreases in the mean of T (down to half of ${\mu }_{T_0}$)) is represented by ${\mathcal{X}}_T=\{0.5,0.55,\dots ,0.9,0.95\}$.

Both are assumed to follow discrete uniform distributions over ${\mathcal{X}}_X$ and ${\mathcal{X}}_T$, respectively, since there is no previous knowledge about the distribution of ${\delta }_X$ and ${\delta }_T$.

We examined the corresponding EATS values under three different circumstances in order to evaluate the monitoring performance of the three TBEA control chart statistics Z₁, Z₂, and Z₃:

• Amplitude shift only (X);
• Time shift only (T);
• Joint shift in X and T.

A methodical comparison of the EATS values for every simulation scenario taken into account in this research. The R code for

Table 3. Performance comparison of TBEA charts in terms of $EAT{S}_X$ based on statistics $Z_1$, $Z_2$, and $Z_3$.

T	X	GW			GLL
	${\mathbf{\sigma }}_{\mathbf{0}}$	1	2	5	1	2	5
Statistic $Z_1$
GW	1	21.5510	14.1099	21.0827	19.6900	15.2781	21.6931
	2	27.9892	16.6258	25.1200	21.4985	15.1495	21.7438
	5	33.1042	20.7142	26.1607	28.1237	19.3543	22.3386
GLL	1	17.7950	10.9025	12.4600	22.2160	16.4947	21.8614
	2	19.9753	11.9716	14.3326	23.7655	17.0123	22.1163
	5	28.9671	16.9695	18.1927	32.7981	21.8704	25.6872
Statistic $Z_2$
GW	1	10.4312	11.4148	18.7882	23.6421	39.4631	75.9221
	2	10.9956	10.7350	15.1434	25.1420	40.2026	76.1981
	5	12.5968	11.9052	11.4375	32.3323	50.1474	71.1973
GLL	1	18.4819	22.6882	29.5228	27.5075	41.8283	78.0663
	2	20.8090	24.5149	31.8933	27.4655	43.5114	76.8857
	5	29.5645	34.2979	38.9396	37.6127	59.9716	81.1843
Statistic $Z_3$
GW	1	10.5936	11.8187	23.6919	23.9461	45.2057	100.9877
	2	11.1630	11.2225	18.9534	24.3150	42.3335	99.8165
	5	13.1390	12.3381	13.8597	31.1905	53.1125	82.5969
GLL	1	17.9513	22.3596	30.4299	27.8660	46.4897	100.4854
	2	20.0293	23.0756	31.0042	26.4268	45.3422	97.1869
	5	28.2391	31.7364	34.7655	34.7621	56.8096	90.4131

,

Table 4. Performance comparison of TBEA charts in terms of $EAT{S}_T$ based on statistics $Z_1$, $Z_2$, and $Z_3$.

T	X	GW			GLL
	${\mathbf{\sigma }}_{\mathbf{0}}$	1	2	5	1	2	5
Statistic $Z_1$
GW	1	43.0516	28.4452	51.7612	38.9605	31.7378	48.5394
	2	57.5864	33.784	61.487	41.2484	30.0360	46.8354
	5	66.2467	41.8893	56.0002	53.3685	38.5007	44.3001
GLL	1	33.2757	20.1557	22.1806	48.1204	35.6363	50.0033
	2	36.3188	21.7279	26.3880	50.2552	36.7846	49.5604
	5	57.2712	33.1263	34.4726	68.5952	45.7229	54.248.
Statistic $Z_2$
GW	1	11.2139	7.2518	12.3845	27.3718	17.5440	20.3819
	2	12.4776	6.5294	9.8990	29.4152	18.3267	21.6359
	5	14.9412	7.6151	7.1165	40.4686	24.4437	25.3666
GLL	1	37.7177	56.0358	94.4464	45.6165	33.1898	53.1667
	2	37.7782	55.3381	92.4662	43.4280	33.9413	51.2362
	5	44.3203	63.7920	79.6923	53.23076	41.8844	50.4228
Statistic $Z_3$
GW	1	10.6157	14.1135	27.9332	10.0001	10.0009	11.0772
	2	12.1199	12.8740	22.8854	10.0000	10.0041	11.4288
	5	13.4004	14.2335	16.2101	10.0195	10.1538	11.8888
GLL	1	26.1076	16.5526	19.7285	45.2155	60.6136	95.3305
	2	28.6396	17.0551	20.2560	43.6382	62.7389	94.7770
	5	41.5562	23.4575	22.3958	54.7746	75.1180	88.3280

and

Table 5. Performance comparison of TBEA charts in terms of $EAT{S}_{XT}$ based on statistics $Z_1$, $Z_2$, and $Z_3$.

T	X	GW			GLL
	${\mathbf{\sigma }}_{\mathbf{0}}$	1	2	5	1	2	5
Statistic $Z_1$
GW	1	9.0116	4.5823	4.2479	10.0018	5.6037	6.2672
	2	9.1367	4.7172	4.6515	10.0191	5.6944	6.4777
	5	9.1306	5.0020	4.6714	10.1593	6.2069	7.3404
GLL	1	10.0183	5.5566	5.8657	10.0086	5.6247	6.2555
	2	10.0676	5.6019	6.0688	10.0161	5.7339	6.5161
	5	10.9804	6.2534	6.8523	10.1270	6.2980	7.5168
Statistic $Z_2$
GW	1	9.8474	5.2677	4.5915	10.0061	5.6591	7.8940
	2	9.5735	5.3518	4.8630	10.0291	5.7439	8.0453
	5	9.0162	5.0785	5.236	10.3752	6.2734	8.7075
GLL	1	10.0083	5.5622	6.1913	10.0173	5.6832	8.1255
	2	10.0310	5.6150	6.3684	10.0494	5.7888	8.3176
	5	10.4515	6.0152	6.9816	10.5664	6.4161	9.1025
Statistic $Z_3$
GW	1	10.0012	10.0077	10.6743	10.0043	5.668	8.4887
	2	10.0002	10.0074	10.51	10.0184	5.7281	8.7130
	5	10.0174	10.0464	10.3101	10.2975	6.2661	8.9588
GLL	1	10.0050	5.5499	6.0662	10.0105	5.6685	8.6374
	2	10.0207	5.5886	6.1897	10.0330	5.7632	8.7957
	5	10.3445	5.9044	6.6064	10.4367	6.2525	9.0303

are given in Appendix A. As seen in Table 3, the statistic Z₁ consistently performs better when the shift solely affects the amplitude X. In particular, Z₁ exhibits the lowest EATS values in 72% of cases. As opposed to Z₁, Z₂ and Z₃ exhibit greater EATS values. For each combination, the average EATS values were roughly 21.45 for Z₁, 33.48 for Z₂, and 39.82 for Z₃. These findings suggest that when amplitude shifts have place, Z₁ is the most effective monitoring statistic. The findings for a change in the interval between events T, shown in Table 4. In 22 out of 36 cases (61%), the Z₃ statistic performs best. Z₂ performs best in 11 cases (31%), while Z₁ performs best in just 3 cases (8%). For Z₁, Z₂, and Z₃, the average EATS values under this shift are 40.72, 28.84, and 19.96, respectively. Due to certain circumstances, Z₁ has a lower average EATS value, making it unsuitable for time shift monitoring statistics.Z₃, on the other hand, produces more reliable outcomes. For tracking time shift, we therefore favor Z₃ statistics. When both amplitude and time shifts occur simultaneously, as shown in Table 5, the Z₁ statistic once more performs better. In 61% of cases, it yields the lowest EATS values; in 39% of cases, Z₂ yields the lowest Eats value. Z₁, Z₂, and Z₃ had average EATS values of 7.49, 7.93, and 8.72, respectively.

These results confirm that Z₁ is the best statistic for identifying simultaneous changes in both X and T, with Z₂ exhibiting intermediate efficiency and Z₃ being the least effective.

4. Application to Real Data

An actual dataset of forest fires in the Provence-Alpes-Côte d’Azur region of southeast France is examined to demonstrate the practicality and utility of the proposed Shewhart-type Time between events and amplitude (TBEA) control charts see https://www.promethee.com/default/incendies or Bountzouklis [42]). This region is extremely vulnerable to wildfires due to its Mediterranean environment, especially during the summer. The dataset, which was taken from an official French forest fire database, documents two important features for every major fire incidence, which is defined as one that burns more than one hectare. These features include X, the amplitude represented by the burned surface area in hectares, and T, the interval in days between subsequent fire episodes.

Between October 2016 and September 2017, there were 92 recorded fire incidents. Two separate phases of the data are separated for analysis purposes. Phase 1 (Low Season) encompasses 47 fire occurrences from October 2016 to mid-June 2017, whereas Phase 2 (High Season) includes 45 fire incidents from mid-June 2017 to the end of September 2017.

The amplitude, or burned areas X, is greater in phase 2, while the interval between events T is shorter. This shows that fire activity and severity have increased.

The independence between T and X was confirmed using the Spearman’s and Kendall’s rank correlation tests, which are listed in

Table 6. Correlation structure for inter-arrival time and amplitude.

Phase	1	2	1 & 2
Spearman
Coefficient	0.26	0.02	0.13
p-value	0.07	0.80	0.19
Kendall
Coefficient	0.18	0.01	0.10
p-value	0.09	0.92	0.20

, prior to creating control charts.

T and X are fitted to the three distributions under consideration. The calculated parameters and Kolmogorov–Smirnov (K-S) distances for the goodness of fit assessment are shown in

Table 7. Parameter estimates and associated standard errors (SE) for the two distributions under consideration, including ${\mu }_{T_0},{\sigma }_{T_0},{\mu }_{X_0}$, and ${\sigma }_{X_0}$.

Variable	Distribution	${\mathit{a}}_0$	${\mathit{b}}_0$	${\mathit{c}}_0$	KS	${\mathit{\mu }}_0$	${\mathit{\sigma }}_0$
X	GW	152.1217	3.8512	0.1901	0.0637	13.5781	22.8687
	SE	6.1782	0.07298	0.01928
	GLL	0.4050	0.0993	2.3077	0.0851
	SE	0.2447	0.0527	0.8408
T	GW	238.0027	4.7618	0.1989	0.2108	5.4681	6.8425
	SE	8.5788	0.1527	0.0213
	GLL	0.0169	12.5155	0.9832	0.1914
	SE	0.0069	4.4335	0.1069

. Based on the lowest K-S distances, the generalized log logistic distribution matched the time t variable the best, whereas the generalized Weibull distribution fit the amplitude.

The top control limits for the TBEA charts utilizing statistics $Z_1$, $Z_2$, and $Z_3$ were computed as follows, assuming an in-control average time to signal (ATS₀) of 730 days (about two years):

$$\begin{array}{cc}\hfill {\mathrm{UCL}}_{Z_1}& =7.9141,\hfill \\ \hfill {\mathrm{UCL}}_{Z_2}& =24.8250,\hfill \\ \hfill {\mathrm{UCL}}_{Z_3}& =18.1545.\hfill \end{array}$$

Figure 2 shows the control charts created with the actual forest fire dataset to demonstrate the effectiveness of the suggested Shewhart-type TBEA control charts based on the three statistics $Z_1$, $Z_2$, and $Z_3$. Several out-of-control signals during the high-risk Phase 2 are identified by the charts based on $Z_1$ and $Z_3$, which show high sensitivity to increases in event frequency and amplitude. Overall, it is evident from the graphical findings how each statistic reacts to modifications in the underlying process.

The outcomes of applying these control limits to Phase 2 data are shown in Figure 2. All three control charts had a number of out-of-control points, particularly with $Z_1$ and $Z_3$, indicating increased danger due to more frequent and severe fires. An out-of-control signal in the actual data indicates a shift in either the magnitude of events or the time between events. Such a signal suggests that the process might be impacted by an assignable cause, and more research is needed to pinpoint and fix the source of the fluctuation. When a signal goes out-of-control, the separate components $\left(T_0\right)$ and $\left(X_0\right)$ can be examined to determine the signal source. A change in the time between events is the cause of the signal if $\left(T_0\right)$ considerably deviates while $\left(X_0\right)$ stays constant. A shift in the event amplitude is responsible for the signal if $\left(X_0\right)$ deviates while $\left(T_0\right)$ stays constant. A simultaneous change in the frequency and magnitude of events is indicated by the signal if both variables diverge. In the forest fire dataset, the two monitored variables are the time between events (T) and the burned area (X). After an out-of-control signal is detected by the proposed statistics $Z_1$, $Z_2$, and $Z_3$, the source of the signal is identified by examining the individual behavior of T and X relative to their in-control patterns. In Phase 2 of the data, it is observed that the time between events (T) shows a noticeable decrease, indicating more frequent occurrences, and the amplitude (X) shows a clear increase, indicating higher severity of events. Thus, both variables deviate from their in-control behavior. If there is a time shift only T deviates. If amplitude shift only X deviates. If there is a joint shift, both T and X deviate. Since in this example both T and X exhibit significant deviations, the out-of-control signal is attributed to a simultaneous shift in both time and amplitude. This illustration shows how the recommended TBEA control charts can detect notable changes in the size and frequency of important events. The method is extensively applicable to various domains where cooperative timing and severity monitoring are crucial, such as public health surveillance, industrial process safety, and environmental hazard tracking.

5. Conclusions

In this work, we use two generalized distributions to track the amplitude and interval between occurrences. We used the generalized Weibull and generalized log-logistics distributions for the three statistics ($Z_1$), ($Z_2$), and ($Z_3$) in the TBEA charts, which are primarily based on traditional distribution. Here, we want to keep an eye on the amplitude of the event as well as the interval between subsequent occurrences. The joint data regarding time T and amplitude X in the process control are taken into account in this TBEA chart. It is highly helpful in manufacturing, environmental monitoring, infrastructure upkeep, and healthcare.

We examined the effects of time, magnitude, and simultaneous time and magnitude shifts in our study. Three distinct statistics are taken into consideration: the hybrid additive statistic ($Z_3$), the ratio statistic ($Z_2$), and the difference statistic ($Z_1$). The consequences of these statistics vary depending on the shift circumstance.

In our investigation, $Z_1$ is the optimal option when amplitude changes take place. When the shift is introduced in time, $Z_3$ performs the best. Once more, $Z_1$ outperforms the other two in the simultaneous shift in both X and T. We use a real-world dataset on forest fire incidents in the Provence-Alpes-Côte d’Azur region to test the models’ practical applicability. The fire activity is detected by the models. The suggested TBEA control charts provide better sensitivity to process shifts and wider applicability across different data types than earlier research that was limited by conventional distributional assumptions.

Even while the suggested Shewhart-type TBEA charts successfully track event frequency and amplitude under the generalized Weibull and generalized log-logistic distributions, there are still a number of areas that need further investigation. To assess the robustness of the suggested statistics, the methodology can be expanded to other flexible lifetime distributions such modified gamma, Burr, and inverse Gaussian models. Future research may use copula-based models to include dependence patterns between the amplitude and time-between-events variables. To enhance the detection of minor shifts, memory-type control charts, such as EWMA or CUSUM versions of the suggested statistics, could be created. Lastly, creating effective computational tools and using the suggested monitoring framework in a variety of real-world applications will increase its practical utility.

All things considered, this study broadens the application and adaptability of TBEA control charts by connecting traditional process monitoring methods with contemporary, generalized distributional models.