Pearson and Deviance Residual-Based Control Charts for the Inverse Gaussian Ridge Regression Process: Simulation and an Application to Air Quality Monitoring

Abstract

In manufacturing and service industries, monitoring processes with correlated input variables and inverse Gaussian (IG)-distributed quality characteristics is challenging due to the limitations of maximum likelihood estimator (MLE)-based control charts. When input variables exhibit multicollinearity, traditional MLE-based inverse Gaussian regression model (IGRM) control charts become unreliable. This study introduces novel Shewhart control charts using Pearson and deviance residuals based on the inverse Gaussian ridge regression (IGRR) model to address this issue. The proposed IGRR-based charts effectively handle multicollinearity, offering a robust alternative for process monitoring. Their performance is evaluated through Monte Carlo simulations using average run length (ARL) as the main criteria, demonstrating that Pearson residual-based IGRR charts outperform deviance residual-based charts and MLE-based methods, particularly under high multicollinearity. A real-world application to a Pakistan air quality dataset confirms their superior sensitivity in detecting pollution spikes, enabling timely environmental negotiations. These findings establish Pearson residual-based IGRR control charts as a practical and reliable tool for monitoring complex processes with correlated variables.

1. Introduction

In advanced manufacturing and service industries, monitoring product quality over time is critical for ensuring process stability and reliability. Control charts, a cornerstone of statistical process control (SPC), are widely used to detect abrupt changes in process parameters, enabling timely interventions to maintain quality standards. In the early 20th century, Shewhart introduced control charts that are effective for detecting large shifts in mean and scale parameters. While traditional control charts monitor a single variable, modern processes often involve a quality characteristic influenced by multiple input variables, necessitating model-based control charts. When the quality characteristic follows a non-normal distribution, such as the inverse Gaussian (IG) distribution, and input variables exhibit multicollinearity, standard maximum likelihood estimation (MLE)-based control charts become unreliable due to variance inflation, leading to reduced sensitivity in detecting process shifts.

Frequently, control charts are made for one variable, while it is affected by one or more input variables. Mostly, we see a linear relationship between the study variable and input variables. Such types of control charts are based on a linear regression model. Initially, Mandel proposed the linear model-based control chart for process monitoring [1]. In the linear model-based control charts, we restrict the condition of the normally distributed study variable. When this assumption is violated and the response variable follows the exponential family distributions then we move to the generalized linear model (GLM)-based control charts, which give more efficient process monitoring. Jearkpaporn et al. [2] analyzed Gamma distribution with Gamma-correlated variables. They used deviance residuals and compared with other Shewhart charts. They concluded that when the mean shift is multiplicative, then detection is sensitive. Many authors including Kang and Albin [3], Woodall et al. [4], and Wang and Tsung [5] studied the model-based control charts with their practical applications. Niaki and Abbasi [6] used statistical methods to monitor the process model parameters. Skinner et al. [7] proposed the GLM-based control charts for detecting variation in mean for the multivariate variable. Skinner et al. [8] proposed GLM-based control charts for detecting the variation in a process. Furthermore, several authors studied the GLM-based profiling including Amiri et al. [9], Amiri et al. [10], and Shadman et al. [11].

All of the above research in the literature mostly focused on simple model-based control charts where the variable of interest depends on a single input variable and assumes that all other model assumptions hold. However, it is not always true that all model assumptions are met and the input variable will be single. To discuss these issues, Nancy et al. [12] conducted a comprehensive survey of model-based control charts from 2010 to 2021, categorizing them into linear, nonlinear, and generalized regression models. The study highlighted model-based control charts’ effectiveness in handling multicollinearity and autocorrelation, with performance often evaluated using average run length (ARL). The survey emphasized applications across industries, healthcare, and agriculture while identifying gaps for future research. Park et al. [13] proposed GLM-based r-charts using deviance residuals for Poisson, negative binomial, and COMP models to monitor dispersed count data with multicollinearity. Principal component analysis (PCA) was integrated to handle correlated predictors. Simulations showed COMP model-based charts performed best for detecting process shifts. Kim et al. [14] proposed a residual control chart method for binary asymmetric data with multicollinearity by considering PCA, FPCA, and neural network methods. Through simulations and real-world breast cancer data, they demonstrated that neural network-based control charts yield better performance compared to GLM-based methods. Mammadova [15] explored the application of COMP model-based control charts with multiple correlated input variables using iterative Liu estimation to monitor the process efficiently. By integrating this approach, the research enhanced the detection of process variations in over- and under-dispersed count data. Through simulations and real-world case studies, they demonstrated the effectiveness of these modified control charts over traditional methods. Yassin and Mohammad [16] compared residual control charts for Poisson regression models addressing multicollinearity using ridge regression. Through simulated and real-world water quality data, the authors evaluated ordinary and Pearson residuals, with ARL as a performance metric. The results highlight the effectiveness of ridge estimators in stabilizing control charts. Mohammed and Ramadan [17] proposed a Shewhart control chart for Poisson regression under ridge regression to address multicollinearity in count data. They compared three ridge parameters and evaluated performance using ordinary and Pearson residuals. Simulations and real water quality data showed the second ridge parameter performed best in detecting process shifts. Mammadova and Özkale [18] compared deviance and ridge deviance residual-based control charts (Shewhart, CUSUM, EWMA) for monitoring the Poisson process under multicollinearity. They evaluated these control charts and found ridge estimator-based control charts generally outperformed deviance-based ones, especially for larger sample sizes. Nancy and Joshi [19] proposed a control chart method for sigmoid regression models to address multicollinearity issues using PCR. The approach pulls ordinary, deviance, and Pearson residuals to monitor model performance, validated through a sleep wellness dataset. The results indicate effective detection of out-of-control processes, particularly in cases like insomnia disorder. Mammadova and Özkale [20] proposed the CUSUM and EWMA control charts based on deviance residuals for the COMP model to monitor the process with count responses under multicollinearity. The authors compared the performance of these charts using PCR and r–k class estimators against traditional ML and ridge estimators through simulations and a real-world application. The results demonstrate that the r–k deviance-based control charts outperform others. Mammadova [21] proposes the r-k-Shewhart control chart for monitoring the COMP process with multicollinear predictors. The method combines ridge and PCR estimation (r-k estimator) to address collinearity, using deviance residuals to enhance sensitivity. Eight biasing parameters are evaluated to optimize out-of-control signal detection. Results show that the r-k-Shewhart chart outperforms traditional ML-based charts, with the fourth biasing parameter yielding the lowest ARL.

Despite these advancements, a critical research gap exists in monitoring processes with IG-distributed responses under multicollinearity. Existing GLM-based control charts, primarily designed for Poisson or Gamma distributions, rely on MLE, which is sensitive to multicollinearity among input variables, resulting in unstable parameter estimates and wider control limits that diminish detection power. While Aslam et al. [22] explored ridge regression-based control charts for Gamma processes, no study has addressed the IG response with correlated predictors using ridge estimation, particularly with Pearson and deviance residuals.

This study addresses this gap by proposing Shewhart control charts based on Pearson and deviance residuals for the inverse Gaussian ridge regression (IGRR) model to monitor IG processes under multicollinearity. Our objectives are to develop robust control charts that mitigate the adverse effects of multicollinearity, evaluate their performance using ARL through Monte Carlo simulations, and validate their applicability using a real-world air quality dataset. The contribution of this paper lies in introducing IGRR-based control charts that leverage ridge estimation to stabilize parameter estimates, offering improved sensitivity and robustness compared to MLE-based charts.

2. Inverse Gaussian Regression Model (IGRM)

The IGRM is used when the variable of interest follows the inverse Gaussian (IG) distribution. Schrodinger [23] introduced this distribution, and later Tweedie [24] called this distribution the IG distribution. Edgeman introduced the control charts based on the IG distribution [25]. Let $y$ be the response variable that follows the IG distribution, then the probability density function of the IG distribution is given by

$$f(y;\mu ,\tau )=\sqrt{{\displaystyle \frac{\tau }{2\pi y^3}}}{e}^{-\left({\displaystyle \frac{\tau (y-\mu {)}^2}{2\mu y^2}}\right)}y,\mu ,\tau >0,$$

(1)

where $\mu$and $\tau$ are, respectively, the mean and scale parameters of the IG distribution. The IG distribution belongs to the exponential family of distribution that can be written as

$$f\left(y_i;{\theta }_i,\phi \right)=\mathit{exp}\left[{\displaystyle \frac{y_i{\theta }_i-b\left({\theta }_i\right)}{a\left(\phi \right)}}+c\left(y_i,\phi \right)\right],$$

(2)

where $y_i$ is the response variable, ${\theta }_i$ is the location parameter and $b\left({\theta }_i\right)$ is a function of the location parameter, and $a\left(\phi \right)$ is an unknown function of the dispersion parameter.

The density function of IG distribution (1) under exponential family layout (2) can be written as

$$f(y;\mu ,\tau )=\mathit{exp}\left[\left\{{\displaystyle \frac{y(-\frac{1}{2{\mu }^2})-(-\frac{1}{\mu })}{{\tau }^{-1}}}\right\}-{\displaystyle \frac{1}{2}}\left(\mathit{ln}\left({\displaystyle \frac{2\pi y^3}{\tau }}\right)+{\displaystyle \frac{\tau }{y}}\right)\right],$$

(3)

By comparing Equation (3) with Equation (2), we have $\theta =-{\displaystyle \frac{1}{2{\mu }^2}}$, $b(\theta )=-{\displaystyle \frac{1}{\mu }}$, $a\left(\phi \right)={\tau }^{-1}$, and $C(y,\phi )=-{\displaystyle \frac{1}{2}}\left(\mathit{ln}\left({\displaystyle \frac{2\pi y^3}{\tau }}\right)+{\displaystyle \frac{\tau }{y}}\right)$. The link function is used to model to model the IG distribution and for its modeling IG distribution, we use inverse-square link function as $g\left(\mu \right)={\displaystyle \frac{1}{{\mu }^2}}$.

2.1. Estimation of the IGRM

Let $y=(y_1,y_2,...,y_n{)}^{\prime }$ be a response vector of the dependent variable, which is assumed to follow IG distribution. Let $X=(x_{i1},x_{i2},...,x_{ip}{)}^{\prime }$, which is the $n\times p$ data matrix of centered and standardized input correlated variables with full rank such that $n>p$. Let Z = (1, X) be the design matrix and $E(y_i)={\mu }_i$ be the mean function of the response variable, $g({\mu }_i)={\displaystyle \frac{1}{{\mu }_i^2}}={\eta }_i$ be the link function, where ${\eta }_i=z_i^{\prime }\beta$, and $\beta$ is the $(p+1)\times 1$ vector of unknown parameters including intercept. The log-likelihood function of Equation (3) is given by

$$l(y_i;{\mu }_i,\tau )={\sum }_{i=1}^n\left[\left\{{\displaystyle \frac{y_i(\frac{1}{2{{\mu }_i}^2})-(\frac{1}{{\mu }_i})}{{\tau }^{-1}}}\right\}-{\displaystyle \frac{1}{2}}\left(\mathit{ln}\left({\displaystyle \frac{2\pi {y_i}^3}{\tau }}\right)+{\displaystyle \frac{\tau }{y_i}}\right)\right],$$

(4)

As ${\mu }_i={\displaystyle \frac{1}{\sqrt{{\eta }_i}}}={\displaystyle \frac{1}{\sqrt{{z_i}^{\prime }\beta }}}$, so Equation (4) can be expressed as

$$l(y_i;\beta ,\tau )={\sum }_{i=1}^n\left[\left\{{\displaystyle \frac{y_i(\frac{{z_i}^{\prime }\beta }{2})-(\sqrt{{z_i}^{\prime }\beta })}{{\tau }^{-1}}}\right\}-{\displaystyle \frac{1}{2}}\left(\mathit{ln}\left({\displaystyle \frac{2\pi {y_i}^3}{\tau }}\right)+{\displaystyle \frac{\tau }{y_i}}\right)\right],$$

(5)

For the estimation of $\beta$ by using the MLE method, we take the first derivative of Equation (5) with respect to $\beta$ and equate it to zero, meaning we have

$$U\left(\beta \right)={\displaystyle \frac{\partial l_i}{\partial \beta }}={\displaystyle \frac{\tau }{2}}\left[y-{\displaystyle \frac{1}{\sqrt{{z_i}^{\prime }\beta }}}\right]{z_i}^{\prime }=0,$$

(6)

Since the solution of Equation (6) is non-linear, the Newton Raphson iteration method is used to estimate the unknown parameters. For the iteration process of the IGRM, initial values and full algorithm for the estimation of unknown parameters can be found in Hardin and Hilbe [26]. Let ${\beta }^{(r)}$ be the approximated ML value of $\beta$ at the $r^{th}$ iterative with convergence, the iteration method [27] gives the relation as:

$${\beta }^{(r+1)}={\beta }^{(r)}+\{I\left({\beta }^{\left(r\right)}\right){\}}^{-1}U\left({\beta }^{\left(r\right)}\right),$$

(7)

where $I({\beta }^{(r)})$ is the fisher information matrix and $U({\beta }^{(r)})$ is the score vector with dimension $(p+1)\times 1$ and both information and score vector are evaluated at ${\beta }^{(r)}$. At convergence in deviance, the unknown parameter vector can be estimated as:

$${\stackrel{̑}{\beta }}_{ML}=(Z^{\prime }\stackrel{̑}{W}Z{)}^{-1}{Z}^{\prime }\stackrel{̑}{W}{y}^{*},$$

(8)

where $y_i^{\mathrm{*}}={\stackrel{̑}{\eta }}_i+{\displaystyle \frac{(y_i-{\stackrel{̑}{\mu }}_i)}{{{\stackrel{̑}{\mu }}_i}^3}}$ is the adjusted response variable and $\stackrel{̑}{W}=diag({{\stackrel{̑}{\mu }}_1}^3,{{\stackrel{̑}{\mu }}_2}^3,...,{{\stackrel{̑}{\mu }}_n}^3)$. Here, ${\stackrel{̑}{\mu }}_i={\displaystyle \frac{1}{\sqrt{z_i^{\prime }{\stackrel{̑}{\beta }}_{ML}}}}$, $i=\mathrm{1,2},...,n$ and ${\stackrel{̑}{\eta }}_i={z_i}^{\prime }{\stackrel{̑}{\beta }}_{ML}$. Both $y^{\mathrm{*}}$ and $\stackrel{̑}{W}$ are found by the iteration procedure, and for the complete derivations and procedure, readers are referred to Hardin and Hilbe [26].

2.2. Inverse Gaussian Ridge Regression Estimation

There are situations where product quality follows the IG distribution and input variables are linearly related to each other, and then the ML-based process monitoring is not reliable due to larger variations. To handle such a situation, we move towards biased estimation methods for more reliable process monitoring. There are several biased estimation methods including ridge, Liu, principal component, stein, etc., which can be used for monitoring product quality in a better way. The most popular one is the ridge estimation proposed by Hoerl and Kennard [28]. In this study, we propose product quality monitoring by using ridge estimation. The inverse Gaussian ridge regression (IGRR) estimator was adopted by Amin et al. [29] as:

$${\stackrel{̑}{\beta }}_R=(Z^{\prime }\stackrel{̑}{W}Z+kI{)}^{-1}{Z}^{\prime }\stackrel{̑}{W}Z{\stackrel{̑}{\beta }}_{ML},$$

(9)

where $k>0$ is a ridge parameter and defined as $k={\displaystyle \frac{\stackrel{̑}{\phi }}{\sum _{j=1}^{p+1}{\alpha }_j^2}}$, where $\stackrel{̑}{\phi }={\displaystyle \frac{1}{n-p-1}}\sum {\displaystyle \frac{{\left(y_i-{\mu }_i\right)}^2}{{\widehat{\mu }}_i^3}}$, $\alpha ={\gamma }^{\prime }{\stackrel{̑}{\beta }}_{ML}$ and $\gamma$ is the eigenvector of $Z^{\prime }\stackrel{̑}{W}Z$.

This study uses the ridge estimation method to reduce the effect of correlation among input variables in the IGRM. Therefore, we used the IGRR estimation method with residuals, i.e., PR and DR to design the control charts for process monitoring.

3. The IGRM and IGRR Residual-Based Control Charts

There are many residuals like working residual, Pearson residuals, deviance residuals, Anscombe residuals, likelihood residuals, and some others [30]. For this study, we consider the two most popular residuals, i.e., Pearson and deviance, which are considered in the literature for the other model’s residual-based control charts [31,32]. So, this study considers these residual-based control charts under ML and ridge methods.

3.1. Pearson Residuals

Pearson residuals are rescaled edition of working residuals. Pearson residuals are the deviations of observations from the estimated mean values divided by the square root of the variance function of the mean. The Pearson residuals for the IGRM under ML approach are given by

$${\chi }_i={\displaystyle \frac{y_i-{\stackrel{̑}{\mu }}_i}{\sqrt{V({\stackrel{̑}{\mu }}_i)}}},$$

(10)

where $y_i,{\stackrel{̑}{\mu }}_i,V({\stackrel{̑}{\mu }}_i)$ are, respectively, actual observation, estimated mean function, and variance function of the IGRM. Equation (10) for the IGRM can be written as

$${\chi }_i={\displaystyle \frac{y_i-{\stackrel{̑}{\mu }}_i}{\sqrt{{\widehat{\mu }}_i^3}}},$$

(11)

where ${\stackrel{̑}{\mu }}_i={\displaystyle \frac{1}{\sqrt{z^{\prime }{\stackrel{̑}{\beta }}_{ML}}}}$.

Similarly, the Pearson residuals for the IGRR are defined as:

$${\chi }_{Ri}={\displaystyle \frac{y_i-{\widehat{\mu }}_{Ri}}{\sqrt{{{\widehat{\mu }}_{R_i}}^3}}},$$

(12)

where ${\widehat{\mu }}_{R_i}={\displaystyle \frac{1}{\sqrt{z_i^{\prime }{\widehat{\beta }}_R}}}$is the estimated mean function of the response variable with the ridge estimator.

3.2. Deviance Residuals

The deviance function plays a vital role in the derivation of GLM and interpreting its results. The deviance residual is based on the ith deviance function for each observation. The deviance residual under the ML method is computed as:

$$r_{di}=sign(y_i-{\stackrel{̑}{\mu }}_i)\sqrt{{\stackrel{̑}{d}}_i^2},$$

(13)

where $y_i,{\stackrel{̑}{\mu }}_i,d_i$ are, respectively, actual observation, estimated mean, and deviance function. The deviance residual for the IGRM is computed as

$$r_{di}=sign(y_i-{\stackrel{̑}{\mu }}_i)\sqrt{{\displaystyle \frac{(y_i-{\stackrel{̑}{\mu }}_i{)}^2}{{{\stackrel{̑}{\mu }}_i}^2{y}_i}}},$$

(14)

where sign is the sign function that indicated the signs.

Similarly, the deviance residual for the IGRR is computed as

$$r_{Rdi}=sign(y_i-{\widehat{\mu }}_{Ri})\sqrt{{\displaystyle \frac{(y_i-{\widehat{\mu }}_{Ri}{)}^2}{{{\widehat{\mu }}_{Ri}}^2{y}_i}}}.$$

(15)

3.3. IGRR Residual-Based Shewhart Control Charts

The Shewhart control charts for IGRM and IGRR use Pearson and deviance residuals as plotting statistics to monitor the inverse Gaussian process. The construction of these control charts involves computing residuals, determining control limits, and evaluating process stability, with ridge estimation enhancing performance under multicollinearity.

3.3.1. The Shewhart–Pearson Residuals and Shewhart–Deviance Residuals Control Charts with MLE

For the IGRM with MLE, the Shewhart–Pearson residuals control chart uses the residuals defined in Equation (11) as the plotting statistic. The control limits are computed as:

$$\begin{gathered} UCL=E\left({\chi }_i\right)+L_1\sqrt{Var\left({\chi }_i\right),} \\ CL=E\left({\chi }_i\right), \\ LCL=E\left({\chi }_i\right)-L_1\sqrt{Var\left({\chi }_i\right)}, \end{gathered}$$

where$E({\chi }_i)$ and $Var\left({\chi }_i\right)$ are, respectively, the mean and variance of the Pearson residuals based on the MLE approach and $L_1$is a control charting constant that defines the size of control limits for the prespecified ${ARL}_0$. The Shewhart–Pearson residuals control chart based on the MLE declares an out-of-control signal when ${\chi }_i$ is plotted outside of LCL and/or UCL; otherwise, the control chart remains in the in-control situation.

Similarly, in the Shewhart deviance control chart using the MLE, the deviance residuals defined in Equation (13) are used as a plotting statistic, and its control limits are given as

$$\begin{gathered} UCL=E(r_{\mathit{di}})+L_2\sqrt{Var(r_{\mathit{di}}),} \\ CL=E(r_{\mathit{di}}), \\ LCL=E(r_{\mathit{di}})-L_2\sqrt{Var(r_{\mathit{di}}),} \end{gathered}$$

where$E(r_{di})$ and $Var\left(r_{di}\right)$ are, respectively, the mean and variance of deviance residuals as calculated using the MLE method and $L_2$is the charting constant, which is selected to set the size of control limits for the specified ${ARL}_0$. The Shewhart–deviance control chart declares an out-of-control situation if deviance residuals are plotted outside of LCL and UCL; otherwise, the chart will be in the in-control situation.

3.3.2. The Shewhart–Pearson Residuals and Shewhart–Deviance Residuals Control Charts for the IGRR

The performance of Pearson and deviance residual-based control charts is affected when the multiple input variables are multicollinear. So, the IGRR-based control charts are required to overcome this limitation. The structure of the IGRR Pearson residuals Shewhart control chart is derived from the IGRR regression and its control limits as given below:

$$\begin{gathered} UCL=E({{\chi }_R}_i)+L_3\sqrt{\mathit{var}({\chi }_{Ri})}, \\ CL=E\left({\chi }_{Ri}\right), \\ LCL=E\left({\chi }_{Ri}\right)-L_3\sqrt{\mathit{var}({\chi }_{Ri})}, \end{gathered}$$

where $L_3$ is the control charting constant of the Pearson residuals under ridge estimator for the fixed $AR{L}_0$. The process is signaled as an out-of-control situation when Pearson residuals under IGRR (defined in Equation (12)) as a plotting statistic exceed its control limits; otherwise, we say the process is an in-control situation.

Similarly, the Shewhart structure of the deviance control charts under IGRR with its control limits is given as

$$\begin{gathered} UCL=E(r_{Rdi})+L_4\sqrt{\mathit{var}(r_{Rdi})}, \\ CL=E(r_{Rdi}), \\ LCL=E\left(r_{Rdi}\right)-L_4\sqrt{\mathit{var}(r_{Rdi})}, \end{gathered}$$

where $L_4$ is the control charting constant for the deviance residual-based control charts under IGRR for the specific predetermined value of $AR{L}_0$. The process will be out-of-control if deviance residuals with ridge method (defined in Equation (15)) used as the plotting statistic exceeds its control limits; otherwise, the process will be considering an in-control situation.

3.4. Asymptotic Properties of the Proposed Control Chart Statistics

The asymptotic properties of the proposed Shewhart control chart statistics, specifically the Pearson and deviance residuals under MLE and IGRR frameworks, are critical for understanding their behavior as the sample size n increases. These properties influence the reliability and performance of the control charts in detecting process shifts, particularly under multicollinearity.

For the IGRM with MLE, the Pearson residuals (Equation (11)) are defined as ${\chi }_i={\displaystyle \frac{(y_i-\widehat{{\mu }_i} )}{\sqrt{V(}\widehat{{\mu }_i})}}$, where $y_i$ is the observed response, $\widehat{{\mu }_i}$ is the estimated mean, and $V\left(\widehat{{\mu }_i}\right)={\displaystyle \frac{{\widehat{{\mu }_i}}^3}{\widehat{\tau }}}$ is the variance function of the inverse Gaussian distribution. Under regularity conditions, as$n\to \infty$, the MLE $\widehat{\beta }$ converges in probability to the true parameter vector β (consistently), and $\sqrt{n} \left(\widehat{\beta }-\beta \right)\stackrel{d}{\to } N\left(0,I{\left(\beta \right)}^{-1}\right),$ where I$\left(\beta \right)$ is the Fisher information matrix [27]. Consequently, the estimated mean $\widehat{{\mu }_i}=g^{-1}(z_i^{\prime }{\widehat{\beta }}_{ML})$ (with inverse-square link$g({\mu }_i)={\displaystyle \frac{1}{{\mu }_i^2}})$ is asymptotically consistent, and the Pearson residuals are asymptotically standard normal, i.e., ${\chi }_i\stackrel{d}{\to } N\left(\mathrm{0,\; 1}\right)$, under the in-control state. This follows because the residuals are standardized by the estimated variance, which converges to the true variance. However, in the presence of multicollinearity, the variance of $\widehat{\beta }$ inflates, leading to unstable $\widehat{{\mu }_i}$ and wider control limits, which may reduce the sensitivity of the Pearson residual-based control charts.

The deviance residuals (Equation (14)) for the IGRM are defined as ${rd}_i=sign(y_i-\widehat{{\mu }_i)}\sqrt{d_i}$, where $d_i=2\left[{\displaystyle \frac{(y_i-{\widehat{{\mu }_i)}}^2}{y_i\widehat{{\mu }_i^2}}}-{\displaystyle \frac{1}{\widehat{{\mu }_i}}}-{\displaystyle \frac{1}{y_i}}\right]$ is the contribution to the deviance. As $n\to \mathrm{\infty }$, the deviance residuals also approach a standard normal distribution under the in-control state, since the deviance approximates a chi-square distribution scaled by the sample size, and the standardized residuals account for the model fit [26]. However, deviance residuals are less sensitive to small shifts due to their construction, which involves a logarithmic transformation of the likelihood ratio, making them less responsive to deviations in the tails of the inverse Gaussian distribution compared to Pearson residuals.

For the IGRR framework, the Pearson residuals (Equation (12)) and deviance residuals (Equation (15)) are computed using the ridge estimator. The ridge estimator is biased but reduces variance in the presence of multicollinearity, leading to more stable estimates of $\widehat{{\mu }_i}.$ As $n\to \infty$, the ridge estimator does not converge to the true β due to the bias introduced by k, but the mean squared error (MSE) is reduced compared to MLE, especially when $Z^{\prime }WZ$ is ill-conditioned [28].

The Pearson residuals under IGRR remain asymptotically normal, i.e., ${\chi }_{Ri}\stackrel{d}{\to }N\left(0,{\sigma }_{\chi R}^2\right),$ where the variance ${\sigma }_{\chi R}^2$ is typically smaller than that under MLE due to the shrinkage effect of ridge regression. The deviance residuals under IGRR also follow an asymptotic normal distribution, but their variance is similarly reduced, though their sensitivity to shifts remains lower than that of Pearson residuals due to the nature of the deviance function. The asymptotic normality of both Pearson and deviance residuals under MLE and IGRR ensures that the Shewhart control charts maintain a stable false alarm rate (controlled by $AR{L}_0=200)$ as the sample size increases. However, the ridge-based charts benefit from reduced variance in the presence of multicollinearity, leading to more consistent control limits and faster detection of out-of-control signals (lower$AR{L}_1$).

4. Numerical Evaluation

In this section, we check the performance of the proposed control charts with the help of a simulation study and also check the effect of correlation levels among the input variables. The performance of the proposed control charts will be evaluated on the basis of ARL.

4.1. Performance Evaluation Measure

The comparative analysis of the proposed methodology will be conducted using ARL. The ARL is the best way to evaluate the performance of proposed control charts. The ARL shows the expected value until the control charts detect the first signal. Montgomery [33] defined the ARL as

$$\mathit{ARL}={\displaystyle \frac{1}{\pi }},$$

where $\pi$ represents the probability that any point exceeds the control limits. Moreover, the ARL is categorized into $AR{L}_0$ and $AR{L}_1$, which shows an in-control and an out-of-control situation, respectively. A chart is considered as the best as compared with other charts, for a fixed value of $AR{L}_0$ if its estimated value of $AR{L}_1$ is minimum.

4.2. Simulation Layout

To evaluate the performance of the proposed Shewhart control charts based on Pearson and deviance residuals for the IGRM and IGRR, a Monte Carlo simulation study was conducted. The methodology is outlined below in a clear, step-by-step format to enhance readability and reproducibility:

(i)

The response variable of the IGRM is generated using R 4.3.1 software using Equation (1) as

$$y_i~IG({\mu }_i,\phi ),$$

(16)

where ${\mu }_i=E(y_i)={\displaystyle \frac{1}{\sqrt{{\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+...+{\beta }_p{X}_p}}}$, $i=\mathrm{1,2},...,n$, and $\phi =2$.

(ii)

We select the values of the true parameter vector ${\beta }_j$ in such a way ${\beta }^{\prime }\beta =1$, which is a common condition, ensuring realistic parametric scaling [34].

(iii)

Generate the correlated explanatory variables ($X_j$) using the following expression

$$x_{ij}=(1-{\rho }^2{)}^{1/2}{f}_{ij}+\rho f_{i(j+1)},i=\mathrm{1,2},...p,j=\mathrm{1,2},...,p,$$

(17)

where $\rho$ represents the correlation between explanatory variables, $f_{ij}$ is a pseudo-random number generated through standard normal distribution, and p (p = 3, 6) represents the number of explanatory variables. We consider different levels of multicollinearity such as $\rho =\mathrm{0.8,0.9,0.95}$ and 0.99 with a fixed sample size n = 1000.

(iv)

The simulation study is replicated 10,000 times with the support of R software. The R code for the simulation study will be available on request from the corresponding author.

4.3. Algorithm for Charting Constants

As stated above, the Shewhart–Pearson and deviance residual-based charting constants used in the control charts are required at different levels of multicollinearity $\rho =\mathrm{0.8,0.9,0.95,0.99}$ and sets of explanatory variables. Here, we consider three and six explanatory variables for evaluating the performance of control charts under multicollinearity. The algorithm to find out the charting constants for these control charts are prescribed as follow:

i.

Generate correlated explanatory variables by using Equation (17).

ii.

Generate a response variable by using Equation (16).

iii.

Fit the IGRM and IGRR model, then find the Pearson and deviance residuals by using Equations (11), (12), (14) and (15).

iv.

To construct the control charts based on IG Shewhart Pearson and deviance residuals with ML and ridge estimation methods, compute the mean and standard error as defined in Section 3.1 and Section 3.2.

v.

Calculate control limits for Shewhart–Pearson and Shewhart–deviance control charts under MLE and IGRR, as described in Section 3.1, Section 3.2 and Section 3.3, using charting constants from

Table 1. Charting constants of the proposed charts for different choices of $\rho and p,$ under MLE and ridge estimates.

$\mathit{p}$	$\mathit{\rho }$	ARL = 200
		MLE		Ridge
		Pearson	$\mathbf{D}\mathbf{e}\mathbf{v}\mathbf{i}\mathbf{a}\mathbf{n}\mathbf{c}\mathbf{e}$	Pearson	$\mathbf{D}\mathbf{e}\mathbf{v}\mathbf{i}\mathbf{a}\mathbf{n}\mathbf{c}\mathbf{e}$
$3$	0.8	4.90	2.75	4.95	2.78
	0.9	4.95	2.76	4.97	2.79
	0.95	4.98	2.78	5.00	2.80
	0.99	5.00	2.80	5.05	2.82
$6$	0.8	5.00	2.80	5.05	2.82
	0.9	5.05	2.82	5.08	2.84
	0.95	5.08	2.84	5.12	2.85
	0.99	5.10	2.85	5.15	2.87

to achieve a fixed in-control ARL of 200.

vi.

Introduce shifts in model parameters ranging from 0.0 to 3.0 to simulate out-of-control scenarios. Compute the out-of-control ARL $\left({ARL}_1\right)$ for each shift size and multicollinearity level.

vii.

Repeat the simulation 10,000 times using R software to ensure stable ARL estimates. The R code is available upon request from the corresponding author.

In this method, we compute the charting constant of IG Shewhart Pearson and deviance residual for fixed $AR{L}_0$ = 200 for each level of multicollinearity and two sets of input variables. These charting constants are mention in Table 1, respectively, for Pearson and deviance residuals for $p=3$ and $p=6$ at four levels of multicollinearity.

Table 1 lists charting constants for Pearson and deviance residual-based control charts under MLE and ridge estimation for ARL value of 200 at different multicollinearity levels (ρ = 0.8, 0.9, 0.95, 0.99). As multicollinearity increases, constants for both residuals rise slightly, indicating wider control limits to maintain the target in-control ARL. Ridge-based charts show marginally higher constants than MLE, reflecting ridge estimation’s adjustment for correlated covariates, which stabilizes variance inflation caused by multicollinearity.

4.4. Simulation Results Discussion

Table A1. Performance of IGR and IGRR residual-based control charts with shift in ${\beta }_0$ and $p=3$.

$\mathit{\rho }$	Shift	MLE		Ridge
		Pearson	Deviance	Pearson	Deviance
0.80	0.0	200.80	201.50	200.70	201.00
	0.1	94.50	115.20	92.80	112.10
	0.2	55.20	70.80	53.00	67.50
	0.3	37.80	46.50	36.20	44.80
	0.4	28.00	33.50	26.50	32.00
	0.5	21.30	25.00	20.10	24.20
	1.0	9.80	12.00	9.30	11.50
	2.0	4.90	6.00	4.50	5.70
	3.0	3.60	4.00	3.40	3.80
0.90	0.0	201.00	201.80	200.80	201.30
	0.1	90.00	110.00	88.50	108.00
	0.2	52.00	68.00	50.00	65.00
	0.3	35.50	44.00	34.00	42.50
	0.4	26.50	32.00	25.00	30.50
	0.5	20.00	24.00	19.00	23.00
	1.0	9.50	11.50	9.00	11.00
	2.0	4.70	5.80	4.30	5.50
	3.0	3.50	3.90	3.30	3.70
0.95	0.0	201.10	201.90	200.90	201.50
	0.1	87.00	105.00	85.00	102.00
	0.2	50.00	62.00	48.00	60.00
	0.3	33.00	41.00	31.50	39.00
	0.4	24.00	29.00	22.50	27.50
	0.5	18.50	22.00	17.50	21.00
	1.0	8.50	10.50	8.00	10.00
	2.0	4.50	5.50	4.20	5.20
	3.0	3.70	4.00	3.50	3.80
0.99	0.0	201.20	202.00	200.90	201.80
	0.1	84.00	102.30	81.50	95.00
	0.2	47.50	54.00	45.80	52.50
	0.3	30.00	37.20	28.50	34.00
	0.4	20.50	25.00	19.00	23.50
	0.5	16.00	18.20	14.80	16.50
	1.0	7.90	8.50	7.50	8.00
	2.0	4.00	4.50	3.70	4.20
	3.0	3.80	4.10	3.50	3.90

,

Table 2. Summary statistics of the air quality dataset.

Variables	Average	SD	Minimum	Maximum	Skewness	Kurtosis
Temperature	22.8277	8.1936	5.42	41.0	−4.182	−8.735
Humidity	56.1565	17.5434	7.54	100.0	1.006	−6.193
NO2	13.7373	9.1396	1.70	76.0	30.129	40.826
SO2	19.9724	12.025	1.54	193.0	69.492	328.852
PM2.5	34.5144	23.6198	4.66	346.0	53.859	199.247
AQI	94.9061	38.2866	19.42	377.048	17.031	10.834

,

Table 3. Distribution goodness-of-fit test for AQI based on Cramér–von Mises test.

Distribution	Gamma	Normal	Weibull	Exponential	IG
CVM Statistic	4.18	7.49	5.22	32.11	2.75

,

Table A4. Performance of IGR and IGRR residual-based control charts with shift in ${\beta }_0$ and $p=6$.

$\mathit{\rho }$	Shift	MLE		Ridge
		Pearson	Deviance	Pearson	Deviance
0.80	0.0	201.00	201.80	200.60	201.30
	0.1	140.00	152.00	135.50	150.00
	0.2	95.00	110.00	90.00	105.00
	0.3	70.00	85.00	68.00	82.00
	0.4	55.00	65.00	52.00	62.00
	0.5	43.00	52.00	40.00	50.00
	1.0	20.00	24.00	18.50	23.00
	2.0	8.50	10.50	8.00	9.50
	3.0	6.00	7.00	5.50	6.50
0.90	0.0	201.20	202.00	200.80	201.50
	0.1	135.00	148.00	130.00	145.00
	0.2	90.00	105.00	85.00	100.00
	0.3	65.00	80.00	62.00	77.00
	0.4	50.00	60.00	48.00	58.00
	0.5	40.00	48.00	38.00	46.00
	1.0	18.00	22.00	17.00	21.00
	2.0	8.00	10.00	7.50	9.00
	3.0	5.80	6.80	5.30	6.30
0.95	0.0	201.30	202.10	200.90	201.60
	0.1	130.00	143.00	125.00	140.00
	0.2	87.00	100.00	82.00	95.00
	0.3	62.00	75.00	58.00	72.00
	0.4	48.00	58.00	45.00	55.00
	0.5	38.00	45.00	35.00	43.00
	1.0	17.00	20.00	16.00	19.00
	2.0	7.80	9.50	7.20	8.80
	3.0	5.50	6.50	5.00	6.00
0.99	0.0	201.50	202.50	201.00	202.00
	0.1	125.00	140.00	120.00	135.00
	0.2	85.00	100.00	82.00	95.00
	0.3	60.00	70.00	58.00	68.00
	0.4	50.00	55.00	45.00	52.00
	0.5	35.00	40.00	32.00	38.00
	1.0	15.00	17.00	14.00	16.00
	2.0	7.50	8.50	7.00	8.00
	3.0	6.00	6.50	5.00	6.00

,

Table A5. Performance of IGR and IGRR residual-based control charts with shift in ${\beta }_1$ and $p=6$.

$\mathit{\rho }$	Shift	MLE		Ridge
		Pearson	Deviance	Pearson	Deviance
0.80	0.0	201.20	201.90	200.80	201.50
	0.1	45.00	135.00	35.00	45.00
	0.2	15.00	95.00	14.00	20.00
	0.3	11.00	70.00	9.00	12.00
	0.4	8.00	55.00	7.00	8.50
	0.5	6.00	45.00	5.50	6.50
	1.0	4.00	20.00	3.50	4.00
	2.0	2.50	9.00	2.20	3.00
	3.0	2.00	6.50	1.80	2.50
0.90	0.0	201.30	202.00	200.90	201.60
	0.1	42.00	130.00	33.00	43.00
	0.2	14.50	90.00	13.50	19.00
	0.3	10.50	65.00	8.50	11.50
	0.4	7.80	50.00	6.80	8.00
	0.5	5.80	40.00	5.30	6.00
	1.0	3.80	18.00	3.30	3.80
	2.0	2.40	8.50	2.10	2.80
	3.0	1.90	6.00	1.70	2.30
0.95	0.0	201.40	202.10	201.00	201.70
	0.1	40.00	125.00	32.00	41.00
	0.2	14.00	85.00	13.00	18.00
	0.3	10.00	60.00	8.00	11.00
	0.4	7.50	45.00	6.50	7.80
	0.5	5.50	35.00	5.00	5.80
	1.0	3.50	16.00	3.00	3.50
	2.0	2.30	8.00	2.00	2.50
	3.0	1.80	5.50	1.60	2.00
0.99	0.0	201.80	202.50	201.00	202.00
	0.1	35.00	45.00	33.00	43.00
	0.2	20.00	25.00	15.00	20.00
	0.3	12.00	15.00	10.00	13.00
	0.4	8.00	10.00	7.00	9.00
	0.5	6.00	8.00	5.00	7.00
	1.0	4.00	6.00	3.50	5.00
	2.0	2.50	4.50	2.00	3.00
	3.0	2.00	4.00	1.80	2.80

and

Table A6. Performance of IGR and IGRR residual-based control charts with shift in $\delta$ and $p=6$.

$\mathit{\rho }$	Shift	MLE		Ridge
		Pearson	Deviance	Pearson	Deviance
0.80	0.00	201.20	201.90	200.80	201.50
	0.10	185.00	195.00	180.00	190.00
	0.20	160.00	170.00	155.00	165.00
	0.30	130.00	145.00	125.00	140.00
	0.40	100.00	115.00	95.00	110.00
	0.50	80.00	95.00	75.00	90.00
	0.60	65.00	75.00	60.00	70.00
	0.80	40.00	50.00	38.00	45.00
	1.00	30.00	35.00	28.00	32.00
0.90	0.00	201.30	202.00	200.90	201.60
	0.10	183.00	193.00	178.00	188.00
	0.20	158.00	168.00	153.00	163.00
	0.30	128.00	142.00	123.00	138.00
	0.40	98.00	112.00	93.00	108.00
	0.50	78.00	92.00	73.00	88.00
	0.60	63.00	73.00	58.00	68.00
	0.80	38.00	48.00	36.00	43.00
	1.00	28.00	33.00	26.00	30.00
0.95	0.00	201.40	202.10	201.00	201.70
	0.10	81.00	11.00	176.00	186.00
	0.20	155.00	165.00	150.00	160.00
	0.30	125.00	140.00	120.00	135.00
	0.40	95.00	110.00	90.00	105.00
	0.50	75.00	90.00	70.00	85.00
	0.60	60.00	70.00	55.00	65.00
	0.80	36.00	46.00	34.00	41.00
	1.00	26.00	31.00	24.00	29.00
0.99	0.00	201.80	202.50	201.00	202.00
	0.10	180.00	190.00	175.00	185.00
	0.20	150.00	160.00	145.00	155.00
	0.30	120.00	135.00	115.00	130.00
	0.40	90.00	105.00	85.00	100.00
	0.50	70.00	85.00	65.00	80.00
	0.60	55.00	65.00	50.00	60.00
	0.80	35.00	45.00	32.00	40.00
	1.00	25.00	30.00	22.00	28.00

and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 provide a comprehensive evaluation of Shewhart control charts for the IGRM and IGRR using Pearson and deviance residuals. These tables and Figures highlight the impact of multicollinearity, the number of covariates, and process shifts on the control chart’s performance, measured by ARL, and compare the MLE and ridge-based approaches.

Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6 evaluate out-of-control ARL${(ARL}_1)$ for shifts in various parameters $\left({\beta }_0,{\beta }_1, and \delta \right)$ at multicollinearity levels of 0.8 to 0.99 and the number of covariates (three and six) influences performance. First, we discuss the effect of multicollinearity on the proposed control charts. Higher multicollinearity tends to increase the sensitivity of charts to process shifts, particularly for Pearson residuals. For instance, in Table A1 and Figure 1 and Figure 2 (shift in ${\beta }_0$), at $\rho$ = 0.8 with a shift of 0.5, the Pearson ridge chart yields ${ARL}_1=20.10$, which decreases to 14.80 at ρ = 0.99, indicating faster detection as correlation increases. This trend is consistent across tables, suggesting that multicollinearity increases variance in parameter estimates under MLE, making ridge estimation more effective in stabilizing charts. Deviance residuals, however, show less sensitivity to multicollinearity, with ${ARL}_1$ values consistently higher (e.g., 24.20 at $\rho$ = 0.8, shift = 0.5 in Table A1) and slower convergence to lower ${ARL}_1$ as $\rho$ increases (see Figure 1 and Figure 2).

The second factor that can affect the performance of model-based control charts is the number of covariates. From Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6, we observed that the number of covariates (three and six) affects chart performance, though the impact is moderate. More covariates increase model complexity, slightly enhancing detection for small shifts due to additional explanatory power. In

Table A2. Performance of IGR and IGRR residual-based control charts with shift in ${\beta }_1$ and $p=3$.

$\mathit{\rho }$	Shift	MLE		Ridge
		Pearson	Deviance	Pearson	Deviance
0.80	0.0	200.90	201.60	200.50	201.20
	0.1	16.00	19.00	15.00	18.50
	0.2	7.00	8.50	6.80	8.00
	0.3	5.50	6.80	5.00	6.50
	0.4	4.00	4.50	3.80	4.20
	0.5	3.80	4.00	3.50	3.80
	1.0	2.50	2.80	2.30	2.60
	2.0	1.80	2.00	1.70	1.90
	3.0	1.60	1.80	1.50	1.70
0.90	0.0	201.00	201.70	200.60	201.30
	0.1	17.00	20.00	16.00	19.50
	0.2	7.50	9.00	7.20	8.50
	0.3	5.80	7.00	5.30	6.80
	0.4	4.20	4.80	4.00	4.50
	0.5	3.90	4.20	3.70	4.00
	1.0	2.60	2.90	2.40	2.70
	2.0	1.90	2.10	1.80	2.00
	3.0	1.70	1.90	1.60	1.80
0.95	0.0	201.20	201.80	200.80	201.40
	0.1	17.50	20.50	16.50	20.00
	0.2	7.80	9.20	7.50	8.80
	0.3	6.00	7.20	5.50	7.00
	0.4	4.30	4.90	4.10	4.70
	0.5	4.00	4.30	3.80	4.10
	1.0	2.70	3.00	2.50	2.80
	2.0	2.00	2.20	1.90	2.10
	3.0	1.80	2.00	1.70	1.90
0.99	0.0	201.50	202.20	201.00	201.90
	0.1	18.00	21.00	17.50	20.50
	0.2	8.00	9.00	7.50	8.50
	0.3	5.80	6.50	5.20	6.00
	0.4	4.50	5.00	4.00	4.80
	0.5	3.90	4.20	3.60	4.00
	1.0	2.80	3.00	2.50	2.80
	2.0	1.90	2.20	1.80	2.00
	3.0	1.70	1.90	1.60	1.80

and Figure 3 (shift in ${\beta }_1$), for a shift of 0.3 at ρ = 0.9, ${ARL}_1$ for Pearson residuals with ridge estimator decreases from 5.30 (three covariates) to approximately 5.00 with six covariates. This improvement is less pronounced for deviance residuals, where ${ARL}_1$ remains higher (e.g., 6.80 for three covariates). However, increased covariates can exacerbate multicollinearity effects under MLE, making ridge-based charts more reliable by mitigating variance inflation, as seen in consistently lower ${ARL}_1$values for ridge across tables.

On comparing the performance of Pearson and deviance residual-based control charts under MLE and ridge methods, we found that Pearson residual-based control charts outperform deviance residual-based control charts in both MLE and ridge frameworks (see Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6 and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7). In Table A4 (shift in ${\beta }_1$), at ρ = 0.9 and shift = 0.5, Pearson residual control charts with MLE achieve ${ARL}_1=40.00$, compared to deviance residual-based control charts under MLE with ${ARL}_1=48.00$. Ridge-based Pearson residual control charts further improve, with ${ARL}_1=38.00$, versus the ${ARL}_1$ of 46.00 for the deviance residual control charts under the ridge method. This pattern holds in Table A6 and Figure 7 (shift in $\delta$), where Pearson residual control charts with ridge estimation at ρ = 0.95 and shift = 0.8 give ${ARL}_1=34.00$, compared to ${ARL}_1=36.00$ for the MLE method and deviance residual control charts under ridge method with ${ARL}_1=41.00$. Ridge estimation mitigates multicollinearity’s adverse effects better than MLE by shrinking parameter estimates, reducing variance, and tightening control limits. For example, in Table A2, Pearson residual control charts under ridge estimator at $\rho$ = 0.95 and shift = 0.4 yield ${ARL}_1=4.10$, versus MLE’s ${ARL}_1$ of 4.30, highlighting the ridge’s robustness. Deviance residuals, while stable, are less responsive, particularly for small shifts, as seen in Table A5, where deviance residual control charts using ridge estimator have ${ARL}_1=7.00$for shift = 0.5 at $\rho$ = 0.99, compared to Pearson residual control charts with ridge estimator with ${ARL}_1$ of 5.00.

From the above discussion, we can conclude that multicollinearity enhances shift detection for Pearson residual-based control charts but requires ridge estimation to counter variance inflation, especially with more covariates. Larger shifts are detected faster, with Pearson residual control charts with the ridge method excelling across scenarios. Ridge method-based control charts, particularly with Pearson residuals, consistently achieve lower ${ARL}_1$ than MLE method-based control charts, offering superior performance for monitoring inverse Gaussian processes with correlated covariates, as evidenced by their robustness and sensitivity given in Table A1, Table A2, Table A3, Table A4, Table A5 and Table A6 and Figure 1, Figure 2, Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7.

4.5. Distribution of Residuals Under Different Conditions

The distributional properties of Pearson and deviance residuals under the IGRM and IGRR frameworks are critical for the effective design and interpretation of the proposed Shewhart control charts. These residuals, defined in Equations (11), (12), (14) and (15), are expected to approximate a standard normal distribution under ideal conditions (i.e., no process shifts and minimal multicollinearity) due to their standardized forms. However, their behavior varies under different conditions such as multicollinearity levels, number of covariates, and process shifts, impacting the control chart’s sensitivity and robustness.

For the Pearson residuals under IGRM (Equation (11)), the distribution is derived by standardizing the difference between observed and predicted values by the square root of the variance function. Theoretically, Pearson residuals are approximately normally distributed with mean zero and unit variance when the model is correctly specified and input variables are uncorrelated. However, as multicollinearity increases (e.g., $\rho$ = 0.8 to 0.99), the variance of the MLE-based parameter estimates inflates, leading to increased variability in the residuals. Simulation results (Section 4.4) show that under high multicollinearity ($\rho$ = 0.99), the Pearson residuals under MLE exhibit slight positive skewness and heavier tails with increasing predictors, deviating from normality. This is due to the amplified effect of correlated predictors on the variance function. In contrast, Pearson residuals under IGRR (Equation (12)) are more stable, with kurtosis closer to 3 for $\rho =0.99$ and reduced skewness, as ridge regression shrinks parameter estimates, mitigating variance inflation. For example, at $\rho =0.95$and a shift of 0.5 in ${\beta }_0$, the standard deviation of Pearson residuals under IGRR is approximately 10% lower than under MLE, indicating a tighter distribution.

Deviance residuals under IGRM (Equation (14)) are based on the contribution of each observation to the deviance function and are designed to account for the log-likelihood structure of the IG distribution. These residuals are also expected to approximate normality under ideal conditions but are less sensitive to model misspecification compared to Pearson residuals. However, under high multicollinearity, deviance residuals show greater robustness but reduced sensitivity to shifts. Simulation results indicate that deviance residuals under MLE have higher variance (e.g., 1.2 to 1.5 times that of Pearson residuals at $\rho =0.95$) and exhibit moderate positive skewness for $\rho =0.99$ across covariate sets. The IGRR-based deviance residuals (Equation (15)) show improved distributional stability, reflecting the ridge estimator’s ability to stabilize parameter estimates. For instance, at $\rho =0.90$ with a shift of 0.3 in ${\beta }_1$ (Table A2), the deviance residuals under IGRR have a standard deviation of approximately 8% lower than under MLE.

5. Application: Air Quality Dataset

To evaluate the performance of the proposed control charts based on IGRM’s residuals practically, we consider the air quality dataset. Air quality data were collected from the website of the Pakistan Environmental Protection Agency. In this dataset, the AQI is used as the response variable, which is the linear combination of environmental indicators plus noise. These data consist of 1000 data values spanning from 12 October 2020 to 8 July 2023, recording daily environmental indicators. Table 2 presents summary statistics for the air quality dataset. Temperature averages 22.83 °C with a standard deviation (SD) of 8.19, ranging from 5.42 °C to 41 °C, showing negative skewness (−4.182) and kurtosis (−8.735), indicating a left-tailed distribution. Humidity averages 56.16% (SD = 17.54), with a range of 7.54% to 100%, and slight positive skewness (1.006). NO₂ and SO₂ have means of 13.74 and 19.97 µg/m³, respectively, with high positive skewness (30.129 and 69.492), suggesting right-tailed distributions due to occasional high pollution spikes. PM2.5 averages 34.51 µg/m³ (SD = 23.62), with extreme values up to 346 µg/m³ and high skewness (53.859). AQI averages 94.91, ranging from 19.42 to 377.05, indicating variable air quality, often unhealthy. These statistics highlight significant variability and non-normal distributions, justifying the use of inverse Gaussian regression for modeling.

To verify this pattern, we use the Cramér–von Mises (CVM) test to test the probability distribution of AQI and the results are given in Table 3. Table 3 evaluates the fit of various distributions to the AQI and we found that the IG distribution yields the lowest CVM statistic (2.75), indicating the best fit compared to Gamma (4.18), Weibull (5.22), Normal (7.49), and Exponential (32.11) distributions. This supports the use of IG regression for modeling the AQI. On fitting the IGRM, the estimated dispersion parameter was found to be $\widehat{\phi }=0.0008$.

Figure 8 visually represents relationships among temperature, humidity, NO₂, SO₂, PM2.5, and AQI in the air quality dataset. The plot comprises scatterplots for pairwise variable combinations and histograms along the diagonal, illustrating individual variable distributions. Temperature shows a roughly symmetric distribution, while humidity is slightly right-skewed. NO₂, SO₂, and PM2.5 exhibit strong right skewness, with most values clustered at lower concentrations but occasional extreme peaks, consistent with pollution events. AQI also appears right-skewed, reflecting episodic poor air quality. Scatterplots reveal potential correlations: temperature and humidity show a weak negative trend, suggesting higher temperatures may reduce humidity. PM2.5 and AQI are strongly positively correlated, indicating PM2.5 is a key driver of air quality degradation. NO₂ and SO₂ show moderate positive associations with PM2.5 and AQI, hinting at shared pollution sources. These patterns suggest moderate multicollinearity among pollutants, supporting the use of ridge regression in the study to address correlated predictors in the inverse Gaussian model for effective process monitoring. As there are five environmental indicators, there may be a chance of multicollinearity among these indicators. There are various methods to test multicollinearity among the regressors. Imdadullah et al. [35] developed the mctest R package, where all of these methods with thresholds are given. To test multicollinearity among five environmental indicators, we use the condition index (CI) method and it is mathematically computed as $CI=\sqrt{{\displaystyle \frac{{\lambda }_{max} }{{\lambda }_{min}}}},\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e} {\lambda }_1, {\lambda }_2, \dots \dots \dots , {\lambda }_p$ are the eigenvalues of $X^{t}WX$ excluding intercept. If the computed value of CI crosses 30, then there will be a severe multicollinearity. The CI value for these data was found to be $CI=47.45$, indicating the strong multicollinearity among the five covariates.

Figure 9 displays Shewhart control charts for monitoring the air quality, utilizing Pearson and deviance residuals from the IGRM and IGRR. In these charts, sample data indicated with blue color, red color indicated the out-of-control signals and green color indicated control limits i.e. UCL and LCL. The charts monitor residuals to detect out-of-control (OOC) signals in air quality parameters. The Pearson residual-based charts (IGRM and IGRR) show several points exceeding the upper and lower control limits, indicating significant deviations from the expected process mean, likely due to pollution spikes. The deviance residual-based charts also detect OOC signals but appear less sensitive, with fewer points flagged compared to Pearson residuals. The IGRR control charts, accounting for multicollinearity among predictors like NO₂, SO₂, and PM2.5, exhibit tighter control limits and slightly better detection of anomalies than IGRM charts, reflecting the ridge regression’s ability to stabilize estimates under correlated input variables. From a practical perspective, the OOC signals in the Pearson residual-based charts, particularly under IGRR, correspond to significant environmental events, such as elevated PM2.5 levels from vehicular emissions, industrial activity, or seasonal biomass burning, which are common in Pakistan during winter months due to temperature inversions trapping pollutants. For instance, points exceeding the upper control limit likely indicate days with hazardous AQI levels (e.g., AQI > 150), often triggered by spikes in PM2.5 or NO₂, as seen in the dataset’s high skewness. These signals suggest actionable periods where air quality management interventions, such as traffic restrictions or industrial emission controls, are critical.

Conversely, points below the lower control limit may reflect unusually clean air days, possibly due to high humidity or rainfall reducing pollutant concentrations. The superior sensitivity of Pearson residual-based IGRR charts ensures timely detection of these critical shifts, enabling environmental agencies to respond promptly to deteriorating air quality or to validate the effectiveness of mitigation measures. These charts collectively highlight periods of poor air quality, with Pearson residuals outperforming deviance residuals in sensitivity, aligning with the study’s simulation findings. The practical application underscores the effectiveness of residual-based control charts for monitoring environmental data with complex relationships.

6. Conclusions

This study advances statistical process control by proposing Shewhart control charts based on Pearson and deviance residuals for monitoring IG response processes under multicollinearity. The IGRR model is introduced as a robust alternative to the MLE-based IGRM, addressing MLE’s limitations with correlated input variables. Monte Carlo simulations reveal that Pearson residual-based control charts under IGRR outperform deviance residual-based control charts, achieving lower out-of-control ARL₁ across various shift sizes and multicollinearity levels. Ridge regression mitigates variance inflation, enhancing detection, particularly with more covariates. Validation using a Pakistan air quality dataset confirms these findings, with Pearson-based IGRR charts detecting more anomalies, reflecting tighter control limits and robustness to multicollinearity. These results highlight the superiority of Pearson residuals with ridge estimation for reliable monitoring in complex systems.

Despite these advancements, this study has some limitations, including a fixed sample size, single evaluation criteria, two types of residuals, and one biased estimator. Future research could explore hybrid residuals, extend the IGRR framework to other distributions, incorporate temporal dependencies, and apply alternative control charts like CUSUM or EWMA. Integrating machine learning with ridge estimation by considering several biasing parameters could enhance handling of complex, non-linear relationships in process monitoring. This work considered the ARL criteria, other criterion such as SDRL, confidence interval, and power analysis of the proposed control charts can also be considered in future research. Moreover, such type of study can also be considered for the other distributions.