Parameter Estimation of Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression

Abstract

Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression is a parsimonious ordinal logistic regression with consideration of the existence of spatial and temporal effects. This model has been developed with the following three considerations: the spatial effect, the temporal effect, and predictor selection. The last point prompted the use of Elastic Net regularization in choosing predictors while handling multicollinearity, which often arises when there are many predictors involved. The Elastic Net penalty combines ridge and LASSO penalties, leading to the determination of the appropriate ${\lambda }_{EN}$ and ${\alpha }_{EN}$. Therefore, the objective of this study is to determine the parameter estimator using Maximum Likelihood Estimation. The estimation process comprises defining the likelihood function, determining the natural logarithm of the likelihood function, and maximizing the function using Berndt–Hall–Hall–Hausman. These steps continue until the estimator converges on the values that maximize the likelihood function. This study contributes by developing an estimation framework that integrates spatial and temporal effects with Elastic Net regularization, allowing for improved model interpretation and stability. The findings provide an advanced methodological approach for ordinal logistic regression models that incorporate spatial and temporal dependencies. This framework is particularly useful for applications in fields such as economic forecasting, epidemiology, and environmental studies, where ordinal responses exhibit spatial and temporal patterns.

1. Introduction

One of the important steps in regression analysis is deciding which predictors are to be included in the model. Often, the result of other research can motivate the use of certain predictors. Exploring more research results increases the number of predictors to use in the model. If the number of predictors increases, the estimated regression model becomes more complicated. Many predictors in the estimate model result in a large amount of data that we must collect to obtain the estimated value. When there are many predictors then there would be overfitting [1] and multicollinearity [2].

In logistic regression, overfitting yields a large value of the odds ratio, and the confidence interval of the parameter might be extremely wide [3]. Multicollinearity also causes the logistic estimators to have a high standard error and, in certain cases, the estimator may have the incorrect sign [4]. Generally, the presence of overfitting and multicollinearity negatively affects the estimated regression model, which leads to the wrong conclusion. As the main point of applying a regression model is to obtain a good estimated model that predicts a new response with minimal error, both violations above must be handled. One approach that can be taken is using the regularization techniques.

Regularization techniques, such as Ridge Regression (RR) [5], Least Absolute Shrinkage and Selection Operator (LASSO) [6], and Elastic Net (EN) [7], can improve prediction accuracy. EN combines the regularization techniques of RR and LASSO. EN has been used in linear regression [8,9,10], and in logistic regression [11,12,13]. In these studies, EN has been successful in choosing predictors with no multicollinearity.

This paper introduces a model, namely the Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression (GTWENOLR) model. There are three features in the GTWENOLR model: the effect of location, the effect of time, and predictor selection. As a part of regression, this model has an estimated model and hypothesis tests. This study will describe the parameter estimator of GTWENOLR.

Many researchers have studied the effect of location in regression modeling, including Purhadi et al. [14] who proposed the Geographically Weighted Ordinal Logistic Regression (GWOLR) model. This model is an extension of Ordinal Logistic Regression (OLR), incorporating spatial effects. The presence of spatial influence leads to the creation of localized OLR regression model estimates for each distinct area. This means that instead of assuming a homogeneous relationship between predictor variables and the response variable across the entire study region, the model estimates a unique set of regression coefficients for each location. As a result, GWOLR requires the estimation of multiple sets of coefficients, reflecting the local variations that exist in the data. Although this approach increases model complexity, it enables a more precise understanding of how localized influences affect the outcome being studied, ultimately providing a more accurate representation of spatial heterogeneity in the data.

As one of the spatial regressions, there is a spatial weight that causes each observation to have a different estimated value, and each value could have different significant independent variables. To determine the parameter estimator model, Purhadi et al. employed the Maximum Likelihood Estimation (MLE) and the Newton–Raphson iteration algorithm to determine the optimal solution [14]. Other methods, such as Fisher scoring [15], can also be used as iteration techniques. The result of these studies showed that the location factor also affected the regression modeling, even when the response has an ordinal scale.

However, parameter estimation in the GWOLR model is complex due to its non-closed form structure, requiring numerical iteration. Unlike models with analytical solutions, methods like MLE and local likelihood approaches depend on iterative procedures to reach stable estimates. These iterations capture local variations in the data while balancing computational efficiency with accuracy. Choosing the appropriate iteration method is critical. Considerations include convergence speed, stability, and computational complexity. The Newton–Raphson method is commonly used for its quadratic convergence but requires computationally expensive second derivatives (the Hessian matrix). In cases where computing the Hessian is difficult or the parameter space is expansive, quasi-Newton methods like Broyden–Fletcher–Goldfarb–Shanno (BFGS) offer an efficient alternative by approximating the Hessian, as well as balancing speed and accuracy [16]. The Expectation-Maximization (EM) algorithm may be helpful when dealing with missing or latent data, as it breaks the estimation into more manageable steps [17]. Ultimately, the choice of iteration method should depend on the model structure, the quality of initial parameter estimates, and the available computational resources, ensuring reliable convergence and efficient parameter estimation.

In 2024, Ohyver et al. [18] developed a Geographically and Temporally Weighted Ordinal Logistic Regression (GTWOLR) model, an extension of the GWOLR model that incorporates the influence of time. The estimator was obtained using MLE and Berndt–Hall–Hall–Hausman (BHHH) as the iteration technique. This study’s findings are consistent with other temporal regression analyses. For example, Ma et al. conclude that Geographically Weighted Regression (GWR) is not suitable to model the transit ridership as there is significant hourly average ridership at certain times [19]. Similarly, Ling et al. highlighted the need for models that capture variation in effects across both time and space when analyzing right-turn lane crashes in Indiana. Their study shows that the Geographically and Temporally Weighted Negative Binomial (GTWNBR) is the best model to capture the space and time instability in crashes [20]. These cases highlight the importance of choosing a model that takes location and time into account. The GTWOLR is one of the regression models that capture those two aspects. The existence of spatial and temporal weights causes every location in different time periods to have a different model.

The final feature in GTWENOLR is its ability to construct a parsimonious predictive regression model. In other words, the predictors in GTWENOLR are selected predictors using EN. EN is a combination of LASSO and RR. The development of EN was conceived as a method that would produce better predictions if the predictors were highly correlated [10]. The successful application of EN in predictor selection has been demonstrated in various studies focusing on spatial logistic regression [21,22]. These studies have shown that incorporating EN leads to models that are both simpler and more robust, with fewer predictors, that still capture the essential information required for accurate predictions. The compelling performance of these models provides a strong rationale for employing the EN method in other regression frameworks, including the GTWENOLR model proposed here. By leveraging EN, the model not only achieves a more streamlined predictor set but also improves overall predictive performance, making it a valuable tool for analyses involving complex spatial and temporal data.

Another important perspective is the challenge of multicollinearity, which is prevalent in spatial–temporal datasets where predictors often exhibit high intercorrelations. Multicollinearity can lead to unstable parameter estimates and inflated variances, thereby undermining the reliability of the model [23]. Traditional methods, such as LASSO, may arbitrarily select one predictor from a group of highly correlated variables, potentially discarding valuable information, while RR only shrinks coefficients without eliminating any variables [24]. EN effectively overcomes these issues by combining the strengths of both approaches; it not only performs predictor selection but also encourages the grouping of correlated predictors. This dual action stabilizes the estimation process, reduces variance, and enhances model robustness—making EN an ideal choice for the GTWENOLR model in handling complex predictor interrelationships.

The highlighted features of the GTWENOLR model can be counted as the contributions of this model. Based on these contributions, we need to construct the estimation model. We will estimate the model parameters using the MLE and BHHH iteration techniques. Therefore, obtaining the parameter estimators for GTWENOLR is the goal of this study. This study has the advantage of supporting researchers in modeling cases that have location and time effects.

2. Elastic Net Ordinal Logistic Regression

Elastic Net Ordinal Logistic Regression (ENOLR) is an OLR model that selects predictors. These predictors have been selected using EN, so the accuracy prediction can be improved [21]. EN is one of the regularization methods that combine LASSO and RR to address the weaknesses of each method.

In 1970, Hoerl and Kennard introduce RR as a regression model that can handle multicollinearity. The RR adds a penalty to the loss function in ordinal logistic regression. This penalty, called an $L_2$-penalty, reduces large coefficient values and helps avoid overfitting. Adding this penalty to the loss function of OLR, then the parameter estimator of ENOLR model obtained by solving the formula in (1).

$$\sum _{i=1}^n\sum _{g=1}^G{y}_{ig}\log \left[{\displaystyle \frac{\exp \left({\alpha }_{g\mathrm{R}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathbf{R}}\right)}{1+\exp \left({\alpha }_{g\mathrm{R}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathbf{R}}\right)}}-{\displaystyle \frac{\exp \left({\alpha }_{g-1\mathrm{R}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathbf{R}}\right)}{1+\exp \left({\alpha }_{g-1\mathrm{R}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathbf{R}}\right)}}\right]-{\lambda }_R{‖{\mathbf{\beta }}_{\mathbf{R}}‖}_2^2$$

(1)

The ${\mathbf{\beta }}_{\mathbf{R}}={\left[{\beta }_{1R},{\beta }_{2R}, \cdots ,{ \beta }_{KR}\right]}^{\prime }$ in (1) is the vector of regression coefficients of ENOLR, and ${\alpha }_{g\mathrm{R}},g=1, 2, \cdots , G-1$ are the intercepts. The ${\lambda }_R$ penalty in (1) is a positive number that defines the relative trade-off between the model variance and bias. When there is no penalty, the error is inflated. As the penalty increases, the bias becomes too large, and the model starts to underfit. While RR shrinks the parameter estimates towards 0, the model does not set the values to absolute 0 for any value of the penalty. Even though some parameter estimates become negligibly small, this model does not conduct predictor selection [7].

Predictor selection refers to the process of choosing the most relevant predictor to include in a regression model. This selection is related to determining which predictors should be included in the model and in what form they should be included [25]. RR is a good regression model when there is a multicollinearity, but it does not produce a parsimonious model [26]. Therefore, there is LASSO regression, which was developed by Tibshirani in 1996. There are two advantages in using LASSO regression, the accuracy of prediction and the convenience in interpretation [6].

LASSO is a penalized least squares method imposing an $L_1$-penalty on the regression coefficients. With the addition of this penalty, ENOLR estimators were obtained by minimizing the following equation.

$$\sum _{i=1}^n\sum _{g=1}^G{y}_{ig}\log \left[{\displaystyle \frac{\exp \left({\alpha }_{g\mathrm{L}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{L}}\right)}{1+\exp \left({\alpha }_{g\mathrm{L}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{L}}\right)}}-{\displaystyle \frac{\exp \left({\alpha }_{g-1\mathrm{L}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{L}}\right)}{1+\exp \left({\alpha }_{g-1\mathrm{L}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{L}}\right)}}\right]-{\lambda }_L{‖{\mathbf{\beta }}_{\mathrm{L}}‖}_2^2.$$

(2)

While the regression coefficients still shrunk towards 0, a consequence of penalizing the absolute values is that some parameters are actually set to 0 for some value of ${\lambda }_L$. Thus, the lasso yields models simultaneously use regularization to improve the model and to conduct predictor selection.

While considering the $‖.‖$, there are three conditions for the solution of (2).

$$\mathrm{Condition} 1:{\mathbf{\beta }}_{\mathrm{L}}>0,\mathrm{then} ‖{\mathbf{\beta }}_{\mathrm{L}}‖={\mathbf{\beta }}_{\mathrm{L}},$$

(3)

$$\mathrm{Condition} 2:{\mathbf{\beta }}_{\mathrm{L}} < 0,\mathrm{then} ‖{\mathbf{\beta }}_{\mathrm{L}}‖=-{\mathbf{\beta }}_{\mathrm{L}},\mathrm{and}$$

(4)

$$\mathrm{Condition} 3:{\mathbf{\beta }}_{\mathrm{L}}=0,\mathrm{then} ‖{\mathbf{\beta }}_{\mathrm{L}}‖=0.$$

(5)

Even though the LASSO regression has two advantages, it also has one drawback, especially compared to RR. Since the main idea of LASSO is to select the predictor, then in the result there is a certain variable that sometimes it is not the important one.

EN is one of the techniques that combines the capabilities of RR and LASSO. It means EN regression will choose predictors while overcoming the effects of multicollinearity. Ref. [7] stated that the advantage of this model is that it enables effective regularization via the ridge-type penalty with predictor selection quality of the LASSO penalty.

The EN regression model combines two penalties, which [26] stated that the EN estimator is a mixture of ridge and LASSO estimators. The estimators will be achieved by minimizing the formula in (6).

$$\sum _{i=1}^n\sum _{g=1}^G{y}_{ig}\log \left[{\displaystyle \frac{\exp \left({\alpha }_{g\mathrm{E}\mathrm{N}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{E}\mathrm{N}}\right)}{1+\exp \left({\alpha }_{g\mathrm{E}\mathrm{N}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{E}\mathrm{N}}\right)}}-{\displaystyle \frac{\exp \left({\alpha }_{g-1\mathrm{E}\mathrm{N}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{E}\mathrm{N}}\right)}{1+\exp \left({\alpha }_{g-1\mathrm{E}\mathrm{N}}+{\mathbf{x}}_i^{\prime }{\mathbf{\beta }}_{\mathrm{E}\mathrm{N}}\right)}}\right]$$

$$-\left[{\lambda }_{EN}\left[\left(1-{\vartheta }_{EN}\right){‖{\mathbf{\beta }}_{\mathrm{E}\mathrm{N}}‖}_1+{{\vartheta }_{EN}‖{\mathbf{\beta }}_{\mathrm{E}\mathrm{N}}‖}_2^2\right]\right].$$

(6)

The regularization parameter ${\lambda }_{EN}$ in (6) is the sum of two nonnegative penalties, ${\lambda }_{EN}={\lambda }_L+{\lambda }_R$. Let ${\vartheta }_{EN}={\displaystyle \frac{{\lambda }_R}{{\lambda }_L+{\lambda }_R}}$ and $1-{\vartheta }_{EN}={\displaystyle \frac{{\lambda }_L}{{\lambda }_L+{\lambda }_R}}$, where $0<{\alpha }_{EN}<1$ [27]. Then,

$${\widehat{\mathbf{\beta }}}_{\mathrm{E}\mathrm{N}}=\left\{\begin{array}{c}{\widehat{\mathbf{\beta }}}_{\mathrm{R}}, {\vartheta }_{EN}=0\\ {\widehat{\mathbf{\beta }}}_{\mathrm{E}\mathrm{N}},0<{\vartheta }_{EN}<1\\ {\widehat{\mathbf{\beta }}}_{\mathrm{L}}{,\vartheta }_{EN}=1\end{array}\right..$$

The values of ${\lambda }_L$ and ${\lambda }_R$ can be determined using Cross Validation (CV) and Generalized Cross Validation (GCV).

3. Geographically and Temporally Weighted Ordinal Logistic Regression

A logistic regression model called the Geographically and Temporally Weighted Ordinal Logistic Regression (GTWOLR) is used to find the relationship between an ordinal dependent variable and independent variables [18]. The observations used are locations, where the locations are observed over time. The logit model gives in (7).

$$\mathrm{logit}\left[P\left({\mathbf{y}}_{iT}\le g|{\mathbf{x}}_{it}\right)\right]={\alpha }_g\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{it}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right); g=\mathrm{1,2},\cdots ,G-1;t=\mathrm{1,2},\cdots T.$$

(7)

where:

$P\left({\mathbf{y}}_{iT}\le g|{\mathbf{x}}_{it}\right)$ is a cumulative probability for $g$ category given ${\mathbf{x}}_{it}$, ${\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)={\left[{\beta }_{1T}\left(u_i,v_i,t_i\right),{\beta }_{2T}\left(u_i,v_i,t_i\right),\cdots {\beta }_{KT}\left(u_i,v_i,t_i\right)\right]}^{\prime }$ is regression coefficient vector at location $i$ and time $t$, $\left\{{\alpha }_g\left(u_i,v_i,t_i\right)\right\}$ is intercept with ${\alpha }_{1T}\left(u_i,v_i,t_i\right)\le {\alpha }_{2T}\left(u_i,v_i,t_i\right)\le \cdots \le {\alpha }_{G-1,T}\left(u_i,v_i,t_i\right)$, and $\left(u_i,v_i,t_i\right)$ is coordinate point at location $i$ and time $t$.

Equation (7) shows ${\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)$ as the model parameter. In order to determine the estimators, suppose there are $n$ vectors of random variables samples ${\mathbf{y}}_{1T},{\mathbf{y}}_{2T},\cdots ,{\mathbf{y}}_{nT}$ where ${\mathbf{y}}_{iT}={[y_{i1T}\hspace{1em}{y}_{i1T}\hspace{1em}\cdots \hspace{1em}{y}_{i,G-1,T}]}^{\prime }$ have multinomial distribution with ${\pi }_{gT}\left({\mathbf{x}}_{iT}\right)$ as the probability of $g$ category. Equation (8) gives the set of parameters.

$${\mathrm{\Omega }}_T=\left\{{\alpha }_{1T}\left(u_i,v_i,t_i\right),{\alpha }_{2T}\left(u_i,v_i,t_i\right),\cdots ,{\alpha }_{G-1,T}\left(u_i,v_i,t_i\right)\right.,$$

$$\left.{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right); i=\mathrm{1,2},\cdots ,n\right\}$$

(8)

The natural logarithm of the likelihood function for every location for GTWOLR is given in (9).

$$\log \left[L\left({\mathrm{\Omega }}_{iT}\right)\right]$$

$$=\sum _{j=1}^n\sum _{g=1}^G\sum _{h=1}^T{w}_{ijh}\left\{y_{jgh}\log \left[{\displaystyle \frac{\exp \left({\alpha }_{gT}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{gT}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{iT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right]\right.$$

$$\left.-{\displaystyle \frac{\exp \left({\alpha }_{g-1,T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{g-1,T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right\}.$$

(9)

In Equation (9), there is $w_{ijh}$, the diagonal element of spatial–temporal matrix $\left(\mathit{W}\left(u_i,v_i,t_i\right)\right)$.

Suppose the dependent variable has three categories, then (9) changes to (10).

$$\log \left[L\left({\mathrm{\Omega }}_{iT}\right)\right]$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{y_{j1h}\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)\right.$$

$$-\left(y_{j1h}+y_{j2h}\right)\log \left(1+\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)\right)\right)$$

$$+y_{j2h}\log \left(\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)\right.$$

$$\left. -\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)\right)$$

$$\left. +\left(y_{j1h}-1\right)\log \left(1+\mathrm{e}\mathbf{x}\mathrm{p}\left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)\right)\right\}.$$

(10)

The next step is taking the first derivative with respect to each of parameter models for $t=T$ using (10). The results are given in (11)–(13).

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{iT}\right)\right) }{\partial {\alpha }_{1T}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{y_{j1h}-\left(y_{j1h}+y_{j2h}\right){\displaystyle \frac{\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right.$$

$$\left.-y_{j2h}{\displaystyle \frac{\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{{\Delta }_1}}\right\}.$$

(11)

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{iT}\right)\right) }{\partial {\alpha }_{2T}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{y_{j2h}{\displaystyle \frac{\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{{\Delta }_1}}\right.$$

$$+\left.\left(y_{j1h}-1\right){\displaystyle \frac{\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right\}.$$

(12)

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{iT}\right)\right) }{\partial {\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{\left(y_{j1h}+y_{j2h}\right){\mathbf{x}}_{jT}^{\prime }\left[{\displaystyle \frac{1}{1+\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right]\right.$$

$$\left.+\left(y_{j1h}-1\right){\mathbf{x}}_{jT}^{\prime }{\displaystyle \frac{\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right\}.$$

(13)

where:

$${\Delta }_1=\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)-\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right).$$

To obtain the estimators, the BHHH iteration is applied and gives ${\widehat{\mathrm{\Omega }}}_T$ as the set of parameter estimators which are displayed in (14).

$${\widehat{\mathrm{\Omega }}}_{iT}=\left\{{\widehat{\alpha }}_{1T}\left(u_i,v_i,t_i\right),{\widehat{\alpha }}_{2T}\left(u_i,v_i,t_i\right),\right.$$

$$\left.{\widehat{\beta }}_{1T}\left(u_i,v_i,t_i\right),{\widehat{\beta }}_{2T}\left(u_i,v_i,t_i\right),\cdots ,{\widehat{\beta }}_{KT}\left(u_i,v_i,t_i\right); i=\mathrm{1,2},\cdots ,n\right\}.$$

(14)

4. Geographically and Temporally Weighted Elastic Net Ordinal Logistic Regression

The GTWENOLR model integrates spatial and temporal weighting into an ordinal logistic regression framework, enhanced by the EN penalization technique. This model assigns distinct weights to each observation based on both spatial and temporal proximity, allowing it to capture heterogeneity across various locations and time periods. The application of EN—which combines the strengths of LASSO and RR—enables effective variable selection and addresses issues of multicollinearity, particularly when predictors are highly correlated. This results in more stable and interpretable parameter estimates. For parameter estimation in the GTWENOLR model, MLE is employed, with optimization performed using the BHHH iterative algorithm.

GTWENOLR is developed by GTWOLR which focuses on selecting the predictor. The regularization term for this model is defined in (15).

$${\lambda }_{EN}\left[{\vartheta }_{EN}{‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖}_1+{\displaystyle \frac{1}{2}}\left(1-{\vartheta }_{EN}\right){‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖}_2^2\right].$$

(15)

The set of parameters for GTWENOLR, ${\mathrm{\Omega }}_{{EN}_T}$, that must be estimated, are displayed in (16).

$${\mathrm{\Omega }}_{{EN}_T}=\left\{{\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right),{\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right),\cdots ,{\alpha }_{{EN}_{G-1,T}}\left(u_i,v_i,t_i\right)\right.,$$

$$\left.{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right); i=\mathrm{1,2},\cdots ,n\right\}.$$

(16)

Suppose the response variable has three categories, then:

$${\mathrm{\Omega }}_{{EN}_T}=\left\{{\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right),{\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right),\right. \left.{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right); i=\mathrm{1,2},\cdots ,n\right\}.$$

(17)

Using Equation (10), the natural logarithm of the likelihood function for every location for GTWENOLR is given in (18).

$$\log \left[L\left({\mathrm{\Omega }}_{i{EN}_T}\right)\right]$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{y_{j1h}\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)\right.$$

$$-\left(y_{j1h}+y_{j2h}\right)\log \left(1+\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)\right)\right)$$

$$+y_{j2h}\log \left(\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)\right.$$

$$\left.-\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)\right)$$

$$\left.+\left(y_{j1h}-1\right)\log \left(1+\mathrm{e}\mathbf{x}\mathrm{p}\left({\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)\right)\right\}$$

$$+{\lambda }_{EN}\left[{\vartheta }_{EN}{‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖}_1+{\displaystyle \frac{1}{2}}\left(1-{\vartheta }_{EN}\right){‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖}_2^2\right].$$

(18)

The estimator of ${\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)$ and ${\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)$ is obtained by solving the formula in (19) and (20). Further, folllowing (3)–(5), the estimator for ${\mathbf{\beta }}_{{EN}_T}$ will determined using three conditions, and defined in (21)–(23).

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right) }{\partial {\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{y_{j1h}-\left(y_{j1h}+y_{j2h}\right){\displaystyle \frac{\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)}}\right.$$

$$\left.-y_{j2h}{\displaystyle \frac{\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)}{{\Delta }_2}}\right\}.$$

(19)

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right) }{\partial {\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{y_{j2h}{\displaystyle \frac{\exp \left({\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)}{{\Delta }_2}}\right.$$

$$+\left.\left(y_{j1h}-1\right){\displaystyle \frac{\exp \left({\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)}}\right\}.$$

(20)

where:

$${\Delta }_2=\exp \left({\alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right)-\exp \left({\alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right).$$

$$\mathrm{Condition} 1:{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)>0, \mathrm{then} ‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖={\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right),\mathrm{and}$$

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right) }{\partial {\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{\left(y_{j1h}+y_{j2h}\right){\mathbf{x}}_{jT}^{\prime }\left[{\displaystyle \frac{1}{1+\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right]\right.$$

$$+\left.\left(y_{j1h}-1\right){\mathbf{x}}_{jT}^{\prime }{\displaystyle \frac{\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right\}$$

$$-{\lambda }_{EN}\left[{\vartheta }_{EN}+\left(1-{\vartheta }_{EN}\right){\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right].$$

(21)

$$\mathrm{Condition} 2:{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)>0, \mathrm{then} ‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖=-{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right),\mathrm{and}$$

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right) }{\partial {\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{\left(y_{j1h}+y_{j2h}\right){\mathbf{x}}_{jT}^{\prime }\left[{\displaystyle \frac{1}{1+\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right]\right.$$

$$+\left.\left(y_{j1h}-1\right){\mathbf{x}}_{jT}^{\prime }{\displaystyle \frac{\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right\}$$

$$-{\lambda }_{EN}\left[{-\vartheta }_{EN}+\left(1-{\vartheta }_{EN}\right){\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)\right].$$

(22)

$$\mathrm{Condition} 3:{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)=0, \mathrm{then} ‖{\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)‖=\mathbf{0},$$

$${\displaystyle \frac{\partial \log \left(L\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right) }{\partial {\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)}}$$

$$=\sum _{j=1}^n\sum _{h=1}^T{w}_{ijh}\left\{\left(y_{j1h}+y_{j2h}\right){\mathbf{x}}_{jT}^{\prime }\left[{\displaystyle \frac{1}{1+\exp \left({\alpha }_{1T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right]\right.$$

$$+\left.\left(y_{j1h}-1\right){\mathbf{x}}_{jT}^{\prime }{\displaystyle \frac{\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}{1+\exp \left({\alpha }_{2T}\left(u_i,v_i,t_i\right)+{\mathbf{x}}_{jT}^{\prime }{\mathbf{\beta }}_T\left(u_i,v_i,t_i\right)\right)}}\right\}$$

$$-\left(1-{\vartheta }_{EN}\right){\mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right).$$

(23)

The addition of a regularization term in determining the estimator causes complicated computation since there are several things to do. When using GTWOLR, one focuses on determining spatial and temporal bandwidth, although this task also takes time to acquire the optimal bandwidth. For GTWENOLR, the task increases by determining ${\lambda }_{EN}$ and ${\vartheta }_{EN}$. The bandwidths, ${\lambda }_{EN}$ and ${\vartheta }_{EN}$, need to be derived first before we determine the estimator [24]. This is also supported by Tay et al. [28], who proposed an algorithm to estimate the parameters of extending EN to all Generalized Linear Model (GLM) families.

Since Equations (19)–(23) are nonanalytical equations, then we applied the BHHH algorithm to obtain the solutions. Here are the algorithms to obtain the estimators of the GTWENOLR model.

• Select a value of ${\vartheta }_{EN}$ dan ${\lambda }_{EN}$, where ${\vartheta }_{EN}\in \left[\mathrm{0,1}\right]$ dan ${\lambda }_{EN1}>{\lambda }_{EN2}>\cdots >{\lambda }_{ENm}$.
• Determine the threshold, $\epsilon$. The $\epsilon$ is a small positive number to determine when to stop iterating.
• Determine the initial values, ${\alpha }_{{EN}_{1T}}^{\left(0\right)}\left(u_i,v_i,t_i\right)$, ${\alpha }_{{EN}_{2T}}^{\left(0\right)}\left(u_i,v_i,t_i\right)$, ${\mathbf{\beta }}_{{EN}_T}^{\left(0\right)}\left(u_i,v_i,t_i\right)$, $\mathbf{H}\left({\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(0\right)}\right)$, and $\mathbf{g}\left({\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(0\right)}\right)$. The estimated values of OLR can be used as the initial values. $\mathbf{H}\left(·\right)$ and $\mathbf{g}\left(·\right)$ are the Hessian matrix and the gradient vector.
• For $k=1,2,\cdots ,m$, start the iteration using:

  $${\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(\hslash +1\right)}={\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(\hslash \right)}-{\mathbf{H}}^{-1}\left({\widehat{\mathrm{\Omega }}}_{i{EN}_T}^{\left(\hslash \right)}\right)\mathbf{g}\left({\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(\hslash \right)}\right),$$

  (24)

  where:

  $${\mathbf{g}}_j\left({\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(\hslash \right)}\right)={\left[\begin{array}{ccc}{\displaystyle \frac{\partial \log \left[\mathcal{l}\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right]}{{\partial \alpha }_{{EN}_{1T}}\left(u_i,v_i,t_i\right)}}& {\displaystyle \frac{\partial \log \left[\mathcal{l}\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right]}{{\partial \alpha }_{{EN}_{2T}}\left(u_i,v_i,t_i\right)}}& {\displaystyle \frac{\partial \log \left[\mathcal{l}\left({\mathrm{\Omega }}_{{iEN}_T}\right)\right]}{{\partial \mathbf{\beta }}_{{EN}_T}\left(u_i,v_i,t_i\right)}}\end{array}\right]}^T,$$

  (25)

  and

  $$\mathbf{H}\left({\widehat{\mathrm{\Omega }}}_{i{EN}_T}^{\left(\hslash \right)}\right)=-\sum _{j=1}^n\left({\mathbf{g}}_j\left({\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(\hslash \right)}\right){\mathbf{g}}_j^T\left({\widehat{\mathrm{\Omega }}}_{{iEN}_T}^{\left(\hslash \right)}\right)\right).$$

  (26)
• Stop the iteration when $‖{\widehat{\mathrm{\Omega }}}_{{EN}_T}^{\left({\mathrm{\hslash }}^{\ast }+1\right)}-{\widehat{\mathrm{\Omega }}}_{{EN}_T}^{\left(\mathrm{\hslash }\right)}‖\le \epsilon .$

The choosing of ${\vartheta }_{EN}$ dan ${\lambda }_{EN}$ is an important task in GTWENOLR. There are several techniques that we can use to choose them. For example, we can use CV [29] and GCV [30].

5. Discussion

The dataset for GTWENOLR consists of the $n$ location which is observed $T$ times. In obtaining the estimator, all these observations are used. In Equation (8), there is a spatial–temporal matrix, $\mathit{W}\left(u_i,v_i,t_i\right)$, that causes the estimated parameter to vary over location and time. In computing $\mathit{W}\left(u_i,v_i,t_i\right)$, we need to determine the spatial and temporal bandwidths. We can use the Fotheringham et al. algorithm [31], or the Huang et al. algorithm [32]. Incorporating the $\mathit{W}\left(u_i,v_i,t_i\right)$ produces multiple regression models. However, these estimated models have not considered the possibility of the existence of multicollinearity. This multicollinearity sometimes arises when there are many predictors involved. This is also related to which or how many predictors we want to use in our model. Adopting the idea of [32], we use EN to select the independent variables and to handle the multicollinearity on GTWOLR.

Using the concept of EN led GTWOLR to transform to GTWENOLR, which has an optimal number of predictors with no multicollinearity. The ${\vartheta }_{EN}$ is known as the EN penalty, which bridges the gap between RR and LASSO. When the ${\vartheta }_{EN}$ approaches zero, the GTWENOLR tends to have RR estimated coefficients. The values of the coefficients shrink towards zero, yet it does not eliminate any coefficients entirely. Because all the predictors remain in the model, the RR does not produce an optimal model. When there are many predictors and we keep them in our model, the overfitting may arise [5]. LASSO yields a simple regression model, where the predictors can be fewer in count. The existence of the LASSO penalty makes some coefficients shrink to zero, that is, make the model simpler. When ${\vartheta }_{EN}=1$, GTWENOLR has LASSO coefficients. The purpose of the use of EN in GTWOLR is to obtain a simple regression model with no multicollinearity. One of the obstacles when using EN is how to choose the appropriate ${\vartheta }_{EN}$ and ${\lambda }_{EN}$. Another obstacle is how to choose the optimal spatial and temporal bandwidths.

In [31], CV is used to determine the optimal bandwidth. To compute the CV, one must know the fitted value for the $-i$ response, with point $i$ excluded from the calibration process. In our proposed model, since the response has an ordinal category, we can use the Leave-One-Out Approach Error Rate (LOOAER).

A relevant case for applying GTWENOLR is the Human Development Index (HDI). The HDI measures how the population can access the outcomes of development in terms of income, health, education, and other aspects. It serves as an important indicator for measuring the success of improvement of human life quality. However, the HDI varies significantly due to socioeconomic disparities, infrastructure quality, and geographical differences. In urban areas, income levels and healthcare access often drive development, while in rural regions, education accessibility and basic infrastructure play a more pivotal role. Additionally, geographic factors such as climate and natural resource availability further shape development patterns, underscoring the need for region-specific policy interventions to address localized challenges effectively. Over time, government policies, economic growth, demographic shifts, and external shocks (such as financial crises or pandemics) contribute to HDI fluctuations. These temporal dynamics underscore the need for region-specific and time-sensitive policy interventions to ensure sustainable and equitable human development. Given these complexities, GTWENOLR provides a robust framework for analyzing the HDI by incorporating both spatial and temporal effects. The model captures spatial dependencies, where the HDI in one region may be influenced by neighboring regions due to economic spillovers or shared infrastructure. Simultaneously, temporal dependencies are considered to reflect long-term policy impacts and socioeconomic trends. Furthermore, GTWENOLR mitigates multicollinearity through EN regularization, ensuring more stable and interpretable estimates.

6. Conclusions

The GTWENOLR model was developed to address the limitations of the GWOLR model, which, despite its ability to account for spatial variation, does not incorporate temporal effects. While there has been research discussing the influence of time in regression modeling, GWOLR does not fully capture the dynamic nature of temporal changes, which can limit its applicability in time-sensitive analyses. Additionally, the GTWENOLR model itself faces challenges, particularly in handling issues such as multicollinearity and overfitting. These problems arise when spatial and temporal variables interact in complex ways, leading to unstable parameter estimates. To get around these problems and make the GTWENOLR model more reliable, more advanced techniques like regularization need to be added.

Using MLE and the BHHH algorithm to estimate the parameters of the GTWENOLR model provides several key advantages. When used together, MLE and the BHHH algorithm make computations faster, especially when working with complicated models that have both spatial and temporal parts. The coefficients obtained from this method enable the development of regression models that vary by location and time, which better capture the unique characteristics of each specific geographic area and temporal period. Additionally, this approach allows for the selection of independent variables that do not exhibit high multicollinearity, improving the stability and interpretability of the model. As a result, the model becomes more adaptive and accurately reflects spatial and temporal variations in the data.

The GTWENOLR model offers several opportunities for future research and development. One potential extension is Bayesian GTWENOLR, which incorporates prior distributions for greater robustness in handling uncertainty and small sample sizes. Additionally, integrating machine learning techniques, such as deep learning, could enhance the model’s ability to capture nonlinear spatial–temporal dependencies. Further improvements may include adaptive bandwidth selection, allowing data-driven adjustments to spatial and temporal weighting. Expanding GTWENOLR to multivariate ordinal outcomes could also enhance its applicability in fields like economic forecasting, healthcare analytics, and climate modeling. Moreover, a formal investigation of the asymptotic properties of the estimators, such as consistency and asymptotic normality, remains an open question. Establishing these properties would strengthen the theoretical foundation of the model and ensure its reliability for large-sample applications. These advancements would further refine GTWENOLR as a versatile and powerful tool for spatial–temporal ordinal classification problems.