Dual Kriging with a Nonlinear Hybrid Gaussian RBF–Polynomial Trend: The Theory and Application to PM_2.5 Estimation in Northern Thailand

Abstract

Accurate spatial interpolation of environmental data requires utilizing flexible models that can capture complex spatial patterns. In this paper, we present two improved dual kriging (DK) models comprising a nonlinear trend function that combines Gaussian radial basis functions with a first-order polynomial. The proposed model, DK–RBFP, and its extension, DK–RBFPGA, which includes k-means clustering and a genetic algorithm for parameter optimization, respectively, exhibit enhanced performance in capturing spatial variation. The complete monotonicity of the covariance function and the strict positive definiteness of the coefficient matrix provide theoretical support for the uniqueness of the DK solution. When applied to datasets of PM_2.5 concentrations for northern Thailand, both models perform better than the conventional DK model using a second-order polynomial trend (DK–POLY), as evidenced by accuracy metrics including the mean absolute percentage error (MAPE), the mean squared error (MSE), and the root mean square error (RMSE). The outcomes indicate that integrating nonlinear trend components with data-driven optimization significantly enhances accuracy and flexibility in environmental spatial predictions.

1. Introduction

Spatial interpolation refers to methods used to estimate the value of a target variable at a specific location based on observed values from nearby sites. Common techniques include inverse distance weighting (IDW), geographically weighted regression (GWR), and kriging. Kriging is a geostatistical approach that provides the best linear unbiased prediction with minimum variance by modeling a spatial process as the sum of a deterministic trend and a stochastic residual. Ordinary kriging (OK), the most widely used variant, assumes a constant mean across the study area. In contrast, kriging with an external drift (KED) incorporates auxiliary variables that are highly correlated with the target variable, defining the trend as a function of these variables to improve estimation accuracy [1].

One of the main computational challenges in KED is that a distinct kriging system must be solved for each interpolation point to calculate the linear estimator weights. To overcome this, DK, which is derived from KED, uses a single system to compute weights for all points while producing identical estimation results [2]. Compared with OK, DK offers greater flexibility by simultaneously modeling global trends and local variations. Its trend component enables the capturing of complex, nonlinear spatial structures, while solving a single system makes DK more computationally efficient for large datasets. Unlike machine learning (ML) methods such as support vector machines (SVMs) or neural networks, DK explicitly incorporates spatial autocorrelation, providing more interpretable spatial predictions. ML approaches, by contrast, require large datasets and are not inherently designed to account for spatial dependencies [3]. Recent studies have confirmed the effectiveness of DK in relevant research areas [4,5,6,7].

The trend component in DK is critical for representing large-scale spatial variation. By explicitly modeling the trend, DK can better capture nonlinear spatial patterns, integrate auxiliary information, and improve prediction accuracy and interpretability. Polynomial functions are commonly used to describe trends in KED and DK; however, they often fail to represent nonlinear interactions between predictors and the response variable. While several studies have explored KED and DK, few focus on nonlinear trend functions. Notable exceptions include Snepvangers et al. [8], who applied a logarithmic trend, and Freier and Lieres [9], who extended universal kriging to include nonlinear trends using Taylor-based linearization. More recently, Baisad et al. [10] employed least squares support vector regression (LSSVR) as the trend in KED. Despite these advances, research on nonlinear trends in DK remains limited. The choice of trend function directly influences the structure of the coefficient matrix in the DK system. This structure determines the existence and uniqueness of the optimal weight solutions. Therefore, selecting an appropriate trend function is a vital step in ensuring that DK provides reliable and consistent estimates.

In this paper, we propose a nonlinear trend function for the DK framework that integrates a radial basis function (RBF) with a first-order polynomial to capture both linear and nonlinear dependencies of the target variable on auxiliary variables. RBFs are widely used in machine learning and geostatistics for interpolation and function approximation, as they model relationships based on distances from a set of centers, enabling flexible representation of nonlinear spatial patterns [11,12,13]. Commonly used RBFs include the thin-plate spline (TPS), the multiquadric function (MQF), the inverse multiquadric function (IMQF), and the Gaussian function (GF) [14]. Among these, the GF, also known as the Gaussian radial basis function (GRBF), is particularly popular for spatial interpolation, regression [15,16,17], and kernel-based ML methods [18,19].

The performance of the GRBF depends on two key parameters: the center, which defines where the function peaks, and the bandwidth, which controls its width. Selecting centers typically involves data reduction methods such as k-means clustering or subset selection methods to identify representative points from the dataset [20,21], while bandwidth is often estimated using cross-validation, maximum likelihood, or rule-of-thumb approaches based on inter-center distances [22,23]. One practical approach is to set the bandwidth proportional to the average distance between centers [24]. In this study, k-means clustering is used to choose centers, and a genetic algorithm (GA) is employed to estimate the bandwidth. GAs are population-based optimization techniques inspired by natural selection [25,26], and they have been increasingly applied to estimate parameters in nonlinear regression models [27,28,29,30].

Finally, we apply the proposed DK approach with nonlinear trend functions to interpolate PM_2.5 concentrations using air pressure and relative humidity as auxiliary variables and to compare its performance against DK with a second-order polynomial trend.

The following summarizes the main contributions of this work: (1) the development of a novel nonlinear trend function within the DK framework by combining GRBFs with a first-order polynomial; (2) a theoretical guarantee of the uniqueness of the DK system solution, established by demonstrating the nonsingularity of the coefficient matrix based on the theorems of complete monotonicity and strict positive definiteness; and (3) the implementation of a hybrid approach in which the centers of the GRBFs are determined using k-means clustering, and the bandwidth parameter is estimated using GA.

The remainder of this paper is organized as follows: Section 2 outlines the mathematical backgrounds of KED and DK, as well as the theoretical concepts related to strictly positive definite matrices and completely monotone functions. In Section 3, we present a hybrid nonlinear trend model that combines the Gaussian radial basis function with the first-order polynomial. Section 4 addresses the requirements necessary to ensure the DK system’s nonsingularity. Section 5 gives a detailed explanation of the dataset and determines the application of distance correlation coefficients in finding appropriate auxiliary variables. Then, the performance of the proposed DK method is tested by estimating PM_2.5 concentrations in Northern Thailand. Conclusions and discussions are given in Section 6.

2. Theoretical Background

2.1. Kriging with an External Drift

Let Z(s) be a random function at spatial locations s ∈ D, where d is the spatial dimension and $D\subset {\mathbb{R}}^d$ is the spatial domain of interest. The random function Z(s) for KED is modeled as

$$\begin{array}{c}\hfill Z(\mathit{s})=\mu (\mathit{s})+ϵ(\mathit{s}),\end{array}$$

(1)

where μ(s) is the drift or trend component and ϵ(s) is a stochastic residual with zero mean and a stationary covariance.

The drift component μ(s) is a deterministic function, often referred to as the trend function. It incorporates external covariates or auxiliary variables, which are predetermined variables having a systematic effect on the underlying spatial process. The trend function μ(s) is expressed as

$$\mu (\mathit{s})=\sum _{\mathcal{l}=0}^L{\alpha }_{\mathcal{l}}{f}_{\mathcal{l}}(X(\mathit{s})),$$

(2)

where α_ℓ denotes an unknown coefficient to be estimated and $X(\mathit{s})={\left[X_1(\mathit{s}),\dots ,X_m(\mathit{s})\right]}^T$ represents the vector of m auxiliary variables at location s. The function f_ℓ denotes a basis function defined in terms of the auxiliary variables. In general, f₀ is defined as the constant function (e.g., f₀(x) = 1), and L + 1 is the total number of basis functions.

We can rewrite Equation (1) in the following form:

$$Z(\mathit{s})={\mathcal{F}}^T\mathit{\alpha }+ϵ(\mathit{s}),$$

(3)

where $\mathcal{F}={\left[f_0(X(\mathit{s})),f_1(X(\mathit{s})),\dots ,f_L(X(\mathit{s}))\right]}^T$ and $\mathit{\alpha }={\left[{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_L\right]}^T$.

The residual term ϵ(s) is assumed to be a second-order stationary random function with a mean of zero and a covariance function C(h) [31], where h is the spatial lag vector between two locations. Specifically, the residual satisfies the following properties:

$$\mathbb{E}(ϵ(\mathit{s}))=0,\mathrm{Cov}(ϵ(\mathit{s}),ϵ(\mathit{s}+\mathit{h}))=C(\mathit{h}),$$

(4)

where s, $\mathit{s}+\mathit{h}\in D\subseteq {\mathbb{R}}^d$.

Typically, the covariance function is modeled using the variogram γ(h) of the residual process and is defined as

$$\begin{array}{cc}\hfill \gamma (\mathit{h})& ={\displaystyle \frac{1}{2}}\mathrm{Var}\left(ϵ(\mathit{s}+\mathit{h})-ϵ(\mathit{s})\right),\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\mathbb{E}{\left(ϵ(\mathit{s}+\mathit{h})-ϵ(\mathit{s})\right)}^2,\hfill \end{array}$$

(5)

which leads to

$$C(\mathit{h})={\sigma }^2-\gamma (\mathit{h}),$$

(6)

where σ² is the variance in the residual process [32].

Given n observed values, $Z({\mathit{s}}_1),\dots ,Z({\mathit{s}}_n)$, at sample locations ${\mathit{s}}_1,{\mathit{s}}_2,\dots ,{\mathit{s}}_n$, the KED estimator at an unobserved location s₀ is expressed as a linear combination of the observed values Z(s_i) for $i=1,\dots ,n$:

$${\widehat{Z}}_{KED}({\mathit{s}}_0)=\sum _{i=1}^n{w}_{i}Z({\mathit{s}}_i)={\mathit{Z}}^T\mathcal{W},$$

(7)

where w_i denotes the KED weight associated with Z(s_i), $\mathbf{Z}={\left[Z({\mathit{s}}_1),\dots ,Z({\mathit{s}}_n)\right]}^T$ is the vector of the observed values of the target variable, and $\mathcal{W}={\left[w_1,w_2,\dots ,w_n\right]}^T$ represents the vector of KED weights. The optimal weights are derived by minimizing the estimation error variance under the constraint of unbiasedness, resulting in the following formulation of the KED system:

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}\mathit{C}& |& \mathit{F}\\ --& +& --\\ {\mathit{F}}^T& |& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathcal{W}\\ --\\ \mathcal{L}\end{array}\right]=\left[\begin{array}{c}{\mathit{C}}_0\\ --\\ {\mathit{F}}_0\end{array}\right],\end{array}$$

(8)

where L = η + m;

0 denotes the (L + 1) × (L + 1) zero matrix;

$\mathcal{L}={\left[{\lambda }_0,{\lambda }_1,\dots ,{\lambda }_L\right]}^T$ represents the vector of Lagrange multipliers;

${\mathit{C}}_0={\left[C({\mathit{s}}_1-{\mathit{s}}_0),C({\mathit{s}}_2-{\mathit{s}}_0),\dots ,C({\mathit{s}}_n-{\mathit{s}}_0)\right]}^T$;

${\mathit{F}}_0={\left[f_0(X({\mathit{s}}_0)),f_1(X({\mathit{s}}_0)),\dots ,f_L(X({\mathit{s}}_0))\right]}^T$;

$\mathit{C}=\left[\begin{array}{ccc}C({\mathit{s}}_1-{\mathit{s}}_1)& \cdots & C({\mathit{s}}_1-{\mathit{s}}_n)\\ C({\mathit{s}}_2-{\mathit{s}}_1)& \cdots & C({\mathit{s}}_2-{\mathit{s}}_n)\\ ⋮& \ddots & ⋮\\ C({\mathit{s}}_n-{\mathit{s}}_1)& \cdots & C({\mathit{s}}_n-{\mathit{s}}_n)\end{array}\right]$;

and

$\mathit{F}=\left[\begin{array}{cccc}{f}_0(X({\mathit{s}}_1))& f_1(X({\mathit{s}}_1))& \cdots & f_L(X({\mathit{s}}_1))\\ f_0(X({\mathit{s}}_2))& f_1(X({\mathit{s}}_2))& \cdots & f_L(X({\mathit{s}}_2))\\ ⋮& ⋮& \ddots & ⋮\\ f_0(X({\mathit{s}}_n))& f_1(X({\mathit{s}}_n))& \cdots & f_L(X({\mathit{s}}_n))\end{array}\right]$.

2.2. Dual Kriging

Using the KED system as presented in Equation (8) and assuming an invertible coefficient matrix, the solution can be formalized as [1,2]

$$\begin{array}{c}\hfill \left[\begin{array}{c}\mathcal{W}\\ --\\ \mathcal{L}\end{array}\right]=\left[\begin{array}{ccc}\mathcal{U}& |& \mathcal{V}\\ --& +& --\\ {\mathcal{V}}^T& |& \mathbf{\Omega }\end{array}\right]\left[\begin{array}{c}{\mathit{C}}_0\\ --\\ {\mathit{F}}_0\end{array}\right],\end{array}$$

(9)

where $\mathcal{U}$, $\mathcal{V}$ and Ω denote matrices of dimensions n × n, n × (L + 1), and (L + 1) × (L + 1), respectively. Since the coefficient matrix in Equation (8) is symmetric, its inverse is also symmetric. Consequently, the matrix $\mathcal{U}$ is symmetric.

Substituting the weight vector $\mathcal{W}$, as given in Equation (9), into the estimator formulation in Equation (7) yields the following estimated value:

$$\begin{array}{c}\hfill {\widehat{Z}}_{KED}({\mathit{s}}_0)={\mathit{Z}}^T\mathcal{V}{\mathit{F}}_0+{\mathit{Z}}^T\mathcal{U}{\mathit{C}}_0.\end{array}$$

(10)

Let α and β be matrices of dimensions L × 1 and n × 1, respectively; they are defined as follows:

$$\begin{array}{c}\hfill {\mathit{\alpha }}^T=\left[{\alpha }_0,\dots ,{\alpha }_L\right]={\mathit{Z}}^T\mathcal{V}\hspace{1em}\mathrm{and}\hspace{1em}{\mathit{\beta }}^T=\left[{\beta }_1,\dots ,{\beta }_n\right]={\mathit{Z}}^T\mathcal{U}={\mathit{Z}}^T{\mathcal{U}}^{\mathit{T}}.\end{array}$$

(11)

As a result, Equation (10) can be reformulated to yield the dual kriging (DK) estimator, which is expressed as follows:

$$\begin{array}{c}\hfill {\widehat{Z}}_{\mathrm{DK}}\left({\mathit{s}}_0\right)={\mathit{\alpha }}^T{\mathit{F}}_0+{\mathit{\beta }}^T{\mathit{C}}_0.\end{array}$$

(12)

Therefore, the vectors α and β represent the weighting coefficients used in the DK estimator. According to Equation (11), these vectors can be expressed in matrix form as follows:

$$\begin{array}{c}\hfill \left[\begin{array}{c}\mathit{\beta }\\ --\\ \mathit{\alpha }\end{array}\right]=\left[\begin{array}{ccc}\mathcal{U}& |& \mathcal{A}\\ --& +& --\\ {\mathcal{V}}^T& |& \mathcal{B}\end{array}\right]\left[\begin{array}{c}\mathit{Z}\\ --\\ {\mathbf{0}}_{L+1}\end{array}\right],\end{array}$$

(13)

where $\mathcal{A}$ and $\mathcal{B}$ denote arbitrary matrices of dimensions n × (L + 1) and (L + 1) × (L + 1), respectively. The term ${\mathbf{0}}_{L+1}$ refers to a zero column vector of size (L + 1) × 1.

By setting $\mathcal{A}$ = $\mathcal{V}$ and $\mathcal{B}$ = Ω, the coefficient vectors α and β are obtained as the solution to the DK system, as shown below.

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}\mathit{C}& |& \mathit{F}\\ --& +& --\\ {\mathit{F}}^T& |& \mathbf{0}\end{array}\right]\left[\begin{array}{c}\mathit{\beta }\\ --\\ \mathit{\alpha }\end{array}\right]=\left[\begin{array}{c}\mathit{Z}\\ --\\ {\mathbf{0}}_{L+1}\end{array}\right].\end{array}$$

(14)

The estimated value of the target variable at the unobserved location s₀ can be computed by substituting the coefficient vectors α and β into Equation (12).

The solution of the individual systems used to determine the optimal weights in dual kriging (DK) depends on the invertibility of the coefficient matrices, which ensures a unique solution. Achieving this property requires appropriate selection of variogram or covariance functions and trend models. Myers [33] noted that these coefficient matrices are nonsingular if the covariance function is expressed in terms of linearly independent, real-valued functions of the trend model and is strictly positive definite with respect to the basis functions. In the following section, we introduce the definitions and theorems of positive definite matrices and completely monotone functions to establish the conditions necessary for the invertibility of the dual kriging system.

2.3. The Relationship Between the Positive Definite Matrix and the Completely Monotone Function

The definitions and theorems presented in this section are adapted from the foundational work [34,35], which provides a comprehensive treatment of positive definite matrices, completely monotone functions, and their applications in approximation theory.

Definition 1.

Let $\mathcal{C}$ be an $n\times n$ symmetric matrix and $u\in {\mathbb{R}}^n$.

1. 

$\mathcal{C}$ is positive definite (denoted by $\mathcal{C}\ge 0$) if

$$u^T\mathcal{C}u\ge 0\mathit{for}\mathit{all}u\ne 0.$$

2. 

$\mathcal{C}$ is strictly positive definite (denoted by $\mathcal{C}>0$) if

$$u^T\mathcal{C}u>0\mathit{for}\mathit{all}u\ne 0.$$

Definition 2.

Let $f:[0,\infty )\to \mathbb{R}$. The function f is called a completely monotone function if

1. 

$f\in C[0,\infty )\cap C^{\infty }(0,\infty )$;

2. 

${(-1)}^k{f}^{(k)}(h)\ge 0$ for all $k\in \mathbb{N}$ and $h\in (0,\infty )$.

To illustrate the concept of complete monotonicity, we now present a specific function that satisfies the conditions outlined above.

Example 1.

The function $f(h)=e^{-\sqrt{h}/a}$ is completely monotone for every a > 0.

We see that the function $f(h)$ is continuous on $[0,\infty )$. We will verify that ${(-1)}^k{f}^{(k)}\ge 0$ for all $k\in \mathbb{N}$ by strong induction. For $h>0$, it is obvious that

$$(-1)f^{\prime }\left(h\right)={\displaystyle \frac{1}{2a\sqrt{h}}}{e}^{-\sqrt{h}/a}\ge 0.$$

Next, we assume that

$${(-1)}^i{f}^{\left(i\right)}\left(h\right)\ge 0\mathit{for}\mathit{all}i=1,2,\dots ,k.$$

Using the Leibniz rule, where

$${\left(fg\right)}^{\left(k\right)}=\sum _{i=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right)f^{(k-i)}{g}^{\left(i\right)},$$

we compute the following:

$$\begin{array}{cc}\hfill {(-1)}^{k+1}{f}^{(k+1)}\left(h\right)& ={\displaystyle \frac{1}{2a}}\sum _{i=0}^k\left({\displaystyle \genfrac{}{}{0pt}{}{k}{i}}\right){\displaystyle \frac{(2i+1)!!}{2^i{h}^{(2i+1)/2}}}{(-1)}^{k-i}{f}^{(k-i)}\left(h\right)\ge 0,\hfill \end{array}$$

where $(2i+1)!!=(2i+1)(2i-1)(2i-3)\dots 3·1$. We then conclude that

$${(-1)}^k{f}^{\left(k\right)}\left(h\right)\ge 0\mathit{for}\mathit{all}k\in \mathbb{N}.$$

The following theorem provides a powerful criterion for ensuring strict positive definiteness of matrices formed from completely monotone functions.

Theorem 1

(Schoenberg’s Theorem). If f is a completely monotone function and not constant on $[0,\infty )$, then for any distinct points ${\mathit{s}}_1,\dots ,{\mathit{s}}_n$, the matrix

$$\mathcal{C}=\left[f(\parallel {\mathit{s}}_i-{\mathit{s}}_j{\parallel }^2)\right]$$

is strictly positive definite.

We now turn to a related result concerning the nonsingularity of a specific matrix structure known as the bordered Gramian matrix.

Theorem 2.

Let $\mathcal{C}$ be a positive definite matrix of size $n\times n$, and let $\mathcal{F}$ be an $n\times (L+1)$ matrix. The bordered Gramian matrix

$$\left[\begin{array}{ccc}\mathcal{C}& |& \mathcal{F}\\ --& +& --\\ {\mathcal{F}}^T& |& \mathbf{0}\end{array}\right]$$

is nonsingular if and only if

$$\mathrm{rank}\left(\mathcal{F}\right)=L+1\mathit{and}\mathcal{C}+\mathcal{F}{\mathcal{F}}^T>0.$$

3. A New Approach to Nonlinear Trend Modeling in Dual Kriging

This section presents the trend functions employed in the DK approach adopted in this study. These include a second-order polynomial function and the proposed nonlinear trend function, which were constructed by integrating GRBFs with a first-order polynomial. Additionally, the parameter estimates obtained from these two distinct trend models are presented.

Let the dataset at each sample location ${\mathit{s}}_i\in {\mathbb{R}}^d$ be denoted by ${\left\{(X({\mathit{s}}_i),Z({\mathit{s}}_i))\right\}}_{i=1}^n$, where $X({\mathit{s}}_i)={\left[X_1({\mathit{s}}_i),\dots ,X_m({\mathit{s}}_i)\right]}^T\in {\mathbb{R}}^m$ is a vector comprising m auxiliary variables and Z(s_i) denotes the observed response at location s_i.

The conventional trend functions employed in KED and DK methods are typically first-order (linear) or second-order (quadratic) polynomial models. In this study, we focus on the second-order polynomial model, which is defined as follows [36,37]:

$$\mu \left(\mathit{s}\right)=b_0+\sum _{j=1}^m{b}_j{X}_j\left(\mathit{s}\right)+\sum _{i=1}^m{b}_{ii}{X}_i{\left(\mathit{s}\right)}^2+\sum _{i=1}^{m-1}\sum _{j=i+1}^m{b}_{ij}{X}_i\left(\mathit{s}\right)X_j\left(\mathit{s}\right).$$

(15)

Given that ${\displaystyle \frac{(m+1)(m+2)}{2}}\le n$, the unknown coefficients in Equation (15) can be estimated using the ordinary least squares (OLS) method [38,39], which is given by

$$\widehat{\mathcal{B}}={\left({\mathcal{X}}^T\mathcal{X}\right)}^{-1}{\mathcal{X}}^T\mathit{Z},$$

(16)

where $\widehat{\mathcal{B}}$ is an ${\displaystyle \frac{(m+1)(m+2)}{2}}\times 1$ matrix of estimated coefficients,

$\mathcal{X}$ is an $n\times {\displaystyle \frac{(m+1)(m+2)}{2}}$ matrix whose rows are the vectors

$\mathcal{X}({\mathit{s}}_i)=[1,X_1({\mathit{s}}_i),\dots ,X_m({\mathit{s}}_i),X_1^2({\mathit{s}}_i),\dots ,X_m^2({\mathit{s}}_i),X_1({\mathit{s}}_i)X_2({\mathit{s}}_i),\dots ,X_{m-1}({\mathit{s}}_i)X_m({\mathit{s}}_i)]$,

and $\mathit{Z}={[Z({\mathit{s}}_1),Z({\mathit{s}}_2),\dots ,Z({\mathit{s}}_n)]}^T$.

• In addition, the matrix $\mathcal{X}$ has full column rank.

The proposed nonlinear trend function is created by integrating GRBFs with a first-order polynomial. GRBFs are widely used for modeling and estimating functions in both spatial and temporal contexts. Their formal definition is provided as follows [15]:

$${\phi }_i(X(\mathit{s}),P_i)={\phi }_i(X(\mathit{s}),{\mathrm{\Psi }}_i,{\sigma }_i)=exp\left[-{\left(\frac{\Vert X(\mathit{s})-{\mathrm{\Psi }}_i\Vert }{{\sigma }_i}\right)}^2\right].$$

(17)

In Equation (17), the symbol $\parallel ·\parallel$ represents some norm on ${\mathbb{R}}^m$, typically the Euclidean norm. Each parameter vector P_i is defined as $P_i=\left[{\mathrm{\Psi }}_i,{\sigma }_i\right]$, where Ψ_i and σ_i denote the center (or mean) and the bandwidth (or scale) parameters, respectively.

Given the effectiveness of GRBFs in capturing highly nonlinear relationships, the proposed trend model is augmented with a first-order polynomial to account for the linear component of the trend. According to Equation (1), the resulting trend function is defined as follows:

$$\mu (\mathit{s})=a_0+\sum _{i=1}^m{a}_i{X}_i(\mathit{s})+\sum _{j=1}^{\eta }{b}_j{\phi }_j(X(\mathit{s}),P_j),$$

(18)

where a_i and b_j denote unknown coefficients for $i=0,1,\dots ,m$ and $j=1,\dots ,\eta$, respectively. The vector $X(\mathit{s})={\left[X_1(\mathit{s}),\dots ,X_m(\mathit{s})\right]}^T$ represents the m auxiliary variables at location s, while φ_j(X(s), P_j) is the radial basis function with parameter P_j. Additionally, η denotes the total number of radial basis functions.

In this study, we consider the parameters of the trend model(Model (18)) in two cases. In the first case, Ψ_j and σ_j are estimated using the method proposed in [15]. Subsequently, the parameters a_i and b_j for $i=0,1,\dots ,m$ and $j=1,\dots ,\eta$ are determined using the least squares method.

In the second case, the trend model is treated as a nonlinear model. The parameter Ψ_j is obtained using the same method as in the first case, while the parameters σ_j, a_i, and b_j for $i=0,1,\dots ,m$ and $j=1,\dots ,\eta$ are estimated using the genetic algorithm. The model parameters are estimated using a genetic algorithm implemented via the ga function in MATLAB (Version 2018a). The algorithm is configured with a population size of 30, a crossover probability of 0.8, a mutation probability of 0.2, and a maximum of 20,000 generations. Stochastic uniform selection is employed as the selection method, combined with scattered crossover and Gaussian mutation strategies.

In the first case, for the purpose of estimating the parameters Ψ_j and σ_j, the dataset is partitioned into η clusters, denoted by $\{G_1,\dots ,G_{\eta }\}$. In this study, k-means clustering is employed to partition the dataset into n distinct clusters. As an unsupervised machine learning algorithm, k-means groups data points based on their similarity by assigning them to the nearest cluster centroid [40]. Following the method proposed in [15], the parameters Ψ_j and σ_j are estimated as follows:

$${\mathrm{\Psi }}_j={\displaystyle \frac{1}{n_j}}\sum _{X_{\alpha }\in G_j}{X}_{\alpha },$$

(19)

and

$${\sigma }_j^2={\displaystyle \frac{1}{n_j}}\sum _{X_{\alpha }\in G_j}\Vert X_{\alpha }-{\mathrm{\Psi }}_j{\Vert }^2,$$

(20)

where n_j denotes the number of observations belonging to the j-th cluster G_j. Subsequently, Equation (18) can be expressed in matrix form as

$$\mu \left(\mathit{s}\right)=Y^T\left(\mathit{s}\right)\mathit{\theta },$$

(21)

where $Y(\mathit{s})={[1,X_1(\mathit{s}),\dots ,X_m(\mathit{s}),{\phi }_1(X(\mathit{s})),{\phi }_2(X(\mathit{s})),\dots ,{\phi }_{\eta }(X(\mathit{s}))]}^T$ and $\mathit{\theta }={[a_0,a_1,\dots ,a_m,b_1,\dots ,b_{\eta }]}^T$.

Assuming that 1 + m + η ≤ n, the parameter matrix θ can be estimated using the least squares method, and it is given by

$$\widehat{\mathit{\theta }}={\left({\mathbf{Y}}^T\mathbf{Y}\right)}^{-1}{\mathbf{Y}}^T\mathit{Z},$$

(22)

where Y represents an n × (1 + m + η) matrix whose rows are the vectors $Y^T\left({\mathit{s}}_i\right)$. Furthermore, Y is of full column rank.

4. Nonsingularity of the Dual Kriging System

The following result combines the preceding theoretical results to establish conditions under which the bordered Gramian matrix, which was constructed from a completely monotone function and a full-rank matrix, is guaranteed to be nonsingular.

Theorem 3.

Let $X({\mathit{s}}_1),X({\mathit{s}}_2),\dots ,X({\mathit{s}}_n)$ be distinct points in ${\mathbb{R}}^m$, and assume $n\ge 1+m+\eta$. Define the matrix $\mathit{F}\in {\mathbb{R}}^{n\times (1+m+\eta )}$ by

$$\mathit{F}=\left[\begin{array}{ccccccc}1& X_1\left({\mathit{s}}_1\right)& \cdots & X_m\left({\mathit{s}}_1\right)& {\phi }_1\left(X\left({\mathit{s}}_1\right)\right)& \cdots & {\phi }_{\eta }\left(X\left({\mathit{s}}_1\right)\right)\\ 1& X_1\left({\mathit{s}}_2\right)& \cdots & X_m\left({\mathit{s}}_2\right)& {\phi }_1\left(X\left({\mathit{s}}_2\right)\right)& \cdots & {\phi }_{\eta }\left(X\left({\mathit{s}}_2\right)\right)\\ ⋮& ⋮& \ddots & ⋮& ⋮& \ddots & ⋮\\ 1& X_1\left({\mathit{s}}_n\right)& \cdots & X_m\left({\mathit{s}}_n\right)& {\phi }_1\left(X\left({\mathit{s}}_n\right)\right)& \cdots & {\phi }_{\eta }\left(X\left({\mathit{s}}_n\right)\right)\end{array}\right],$$

where for $j=1,\dots ,\eta$,

$${\phi }_j\left(X\left(\mathit{s}\right)\right)=exp\left[-{\left({\displaystyle \frac{\parallel X\left(\mathit{s}\right)-{\mathrm{\Psi }}_j\parallel }{{\sigma }_j}}\right)}^2\right],{\sigma }_j>0,$$

are Gaussian radial basis functions with distinct centers ${\mathrm{\Psi }}_j\in {\mathbb{R}}^m$. Then $\mathrm{rank}(\mathit{F})=1+m+\eta$.

Proof. 

We will show that the columns of F are linearly independent. To establish linear independence, it suffices to show that the homogeneous system

$$\mathit{F}\mathbf{c}=0$$

has only the trivial solution c = 0, where

$$\mathbf{c}={\left[c_0,c_1,\dots ,c_m,c_{m+1},c_{m+2},\dots ,c_{m+\eta }\right]}^T\in {\mathbb{R}}^{1+m+\eta }.$$

For each $i=1,\dots ,n$, the relation Fc = 0 can be written as

$$c_0+\sum _{k=1}^m{c}_k{X}_k\left(s_i\right)+\sum _{j=1}^{\eta }{c}_{m+j}{\phi }_j\left(X\left(s_i\right)\right)=0.$$

(23)

We decompose the matrix F as

$$\mathit{F}=\left[\mathbf{1}\left|P\right|\mathrm{\Phi }\right],$$

where $\mathbf{1}\in {\mathbb{R}}^n$ is the all-ones column, $P\in {\mathbb{R}}^{n\times m}$ with P_ik = X_k(s_i), and $\mathrm{\Phi }\in {\mathbb{R}}^{n\times \eta }$ with Φ_ij = φ_j($\mathcal{X}$(s_i)).

For distinct sites X(s_i) and GRBFs with distinct centers and σ_j > 0, Φ has full column rank when n ≥ η. Furthermore, for distinct sites with n ≥ η + m + 1 > m + 1, the polynomial block $\left[\mathbf{1}\right|P]$ has rank m + 1, since it spans the affine-linear functions in ${\mathbb{R}}^m$.

Case 1: Suppose some $c_j\ne 0$ for a $j\in \{m+1,\dots ,m+\eta \}$. Then by (23), the nontrivial RBF combination ${\sum }_{j=1}^{\eta }{c}_{m+j}{\phi }_j$ coincides with an affine-linear function on the finite set ${\left\{X(s_i)\right\}}_{i=1}^n$. Since Φ and $\left[\mathbf{1}\right|P]$ are of full rank and span disjoint subspaces under the stated conditions, this leads to a contradiction. Hence this case is impossible.

Case 2: Assume $c_{m+1}=\dots =c_{m+\eta }=0$. Then (23) reduces to

$$c_0+\sum _{k=1}^m{c}_k{X}_k\left(s_i\right)=0,i=1,\dots ,n.$$

Because n > m + 1 and $\left[\mathbf{1}\right|P]$ has full column rank 1 + m, it follows that $c_0=c_1=\cdots =c_m=0$.

Since both cases imply c = 0, we conclude that the columns of F are linearly independent.

□

Remark 1.

This theorem guarantees uniqueness in interpolation schemes that combine Gaussian radial basis functions with an affine polynomial tail. The assumptions of distinct centers and sufficiently many sample sites are essential for ensuring full rank in both the GRBF and polynomial blocks.

On the other hand, this study assumes isotropy, implying that the variogram is a function of the distance (i.e., the magnitude of the lag vector h) rather than its direction. The exponential variogram is used in its isotropic form without a nugget effect. This model is commonly applied in a wide range of spatial analysis contexts and is defined as follows [31]:

$$\gamma \left(h\right)=k_2\left(1-e^{-h/k_1}\right),$$

(24)

where h = ∥h∥ is the separation distance, k₁ is the range parameter, and k₂ represents the sill, corresponding to the variance in the spatial process.

The variogram model is developed using spatial data in conjunction with an empirical variogram estimator, which captures the degree of spatial dependence between observations across different distances. In this study, we adopt the empirical variogram estimator proposed by Matheron [41], a widely recognized and commonly applied approach in geostatistics.

Theorem 4.

The DK system in Equation (14) admits a unique solution.

Proof. 

From Equation (6), the covariance function can be derived as follows:

$$C\left(h\right)=k_2-\left[k_2(1-e^{-h/k_1})\right]=k_2{e}^{-h/k_1}.$$

(25)

According to Example 1, the function

$$f\left(h\right)=k_2{e}^{-\sqrt{h}/k_1}$$

is a completely monotone function and not constant on $[0,\infty )$. According to Schoenberg’s Theorem, for any distinct points ${\mathit{s}}_1,\dots ,{\mathit{s}}_n$, the matrix

$$\mathit{C}=\left[f(\parallel {\mathit{s}}_i-{\mathit{s}}_j{\parallel }^2)\right]=\left[C({\mathit{s}}_i-{\mathit{s}}_j)\right]$$

in Equation (14) is strictly positive definite. Since the matrix FF^T is always positive definite, it follows that the sum C + FF^T is strictly positive definite. Furthermore, Theorem 3 assures that rank(F) = 1 + m + η = L + 1. Combining these two facts, Theorem 2 applies and ensures that the coefficient matrix in the dual kriging system (Equation (14)) is nonsingular.

Consequently, the DK system admits a unique solution. □

5. Application to PM_2.5 Spatial Estimation in Northern Thailand

5.1. Study Area and Data Description

In this study, fine particulate matter (PM_2.5) pollution over northern Thailand is investigated. This region is characterized by its mountainous landscape and persistent air quality issues, particularly during the dry season [42,43]. Northern Thailand consists of 15 provinces and is influenced by a mix of complex topography, intensive agricultural practices, and seasonal burning activities, which together contribute to severe haze events and transboundary air pollution [44,45]. As illustrated in Figure 1, the study area encompasses several provinces in the northern part of the country. The inset map provides a national reference, while the enlarged panel highlights the specific provinces included in the analysis, shown in orange. Fifteen monitoring stations, each corresponding to a province and marked with green circles, were employed to collect data on PM_2.5 concentrations along with associated meteorological variables.

Model accuracy was assessed using data collected during March and April of both 2023 and 2024. PM_2.5 concentration data (μg/m³) were obtained from the Pollution Control Department (PCD), while meteorological variables, including air pressure (hPa) and relative humidity (%), were retrieved from the Thai Meteorological Department (TMD). All datasets were aggregated weekly to reduce short-term variability and highlight broader spatial and temporal patterns.

The relationship between PM_2.5 concentrations and meteorological conditions was investigated using the distance correlation coefficient (DCC). This method enabled the identification of spatial dependencies between air pollution levels and meteorological factors across the study region.

For model validation, the 15 monitoring stations were divided into two subsets. Twelve stations were used for model training, while the remaining three served as an independent test set. This partitioning allowed for an objective evaluation of the model’s predictive performance and ensured a reliable basis for spatial interpolation of PM_2.5 in areas lacking direct observations.

5.2. Performance Metrics and Quantitative Evaluation

This study compares the accuracy of the DK constructed using three different trend functions. The first model (DK–POLY) utilizes a second-order polynomial trend, with its parameters estimated via ordinary least squares (OLS). The second approach (DK–RBFP) incorporates a non-linear trend function based on the GRBF combined with a first-order polynomial function. In this method, the mean and scale parameters of the GRBF are estimated using the methods described in Equations (19) and (20), respectively, while the remaining parameters are estimated using OLS. The third model (DK–RBFPGA) also employs a GRBF-based trend combined with a first-order polynomial function, where the mean parameters are estimated using the approach in Equation (19), and the remaining parameters are optimized using the GA.

For the comparison of the performance of the three models, a k-fold cross-validation technique was used [46]. The dataset was divided into five equal sets randomly (folds). For each of the five iterations, one fold was used as the validation set, while the remaining four were used for model training. This procedure guaranteed that each fold was used as the validation set once. Model accuracy was then computed by averaging over all iterations. Three metrics of performance were used to assess model performance, namely the mean absolute percentage error (MAPE) [10], the mean squared error (MSE) [47], and the root mean square error (RMSE) [10], as described below:

$$\mathrm{MAPE}={\displaystyle \frac{100\%}{n}}\sum _{i=1}^n\left|{\displaystyle \frac{Z\left({\mathit{s}}_i\right)-\widehat{Z}\left({\mathit{s}}_i\right)}{Z\left({\mathit{s}}_i\right)}}\right|,$$

(26)

$$\mathrm{MSE}={\displaystyle \frac{1}{n}}\sum _{i=1}^n{(Z\left({\mathit{s}}_i\right)-\widehat{Z}\left({\mathit{s}}_i\right))}^2,$$

(27)

$$\mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{n}}\sum _{i=1}^n{(Z\left({\mathit{s}}_i\right)-\widehat{Z}\left({\mathit{s}}_i\right))}^2},$$

(28)

where Z(s_i) denotes the observed value at location s_i, $\widehat{Z}({\mathit{s}}_i)$ is the estimated value at location s_i, and n represents the number of observations.

5.3. The Distance Correlation Coefficient

The distance correlation coefficient (DCC), which was introduced by Székely et al. [48], measures the statistical dependence between two random vectors, $U\in {\mathbb{R}}^p$ and $V\in {\mathbb{R}}^q$, and is designed to detect both linear and nonlinear relationships. The distance correlation coefficient has proven to be a flexible and effective statistical measure, with recent research highlighting its broad applicability and extension across various fields [49,50,51,52,53]. Its strength lies in its ability to detect both linear and nonlinear relationships, making it particularly well-suited for analyzing complex and high-dimensional data [54].

Given a sample of n paired observations ${\left\{(U_i,V_i)\right\}}_{i=1}^n$, the calculation is carried out as follows [48,55]:

First, the empirical distance covariance, $d\mathcal{C}(U,V)$, is computed as:

$$d\mathcal{C}(U,V)={\displaystyle \frac{1}{n}}\sqrt{\sum _{i=1}^n\sum _{j=1}^n{A}_{ij}{B}_{ij}}$$

(29)

where A_ij and B_ij are the doubly centered Euclidean distances within the samples of U and V, respectively. The centered distance matrices A_ij and B_ij are computed by double-centering the pairwise Euclidean distance matrices of the variables U and V, respectively. Specifically, the entries of the centered distance matrix for U are given by

$$A_{ij}=\parallel U_i-U_j{\parallel }_{{\mathbb{R}}^p}-{\displaystyle \frac{1}{n}}\sum _{l=1}^n\parallel U_i-U_l{\parallel }_{{\mathbb{R}}^p}-{\displaystyle \frac{1}{n}}\sum _{k=1}^n\parallel U_k-U_j{\parallel }_{{\mathbb{R}}^p}+{\displaystyle \frac{1}{n^2}}\sum _{k=1}^n\sum _{l=1}^n{\parallel U_k-U_l\parallel }_{{\mathbb{R}}^p}$$

(30)

Similarly, the centered distance matrix for V is defined as

$$B_{ij}=\parallel V_i-V_j{\parallel }_{{\mathbb{R}}^q}-{\displaystyle \frac{1}{n}}\sum _{l=1}^n\parallel V_i-V_l{\parallel }_{{\mathbb{R}}^q}-{\displaystyle \frac{1}{n}}\sum _{k=1}^n\parallel V_k-V_j{\parallel }_{{\mathbb{R}}^q}+{\displaystyle \frac{1}{n^2}}\sum _{k=1}^n\sum _{l=1}^n{\parallel V_k-V_l\parallel }_{{\mathbb{R}}^q}$$

(31)

In these equations, the notations ${\parallel ·\parallel }_{{\mathbb{R}}^p}$ and ${\parallel ·\parallel }_{{\mathbb{R}}^q}$ denote the Euclidean norm in p-dimensional and q-dimensional spaces, respectively.

Next, the empirical distance variances for U and V are calculated as

$$d\mathcal{V}\left(U\right)={\displaystyle \frac{1}{n}}\sqrt{\sum _{i=1}^n\sum _{j=1}^n{A}_{ij}^2}$$

(32)

and

$$d\mathcal{V}\left(V\right)={\displaystyle \frac{1}{n}}\sqrt{\sum _{i=1}^n\sum _{j=1}^n{B}_{ij}^2}.$$

(33)

Finally, the empirical distance correlation coefficient is given by

$$d\mathcal{R}(U,V)=\left\{\begin{array}{cc}{\displaystyle \frac{d\mathcal{C}(U,V)}{\sqrt{d\mathcal{V}\left(U\right)d\mathcal{V}\left(V\right)}}},\hfill & \mathrm{if}d\mathcal{V}\left(U\right)>0\mathrm{and}d\mathcal{V}\left(V\right)>0,\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

(34)

The distance correlation coefficient $d\mathcal{R}(U,V)$ takes values between 0 and 1, where a value of 0 indicates complete independence between U and V, while a value of 1 indicates perfect dependence, regardless of whether the relationship is linear or nonlinear.

5.4. Results

The trend function of DK relies on auxiliary variables, which must exhibit a relationship with the target variable, in this case, the PM_2.5 concentration. As such, it is essential to quantify the relationship between the candidate auxiliary variables and the PM_2.5 concentration. In this study, the distance correlation coefficient was employed to evaluate this relationship, with air pressure and relative humidity identified as the candidate auxiliary variables.

Table 1. Weekly distance correlation coefficients for PM_2.5 concentrations with air pressure and relative humidity during March and April 2023, based on the complete dataset.

Auxiliary Variable	Month	Week 1	Week 2	Week 3	Week 4	Average
Air Pressure	March 2023	0.522	0.470	0.449	0.414	0.464
	April 2023	0.458	0.393	0.338	0.737	0.481
Relative Humidity	March 2023	0.424	0.440	0.401	0.345	0.402
	April 2023	0.428	0.490	0.417	0.477	0.453

and

Table 2. Weekly distance correlation coefficients for PM_2.5 concentrations with air pressure and relative humidity during March and April 2024, based on the complete dataset.

Auxiliary Variable	Month	Week 1	Week 2	Week 3	Week 4	Average
Air Pressure	March 2024	0.446	0.511	0.604	0.574	0.534
	April 2024	0.368	0.542	0.459	0.440	0.452
Relative Humidity	March 2024	0.546	0.587	0.762	0.905	0.700
	April 2024	0.744	0.499	0.424	0.445	0.528

present the weekly distance correlations between PM_2.5 concentrations and two auxiliary meteorological variables, namely air pressure and relative humidity, during March and April for the years 2023 and 2024, respectively.

Table 1 shows the correlation results for 2023. It shows that air pressure had a moderate effect on PM_2.5 concentrations, with average correlation values of 0.464 in March and 0.481 in April. In Week 4 of April, there was a peak, with the correlation reaching 0.737. This suggests that air pressure may have a short-term effect on PM_2.5 levels. In contrast, relative humidity demonstrated weaker correlations, with average values of 0.402 in March and 0.453 in April, reflecting a comparatively lower association with PM_2.5 during this period.

In comparison, Table 2 shows that the 2024 results exhibited generally stronger correlations, particularly for relative humidity. In March 2024, the correlation between relative humidity and PM_2.5 climbed progressively from Week 1 to Week 4, reaching a high of 0.905 and an overall monthly average of 0.700. Although the average correlation dropped slightly to 0.528 in April, it remained higher than the values observed in 2023. Air pressure also showed stronger correlations in March 2024, averaging 0.534, though it decreased to 0.452 in April. These findings highlight that the influence of meteorological factors on PM_2.5 concentrations can vary from year to year and suggest that relative humidity may play a more significant role under certain atmospheric conditions.

The results indicate that the correlations between PM_2.5 concentrations and the auxiliary variables, air pressure and relative humidity, vary across different periods and years. Relative humidity generally shows a stronger and more consistent association with PM_2.5, while air pressure demonstrates a moderate relationship that fluctuates more noticeably over time.

Figure 2, Figure 3 and Figure 4 present a comparative evaluation of the predictive capabilities of DK with three different trend models: DK–POLY, DK–RBFP, and DK–RBFPGA. The assessment focuses on their performance during March and April of both 2023 and 2024. Each figure highlights a specific error metric, including the MAPE, MSE, and RMSE, to assess the accuracy of each model.

Figure 2 presents a comparison of the MAPE values for the three models across the specified months and years. In 2023, DK–POLY recorded the highest MAPE, indicating lower predictive accuracy compared to the other models. Conversely, both DK–RBFP and DK–RBFPGA demonstrated superior performance, with DK–RBFPGA showing a marginal advantage. A consistent trend of improvement was observed in 2024, as the MAPE values decreased for all three models. Throughout this period, DK–RBFPGA consistently yielded the lowest MAPE, indicating its enhanced reliability in prediction.

A comparison of the MSE for the same techniques and time periods is shown in Figure 3. DK–POLY in 2023 showed a significantly larger MSE, which was consistent with the MAPE results and confirmed its lower accuracy. DK–RBFPGA, on the other hand, had the lowest MSE, suggesting the most accurate predictions. The DK–POLY approach continued to exhibit the largest error, whereas DK–RBFPGA retained the lowest error, further demonstrating its superior prediction accuracy in 2024, even though the MSE dropped for all methods.

The RMSE for the three DK approaches over the same time periods is also compared in Figure 4. DK–POLY had the greatest RMSE in 2023, while the DK–RBFP and DK–RBFPGA approaches demonstrated reduced errors, with DK–RBFPGA performing slightly better. These results are consistent with the MAPE and MSE studies. The trend of reduced error continued in 2024, with all methods exhibiting lower RMSE values. However, DK–POLY consistently presented the highest RMSE, while DK–RBFPGA again achieved the lowest RMSE, highlighting its improved accuracy.

In summary, Figure 2, Figure 3 and Figure 4 consistently demonstrate that DK–POLY produced the least accurate predictions, with significantly higher MAPE, MSE, and RMSE values, particularly in 2023. In contrast, DK–RBFPGA achieved the highest prediction accuracy, consistently yielding the lowest error metrics for both 2023 and 2024. Furthermore, over the two-year period, the average MAPE, MSE, and RMSE values for DK–RBFPGA improved by 9.0%, 19.8%, and 8.7%, respectively, compared to DK–RBFP. Nevertheless, the MAPE values for both DK–RBFP and DK–RBFPGA, ranging from 18.5% to 30.7%, indicate relatively high prediction errors. This may be attributed to the limited number of data points, which restricts the development of reliable trend and variogram models.

Based on the results indicating that the DK–RBFPGA method provides the highest prediction accuracy, its estimates were subsequently used to generate maps illustrating the spatial distribution of PM_2.5 for Weeks 1 to 4 of March and April in both 2023 and 2024. The data used for this mapping were taken from fold 2. The maps were produced using QGIS (Quantum Geographic Information System) software, version 3.40.4, with the study area divided into a grid of square cells, each measuring 0.01 degrees per side. The resulting spatial distribution patterns of weekly mean PM_2.5 concentrations are presented in Figure 5, Figure 6, Figure 7 and Figure 8.

Figure 5 illustrates the spatial distribution of weekly mean PM_2.5 concentrations across northern Thailand for March 2023. During the first two weeks (Figure 5a,b), moderate concentration levels were observed in most areas, with slightly elevated values in the central region during Week 1 and in both the central and northwestern regions during Week 2. In Week 3, there was a significant reduction in PM_2.5 concentrations (Figure 5c). However, concentrations increased again in Week 4 (Figure 5d), particularly in the northwestern part of the region.

The spatial distribution of weekly mean PM_2.5 concentrations for April 2023 is depicted in Figure 6. High concentrations were observed in the northern and northeastern areas during Week 1 (Figure 6a). PM_2.5 levels declined in Week 2 (Figure 6b), and this downward trend continued through Weeks 3 and 4 (Figure 6c,d). By Week 4, concentrations had significantly decreased across the region.

Figure 7 presents the spatial distribution of weekly mean PM_2.5 concentrations for March 2024. During Week 1 (Figure 7a), concentrations were generally low across the region, although some moderate hotspots appeared in the central part. Week 2 (Figure 7b) shows an increase in PM_2.5 levels, particularly in the central and northwestern regions. In Week 3 (Figure 7c), concentrations rose slightly compared to Week 2. However, by Week 4 (Figure 7d), levels had returned to lower values.

Figure 8 shows the spatial distribution of weekly mean PM_2.5 concentrations over northern Thailand for April 2024. In Week 1 (Figure 8a), concentrations were mostly moderate, with elevated values particularly in the northwestern region. A decrease in PM_2.5 concentrations was observed across most areas in Week 2 (Figure 7b), and this trend persisted through Week 3 (Figure 7c). However, in Week 4 (Figure 7d), concentrations increased slightly compared to the preceding week.

Based on Figure 5 and Figure 6, PM_2.5 concentrations were significantly elevated during Week 4 of March 2023, as well as Weeks 1 and 2 of April 2023. These increases were likely influenced by forest fires and cross-border agricultural burning. This interpretation aligns with previous studies that identified March and April as peak periods for biomass burning and transboundary haze transport in northern Thailand [56,57]. In contrast, as shown in Figure 7 and Figure 8, air quality in 2024 showed notable improvement compared to 2023. This improvement is primarily attributed to the systematic implementation of the national action plan on combating airborne dust. Key measures included reducing open burning, monitoring black smoke emissions, strengthening the enforcement of environmental laws, and enhancing regional cooperation to address transboundary haze. These coordinated efforts contributed to more tangible progress in controlling air pollution [58].

6. Conclusions and Discussion

This study improves the dual kriging (DK) method by introducing a novel nonlinear trend function that combines the Gaussian radial basis functions (GRBFs) with the first-order polynomial, resulting in two enhanced models: DK–RBFP and DK–RBFPGA. The proposed trend structure enhances the model’s ability to capture complex spatial patterns. Theoretical support for the uniqueness of the DK solution is provided by the complete monotonicity of the covariance function and the positive definiteness of the coefficient matrix. To further increase flexibility and accuracy, the DK–RBFPGA model employs k-means clustering to identify GRBF centers and utilizes a genetic algorithm to optimize the bandwidth and other trend parameters. When tested on the spatial interpolation of PM_2.5 concentrations in northern Thailand, both enhanced models consistently outperformed the standard polynomial-based DK model, achieving significantly lower MAPE, MSE, and RMSE values. These results highlight the advantages of incorporating nonlinear trend structures and data-driven parameter optimization, which together provide a more accurate and flexible framework for environmental data analysis and spatial prediction.

Although the proposed trend model in the DK method has demonstrated strong performance in this application, its effectiveness relies on the appropriate selection of the number of clusters, which determines the number of GRBFs used in the trend function. This selection should be suited to the characteristics and quantity of the data. Moreover, accurate estimation of the GRBF parameters, including the centers and bandwidths, is crucial for achieving effective spatial interpolation.

As we have seen in this case study, DK–RBFPGA has greater predictive accuracy in comparison to DK–RBFP. However, employing the GA to estimate function parameters results in much higher computational expenses in comparison to the least squares method used in DK–RBFP, especially in the presence of large volumes of data. The least squares method benefits from performing direct matrix calculations, which are efficient and require less memory when compared to the GA’s iterative population-based searches with recurring fitness evaluations, where the algorithm assesses how well each solution solves the problem, making it much slower and more costly in terms of resources. As a result, the conflict between the two methods stems from the decision to be made in balancing higher predictive accuracy with computational efficiency.

Furthermore, the proposed models may be adapted for real-time prediction by using a local neighborhood approach that decreases the computational complexity for big datasets. In this method, the prediction for a given case is based on the closest observations, which drastically reduces the amount of data that needs to be processed. This approach helps the models remain responsive even with the large volumes and velocity requirements of data, which are typical in real-time systems.

In future work, we aim to explore alternative variogram models and nonlinear trend functions that guarantee the uniqueness of the DK solution and to extend our approach for broader applicability across diverse data types. In addition, further research will focus on the computational complexity and asymptotic properties of parameter estimation for the trend function.