Spatial modeling method based on parameter reconstruction unified geographically weighted regression model

Abstract

The invention belongs to the technical field of spatial modeling, and discloses a spatial modeling method for reconstructing a unified geographically weighted regression model based on parameters. According to the method, a parameter reconstruction technology is introduced into a unified geographically weighted regression model, and in Bayesian inference based on the model, simplified RP prior is constructed, and posteriori distribution in a closed form is derived, so that dependence on a traditional MCMC method is eliminated, and parameter dimension and calculation complexity are remarkably reduced. According to the method, the adaptability and calculation efficiency of the model under high-dimensional spatial data are improved while the uncertainty quantification advantage of Bayesian inference is kept, and the method is particularly suitable for processing large-scale data sets with high spatial heterogeneity; the method is suitable for environment monitoring data modeling, urban traffic behavior prediction, epidemic spatial propagation modeling, regional resource distribution modeling and other data scenes with spatial correlation and local variation characteristics, and efficient modeling, parameter inference and prediction analysis are achieved.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of spatial modeling, and particularly relates to a spatial modeling method based on parameter reconstruction of a unified geographic weighted regression model. BACKGROUND

[0002] Spatial modeling is a process of abstracting and simulating geographical spatial phenomena. It focuses on constructing a model that can reflect geographical spatial entities, phenomena and their mutual relationships. The general steps of spatial modeling include: defining the modeling target, determining the requirements such as optimizing land use layout for urban planning or evaluating pollutant diffusion for environmental simulation, and defining the geographical range; then collecting relevant spatial data such as land use status map and terrain data, performing data preprocessing and integrating into a unified framework; selecting a suitable modeling method such as geographic weighted regression and geographic spatial network model according to the target and data characteristics, determining the corresponding algorithm and parameters, constructing the model and verifying it; finally analyzing the model results, visualizing and applying them to actual decision support, and optimizing and updating the model according to the verification feedback and data changes.

[0003] With the development of remote sensing monitoring, geographic information systems, Internet of Things and big data platforms, spatial data has been widely collected and applied in the fields of environmental science, urban planning, public health, transportation, etc. These data have the characteristics of high dimensionality, high heterogeneity and strong spatial correlation, which puts higher requirements on modeling methods.

[0004] Current spatial modeling often uses fixed effect models and traditional geographic weighted regression (GWR) models. Although these methods can handle spatial autocorrelation and local variability to some extent, the models are rigid and difficult to adapt to the changing characteristics of different regions. Especially when dealing with large samples or high-dimensional covariates, there is a serious dimension disaster problem, which leads to a significant decline in model performance. Therefore, in practical applications, how to improve the computability and efficiency of large-scale spatial modeling while ensuring the modeling ability and uncertainty expression of the model has become a key technical problem that needs to be solved in the field of statistical modeling and geographic modeling. SUMMARY

[0005] In view of the deficiencies of the prior art, the present application provides a spatial modeling method based on parameter reconstruction of a unified geographic weighted regression model. The present application not only improves the adaptability and computational efficiency of the model under high-dimensional spatial data, but also maintains the advantages of Bayesian inference uncertainty quantification, and is particularly suitable for fast prediction and high-precision modeling of large-scale spatial data with strong spatial heterogeneity.

[0006] To achieve the above object, the present application provides the following technical scheme:

[0007] The method comprises the following steps: defining a modeling target, defining a geographical or spatial range, setting a response variable and defining a spatial position set; collecting relevant spatial data, and performing data preprocessing and integration to form a sample set; selecting a modeling method, constructing a model and verifying the model; analyzing the model results, applying the model results to actual decision support, and optimizing and updating the model according to verification feedback and data changes;

[0008] The method for constructing the model comprises the following steps:

[0009] Step 1: defining a spatial position set S = {s1, s2,..., sn} and setting a response variable y and a covariate matrix X = (x1, x2,..., xn)T. n} and setting a response variable and a covariate matrix Constructing a unified geographical weighted regression model:

[0010]

[0011] ε i ~ N (0, σ 2 ).

[0012] i = 1, 2,..., n

[0013] Wherein:

[0014] X represents a covariate with a global effect, Z represents a covariate with a spatial variation effect, n represents the number of samples, p is the number of global covariates, and q is the number of covariates with a spatial variation coefficient;

[0015] y i : the response value of the ith sample point;

[0016] ρ1: a spatial autocorrelation coefficient, controlling the spatial dependence degree between samples of the response variable y;

[0017] ρ2: a spatial error autocorrelation coefficient, controlling the parameter of the spatial error term μ autocorrelation degree;

[0018] W 1i : the ith column vector of the spatial weight matrix W1, used for weighting the y value of the surrounding points;

[0019] W 2i : the ith column vector of the spatial weight matrix W2, used for weighting the error value of the surrounding points;

[0020] x i : the global covariate row vector of the ith sample;

[0021] α: the regression coefficient of the global covariate, which is a self-defined row vector;

[0022] z i: the covariate row vector with spatially varying coefficients for the i th sample;

[0023] β(s i ): the local regression variable coefficient at position s i , a row vector varying with position;

[0024] μ i : the spatial structure error term for the i th sample;

[0025] ε i : the independent and identically distributed random disturbance term for the i th sample, following a normal distribution with zero mean and variance σ 2 ;

[0026] Step 2: modeling the local regression variable coefficient function β(s i ) using parametric reconstruction techniques, comprising:

[0027] Step 2.1: setting a distribution mode, selecting m = log2(n) interpolation centers from the original n sample points;

[0028] Step 2.2: selecting an interpolator, constructing an interpolation basis function using the interpolation centers, and reconstructing β(s i ):

[0029]

[0030] where γ is the set of all variable coefficient function parameters; is the product of the q variable coefficient covariates of each sample point and the interpolation basis function; is the RP reconstructed covariate matrix;

[0031] Step 2.3: based on the RP reconstructed covariate matrix rewriting the unified geographically weighted regression model matrix expression:

[0032]

[0033] where A = I n - ρ1W1, B = I n - ρ2W2I n , I n is an n-dimensional identity matrix;

[0034] ∈ = [ε1, ε2, …, ε i …, ε n ] T , represents a vector of random disturbance terms for all sample points, and ε i is the i th element thereof.

[0035] Preferably, the distribution mode set in step 2.1 is uniform distribution.

[0036] Preferably, the interpolator selected in step 2.2 is a Gaussian kernel interpolator.

[0037] Preferably, the Bayesian estimation method for reconstructing the unified geographically weighted regression model based on parameters comprises:

[0038] Step 3.1: Assign a prior to the parameters to be estimated (γ, α, σ 2 ):

[0039]

[0040] Define the covariance function d is the dimension of s i ;

[0041] wherein,

[0042] V = diag(R1, R2, …, R q ), is the prior covariance matrix of the interpolation parameter γ, block diagonal structure; wherein, Rl is the lth covariance matrix, constructed based on the interpolation nodes a i ,a j ;

[0043] μ γ : prior mean of the parameter γ;

[0044] Step 3.2: Derive the posterior distribution of the parameters to be estimated (γ, α, σ 2 ) based on the observed data:

[0045]

[0046] wherein,

[0047]

[0048] ∑ α|y = (X T B T H1BX) -1 ,

[0049]

[0050] s1 = I n -BX(X T B T BX) -1 X T B T ,

[0051]

[0052] Q1 is used to estimate σ2 Q2 is used to estimate a;

[0053] Step 3.3: Estimate parameters (p1, p2, m γ , q) using maximum likelihood method:

[0054] Define the log-likelihood function

[0055]

[0056] where,

[0057] Solve the maximum likelihood estimates of (p1, p2, m 2 , q) by numerical optimization algorithm. γ

[0058] Further preferably, the Bayesian estimation method for reconstructing the unified geographically weighted regression model based on parameters further comprises:

[0059] Step 4: Use the derived posterior distribution to predict the response value y * of a new location s * , and construct a confidence interval combining the posterior variance to realize the uncertainty expression of the prediction output:

[0060] The posterior distribution of y * at (s * , x * , z * ) is a t-distribution:

[0061]

[0062]

[0063] where,

[0064] w *|y : prediction point variance or uncertainty estimate;

[0065] m *|y : mean of the prediction distribution of the model for new locations;

[0066] Q4: represents the fitting error of the model on known data;

[0067] where,

[0068]

[0069] m α|f = å α|f (y * b3 + b4),

[0070] ​

[0071] b1=[(IA)1x * +(IA)2X-Ω *y Ω y -1 ((IA)3x * +(IA)4X)] T ,

[0072]

[0073] c1 = Ω * Ω y -1 y,

[0074] A joint =I n+1 -W 1joint ,

[0075] B joint =I 2+1 -W 2joint '

[0076]

[0077] Compared with the prior art, the beneficial effects of the present invention are:

[0078] 1. This invention constructs a unified geographically weighted regression framework, which enhances the model's adaptability to spatially heterogeneous data, enabling the model to flexibly capture the local non-stationarity and irregular variation of spatial data, thereby improving the accuracy and robustness of modeling.

[0079] 2. This invention introduces parameter reconstruction technology to construct the model. By compressing and reconstructing the model parameter space, it effectively improves the computational efficiency in the large sample modeling process, significantly reduces the estimation dimensionality, and alleviates the computational burden.

[0080] 3. This invention uses a simplified RP prior structure to derive a posterior expression with a closed form, thereby enabling the model to complete Bayesian inference and variable prediction without relying on the traditional Markov chain Monte Carlo (MCMC) sampling algorithm, which greatly improves the inference efficiency of the model.

[0081] 4. By combining structured prior design with parameter reconstruction mechanism, this invention effectively preserves the key relationships and spatial interpretability between parameters, ensuring that the model still has good predictive performance and interpretability in the face of high-dimensional spatial data and complex sample structures.

[0082] 5. This invention has good scalability and adaptability at the engineering implementation level, supports rapid training and prediction in high-performance computing environments, and can be widely applied to practical large-scale spatial modeling and decision support tasks such as remote sensing, urban planning, and environmental monitoring. It has high application value and promotion potential. Attached Figure Description

[0083] Figure 1 The residuals and residual histograms of the UGWR model in this embodiment of the invention;

[0084] Figure 2 This is a graph showing the variable coefficients of the UGWR model in an embodiment of the present invention.

[0085] In the figure: Residual plot; Residual; Observation Order; Histogram of residuals; Frequency; UGWR Coefficientestimation; Normalized Solar Radiation. Detailed Implementation

[0086] To make the technical means, creative features, objectives and effects of this invention easier to understand, the following embodiments are provided to illustrate this invention in detail. It should be noted that the description of these embodiments is for the purpose of helping to understand this invention, but does not constitute a limitation of this invention.

[0087] This invention presents a spatial modeling method based on a parameter-reconstructed unified geographic weighted regression model. It proposes a unified geographic weighted regression modeling framework, extending the traditional variable-coefficient autoregressive model, aiming to more reliably model complex structures in spatial data and improve model performance in spatial heterogeneity modeling and prediction tasks. Addressing the common technical challenges in current spatial modeling, such as strong data heterogeneity, drastic local variations, high dimensionality, and computational complexity, this method systematically innovates in multiple aspects, from model structure and inference methods to engineering implementation, exhibiting significant technical advantages. This method integrates geographic weighted regression and Bayesian inference, systematically introducing spatial response autocorrelation, residual spatial autoregressive structures, and spatial variable-coefficient modeling mechanisms, making it applicable to various data scenarios with spatial dependence and local variability.

[0088] The overall structure of the model includes three core mechanisms: first, it introduces an autoregressive term for the response variable to capture the spatial dependence between observations; second, it sets up a spatial autocorrelation structure for the residual term to model the spatial continuity not fully explained by the autoregressive term of the response variable; and third, it introduces a spatially location-dependent variable coefficient function to characterize the heterogeneous changes of covariate effects in different regions, thereby achieving fine modeling of local structural specificity.

[0089] To express how spatial coefficients change continuously with location, the method uses RP prior interpolation to model the coefficient function and uses spatial distance to define its covariance structure, flexibly characterizing spatial smoothness and correlation.

[0090] Considering the computational bottleneck faced by Gaussian processes in large-sample scenarios, this method selects a Gaussian interpolator (corresponding to step 2 below). By selecting m = log₂n nodes instead of all samples, the number of parameters required for the variable coefficient function is reduced, simplifying the solution model and lowering the complexity from the traditional O(n log₂n) to... 3 The time complexity is reduced to O((log₂n)). 3 This significantly reduces the computational complexity of inverting the covariance matrix and improves the scalability of the model under big data conditions.

[0091] In terms of parameter inference, the method introduces a Bayesian inference framework. Generally, the posterior distribution in Bayesian inference often involves complex high-dimensional integrals, which are difficult to solve analytically. Therefore, sampling algorithms such as MCMC are required for approximate inference. This invention also proposes a simplified RP prior parameter estimation scheme that, while retaining modeling flexibility, gives the posterior distribution a closed analytical form. This eliminates the need for Markov Chain Monte Carlo (MCMC) sampling, enabling efficient parameter estimation and uncertainty quantification.

[0092] Overall, this invention not only improves the model's adaptability and computational efficiency in high-dimensional spatial data but also maintains the advantage of Bayesian inference uncertainty quantification, making it particularly suitable for rapid prediction and high-precision modeling of large-scale, spatially heterogeneous spatial data. This method achieves a good balance between theoretical modeling and practical application, not only improving the model's ability to express complex spatial structures but also significantly enhancing its adaptability and predictive performance on large-scale data. Simulation experiments and empirical applications have both verified the significant advantages of this method in terms of prediction accuracy, inference efficiency, and model stability.

[0093] This invention is applicable to various practical fields with high requirements for spatial modeling. For example, in environmental science, it can be used for air pollution concentration prediction and spatial estimation of water quality; in urban planning and traffic analysis, it can be applied to tasks such as shared bicycle travel heat analysis and urban functional zone identification; in public health and epidemiological modeling, it can support spatial modeling and prediction of infectious disease transmission paths; furthermore, in natural resource management, this method can also be used for land use type estimation and regional resource allocation optimization. A common characteristic of these applications is the high demand for modeling efficiency, uncertainty representation ability, and prediction accuracy for large-scale spatial data, and this invention provides effective technical support in these aspects.

[0094] I. Modeling Methods and Steps

[0095] A spatial modeling method based on a parameter-reconstructed unified geographic weighted regression model according to the present invention includes the following steps:

[0096] (1) Establish a unified geographically weighted regression model, which is used to define the statistical relationship between the response variable and the covariate. The response variable is the measured value of the research object, and the covariate is the factor that affects the response variable.

[0097] (2) Utilize RP priors to reduce model parameters;

[0098] (3) Use the simplified RP prior to derive the closed-form posterior distribution;

[0099] (4) Use closed-form posterior distributions for model inference and prediction;

[0100] (5) Performance verification and model optimization enable the model to achieve efficient computation on large-scale datasets while maintaining prediction accuracy.

[0101] Specifically as follows:

[0102] Step 1: Construct a Uniform Geographically Weighted Regression Model (UGWR)

[0103] First, define the set of spatial locations S = {s1, s2, ..., s}. n}, and set the response variable. and covariate matrix That is, y is an n-dimensional vector in the real number field, and X and Z are n×p and n×q matrices in the real number field, where X represents a covariate with global effect, Z represents a covariate with spatial variation effect, n represents the sample size, p is the number of global covariates, and q is the number of covariates with spatial variation coefficients.

[0104] The Unified Geographically Weighted Regression Model (UGWR) of this invention is constructed as follows:

[0105]

[0106] ε i ~N(0,σ 2 ).

[0107] i = 1, 2, ..., n

[0108] in:

[0109] : Represents the mathematical symbol "transpose", meaning that matrix transpose is the process of swapping the rows and columns of a matrix;

[0110] y i : The response value of the i-th sample point;

[0111] ρ1: Spatial autocorrelation coefficient, which controls the degree of spatial dependence between samples of the response variable y;

[0112] ρ2: Spatial error autocorrelation coefficient, a parameter that controls the degree of autocorrelation of the spatial error term μ;

[0113] W 1i : The i-th column vector of the spatial weight matrix W1 (used to weight the y-values ​​of surrounding points);

[0114] W 2i : The i-th column vector of the spatial weight matrix W2 (used to weight the error values ​​of surrounding points);

[0115] x i : The global covariate row vector of the i-th sample;

[0116] α: Regression coefficients of global covariates, row vector (fixed value, does not change with position);

[0117] z i The row vector of covariates with spatial variation coefficients for the i-th sample;

[0118] β(s i ): Position s i The local regression coefficients at a given location are row vectors that vary with position.

[0119] μ i The spatial structure error term of the i-th sample has spatial correlation, i = 1, 2, ..., n;

[0120] ε i The independent and identically distributed random disturbance term of the i-th sample, with zero mean and variance σ. 2 The normal distribution, i.e.

[0121] The Unified Geographically Weighted Regression (UGWR) model of this invention makes three key improvements on the existing Geographically Weighted Regression (GWR) model: First, it introduces a spatial lag term for the response variable. To characterize the mutual influence between spatially adjacent regions; secondly, to introduce spatial error structures. The error term is modeled as a spatial autoregressive process, thereby better controlling the interference caused by spatial autocorrelation; thirdly, spatially variable coefficient β(s) is used. i The modeling method incorporates regression coefficients for some covariates that vary with geographical location, enhancing the model's adaptability to spatial heterogeneity. Through these three modifications, the model outperforms the traditional GWR method in explaining spatial dependence, reflecting regional differences, and improving predictive ability.

[0122] Step 2: Select an interpolator and model the locally regressive variable coefficient function β(s) using parametric reconstruction (RP) techniques. i );

[0123] The Gaussian kernel-based interpolator is defined as follows:

[0124]

[0125] The interpolation basis function vector b(x) is defined as follows:

[0126]

[0127] Its components are defined as follows:

[0128]

[0129] The Gaussian kernel correlation function is defined as follows:

[0130]

[0131] in

[0132] g(x) is a polynomial basis function vector. Different forms of g(x) affect the fitting ability and smoothness of the interpolation function. Generally, the choice can be based on the spatial variation trend of the coefficients to be fitted: if the local regression coefficients change smoothly, a low-order polynomial (such as a constant term or a first-order term) can be used; if the coefficients change drastically with position, a higher-order polynomial or a more flexible expression such as a radial basis function can be introduced. In this embodiment, g(x) is chosen to be g(x) = [1, x, x...]. 2 The form is ,...].

[0133] in, The set of selected interpolation nodes, consisting of m = log2(n) nodes, is called the interpolation centers. These nodes are used to construct the interpolation basis function, which in turn reconstructs the variable coefficient function. Nodes can be selected using a uniform filling strategy or other representative distribution methods, such as K-means cluster centers.

[0134] To reduce model complexity, this invention employs the aforementioned Gaussian kernel-based interpolator for the variable coefficient β(s) i Interpolation is performed to represent the variable coefficient function in another way. This method reconstructs the variable coefficient function through parameter substitution, which is defined as the RP (Reconstruction Parameterization) method in this embodiment.

[0135] In the implementation, we select m = log2(n) representative nodes (i.e., interpolation centers) from the original n sample points. The node selection employs a uniform filling strategy to cover the overall distribution of the input space. This method significantly reduces the number of parameters required to estimate the variable coefficient function, thereby improving the computational efficiency and generalization ability of the model.

[0136]

[0137] in:

[0138] γ: The set of all parameters of the variable coefficient function, representing the variable coefficient function β(s) i The overall parameter vector;

[0139] The product of q variable coefficient covariates for each sample point and the interpolation basis function;

[0140] The RP reconstruction covariate matrix is ​​the covariate matrix with spatial variation effect of the RP reconstruction of the entire sample set.

[0141] The Unified Geographically Weighted Regression (UGWR) model can be rewritten in the following matrix form:

[0142]

[0143] Among them, A=I n -ρ1W1,B=I n -ρ2W2I n I n Let be an n-dimensional identity matrix, where = [ε1, ε2, ..., ε i ,ε n ] represents the vector consisting of the random perturbation terms of all sample points, ε i It is its i-th element.

[0144] Step 3: Derive the Bayesian estimation algorithm based on the Unified Geographically Weighted Regression (UGWR) model to infer the parameters to be estimated (α, γ, σ). 2 ,ρ1,ρ2):

[0145] The Bayesian inference part (Bayesian estimation) is based on the structure proposed in the Unified Geographically Weighted Regression (UGWR) model. It further treats the parameters to be estimated as random variables and constructs their prior and posterior distributions, thereby quantifying and inferring the uncertainty of the model parameters. In other words, the Bayesian part is an extension of the inference and estimation aspects of the UGWR model.

[0146] Bayesian methods combine the prior distribution of parameters designed by the experimenter and the data distribution represented by the likelihood function to generate the posterior distribution of the parameters. This posterior distribution contains both information from the sample data of the likelihood function and prior or subjective information about the parameter distribution. Traditional Bayesian methods require numerical integration to calculate the mean and variance of the parameters. In practical problems, it is often difficult to obtain a Bayesian estimate of the parameters or an explicit representation of the posterior distribution. When the model contains two parameters, the integration in the above equation becomes a double integral. As the number of parameters in a model increases, the integration becomes extremely difficult. An alternative method is to obtain the posterior density of the parameters by randomly sampling the posterior distribution of the parameters. This method is called the Markov Chain Monte Carlo (MCMC) method, and the sampling method used is Gibbs sampling. The core of sampling estimation is to determine the posterior distribution of the parameters to be estimated in the model.

[0147] In summary, in traditional Bayesian inference, when the posterior distribution cannot be solved analytically, the MCMC sampling method is needed for approximation.

[0148] This invention constructs a simplified Bayesian prior structure (RP prior construction) based on the Unified Geographically Weighted Regression (UGWR) model using parameter reconstruction (RP). This allows the posterior distribution to have a closed form. A closed-form posterior distribution means that the posterior distribution expression of the parameters can be directly given without approximation through MCMC sampling, thus eliminating complex sampling steps, improving the efficiency of large-scale modeling, significantly reducing computational complexity and inference time, and enhancing the model's practicality in big data and real-time application scenarios. A "closed-form" posterior distribution means that, in Bayesian inference, the posterior distribution of the parameters can be analytically written in the form of a certain standard probability distribution. Traditional GWR does not predefine the covariance form, directly representing it with a single symbol, which cannot be decomposed in subsequent calculations, resulting in the model's posterior distribution being unsolvable analytically. The model setting of this invention allows σ to be included in the posterior distribution calculation. 2The method explicitly proposes that the error variance can be extracted uniformly during derivation, which facilitates the integration of the squared residual terms in the likelihood function to obtain an analytical solution, thereby significantly improving inference efficiency and model scalability. This method is the key innovation of this invention in Bayesian modeling strategy.

[0149] Step 3.1: Parameters to be estimated (α, γ, σ) 2 Each of them specifies the prior.

[0150] It should be noted that the priors of the two parameters ρ1 and ρ2 both follow a uniform distribution of U(-1, 1), but the posteriors of these two parameters are not closed-form solutions and are estimated using maximum likelihood estimation. Therefore, priors for these two parameters will not be assigned here.

[0151] Here we define (γ, α, σ) 2 Assignment Priors:

[0152]

[0153] Here, V = diag(R1, R2, ..., R q ), d is s i dimensionality

[0154]

[0155] C l (s1,s2) is a covariance function used to characterize the similarity of variable coefficient functions between different spatial locations. It indicates that the closer the locations are, the stronger the correlation and the larger the covariance; while the farther the locations are, the weaker the correlation and the covariance decays exponentially with distance.

[0156] in:

[0157] μ γ : The prior mean of parameter γ;

[0158] V: The prior covariance matrix of the interpolation parameter γ, with a block diagonal structure;

[0159] R l The l-th covariance matrix, based on interpolation node a i ,a j structure.

[0160] The RP prior uses a Gaussian correlation function to model the similarity between input data points; it captures the local variation characteristics of covariates and establishes spatial dependence through a covariance function, thereby enabling the variable coefficient model to reflect the local variation and spatial correlation of spatial data, improving the prediction accuracy and adaptability of the model in large-scale spatial datasets with significant regional differences and dense data; it constructs a smooth function model through interpolation, accurately matching at known points and estimating at unknown points, so as to reconstruct and estimate unknown smooth functions from discrete data points when sampling points are limited.

[0161] Based on the above priors, we can derive the closed-form posterior distribution. Unlike the traditional Bayesian GWR method which relies on MCMC sampling, this invention combines RP technology to reconstruct spatially variable coefficients in a low dimension and introduces structured RP priors, making the posterior analytically derivable without the need for MCMC sampling, thus significantly simplifying the computation process.

[0162] Step 3.2: Derive the parameters to be estimated (α, γ, σ) based on the observation data 2 The posterior distribution of ).

[0163] Specific parameter distribution:

[0164]

[0165] Σ α|y =(X T B T H1BX) -1 ,

[0166]

[0167] This part involves Bayesian inference, estimating unknown parameters (such as regression coefficient α, variable coefficient parameter γ, and error variance σ) based on observed data y. 2 The posterior distribution of (etc.). Due to the adoption of a reasonable prior structure, σ is included in the calculation of the posterior distribution. 2 The method is explicitly proposed, which allows for the unified extraction of error variance during derivation. This facilitates the integration of the squared residual terms in the likelihood function, ultimately yielding a closed-form posterior distribution such as the multidimensional t-distribution or the inverse gamma distribution. This avoids the complex MCMC sampling process and significantly improves inference efficiency.

[0168] in:

[0169] Multidimensional t-distribution;

[0170] IG(·,·): Inverse gamma distribution, which is σ 2 The posterior distribution of;

[0171] Q1: A quantity related to the sum of squared residuals, used to estimate σ. 2;

[0172] Q2: The quantity used to estimate α, taking into account prior and observational errors;

[0173] ~: indicates that it "follows" a certain distribution;

[0174] μ: The mean of the posterior distribution of each parameter, distinguished by subscripts;

[0175] Σ; the covariance of the posterior distribution of each parameter, distinguished by subscripts.

[0176] Step 3.3: Estimate the parameters (ρ1, ρ2, μ) using the maximum likelihood method. γ ,θ)

[0177] We use the maximum likelihood method to estimate (ρ1, ρ2, μ) γ The idea of ​​maximum likelihood is: given the observed data, find a set of parameters such that the model is "most likely" to generate the currently observed data under these parameters.

[0178] Given the parameters, the response variable y follows a multivariate normal distribution:

[0179]

[0180] in, Therefore, the log-likelihood function is:

[0181]

[0182] (ρ1,ρ2,σ 2 ,α,μ γ The maximum likelihood estimate of θ can be solved using a numerical optimization algorithm.

[0183] Step 4: Model Prediction and Uncertainty Quantification

[0184] Using the derived posterior distribution to the new position s * response value y * Make predictions and construct confidence intervals by combining posterior variance to express the uncertainty of the prediction output.

[0185] Next, y * In (s * ,x * ,z * The posterior distribution at position () is a t-distribution.

[0186]

[0187] And there is,

[0188]

[0189] in:

[0190]

[0191] μ α|f =∑ α|f (y * b3+b4),

[0192]

[0193] b1=[(IA)1x * +(IA)2X-Ω *y Ω y -1 (IA)3x * +(IA)4X)] T ,

[0194]

[0195] c1 = Ω *y Ω y -1 y,

[0196] A joint =I n+1 -W 1join ,

[0197] B joint =I 2+1 -W 2joint '

[0198]

[0199] here,

[0200] ω *|y : Estimation of the variance or uncertainty of the prediction point;

[0201] μ *|y : The mean of the model's predicted distribution for the new location;

[0202] Q4: Indicates the model's fitting error on the known data;

[0203] Σ: Posterior covariance of different parameters, distinguished by subscripts;

[0204] b1 to b6: Various intermediate auxiliary quantities used to simplify the representation of long formulas, combining position, model structure and uncertainty information;

[0205] I n+1 : n+1 dimensional identity matrix

[0206] A joint Bjoint A matrix that combines new prediction points with existing data, facilitating joint modeling.

[0207] Ω: Spatial correlation matrix, used to express the similarity relationship between different locations, distinguished by subscripts.

[0208] Step 5: Performance Validation and Model Optimization

[0209] The model's performance was tested through simulations and real-world data to evaluate its prediction accuracy and convergence stability under different data scales. Further improvements were made to the model's training and inference efficiency by incorporating distributed computing or stochastic feature methods to meet the engineering needs of large-scale spatial modeling.

[0210] II. Implementation Examples

[0211] To verify the effectiveness of our research method in real-world scenarios, we applied it to the modeling and analysis of shared bicycle demand data. This dataset records the hourly number of bicycle rentals and related meteorological observation indicators from December 2017 to November 2018, totaling 8,760 records, exhibiting a typical spatiotemporal dynamic structure and data heterogeneity.

[0212] (1) Data collection content (specific variables):

[0213] Each observation contains the following 12 variables:

[0214] Time information: hours (0–23)

[0215] Response variable: Rented Bike Count (RBC, unit: vehicles)

[0216] Meteorological variables:

[0217] Temperature (TEMP, °C)

[0218] Humidity(HUM,%)

[0219] Wind Speed ​​(WS, m / s)

[0220] Visibility(VIS,m)

[0221] Solar Radiation (SR, MJ / m) 2 )

[0222] Rainfall(RAIN,mm)

[0223] Snowfall(SNOW,cm)

[0224] Dew Point Temperature(DPT,℃)

[0225] (2) Data processing flow:

[0226] 1. Missing value handling: Remove samples with missing records in the observations.

[0227] 2. Standardization: Z-score standardization is applied to continuous variables to improve model stability.

[0228] 3. Training-test split: Randomly select 1,500 samples as the training set and the rest as the test set to ensure the reliability of the model's generalization evaluation.

[0229] 4. Variable selection: Based on scatter plots and correlation analysis, six variables were selected as the main explanatory variables: temperature, humidity, wind speed, precipitation, solar radiation, and visibility.

[0230] (3) Model fitting and result data:

[0231] We fitted three types of models based on the training set data: OLS (Ordinary Least Squares Regression), GWR (Geographically Weighted Regression), and UGWRM (the Unified Geographically Weighted Regression Model of this Invention). The resulting data includes the following two categories:

[0232] Estimation index: R 2 (Goodness of fit), Root Mean Square Error (RMSE), Relative Error (MRE) Prediction Metrics: Mean Square Prediction Error (RMSPE), Relative Prediction Error (MRPE), Bayesian R 2 Bayesian fit index based on posterior distribution

[0233] Among them, the UGWRM (UGWR model) performed best on all indicators, estimating Rw. 2 =0.972, Bayesian R 2 =0.872, RMSE is only 0.0058, which is much lower than other models.

[0234] This invention demonstrates two significant advantages in modeling rental bicycle data: strong uncertainty quantification capability and strong flexible modeling capability.

[0235] First, regarding uncertainty quantification, UGWRM, based on a Bayesian inference framework, selects our proposed prior structure, ensuring that all parameters have a closed-form posterior distribution, thus enabling direct uncertainty quantification. Compared to traditional frequentist models, this method not only provides point estimates but also gives the confidence intervals of parameters and prediction results, which is helpful for risk control and decision support. Furthermore, this embodiment introduces Bayesian R...2 As a model fit index in the posterior sense, UGWRM obtained the highest value (0.872), indicating that it has stronger credibility in posterior prediction. Figure 1 The residual plot and residual histogram shown further demonstrate that the error distribution is approximately normal, indicating that the model structure performs well in capturing uncertainty.

[0236] Secondly, in terms of flexible modeling, UGWRM can simultaneously handle spatial heterogeneity, spatial error structure, and nonlinear response relationships of input variables. Specifically, the model captures spatial heterogeneity by introducing a variable coefficient function to characterize the changes in covariate effects at different time points; it also introduces a spatial autoregressive structure of the residual terms to describe the spatial dependence between error terms; furthermore, for variables such as solar radiation, which have obvious nonlinear relationships with response variables, UGWRM can effectively model them through nonlinear structures. Figure 2 The variable coefficient curves shown indicate that the impact of solar radiation on rental volume varies significantly under different conditions. In particular, the response effect is significantly enhanced after the radiation intensity exceeds 0.5, demonstrating the model's good adaptability in characterizing complex nonlinear relationships.

[0237] In summary, UGWRM not only outperforms existing methods in prediction accuracy, but also excels in uncertainty quantification and structural modeling flexibility, demonstrating good practicality and promising prospects for widespread application.

[0238] The above embodiments are preferred embodiments of the present invention and are not intended to limit the scope of protection of the present invention. Various modifications or variations that can be made by those skilled in the art without creative effort within the scope of the appended claims are still within the scope of protection of this patent.

Claims

1. A spatial modeling method based on parameter reconstruction of a unified geographic weighted regression model, comprising: Define the modeling objectives and the geographic or spatial extent to set response variables and define the set of spatial locations; Collect relevant spatial data, preprocess and integrate the data to form a sample set; select a modeling method, build and validate the model; analyze the model results, apply them to practical decision support, and optimize and update the model based on validation feedback and data changes. The method for constructing the model is characterized by comprising the following steps: Step 1: Define the set of spatial locations S = {s1, s2, ..., s} n }, and set the response variable. and covariate matrix Constructing a unified geographic weighted regression model: in: X represents a covariate with a global effect, Z represents a covariate with a spatial variation effect, n represents the sample size, p is the number of global covariates, and q is the number of covariates with spatial variation coefficients. y i : The response value of the i-th sample point; ρ1: Spatial autocorrelation coefficient, which controls the degree of spatial dependence between samples of the response variable y; ρ2: Spatial error autocorrelation coefficient, a parameter that controls the degree of autocorrelation of the spatial error term μ; W 1i : The i-th column vector of the spatial weight matrix W1, used to weight the y-values ​​of surrounding points; W 2i : The i-th column vector of the spatial weight matrix W2, used to weight the error values ​​of surrounding points; x i : The global covariate row vector of the i-th sample; α: Regression coefficients of global covariates, which are user-defined row vectors; z i The row vector of covariates with spatial variation coefficients for the i-th sample; β(s i ): Position s i The local regression coefficients at a given location are row vectors that vary with position. μ i : Spatial structure error term of the i-th sample; ε i The independent and identically distributed random disturbance term of the i-th sample, with zero mean and variance σ. 2 The normal distribution; Step 2: Model the local regression variable coefficient function β(s) using parameter reconstruction techniques. i ),include: Step 2.1: Set the distribution method and select m = log2(n) interpolation centers from the original n sample points; Step 2.2: Select an interpolator, construct the interpolation basis function using the interpolation center, and apply the interpolation basis function to β(s). i Refactoring: Where: γ is the set of all parameters of the variable coefficient function; The product of q variable coefficient covariates for each sample point and the interpolation basis function; Reconstruct the covariate matrix for RP; Step 2.3: Reconstruct the covariate matrix based on the RP. Rewrite the matrix expression of the unified geographic weighted regression model: Among them, A=I n -ρ1W1,B=I n -ρ2W2I n I n It is an n-dimensional identity matrix; ε represents the vector consisting of the random perturbation terms of all sample points. i It is its i-th element.

2. The spatial modeling method based on a parameter-reconstructed unified geographic weighted regression model according to claim 1, characterized in that, The distribution method set in step 2.1 is uniform distribution.

3. The spatial modeling method based on a parameter-reconstructed unified geographic weighted regression model according to claim 1, characterized in that, The interpolator selected in step 2.2 is the Gaussian kernel interpolator.

4. A spatial modeling method based on a parameter-reconstructed unified geographic weighted regression model according to any one of claims 1 to 3, characterized in that, The Bayesian estimation method based on parameter reconstruction of the unified geographic weighted regression model includes: Step 3.1: For the parameters to be estimated (γ, α, σ) 2 Assignment Priors: Define covariance function l = 1, ..., q; d is s i dimensionality; In the formula, V = diag(R1,R2,…,R) q ), where is the prior covariance matrix of the interpolation parameter γ, with a block diagonal structure; where, Let l be the l-th covariance matrix, based on the interpolation node a i ,a j structure; μ γ : The prior mean of parameter γ; Step 3.2: Derive the parameters to be estimated (γ, α, σ) based on the observed data. 2 The posterior distribution of ) In the formula, Q1 is used to estimate σ 2 Q2 is used to estimate α; Step 3.3: Estimate the parameters (ρ1, ρ2, μ) using the maximum likelihood method. γ ,θ): Define the log-likelihood function in, The solution (ρ1, ρ2, σ) is obtained through a numerical optimization algorithm. 2 ,α,μ γ The maximum likelihood estimate of θ).

5. A spatial modeling method based on a parameter-reconstructed unified geographic weighted regression model according to claim 4, characterized in that, The Bayesian estimation method based on parameter reconstruction of the unified geographic weighted regression model also includes: Step 4: Apply the derived posterior distribution to the new position s * response value y * Make predictions and construct confidence intervals based on posterior variance to express the uncertainty of the prediction output: Derivation of y * In (s * ,x * ,z * The posterior distribution at position () is a t-distribution. in, ω* |y : Estimation of the variance or uncertainty of the prediction point; μ* |y : The mean of the model's predicted distribution for the new location; Q4: Indicates the model's fitting error on the known data; In the formula, μ α|f =∑ α|f (y * b3+b4), c1=Ω *y Oh y -1 y, A joint =I n+1 -W 1joint , B joint =I n+1 -W 2joint ,

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