Statistical inference method and system for non-stationary features of regression relationship of space-time variable coefficient model, and related device

Abstract

The invention provides a statistical inference method and system for non-stationary features of a regression relation of a space-time variable coefficient model and a related device. The statistical inference method comprises the following steps: acquiring a vector form of the space-time variable coefficient model; setting each variable coefficient item in the vector form of the space-time variable coefficient model as a smooth interaction item of an explanatory variable and a space-time coordinate in the space-time variable coefficient model to obtain a generalized model expression of the space-time variable coefficient model; obtaining different forms of each additive item in a generalized additive model expression based on four non-stationary feature types of a regression relationship; performing statistical inference on the non-stationary features of each regression relationship in the space-time variable coefficient model based on different forms of each additive item; according to the method, the basic features of the spatio-temporal data can be processed in a scale adaptive mode, and accurate recognition of the regression relation features of the spatio-temporal variable coefficient model is achieved.

Description

Technical Field

[0001] The present invention relates to, in particular, a statistical inference method, system and related device for the non-stationary characteristics of the regression relationship of a spatiotemporal variable coefficient model. Background Art

[0002] Spatiotemporal data refers to comprehensive data that combines both temporal (temporal dimension) and spatial (spatial dimension) information. Spatiotemporal data is widely found in numerous fields, including geographic information science, environmental science, climatology, econometrics, epidemiology, ecology, and demography. Spatiotemporal data has three fundamental characteristics: spatiotemporal autocorrelation (dependence), spatiotemporal nonstationarity, and spatiotemporal multiscalarity.

[0003] Regression modeling based on the autocorrelation, nonstationarity, and multi-scale characteristics of spatiotemporal data has broad application value in fields such as environmental science, transportation planning, epidemiology, and weather forecasting, and has garnered significant interdisciplinary attention in recent years. With the increasing complexity of spatiotemporal data and the increasing sophistication of decision-making requirements, developing spatiotemporal regression methods that balance theoretical rigor with computational efficiency will continue to be a key focus in both academia and industry. Models that ignore spatiotemporal characteristics can not only lose predictive accuracy but also mislead decision-making. Therefore, research on regression modeling and its applications for spatiotemporal data is of great practical significance.

[0004] Brunsdon et al., in 1996, proposed the use of geographically weighted regression (GWR) to fit spatially varying coefficient models with different regression relationships at different geographic locations. This method represents an early study of regression modeling for the spatially nonstationary characteristics of spatiotemporal data and remains one of the most widely used methods. Its simple estimation process and ease of implementation have led to its widespread application in fields such as geography, environmental science, ecology, meteorology, climatology, epidemiology, econometrics, and computational science. Space and time are the two fundamental dimensions that provide a framework for human activities, social events, and environmental processes. Due to the interactive nature of time and space, regression modeling around the fundamental characteristics of spatiotemporal data is relatively complex. Building on their research on the GWR model, Huang et al. in 2010 first incorporated time into the coefficient function of the GWR model, proposing a method for fitting spatiotemporally varying coefficient models that simultaneously captures both the temporal and spatial nonstationary characteristics of the regression relationship. This method, termed Geographic Time Weighted Regression (GTWR), is known as the Geographic Time Weighted Regression (GTWR). In 2017, Liu et al. proposed a semi-parametric form of the GTWR model, called the mixed GTWR model. The regression coefficients of the GTWR model are divided into two forms: global stationary and spatiotemporal non-stationary. This allows the model to simultaneously capture the global stationary and spatiotemporal non-stationary characteristics of the regression relationship. In this paper, the F-distribution approximation method is used to identify the global stationary characteristics of the regression relationship. Appropriate identification of spatiotemporal nonstationary characteristics of data can provide a more reliable theoretical framework for comprehensive analysis and fully revealing the basic characteristics of spatiotemporal data. Current research on the identification of nonstationary characteristics of the GTWR model regression relationship mainly includes: Huang et al. (2010), Liu et al. (2017), and Xuan et al. (2015) compared the goodness of fit between the null model and the GTWR model, and proposed test methods for the global stationarity, spatial nonstationarity, and temporal nonstationarity of the GTWR model regression relationship using variance analysis techniques. The p-values ​​of the tests were calculated using the F distribution approximation method. Fotheringham et al. (2015) tested the spatiotemporal nonstationarity of the GTWR model regression relationship using the interquartile moments of the coefficient estimates. Xiao et al. (2014) constructed a generalized likelihood ratio test statistic to infer the temporal and spatial nonstationarity of the GTWR model regression relationship, respectively, and calculated the p-values ​​of the tests using the bootstrap method. However, these identification methods focus only on the spatiotemporal nonstationary characteristics of all regression relationships in the GTWR model. These methods correspond to extreme cases of the GTWR model and may rarely occur in reality. To this end, Hong et al. proposed bootstrap test methods in 2021 and 2022, respectively, to identify globally stationary and spatially nonstationary regression relationships in spatiotemporally variable coefficient models. In 2023, Zhang et al. generalized the bootstrap test method to simultaneously identify globally stationary, temporally nonstationary, and spatially nonstationary characteristics of regression relationships in spatiotemporally variable coefficient models.In 2024, Chen et al. used the bootstrap test method to identify the global stationary characteristics of the regression relationship in the multiscale geographically weighted regression (MGWR) model. However, in order to save computing time when calculating the p-value of the test statistic, this method still used the traditional GWR method with a unified bandwidth parameter to fit the alternative model.

[0005] At present, the methods used for statistical inference of the non-stationary characteristics of the regression relationship of both the GWR model and the GTWR model can be divided into two types: the F distribution approximation method and the boostrap test method. Among them, the F distribution approximation method is easy to implement and has low computational cost, but the test power of this method depends on the normal distribution assumption of the model error term. When the error term does not conform to the normal distribution, the F distribution approximation method generally has low power; the boostrap test method is a method based on resampling. This method selects the same test statistic as the F distribution approximation method, constructs new sample data by repeatedly sampling the residuals of the alternative model, and then updates the observed value of the test statistic. The advantage of this method is that it does not assume the probability distribution type of the model error term, but because it involves repeated sampling, the computational cost of this method is high.

[0006] In addition, the statistical inferences of the non-stationary regression relationships of the GWR or GTWR models currently proposed are all implemented based on a unified operating scale. For the two more popular multi-scale spatial or spatiotemporal variable coefficient models, namely the multiscale geographically weighted regression (MGWR) model and the multiscale geographically time-weighted regression (MGTWR) model, there is currently no corresponding statistical inference method research on the non-stationary regression relationship. Summary of the Invention

[0007] The purpose of the present invention is to provide a statistical inference method, system and related devices for the non-stationary characteristics of the regression relationship of a spatiotemporal variable coefficient model, which solves the limitation of high computational cost in the commonly used statistical inference methods for the spatiotemporal non-stationary characteristics of the regression relationship.

[0008] In order to achieve the above object, the technical solution adopted in the present invention is:

[0009] In a first aspect, the present invention provides a statistical inference method for non-stationary characteristics of a spatiotemporal variable coefficient model regression relationship, comprising the following steps:

[0010] Get the vector form of the spatiotemporal variable coefficient model;

[0011] By setting each variable coefficient term in the vector form of the spatiotemporal variable coefficient model as the smooth interaction term between the explanatory variables of the generalized additive model and the spatiotemporal coordinates, the generalized additive model expression of the spatiotemporal variable coefficient model is obtained;

[0012] Based on the four types of non-stationary characteristics of the regression relationship, different forms of each additive term in the generalized additive model expression are obtained;

[0013] Statistical inference is performed on the non-stationary characteristics of each regression relationship in the spatiotemporal varying coefficient model based on the different forms of each additive term.

[0014] Preferably, the vector form of the spatiotemporal variable coefficient model is:

[0015]

[0016] Among them, Y is the response variable;

[0017] β k (u,v,t)·X k =(β k (u1,v1,t1)x 11k ,…,β k (u n ,v n ,t1)x n1k ,…,β k (u1,v1,t T )x 1Tk ,…,β k (u n ,v n ,t T )x nTk ) T ;

[0018] x ij1 ,x ij2 ,…,x ijp (i=1,2,…,n;j=1,2,…T) are the explanatory variables X1,X2,…,X p At the time and space sampling point (u i ,v i ,t j ) observation value; β k (u i ,v i ,t j )(k=1,2,…,p) is the model in (u i ,v i ,t j ) is the estimated regression coefficient at ; n is the number of spatial locations in the model; T is the number of time points in the model; ε is the model error vector.

[0019] Preferably, the generalized additive model expression of the spatiotemporal variable coefficient model is:

[0020]

[0021] Among them, f k (u,v,t,X k )=β k (u,v,t)·X k represents the kth additive term of the generalized additive model; β k (u,v,t)·X k =(β k (u1,v1,t1)x 11k ,…,β k (u n ,v n ,t1)x n1k ,…,β k (u1,v1,t T )x 1Tk ,…,β k (u n ,v n ,t T )x nTk ) T ;

[0022] x ij1 ,x ij2 ,…,x ijp (i=1,2,…,n;j=1,2,…T) are the explanatory variables X1,X2,…,X p At the time and space sampling point (u i ,v i ,t j ) observation value; β k (u i ,v i ,t j )(k=1,2,…,p) is the model in (u i ,v i ,t j ) is the estimated regression coefficient at ; n is the number of spatial locations in the model; T is the number of time points in the model; ε is the model error vector.

[0023] Preferably, based on the four non-stationary feature types of the regression relationship, different forms of each additive term in the generalized additive model expression are obtained, specifically by:

[0024] Representing spatiotemporally varying coefficients through smooth functions of the generalized additive model;

[0025] The four types of non-stationary features based on regression relationships are used to obtain different forms of spatiotemporal coefficients;

[0026] The four representations of spatiotemporal coefficients are used to obtain different forms of each additive term in the generalized additive model expression. The four types of non-stationary features based on regression relationships are spatial non-stationary features, temporal non-stationary features, and spatiotemporal non-stationary features.

[0027] Different forms of time-space varying coefficients are:

[0028]

[0029] The different forms of each additive term in the generalized additive model expression are:

[0030]

[0031] Among them, g k is the smoothing function of the generalized additive model; α 0k X k is a fixed effect term; s(u,v)·X k ,s(t)·X k and te(u,v,t)·X k All are smoothing effect terms; α 0k is a constant that does not change with space and time; ω k and are two constants with values ​​of 0 and 1.

[0032] Preferably, statistical inference is performed on the non-stationary characteristics of each regression relationship in the spatiotemporal variable coefficient model based on the different forms of each additive term. The specific method is:

[0033] Split the generalized additive model to obtain the generalized additive model expression including the mth additive term;

[0034] Set test problem 1 to test the non-stationarity of the regression relationship over time:

[0035]

[0036] Based on test question 1, define the expression of the mth additive term, fit the generalized additive model containing the mth additive term, and obtain the fitting result;

[0037] When the null hypothesis of test question 1 is established, the spatiotemporal interaction smoothing effect term in the mth additive term in the generalized additive model is zero; if the spatiotemporal interaction smoothing effect term in the mth additive term in the fitting result is significantly non-zero, then the mth regression relationship in the spatiotemporal variable coefficient model is time non-stationary;

[0038] Set test problem 2 to test the non-stationarity of the regression relationship in space:

[0039] H 02 :ωm =0

[0040] H 12 :ω m =1

[0041] Based on test question 2, define the expression of the mth additive term, fit the generalized additive model containing the mth additive term, and obtain the fitting result;

[0042] When the null hypothesis of test question 2 is established, the spatiotemporal interaction smoothing effect term in the mth additive term in the generalized additive model is zero; if the spatiotemporal interaction smoothing effect term in the mth additive term in the fitting result is significantly non-zero, then the mth regression relationship in the spatiotemporal variable coefficient model is spatially non-stationary.

[0043] In a second aspect, the present invention provides a statistical inference system for non-stationary characteristics of regression relationships in a spatiotemporal variable coefficient model, comprising:

[0044] A model vector form acquisition unit, used to acquire the vector form of the spatiotemporal variable coefficient model;

[0045] A generalized additive model acquisition unit is used to set each variable coefficient term in the vector form of the spatiotemporal variable coefficient model as a smooth interaction term between the explanatory variable of the generalized additive model and the spatiotemporal coordinates, thereby obtaining a generalized additive model expression of the spatiotemporal variable coefficient model;

[0046] An additive term form acquisition unit is used to obtain different forms of each additive term in the generalized additive model expression based on the four non-stationary feature types of the regression relationship;

[0047] The non-stationary feature statistical unit is used to perform statistical inference on the non-stationary features of each regression relationship in the spatiotemporal varying coefficient model based on the different forms of each additive term.

[0048] In a third aspect, the present invention provides an electronic device, characterized in that it includes a processor and a memory, wherein computer instructions are stored in the memory, and when the computer instructions are executed by the processor, the electronic device executes the method described in any one of claims 1 to 5.

[0049] In a fourth aspect, the present invention provides a computing device cluster, comprising at least one computing device, each computing device comprising a processor and a memory;

[0050] The processor of the at least one computing device is configured to execute instructions stored in the memory of the at least one computing device, so that the computing device cluster performs the method.

[0051] In a fifth aspect, the present invention provides a computer program product, which includes computer-executable instructions, and the computer-executable instructions implement the method when executed.

[0052] In a sixth aspect, the present invention provides a computer-readable storage medium, wherein the computer-readable storage medium stores computer-executable instructions, and the computer-executable instructions implement the described method when executed by a processor.

[0053] Compared with the prior art, the present invention has the following beneficial effects:

[0054] The statistical inference method for the non-stationary characteristics of the regression relationship of the spatiotemporal variable coefficient model provided by the present invention is a method for identifying the regression relationship of different non-stationary characteristics of the spatiotemporal variable coefficient model based on the generalized additive model of the Gaussian process. The generalized additive model is a modeling method that can fit nonlinear regression relationships. The model allows independent fitting of the influence of different explanatory variables on the response variable, and thus can reduce the effect of multicollinearity between explanatory variables to a certain extent. The Gaussian process can capture the spatiotemporal dependency structure of spatiotemporal data through the covariance function, and the sum function of the Gaussian process can also be adjusted according to the scale of the data to capture the local or global correlation of the data. It is a non-parametric Bayesian method that can provide flexible and accurate estimates for nonlinear problems. The Gaussian process and the generalized additive model are combined to form a Gaussian process generalized additive model, which can scale-adaptively process the basic characteristics of spatiotemporal data and realize accurate identification of the regression relationship characteristics of the spatiotemporal variable coefficient model. BRIEF DESCRIPTION OF THE DRAWINGS

[0055] Figure 1 It is the power function curve of the four-group coefficient test when the significance α=0.05 and the collinearity level ρ(X2,X3)=0.0;

[0056] Figure 2 It is the power function curve of the four-group coefficient test when the significance α=0.05 and the collinearity level ρ(X2,X3)=0.5;

[0057] Figure 3 It is the power function curve of the four-group coefficient test when the significance α=0.05 and the collinearity level ρ(X2,X3)=0.9;

[0058] Figure 4 These are the true and estimated surfaces of the power function curves for the four sets of coefficient tests when the significance α is 0.05 and the collinearity level ρ(X2,X3) is 0.99. DETAILED DESCRIPTION

[0059] In the following description, specific details such as specific system structures and techniques are provided for purposes of illustration rather than limitation to facilitate a thorough understanding of the embodiments of the present application. However, it will be apparent to those skilled in the art that the present application may be implemented in other embodiments without these specific details. In other cases, detailed descriptions of well-known systems, devices, circuits, and methods are omitted to avoid obscuring the description of the present application with unnecessary detail.

[0060] Example 1

[0061] The statistical inference method for the non-stationary characteristics of the regression relationship of the spatiotemporal variable coefficient model provided in this embodiment includes the following steps:

[0062] Express the sample expression of the spatiotemporal variable coefficient model in vector form;

[0063] Each variable coefficient term in the vector expression of the time-space variable coefficient model is set as a smooth interaction term between the explanatory variable and the time-space coordinate, and the generalized additive model expression of the time-space variable coefficient model is obtained, and the time-space variable coefficient is represented by a smooth function;

[0064] Based on the four types of non-stationary characteristics of the regression relationship, the spatiotemporal coefficients are expressed as specific smooth function forms to obtain different forms of each additive term in the generalized additive model expression;

[0065] For the four types of non-stationary features, two hypothesis testing problems are designed to realize the statistical inference of the non-stationary features of each regression relationship in the spatiotemporal varying coefficient model one by one.

[0066] Example 2

[0067] As an effective spatiotemporal data regression analysis method that can simultaneously handle spatial and temporal non-stationary characteristics, the spatiotemporal variable coefficient model has received widespread attention and application in many fields such as statistics, ecology, environmental science, epidemiology, and econometrics. Spatiotemporal non-stationarity and multi-scale are two important basic characteristics of spatiotemporal data. Accurately identifying these two characteristics is crucial for a deep understanding and reasonable interpretation of the fitting results of the spatiotemporal variable coefficient model. Currently, the more commonly used statistical inference method for the spatiotemporal non-stationary characteristics of regression relationships is the bootstrap technique. One of the advantages of this technique is that the stability of the inference results does not depend on the probability distribution type of the model error term. However, since the bootstrap test method achieves statistical inference by resampling samples to simulate the probability distribution of the null model, this test method has the limitation of high computational cost, especially when considering the operating scale information of the regression relationship at the same time. To this end, this application constructs a smoothing function of the generalized additive model by establishing parameterized Gaussian process splines and cubic regression splines in geographic location and time, respectively, and proposes an efficient fitting method for the spatiotemporal variable coefficient model and a statistical inference method for globally stationary, temporally non-stationary, and spatially non-stationary regression relationships. Simulation results demonstrate that the proposed method not only accurately identifies different non-stationary characteristics in regression relationships but also provides more accurate estimates of model coefficients. Compared to the bootstrap test, the proposed method offers lower computational cost and better test power, and is more robust to the probability distribution of the model error term and multicollinearity among explanatory variables.

[0068] The statistical inference method for the non-stationary characteristics of the regression relationship of the spatiotemporal variable coefficient model provided in this embodiment includes the following steps:

[0069] Step 1: Obtain the spatiotemporal variable coefficient model, whose sample expression is:

[0070]

[0071] Among them, (y ij ,x ij1 ,x ij2 ,…,x ijp ) is the response variable Y and the explanatory variables X1, X2,…, X p At the time and space sampling point (u i ,v i ,t j ) observations at the location; the model has n spatial locations and T time points, which are respectively denoted as and β k (u i ,v i ,t j )(k=1,2,…,p) is the model in (u i,v i ,t j ) at the estimated regression coefficient; model error ε ij (i=1,2,…,n;j=1,2,…,T) are independent and identically distributed random variables with mean zero. In general, let X1≡1 so that the model includes an intercept term.

[0072] The vector form of the time-space varying coefficient model corresponding to the sample expression (1) is:

[0073]

[0074] in,

[0075] β k (u,v,t)·X k =(β k (u1,v1,t1)x 11k ,…,β k (u n ,v n ,t1)x n1k ,…,β k (u1,v1,t T )x 1Tk ,…,β k (u n ,v n ,t T )x nTk ) T ;ε is the model error vector.

[0076] Step 2: Each variable coefficient term β in the spatiotemporal variable coefficient model (2) k (u,v,t)·X k (k=1,2,…,p) is defined as the explanatory variable X k The smooth interaction term with the space-time coordinates (u, v, t), the space-time variable coefficient model can be expressed as the following generalized additive model form:

[0077]

[0078] Among them, f k (u,v,t,X k )=β k (u,v,t)·X k , the additive term f k (u,v,t,X k ) are u, v, t and explanatory variables X k function.

[0079] Time-space variation coefficient β k(u, v, t) is a function of the space-time coordinates. Its form reflects the space-time non-stationary characteristics of the corresponding regression relationship, which is the corresponding space-time variable coefficient β k Different forms of (u, v, t) can be obtained by the smoothing function g of the generalized additive model k (·) β k (u,v,t) is expressed as follows:

[0080]

[0081] Among them, g k is the smooth function of the generalized additive model; ω k and are two constants with values ​​of 0 and 1.

[0082] Then the corresponding additive term can be expressed as:

[0083]

[0084] Step 3, the variable coefficient β in the spatiotemporal variable coefficient model (3) k Different forms of (u, v, t) correspond to regression relationships with different non-stationary characteristics, including global stationary characteristics, spatial non-stationary characteristics, temporal non-stationary characteristics, and spatiotemporal non-stationary characteristics. To this end, the parameter ω in formula (4) can be used to k and The value of β changes k The different forms of (u, v, t) represent the non-stationary characteristics of the regression relationship. The specific forms can be divided into:

[0085] (i) Global stationary characteristics: ω k =0, That is β k (u,v,t)=g k ;

[0086] (ii) Spatial non-stationary characteristics: ω k =1, That is β k (u,v,t)=g k (u,v);

[0087] (iii) Temporal non-stationary characteristics: ω k =0, That is β k (u,v,t)=g k (t);

[0088] (iv) Spatiotemporal non-stationary characteristics: ω k =1, That is β k (u,v,t)=g k(u,v,t).

[0089] For the above four characteristics, in the generalized additive model (3), the smoothing function g k The four forms of (·) define the time-space varying coefficient β k The four specific forms of (u,v,t) are:

[0090]

[0091] Among them, α 0k is a constant that does not change with space and time; s(u,v) is a two-dimensional isotropic smooth function of the spatial coordinates (u,v) (assuming u and v have the same scale), which changes with the change of spatial position; s(t) is a smooth function of time t, which changes with time; te(u,v,t) is a smooth function of the tensor product of (u,v) and t (allowing (u,v) and t to have different scales), which changes with the change of space and time.

[0092] From formula (6), we can see that the additive term f k (u,v,t,X k ) are as follows:

[0093]

[0094] Among them, α 0k X k is a fixed effect term, s(u,v)·X k ,s(t)·X k and te(u,v,t)·X k is called the smoothing effect term.

[0095] Step 4, by identifying each additive term f in the generalized additive model (3) k (u,v,t,X k ) in different forms, to achieve the statistical inference of the non-stationary characteristic type of each regression relationship in the spatiotemporal variable coefficient model (2). This process is equivalent to the smooth function Chinese k and Tests of different values.

[0096] Here we take the statistical inference method of the mth regression relationship as an example. When m ranges from 1 to p, the statistical inference of all regression relationships in model (2) can be realized. The specific implementation method is as follows:

[0097] Model (3) is split into an expression containing the mth additive term:

[0098]

[0099] in,

[0100]

[0101] The statistical inference of the non-stationary characteristics of each regression relationship in the spatiotemporal variable coefficient model (2) is achieved through the following steps:

[0102] (1) To test the non-stationarity of the regression relationship over time, we propose test question 1:

[0103]

[0104] Among them, H 01 and H 11 They are used to test the null hypothesis and alternative hypothesis of question 1 respectively.

[0105] The mth additive term in the generalized additive model (8) is expressed as (other terms remain unchanged) and fit the model.

[0106] When the null hypothesis H 01 When it is established, the mth additive term can be obtained from formula (7) as follows:

[0107]

[0108] That is, when the null hypothesis of test question 1 is true, The spatiotemporal interaction smoothing effect term te(u,v,t)·X m Should be 0. Therefore, if the model fitting results show that the time-space interaction smoothing effect term te(u,v,t)·X m Significant, that is, te(u,v,t)·X m If it is significantly different from 0, the null hypothesis H in question 1 should be rejected. 01 , that is, the mth regression relationship of the spatiotemporal variable coefficient model (2) is considered to be time non-stationary.

[0109] (2) To test the spatial non-stationarity of the regression relationship, we propose test question 2:

[0110] H 02 :ω m =0;H 12 :ω m =1

[0111] Among them, H 02 and H 12 They are used to test the null hypothesis and alternative hypothesis of question 2 respectively.

[0112] The mth additive term in the generalized additive model (8) is expressed as (other terms remain unchanged) and fit the model. 02 When it is established, it can be seen from formula (7) that

[0113]

[0114] That is, if the null hypothesis of test question 2 is true, then the spatiotemporal interaction smoothing effect term te(u,v,t)·X m Should be 0. Therefore, if the model fitting results show that the time-space interaction smoothing effect term te(u,v,t)·X m If the value is significant, the null hypothesis H in question 2 should be rejected. 02 , that is, the mth regression relationship of the spatiotemporal variable coefficient model (2) is considered to be spatially non-stationary.

[0115] (3) m is changed from 1 to p in sequence, and the above tests (1) and (2) are performed respectively, and finally the statistical inference of the non-stationary characteristics of all regression relationships of the spatiotemporal variable coefficient model (2) is achieved.

[0116] In this application, statistical inference provides theoretical support for revealing the spatiotemporal non-stationary characteristics of the correlation between spatiotemporal data. Non-stationarity can reflect the temporal evolution or spatial differentiation of the underlying mechanism. The inference of non-stationary characteristics can optimize the accuracy of decision-making and help identify the boundary conditions that affect the relationship. For example, in environmental governance, the impact of air pollution on human health may vary depending on the season (time) or the level of urban development (space). Statistical inference can identify high-risk periods and regions for such impacts, thereby guiding differentiated emission reduction policies. In disease transmission, the effectiveness of prevention and control measures (such as blockades) may vary depending on regional population density or time stage (such as the early vs. late stages of the epidemic), requiring dynamic adjustment of strategies. In agricultural production practice, the impact of fertilization on crop yields may vary depending on soil type (space) or growth stage (time). Statistical inference can provide methodological support for precise agricultural assistance. In urban traffic planning, the relationship between road congestion and economic activities may vary depending on urban functional areas (space) or time periods (morning and evening peaks), requiring differentiated planning.

[0117] In addition, statistical inference can accurately capture the temporal and spatial variation patterns of coefficients, enabling spatiotemporal variable coefficient models to more accurately predict future states or spatial extrapolation.

[0118] Example 3

[0119] This embodiment designs four sets of simulation experimental cases to verify the validity and accuracy of the statistical inference method and model coefficient estimation proposed in this application. At the same time, this application also designs four different levels of multicollinearity between explanatory variables to verify the stability of the method proposed in this application.

[0120] In the three-dimensional Cartesian coordinate system, the space-time region is designed as a unit cube [0,1]×[0,1]×[0,1]. In the horizontal and vertical directions of [0,1]×[0,1], there are n sampling points Each spatial location (u i ,v i ). In the vertical direction of [0,1], set T equally spaced time points t j =(j-1) / (T-1). Therefore, there are a total of nT spatiotemporal sampling points (u i ,v i ,t j )(i=1,2,…,n;j=1,2,…,T). In the simulation experiment, the present application takes n=300, T=8, and the total sample size is nT=2400.

[0121] The following spatiotemporal coefficient model containing three explanatory variables X1, X2 and X3 is established:

[0122] y ij =β1(u i ,v i ,t j )x ij1 +β2(u i ,v i ,t j )x ij2 +β3(u i ,v i ,t j )x ij3 +ε ij ,

[0123] i=1,2,…,n; j=1,2,…,T.(11)

[0124] In the simulation, the explanatory variable X1≡1, so that the model contains an intercept term. 30 The observations of are drawn independently from the normal distribution N(0,1) and the uniform distribution U(-1,1). In order to verify the robustness of the proposed method to multicollinearity between explanatory variables, the explanatory variables are set Then ρ(X2,X3)=γ, and γ=0.1, 0.5, 0.9 and 0.99 respectively. When γ=0.99, it indicates that there is extreme collinearity between the explanatory variables. In order to prove the robustness of the proposed statistical inference method to the model error distribution, this application selects the error ε ij There are two types of probability distributions: standard normal distribution N(0,1) and uniform distribution

[0125] Design the following five function types:

[0126]

[0127] The above five groups of functions are used to construct the following four groups of coefficient types:

[0128] G1:

[0129] G2:

[0130] G3:

[0131] G4:

[0132] Where c is a constant. Different values ​​of c correspond to different types of coefficients β3(u, v, t). The regression relationships corresponding to these coefficients have different non-stationary characteristics. The specific situations are as follows:

[0133] G1: When c = 0, the regression relationship corresponding to β3(u, v, t) is globally stable; when c ≠ 0, the regression relationship corresponding to β3(u, v, t) is spatiotemporally non-stationary, and the larger the absolute value of c, the more obvious the spatiotemporal non-stationary characteristics.

[0134] G2: When c = 0, the regression relationship corresponding to β3(u, v, t) is time-nonstationary; when c ≠ 0, the regression relationship corresponding to β3(u, v, t) is space-time nonstationary, and the larger the absolute value of c, the more obvious the space-time nonstationary feature.

[0135] G3: When c = 0, the regression relationship corresponding to β3(u, v, t) is spatially non-stationary; when c ≠ 0, the regression relationship corresponding to β3(u, v, t) is spatiotemporally non-stationary, and the larger the absolute value of c, the more obvious the spatiotemporal non-stationary characteristics.

[0136] G4: When c = 0, the regression relationship corresponding to β2(u, v, t) is time-nonstationary, and the regression relationship corresponding to β3(u, v, t) is space-nonstationary. When c ≠ 0, the regression relationships corresponding to β2(u, v, t) and β3(u, v, t) are both time-space nonstationary, and the larger the absolute value of c, the more obvious the time-space nonstationary characteristics.

[0137] Draw observations and errors of explanatory variables from the specified distribution (1, x ij2 ,x ij3 ,ε ij ), corresponding to each set of coefficients, the observed value y of the response variable can be obtained through model (11) ij. According to the rate of change of the non-stationary characteristics of the coefficients, the c values ​​are set at equal intervals and are symmetric about 0. In order to evaluate the probability of the proposed test method making a type I error and the power of the test, the simulation experiment is repeated 200 times for each c value, and the rejection rate under a given significance level α is calculated. In order to comprehensively evaluate the effectiveness of the statistical inference method, the rejection rates under three commonly used significance levels α = 0.01, 0.05, and 0.1 are calculated in the simulation.

[0138] When the regression relationship corresponding to the coefficients is globally stationary (e.g., G1: c = 0), spatially stationary (e.g., G2: c = 0 or G4(β2): c = 0), and temporally stationary (e.g., G3: c = 0 or G4(β3): c = 0), the rejection rate is an estimate of the probability of making a Type I error and should be close to the significance level α. When the regression relationship corresponding to the coefficients is nonstationary, that is, when the value of c in the four groups is nonzero, the rejection rate is the power of the test, and in this case the rejection rate should increase rapidly with the absolute value of c. If the nonstationary nature of the test is independent of the value of c, then the rejection rate should be equal to 1 regardless of the value of c, as in the inferences about test question 1 for groups G2 and G4, and about test question 2 for groups G3 and G4.

[0139] Tables 1 to 4 list the rejection rates for four simulations under two model error distributions with four different collinearity levels (c = 0). The results show that when the regression relationships tested exhibit global stationary characteristics (e.g., β3 for group G1), temporal nonstationary characteristics (e.g., β3 for group G2 and β2 for group G4), and spatial nonstationary characteristics (β3 for groups G3 and G4), the rejection rates in all four experimental settings are close to their respective significance levels α, demonstrating that the proposed test method can generate valid Type I errors and is robust to multicollinearity among the explanatory variables. In particular, the rejection rates under normal and nonnormal error distributions are comparable, indicating that the proposed statistical inference method is robust to model error distributions. When the nonstationary characteristics tested are independent of the value of c, the rejection rates in all four experimental settings are equal to 1, except for the extreme collinearity level (ρ(X2,X3) = 0.99). When ρ(X2,X3)=0.99, the rejection rates for test question 2 in groups G3 and G4 are lower than those at a significance level of α=0.01. When the significance level α≥0.05, the rejection rates for test question 1 in groups G2 and G4 are both equal to or close to 1. Of course, in practical problems, this extreme collinearity situation is generally rare.

[0140] Table 1 Rejection rate of each group when ρ(X2,X3)=0, c=0

[0141]

[0142] Table 2 Rejection rate of each group when ρ(X2,X3)=0.5,c=0

[0143]

[0144]

[0145] Table 3 Rejection rate of each group when ρ(X2,X3)=0.9,C=0

[0146]

[0147]

[0148] Table 4 Rejection rate of each group when ρ(X2,X3)=0.99,C=0

[0149]

[0150] Figures 1 to 4 The rejection rates for different c values ​​for four sets of simulated coefficients and two error distributions at a significance level of α = 0.05 are described. The results show that the statistical inference method proposed in this application is very effective in identifying special types of regression relationships in spatiotemporal variable coefficient models. In addition, there is no significant difference in the test power under the two error distributions, indicating that the test power is insensitive to changes in the model error distribution. In addition, the comparison Figure 1 and Figure 2 As can be seen, there is no significant difference in the power of the tests under moderate multicollinearity. Figure 3 and Figure 4 It can be seen that as the level of collinearity increases (from moderate to severe and extreme), the power of the test gradually decreases. Figure 1-Figure 4 From the results in Tables 1 to 4, it can be concluded that the test method proposed in this application is highly robust to both the model error distribution and the multicollinearity between explanatory variables.

[0151] Example 5

[0152] The statistical inference system for non-stationary characteristics of regression relationships in a spatiotemporal variable coefficient model provided in this embodiment is characterized by including:

[0153] A model vector form acquisition unit, used to acquire the vector form of the spatiotemporal variable coefficient model;

[0154] A generalized additive model acquisition unit is used to set each variable coefficient term in the vector form of the time-space variable coefficient model as a smooth interaction term between the explanatory variable and the time-space coordinate in the time-space variable coefficient model, thereby obtaining a generalized additive model expression of the time-space variable coefficient model;

[0155] An additive term form acquisition unit is used to obtain different forms of each additive term in the generalized additive model expression based on the four non-stationary feature types of the regression relationship;

[0156] The non-stationary feature statistical unit is used to perform statistical inference on the non-stationary features of each regression relationship in the spatiotemporal varying coefficient model based on the different forms of each additive term.

[0157] Example 6

[0158] This embodiment also provides a computing device. The computing device includes a bus, a processor, a memory, and a communication interface. The processor, the memory, and the communication interface communicate with each other via the bus. The computing device can be a server or a terminal device. It should be understood that this application does not limit the number of processors and memories in the computing device.

[0159] A bus can be a Peripheral Component Interconnect (PCI) bus or an Extended Industry Standard Architecture (EISA) bus. Buses can be categorized as address buses, data buses, control buses, and so on. For ease of presentation, a bus can include the pathways that transmit information between various components of a computing device (e.g., memory, processor, and communication interfaces).

[0160] The processor may include any one or more of a central processing unit (CPU), a graphics processing unit (GPU), a tensor processing unit (TPU), an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a microprocessor (MP) or a digital signal processor (DSP).

[0161] The memory may include volatile memory, such as random access memory (RAM). The processor may also include non-volatile memory, such as read-only memory (ROM), flash memory, hard disk drive (HDD), or solid state drive (SSD).

[0162] The memory stores executable program code, and the processor executes the executable program code to implement the functions of the aforementioned units, thereby implementing, for example, the method described in Example 1. That is, the memory may store instructions for the methods and functions of the computing device described in any of the above embodiments.

[0163] The communication interface uses a transceiver module such as, but not limited to, a network interface card or a transceiver to implement communication between the computing device and other devices or a communication network.

[0164] Example 7

[0165] This embodiment also provides a computing device cluster. The computing device cluster includes at least one computing device. The computing device can be a server, such as a central server, an edge server, or a local server in a local data center. In some embodiments, the computing device can also be a terminal device such as a desktop computer, a laptop computer, or a smartphone.

[0166] The computing device cluster includes at least one computing device. The memory of one or more computing devices in the computing device cluster may store the same instructions for executing the method and functions related to the computing device in any of the above embodiments.

[0167] In some possible implementations, the memory of one or more computing devices in the computing device cluster may also store partial instructions for executing the methods and functions related to the computing devices in any of the above embodiments. In other words, the combination of one or more computing devices can jointly execute instructions for executing the methods and functions of the computing devices.

[0168] It should be noted that the memories in different computing devices in the computing device cluster may store different instructions, each for executing part of the functions of the apparatus.

[0169] In some possible implementations, one or more computing devices in a computing device cluster may be connected via a network. The network may be a wide area network (WAN) or a local area network (LAN). Two computing devices are connected via the network. Specifically, the connection to the network is achieved via a communication interface in each computing device.

[0170] An embodiment of the present disclosure further provides a computer program product comprising instructions, which, when executed on a computer, enables the computer to execute the method and functions involving a computing device in any of the above embodiments.

[0171] Example 8

[0172] This embodiment further provides a computer-readable storage medium having computer instructions stored thereon. When a processor executes the instructions, the processor executes the methods and functions related to the computing device in any of the above embodiments.

[0173] In general, various embodiments of the present disclosure may be implemented in hardware or dedicated circuitry, software, logic, or any combination thereof. Some aspects may be implemented in hardware, while other aspects may be implemented in firmware or software, which may be executed by a controller, microprocessor, or other computing device. Although various aspects of the embodiments of the present disclosure are shown and described as block diagrams, flow charts, or using some other pictorial representation, it should be understood that the blocks, devices, systems, techniques, or methods described herein may be implemented as, by way of non-limiting example, hardware, software, firmware, dedicated circuitry or logic, general-purpose hardware or a controller or other computing device, or some combination thereof.

[0174] Example 9

[0175] The present embodiment provides at least one computer program product tangibly stored on a non-transitory computer-readable storage medium. The computer program product includes computer-executable instructions, such as instructions included in program modules, which are executed in a device on a real or virtual processor of a target to perform the process / method described above with reference to the accompanying drawings. Generally, program modules include routines, programs, libraries, objects, classes, components, data structures, etc. that perform specific tasks or implement specific abstract data types. In various embodiments, the functionality of program modules can be combined or divided between program modules as needed. The machine-executable instructions for the program modules can be executed in local or distributed devices. In distributed devices, program modules can be located in local and remote storage media.

[0176] The computer program code for implementing the disclosed method can be written in one or more programming languages. These computer program codes can be provided to the processor of a general-purpose computer, a special-purpose computer or other programmable data processing device so that the program code, when executed by the computer or other programmable data processing device, causes the functions / operations specified in the flow chart and / or block diagram to be implemented. The program code can be executed entirely on a computer, partially on a computer, as an independent software package, partially on a computer and partially on a remote computer or entirely on a remote computer or server.

[0177] In the context of the present disclosure, computer program code or related data may be carried by any suitable carrier to enable a device, apparatus, or processor to perform the various processes and operations described above. Examples of carriers include signals, computer-readable media, and the like. Examples of signals may include electrical, optical, radio, acoustic, or other forms of propagated signals, such as carrier waves, infrared signals, and the like.

[0178] A computer-readable medium may be any tangible medium containing or storing a program for or relating to an instruction execution system, apparatus, or device, or a data storage device such as a data center containing one or more available media. A computer-readable medium may be a computer-readable signal medium or a computer-readable storage medium. A computer-readable medium may include, but is not limited to, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus, or device, or any suitable combination thereof. More detailed examples of computer-readable storage media include an electrical connection with one or more wires, a portable computer disk, a hard disk, a random access memory (RAM), a read-only memory (ROM), an erasable programmable read-only memory (EPROM or flash memory), an optical storage device, a magnetic storage device, or any suitable combination thereof.

[0179] The above-described embodiments are only used to illustrate the technical solutions of the present application, rather than to limit them. Although the present application has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. These modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the various embodiments of the present application, and should all be included in the scope of protection of the present application.

Claims

1. A statistical inference method for the non-stationary characteristics of the regression relationship of the spatiotemporal variable coefficient model, characterized by: The following steps are involved: Get the vector form of the spatiotemporal variable coefficient model; By setting each variable coefficient term in the vector form of the spatiotemporal variable coefficient model as the smooth interaction term between the explanatory variable and the spatiotemporal coordinate in the spatiotemporal variable coefficient model, the generalized additive model expression of the spatiotemporal variable coefficient model is obtained; Based on the four types of non-stationary characteristics of the regression relationship, different forms of each additive term in the generalized additive model expression are obtained; Statistical inference is performed on the non-stationary characteristics of each regression relationship in the spatiotemporal varying coefficient model based on the different forms of each additive term.

2. The statistical inference method for non-stationary characteristics of regression relationship of spatiotemporal variable coefficient model according to claim 1 is characterized in that: The vector form of the spatiotemporal variable coefficient model is: Among them, Y is the response variable; β k (u,v,t)·X k = (b k (u1,v1,t1)x 11k ,…,b k (u n ,v n ,t1)x n1k ,…,b k (u1,v1,t T )x 1Tk ,…,b k (u n ,v n ,t T )x nTk ) T ; x ij1 ,x ij2 ,…,x ijp (i=1,2,…,n;j=1,2,…T) are the explanatory variables X1,X2,…,X p At the time and space sampling point (u i ,v i ,t j ) observation value; β k (u i ,v i ,t j )(k=1,2,…,p) is the model in (u i ,v i ,t j ) is the estimated regression coefficient at ; n is the number of spatial locations in the model; T is the number of time points in the model; ε is the model error vector.

3. The statistical inference method for non-stationary characteristics of regression relationship of spatiotemporal variable coefficient model according to claim 1 is characterized in that: The generalized additive model expression of the spatiotemporal variable coefficient model is: Among them, Y is the response variable; f k (u,v,t,X k )=β k (u,v,t)·X k represents the kth additive term of the generalized additive model, β k (u,v,t)·X k = (b k (u1,v1,t1)x 11k ,…,b k (u n ,v n ,t1)x n1k ,…,b k (u1,v1,t T )x 1Tk ,…,b k (u n ,v n ,t T )x nTk ) T ; x ij1 ,x ij2 ,…,x ijp (i=1,2,…,n;j=1,2,…T) are the explanatory variables X1,X2,…,X p At the time and space sampling point (u i ,v i ,t j ) observation value; β k (u i ,v i ,t j )(k=1,2,…,p) is the model in (u i ,v i ,t j ) is the estimated regression coefficient at ; n is the number of spatial locations in the model; T is the number of time points in the model; ε is the model error vector.

4. The statistical inference method for non-stationary characteristics of regression relationship of spatiotemporal variable coefficient model according to claim 1, characterized in that: Based on the four types of non-stationary features of the regression relationship, different forms of each additive term in the generalized additive model expression are obtained. The specific method is: Representing spatiotemporally varying coefficients through smooth functions of the generalized additive model; The four types of non-stationary features based on regression relationships are used to obtain different forms of spatiotemporal coefficients; The four representations of spatiotemporal coefficients are used to obtain different forms of each additive term in the generalized additive model expression. The four types of non-stationary features based on regression relationships are spatial non-stationary features, temporal non-stationary features, and spatiotemporal non-stationary features. Different forms of time-space varying coefficients are: The different forms of each additive term in the generalized additive model expression are: Among them, g k is the smoothing function of the generalized additive model; α 0k X k is a fixed effect term; s(u,v)·X k ,s(t)·X k and te(u,v,t)·X k All are smoothing effect terms; α 0k is a constant that does not change with space and time; ω k and are two constants with values ​​of 0 and 1.

5. The statistical inference method for non-stationary characteristics of regression relationship of spatiotemporal variable coefficient model according to claim 1, characterized in that: Based on the different forms of each additive term, the non-stationary characteristics of each regression relationship in the spatiotemporal variable coefficient model are statistically inferred. The specific method is: Split the generalized additive model to obtain the generalized additive model expression including the mth additive term; Set test problem 1 to test the non-stationarity of the regression relationship over time: Based on test question 1, define the expression of the mth additive term, fit the generalized additive model containing the mth additive term, and obtain the fitting result; When the null hypothesis of test question 1 is established, the spatiotemporal interaction smoothing effect term in the mth additive term in the generalized additive model is zero; if the spatiotemporal interaction smoothing effect term in the mth additive term in the fitting result is significantly non-zero, it is inferred that the mth regression relationship in the spatiotemporal variable coefficient model is time non-stationary; Set test problem 2 to test the non-stationarity of the regression relationship in space: H 02 :oh m =0 H 12 :oh m =1 Based on test question 2, define the expression of the mth additive term, fit the generalized additive model containing the mth additive term, and obtain the fitting result; When the null hypothesis of test question 2 is established, the spatiotemporal interaction smoothing effect term in the mth additive term in the generalized additive model is zero; if the spatiotemporal interaction smoothing effect term in the mth additive term in the fitting result is significantly non-zero, it is inferred that the mth regression relationship in the spatiotemporal variable coefficient model is spatially non-stationary.

6. A statistical inference system for the non-stationary characteristics of the regression relationship of a spatiotemporal variable coefficient model, characterized by: include: A model vector form acquisition unit, used to acquire the vector form of the spatiotemporal variable coefficient model; A generalized additive model acquisition unit is used to set each variable coefficient term in the vector form of the spatiotemporal variable coefficient model as a smooth interaction term between the explanatory variable and the spatiotemporal coordinate in the generalized additive model, thereby obtaining a generalized additive model expression of the spatiotemporal variable coefficient model; An additive term form acquisition unit is used to obtain different forms of each additive term in the generalized additive model expression based on the four non-stationary feature types of the regression relationship; The non-stationary feature statistical unit is used to perform statistical inference on the non-stationary features of each regression relationship in the spatiotemporal varying coefficient model based on the different forms of each additive term.

7. An electronic device, characterized in that: The electronic device comprises a processor and a memory, wherein computer instructions are stored in the memory. When the computer instructions are executed by the processor, the electronic device executes the method according to any one of claims 1 to 5.

8. A computing device cluster, characterized in that: comprising at least one computing device, each computing device including a processor and a memory; The processor of the at least one computing device is configured to execute instructions stored in a memory of the at least one computing device, so that the computing device cluster performs the method according to any one of claims 1 to 5.

9. A computer program product, characterized in that The computer program product contains computer-executable instructions, which implement the method according to any one of claims 1 to 5 when executed.

10. A computer-readable storage medium, characterized in that The computer-readable storage medium stores computer-executable instructions, which implement the method according to any one of claims 1 to 5 when executed by a processor.

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