Parameter acquisition method and system of hybrid spatiotemporal weighted regression model based on scale adaptation

Abstract

The scale-adaptive hybrid spatiotemporal weighted regression model parameter acquisition method and system provided by the present invention include the following steps: step 1, respectively obtaining initial estimated values ​​of constant coefficients and spatiotemporal variable coefficients in the hybrid spatiotemporal weighted regression model; step 2, iteratively updating the spatiotemporal variable coefficients according to the initial estimated values ​​of the constant coefficients and the spatiotemporal variable coefficients obtained in step 1, to obtain final estimated values ​​of the spatiotemporal variable coefficients; step 3, updating the constant coefficients according to the final spatiotemporal variable coefficients obtained in step 2, to obtain final estimated values ​​of the constant coefficients; the multi-scale problem of the spatiotemporal weighted regression method is solved, and the constant coefficients and the spatiotemporal variable coefficients are identified at the same time, and the computational burden is reduced compared with the MGTWR proposed by Wu et al., providing a more flexible perspective for analyzing data containing spatiotemporal information.

Description

Technical Field

[0001] The present invention belongs to the field of spatiotemporal information data analysis, and specifically relates to a method and system for acquiring parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation. Background Art

[0002] In recent years, with the development of computers, more and more data containing spatiotemporal information have been captured for analysis and prediction. For such data containing geographic information and time information, local regression methods are the main means to achieve effective analysis. However, with the surge in data volume, how to construct an efficient local regression method is particularly important.

[0003] In the field of local regression methods, the most widely used one is Geographically Weighted Regression (GWR) and its extended model. In recent years, it has been widely used in spatiotemporal information data analysis in various fields. It has been widely valued by scholars at home and abroad and has achieved fruitful research results.

[0004] The original source of the GWR model is based on the first law of geography (everything is interconnected, but things that are close are more closely connected than things that are far away). It is assumed that the regression coefficients vary with the spatial location, and each variable affects the dependent variable at a uniform spatial scale. As people become more and more interested in the topic of how geographic information data changes over time. For example, in 2010, the time factor was first included in the coefficient function of GWR, and the time-space weighted regression (GTWR) model was proposed to obtain the time-space heterogeneity of the regression relationship. The GWR that has been proposed and its various forms of extension can be applied not only to traditional time-space fields such as agriculture, meteorology and environment, but also to emerging time-space fields such as real estate, economy and disease.

[0005] The GWR and its extended forms that have been proposed currently include: geographically weighted logistic regression, geographically weighted Poisson regression, geographically weighted regression based on traffic accessibility, geographically weighted ridge regression, geographically weighted LASSO regression, and space-time weighted regression.

[0006] Defects and shortcomings of the existing technology:

[0007] Although many GWRs and their extended forms have been proposed, in fact, in practice, there may be situations where the effects of some independent variables on the dependent variable are globally stable, while the effects of other independent variables are spatially non-stationary. To this end, Brunsdon et al. proposed a semi-parametric spatially variable coefficient method, namely the hybrid GWR model, in 1999. In this method, some coefficients are assumed to be constants, while other coefficients vary in space. The parameter estimation of the model at two spatial scales (the scale in the spatial dimension can be divided into two types, namely the local scale and the global scale) is achieved through a two-step correction method. However, since the hybrid GWR uses a uniform spatial window width value when estimating the spatially variable coefficient, this method cannot detect the spatial heterogeneity of the regression relationship at different scales. In order to solve the scale effect of the regression relationship, Fotheringham et al. proposed a multi-scale GWR (MGWR) model in 2017. The model considers that each explanatory variable may affect the dependent variable at different spatial scales, thereby improving the accuracy of the model estimation and making it more explanatory. The MGWR method has more advantages than the traditional GWR method. In 2020, Chen and Mei proposed another multi-scale GWR method, namely the scale-adaptive GWR method (SAGWR). This method divides the coefficients into spatially stationary constant coefficients and spatially non-stationary variable coefficients, and then performs post-term fitting on the variable coefficients and selects the spatial window width of each variable coefficient. This method not only combines the interpretability of the MGWR model, but also reduces the iterative complexity of the model, thereby improving computational efficiency.

[0008] The research on spatiotemporal weighted regression methods is mainly: the GTWR method proposed by Huang et al. in 2010 and another GTWR method proposed by Fotheringham et al. in 2015 are the most well-known. Among them, the GTWR method proposed by Huang et al. in 2010 is often used because it proposes a spatiotemporal kernel function that emphasizes the concept of spatiotemporal distance. It is easy to understand and implement, but the method uses all the data of the past, present and future, which is not easy to explain in the time dimension. Fotheringham et al. used another method, that is, to quantify the spatial relationship that changes over time by designing a new spatiotemporal weight matrix. This method separates the selection of the optimal spatial bandwidth from the selection of the temporal bandwidth to provide a better explanation in the time dimension. In 2017, Liu et al. proposed a hybrid GTWR, which divides the explanatory variables into spatiotemporal stationary variables and non-stationary variables; since the influence of the explanatory variables on the explained variables may be at different spatiotemporal scales, taking housing prices as an example, some variables are measured at the micro level, such as window orientation, green space landscape and floor height, and others are measured at the macro level, such as economic development level and population size. Therefore, identifying the spatiotemporal scales of different explanatory variables is crucial to construct more flexible spatiotemporal methods and thus obtain more accurate estimates. To this end, Wu et al. (2019) extended the GTWR method of Huang et al., established a multiscale GTWR (MGTWR) and explored the spatiotemporal changes in the impact of factors related to Shenzhen housing prices from 2010 to 2017. Although a multiscale framework has been established to provide different window widths for each independent variable, using the same spatiotemporal window width for a single variable at different times may not be optimal, because the spatiotemporal heterogeneity of the regression relationship may change significantly over time. At the same time, it may be more reasonable to consider only past and present data when modeling. Summary of the invention

[0009] The purpose of the present invention is to provide a method and system for obtaining parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation, which solves the multi-scale problem of the existing spatiotemporal weighted regression model.

[0010] In order to achieve the above object, the technical solution adopted by the present invention is:

[0011] The method for obtaining parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation provided by the present invention comprises the following steps:

[0012] Step 1, respectively obtain the initial estimated values ​​of the constant coefficients and the spatiotemporal variable coefficients in the mixed spatiotemporal weighted regression model;

[0013] Step 2, iteratively updating the time-space variable coefficient according to the initial estimated values ​​of the constant coefficient and the time-space variable coefficient obtained in step 1 to obtain the final estimated value of the time-space variable coefficient;

[0014] Step 3, update the constant coefficient according to the final spatiotemporal variable coefficient obtained in step 2 to obtain the final constant coefficient estimate.

[0015] 2. The method for acquiring parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation according to claim 1 is characterized in that, in step 1, a two-step estimation method is used to respectively obtain initial estimated values ​​of the constant coefficients and spatiotemporal variable coefficients of the hybrid spatiotemporal weighted regression model.

[0016] Preferably, a two-step estimation method is used to obtain initial estimates of the constant coefficients and the spatiotemporal-variant coefficients of the hybrid spatiotemporal weighted regression model, respectively, including the following steps:

[0017] S11, setting the expression of the mixed spatiotemporal weighted regression model;

[0018] S12, combining the expression of the hybrid space-time weighted regression model to obtain the function expression of the constant coefficient on the space-time window width; combining the expression of the hybrid space-time weighted regression model to obtain the function expression of the space-time variable coefficient on the space-time window width;

[0019] S13, using the corrected Akaike information criterion, combined with the expression of the hybrid spatiotemporal weighted regression model; combining the function expression of the constant coefficient on the spatiotemporal window width and the function expression of the spatiotemporal variable coefficient on the spatiotemporal window width, to calculate the optimal spatiotemporal window width;

[0020] S14, according to the optimal spatiotemporal window width combined with the function expression of the constant coefficient with respect to the spatiotemporal window width and the function expression of the spatiotemporal variable coefficient with respect to the spatiotemporal window width, an initial estimate of the constant coefficient and an initial estimate of the spatiotemporal variable coefficient are obtained.

[0021] Preferably, in step 2, the time-space varying coefficient of each explanatory variable is estimated based on the initial estimated values ​​of the constant coefficient and the time-space varying coefficient obtained in step 1, combined with the post-term fitting method, to obtain the final estimated value of the time-space varying coefficient.

[0022] Preferably, the spatiotemporal coefficient of each explanatory variable is estimated by combining the post-term fitting method to obtain the final spatiotemporal coefficient estimate. The specific method is:

[0023] S21, taking the obtained initial estimate of the constant coefficient as the initial estimate of the SAGTWR method, then constructing the GTWR model;

[0024] S22, the obtained initial estimate of the variable coefficient is used as the initial estimate of the SAGTWR method, and then the single variable GTWR model;

[0025] S23, based on the obtained single variable The GTWR model is used to update the time-space variable coefficients to obtain the functional relationship between the updated time-space variable coefficients and the time-space window width;

[0026] S24, using the corrected Akaike information criterion and combining the updated functional relationship of the spatiotemporal variable coefficient with the spatiotemporal window width to calculate the optimal spatiotemporal window width, obtain the corresponding optimal spatiotemporal window width, and then obtain the updated spatiotemporal variable coefficient estimate;

[0027] S25, repeatedly executing S22 to S24 until the iterative update of all spatiotemporal coefficient estimates corresponding to each explanatory variable is completed, and an updated spatiotemporal coefficient estimate corresponding to each explanatory variable is obtained;

[0028] S26, repeat S22 to S25 until the convergence criterion is met to obtain the final estimated value of the spatiotemporal coefficient.

[0029] The scale-adaptive hybrid spatiotemporal weighted regression model parameter acquisition system includes:

[0030] The coefficient initial estimation unit is used to obtain initial estimation values ​​of the constant coefficient and the time-space variable coefficient of the mixed time-space weighted regression model respectively;

[0031] A variable coefficient updating unit, used for iteratively updating the time-space variable coefficient according to the obtained constant coefficient and the initial estimated value of the time-space variable coefficient to obtain the final time-space variable coefficient;

[0032] The constant coefficient updating unit is used to update the constant coefficient according to the final spatiotemporal variable coefficient to obtain the final constant coefficient.

[0033] Preferably, the coefficient initial estimation unit comprises:

[0034] A model setting unit, used to set the expression of the mixed spatiotemporal weighted regression model;

[0035] A function construction unit, used to obtain a function expression of a constant coefficient on a spatiotemporal window width by combining an expression of a hybrid spatiotemporal weighted regression model; and to obtain a function expression of a spatiotemporal variable coefficient on a spatiotemporal window width by combining an expression of a hybrid spatiotemporal weighted regression model;

[0036] An optimal spatiotemporal window width calculation unit is used to calculate the optimal spatiotemporal window width by using the corrected Akaike information criterion and combining the function expression of the constant coefficient on the spatiotemporal window width and the function expression of the spatiotemporal variable coefficient on the spatiotemporal window width;

[0037] The initial estimation value calculation unit is used to obtain the initial estimation value of the constant coefficient and the initial estimation value of the space-time variable coefficient according to the optimal space-time window width combined with the function expression of the constant coefficient on the space-time window width and the function expression of the space-time variable coefficient on the space-time window width.

[0038] Preferably, it is characterized in that the variable coefficient updating unit comprises:

[0039] A constant coefficient model building unit is used to use the obtained initial estimate of the constant coefficient as the initial estimate of the SAGTWR method to build a GTWR model;

[0040] The variable coefficient model building unit is used to use the obtained variable coefficient initial estimate as the initial estimate of the SAGTWR method, and then build a single variable GTWR model;

[0041] Functional relationship building unit, used to obtain a single variable The GTWR model is used to update the time-space variable coefficients to obtain the functional relationship between the updated time-space variable coefficients and the time-space window width;

[0042] A variable coefficient estimation value acquisition unit is used to calculate the optimal spatiotemporal window width by using the corrected Akaike information criterion and combining the updated spatiotemporal variable coefficient with the functional relationship of the spatiotemporal window width to obtain the corresponding optimal spatiotemporal window width, and then obtain the updated spatiotemporal variable coefficient estimation value;

[0043] The iterative updating unit is used to complete the iterative update of all spatiotemporal coefficient estimates corresponding to each explanatory variable, and obtain the updated spatiotemporal coefficient estimates corresponding to each explanatory variable; iterative update is performed until the convergence criterion is met to obtain the final spatiotemporal coefficient estimates.

[0044] Compared with the prior art, the present invention has the following beneficial effects:

[0045] The present invention provides a method and system for obtaining parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation. It takes complex spatiotemporal phenomena as the target object, studies how different variables operate at different spatiotemporal scales, and uses the basic GWR as a reference method. It is expanded in the time dimension and multi-scale. It can not only be regarded as a multi-scale promotion of GTWR, but also a natural extension of SAGWR in the time dimension. Therefore, a scale-adaptive method is proposed to deal with multi-scale problems in spatiotemporal methods. It has the following advantages:

[0046] First: SAGTWR provides a more reasonable explanation for the relationship between variables than ordinary GTWR;

[0047] Second: SAGTWR is more friendly to constant coefficients and has higher accuracy than MGTWR proposed by Wu et al.;

[0048] Third: This method not only improves the estimation accuracy of spatiotemporal coefficients, especially the better fitting effect at spatiotemporal boundaries, but also provides information on the spatiotemporal scale and spatial non-stationarity level of each explanatory variable;

[0049] Fourth: In terms of computational efficiency, the SAGTWR method reduces the computational burden compared to the existing multi-scale GTWR method. It only needs to iteratively calculate the variable coefficient part, eliminating the process of fitting the post-term for the constant coefficient, and is more efficient.

[0050] Fifth: In the time dimension, only previous and current data are included, and there is no need to expand the size of the weight matrix over time as mentioned by Fotheringham et al. in 2015;

[0051] Sixth: This method is easy to understand and implement, and can be implemented in popular programming software, such as: R, MATLAB, Python, etc.;

[0052] Seventh: This method can be used as a basic method and expanded according to actual data. BRIEF DESCRIPTION OF THE DRAWINGS

[0053] Figure 1 It is a box plot of the optimal window width size of the three groups of coefficients in the simulation experiment repeated 100 times;

[0054] Figure 2 It is a box plot of the optimal spatial window width size of the three groups of coefficients in the simulation experiment repeated 100 times;

[0055] Figure 3 It is a box plot of the root mean square error (RMSE) of the three groups of estimated coefficients in the simulation experiment repeated 100 times. DETAILED DESCRIPTION

[0056] The present invention will be further described in detail below in conjunction with the accompanying drawings.

[0057] The purpose of the present invention is to overcome the defects of the prior art and provide a method for obtaining parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation, which reduces the calculation difficulty, reduces the calculation burden, and improves the correction accuracy of the model.

[0058] This method first uses a two-step estimation method to obtain the initial estimates of the constant coefficients and variable coefficients of the mixed spatiotemporal weighted regression model, and then uses the post-term fitting method to estimate the variable coefficients one by one, where the optimal window width of each variable coefficient is selected by the commonly used AICc criterion, and finally the estimated variable coefficients are used to repeatedly estimate the constant coefficient.

[0059] Specifically, the method for obtaining parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation provided by the present invention comprises the following steps:

[0060] Step 1: Assuming that some of the coefficients of the hybrid spatiotemporal weighted regression model are constants and some are time-varying and space-varying, the general expression of the hybrid spatiotemporal weighted regression model is:

[0061]

[0062] Among them, y i Indicates that the explained variable is at the i-th spatiotemporal position (u i ,v i ,t i ) observations; Indicates that the k1th constant coefficient corresponds to the explanatory variable at the i-th spatiotemporal position (u i ,v i ,t i ) observations; Indicates that the k2th spatiotemporal coefficient corresponds to the explanatory variable at the ith spatiotemporal position (u i ,v i ,t i ) observations; is a constant coefficient; is the spatiotemporal coefficient; n is the spatiotemporal sample size; q is the number of constant coefficients; p is the number of explanatory variables; ε i Represents the model random error term; in general, set z1 = 1 or x q+1 =1 means that the model includes a constant intercept term or an intercept term that varies with time and space.

[0063] Step 2, the estimation of its constant coefficient is as follows:

[0064]

[0065] in, is the k1th constant coefficient estimate; I is the unit matrix, Y = (y1, y2, ..., y n ) T is the vector of observed values ​​of the explained variable, and Z is the matrix of observed values ​​of the explanatory variables corresponding to the constant coefficients; is the vector of constant coefficient estimates;

[0066]

[0067] in, is the k3th row of the matrix X (k3 = 1, 2, ..., n); X represents the matrix of observed values ​​of the explanatory variables corresponding to the spatiotemporal coefficients, W(u i ,v i ,t i )=Diag(w i1 ,w i2 ,…,w in ) is about the space-time position (u i ,v i ,t i ), where:

[0068] w ij =K s (ds ij ,h s )×K t (dt ij ,h t )

[0069]

[0070]

[0071] Among them, ds ij represents the Euclidean distance from spatial position point j to regression point i; h s is the window width; dt ij represents the time distance from time position point j to regression point i; h t is the window width; j = 1, 2,…, n.

[0072] Step 3, regarding the space-time position (u i ,v i ,t i The estimated value of the time-varying coefficient of ) is derived from the following formula:

[0073]

[0074] in, is the time-varying coefficient estimate vector; is the estimated value of the k2th spatiotemporal coefficient.

[0075] Step 4: Use the corrected Akaike information criterion (AIC c ) Calculate the optimal spatiotemporal window width:

[0076]

[0077]

[0078] Among them, RSS m =Y T (IL m ) T (IL m )Y is the residual sum of squares; L m =L v +(IL v )Z(Z T (IL v ) T (IL v )Z) - 1 Z T (ILv ) T (IL v );tr(·) is the trace of the matrix.

[0079] Based on the optimal spatiotemporal window width obtained by the corrected Akaike information criterion, the initial estimates of the constant coefficients and the initial estimates of the spatiotemporal variable coefficients are obtained in combination with steps 2 and 3, respectively.

[0080] Step 5: Set the initial estimate of the constant coefficient obtained in step 4 as the initial estimate of the SAGTWR method, and use α old Indicates that The GTWR model is constructed by the following formula:

[0081]

[0082] Among them, ε is the model random error vector.

[0083] Step 6: Set the initial estimate of the spatiotemporal coefficient obtained in step 4 as the initial estimate of the SAGTWR method, denoted as Then construct a single variable The GTWR model:

[0084]

[0085] Among them, k2=q+1,q+2,…,p.

[0086] According to the single variable The GTWR model expression is combined with the following formula for the time-space variable coefficient Perform the first update to obtain the time-space variable coefficients after the first update

[0087]

[0088]

[0089] in, is the observed value vector of the explanatory variable corresponding to the k2th spatiotemporal coefficient at n spatiotemporal locations; is about the space-time position (u i ,v i ,t i )’s weight matrix;

[0090] Using the corrected Akaike information criterion, combined with the spatiotemporal varying coefficients Calculate the optimal space-time window width and obtain the corresponding optimal space-time window width and

[0091]

[0092]

[0093] The residual sum of squares is

[0094] The optimal spatiotemporal window width obtained based on the corrected Akaike information criterion is combined with the expression of the spatiotemporal varying coefficient after the first update to obtain the estimated value of the spatiotemporal varying coefficient after the first update.

[0095] Step 7: For each explanatory variable, repeat step 6 to complete the first iterative update of all spatiotemporal coefficients. and the corresponding optimal spatiotemporal window width and Complete the first update of the explained variables of the univariate GTWR model k2=q+1,q+2,…,p.

[0096] Step 8: Repeat steps 6 and 7 until the convergence criterion is met, and finally obtain

[0097] Step 9, convergence criteria:

[0098]

[0099] in,

[0100] Respectively represent the results of this iteration and the previous iteration; Following the method of Fotheringham (2017) and Chen and Mei (2020), let δ = 10 -3 .

[0101] Step 10, finally update the constant coefficient:

[0102]

[0103] Simulation verification:

[0104] The estimation accuracy and computational time of the two-step estimation method of hybrid GTWR, MGTWR, GTWR and our proposed SAGTWR method are evaluated by simulating data with different temporal and spatial heterogeneity. The simulation is carried out in R 4.0.3 environment.

[0105] We use a square as the spatial region of study, in which time T is generated equidistantly with an increment of 1. The three-dimensional space-time position used to generate the sample can be obtained by the following method:

[0106]

[0107] where mod(a,b) and int(a,b) are the remainder and integer part of a divided by b, respectively. The value of m is set to 60, and t i The value is 1, 2, ... 8, using the time layers corresponding to T1, T2, ... T8, and the sample size n = m 2 The data generation process is based on step 1. The explanatory variable data of the first and second groups are randomly drawn from the uniform distribution U(0,1). The explanatory variable data of the third group are randomly drawn from the standard normal distribution N(0,1). i are independent error terms that follow a standard normal distribution N(0,1).

[0108] In order to evaluate the performance of the SAGTWR method in estimating regression coefficients of varying complexity, we therefore designed three sets of simulation coefficients whose exact equations have different degrees of spatial and temporal heterogeneity, as follows:

[0109]

[0110]

[0111]

[0112] In order to make a consistent comparison among the hybrid GTWR, MGTWR, GTWR and the SAGTWR proposed in this method, the same kernel function is used to generate the weight matrix in the correction process of the four GTWR methods, and the spatiotemporal window width h is selected using the AICc criterion. s ,h t and the spatiotemporal window width of each univariate

[0113] Considering the randomness of data generation, the simulation experiment was repeated N = 100 times. For constant coefficients, we used the commonly used mean, standard deviation (SD) and root mean square error (RMSE) as indicators to measure the estimation accuracy. The root mean square error is a common method to measure the difference between the estimated value and the observed value. Since the so-called constant term of GTWR and MGTWR is also random (u i ,v i ,t i ), which is measured by the average root mean square error (ARMSE). Therefore, we compare the ARMSE values ​​of these two methods with the RMSE of the other two methods.

[0114] make is the result of the k1th constant coefficient in the rth simulation experiment. The mean after N repeated experiments is:

[0115]

[0116] The standard deviation is:

[0117]

[0118] The root mean square error is:

[0119]

[0120] The k2th spatiotemporal coefficient in GTWR and MGTWR At the position point (u i ,v i ,t i ) after N repeated experiments is:

[0121]

[0122] Its average root mean square error is:

[0123]

[0124] At the same time, the fitting effects of the four methods are measured by the average AICc (MAICc). MAICc represents the average value of AICc in N repeated experiments.

[0125] Table 1. Minimum, maximum, mean, standard deviation and (average) root mean square error of the constant coefficient estimates of the three sets of simulated data in 100 repeated experiments

[0126]

[0127] Table 2. (Average) root mean square error, average AICc, and model time consumption of the three sets of simulated data in 100 repeated experiments

[0128]

[0129] The results of the four estimation methods are shown in Table 1, where "scale-adaptive" represents the SAGTWR method, "two-step" represents the hybrid GTWR two-step estimation method, "multiscale" represents the MGWTR method, and "GTWR" represents the GWTR method. It can be seen from Table 1 that no matter what kind of spatiotemporal heterogeneity the variable coefficient part exhibits, the two-step estimation and SAGTWR methods perform almost the same in estimating the constant coefficients. In addition, it can be observed that the (A)RMSE values ​​of the two-step estimation and SAGTWR methods are smaller than those of the other two methods, especially much smaller than the GTWR method. This shows that the difference between the estimated value and the observed value of the constant coefficient in the SAGTWR method is smaller than that of the MGTWR and GTWR methods. In some cases, such as in group 1 and 3, the estimation of the constant coefficients has not been significantly improved compared with the two-step method with the iterative method in SAGTWR. Combined with Figure 3 We can also see that although there is almost no difference in RMSE between the two-step estimation method and the SAGTWR method in 100 experiments, they have significantly improved the constant coefficients compared with the other two methods.

[0130] Figure 1 The box plots of the optimal spatial window widths of the three groups of coefficients in the four methods are shown, where hs_β1, hs_β2 and hs_β3 represent the optimal spatial window widths of the corresponding variable coefficients β1(u,v,t), β2(u,v,t), and β3(u,v,t). Since the GTWR and two-step estimation methods only have a single spatial window width, they are the same under each variable coefficient for the convenience of comparison. The scatter plots are the scatter plots of each variable coefficient in 100 experiments. From the leftmost group 1 figure, it can be observed that β2(u,v,t) is smaller than β1(u,v,t), while hs_β1 in the two-step estimation method and GTWR method is smaller than hs_β2 in the two-step estimation method and GTWR method. s It is more like an average window width, which is between the maximum and minimum values ​​of the window widths of each variable coefficient. Moreover, from the perspective of parameter construction, β2(u,v,t) in group 1 has higher spatial heterogeneity than β1(u,v,t). Similarly, the optimal spatiotemporal window widths of each variable in groups 2 and 3 also match the spatial heterogeneity level of their simulated data. This result is consistent with the common findings in nonparametric smoothing techniques, where functions with high heterogeneity require smaller window widths to obtain effective estimates. MGTWR and the SAGTWR proposed in the present invention perform similarly in selecting the optimal window width for each variable coefficient. Combined with the time consumption of the method in the last column of Table 2, SAGTWR has higher computational efficiency. Therefore, in summary, SAGTWR can select appropriate window widths for variable coefficients with different levels of heterogeneity.

[0131] Figure 2The box plots of the optimal time window widths of the three groups of coefficients in the four methods are shown. The results show that group 2 and 3 can observe that β1(u,v,t), β2(u,v,t), and β3(u,v,t) all have their own corresponding optimal time and space window widths, while the time window width in the two-step estimation is more like an average window width. In addition, the optimal window width in the two-step estimation seems to be mainly controlled by the coefficient with high temporal heterogeneity, and its single time window width is closer to the one with the smallest window width among β1(u,v,t), β2(u,v,t), and β3(u,v,t), that is, the time window width of the coefficient with higher heterogeneity. Combined with the simulated data, the optimal time and space window widths of each variable in the three groups also match the temporal heterogeneity level of its simulated data. It is worth noting that from the scatter point point, the time window width distribution of some variables is not very concentrated, especially β3(u,v,t) in group 3. The reason for this may be that the coefficient functions with different heterogeneity used in the simulation have obtained similar values ​​at certain spatial positions at different times. SAGTWR and MGTWR are almost the same in the choice of time window width. In general, the time window width of each variable coefficient observed through its box plot is consistent with the strength of its temporal heterogeneity.

[0132] In summary, the SAMGTWR method we proposed can not only effectively select the spatial window width of each variable coefficient, but also effectively select its respective time window width, which is more time-saving than MGTWR.

[0133] Table 2 shows the average root mean square error (ARMSE), average AICc (MAICc) and average time consumption of the spatiotemporal coefficient estimates of three sets of simulation data using the SAGTWR method, two-step estimation method, MGWTR method and GTWR method. By observing the MAICc of each variable coefficient, it can be found that the difference in the average AICc values ​​of the four methods is between 3, which is not very different, that is, the performance in terms of goodness of fit measurement is equivalent. However, the average AICc values ​​of these three methods are all smaller than those of the GTWR method. In most cases, the effect of the SAGTWR method is almost the same as that of the MGTWR method. The MAICc of MGTWR is smaller than that of SAGTWR, which may be the result of MGTWR estimating the constant coefficient using a local method.

[0134] The average root mean square error results of each variable coefficient show that in the three groups of experiments, the ARMSE values ​​of the SAGTWR method are mostly smaller than the ARMSE values ​​of the other three methods or the difference is small, but more often, it is significantly better than the GTWR and two-step estimation methods, which further indicates that this method can improve the estimation accuracy of spatiotemporal variable coefficients. For the variable coefficients whose optimal window width of each coefficient in the SAGTWR method is significantly different from the window width of the two-step estimation method, a more significant improvement has been achieved. For example, for the coefficient β3(u,v,t) in group3, it is observed that Figure 1 and Figure 2 From the window width values ​​of group 3, we can find that the spatial and temporal window widths of the variable coefficient β3(u,v,t) are quite different from the single window width. The ARMSE value of β3(u,v,t) is 0.0361 in SAGTWR, 0.0742 in the two-step estimation method, and 0.0673 in GTWR, which means that the accuracy of SAGTWR estimation is higher than that of the two-step estimation method and GTWR method. Figure 3 The box plot on the far right for group 3 can also further prove that the SGTWR method can improve the estimation accuracy of spatiotemporal coefficients. The last column of Table 2 shows the average time consumption of 100 experiments. We can observe that SAGTWR saves about 6% to 33% time compared with MGTWR. Therefore, in summary, SAGTWR improves the estimation accuracy of spatiotemporal coefficients and reduces the computational burden of MGTWR to a certain extent.

Claims

1. A method for obtaining parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation, characterized in that: The following steps are involved: Step 1: Assuming that some of the coefficients of the hybrid spatiotemporal weighted regression model are constants and some are time-varying and space-varying, the expression of the hybrid spatiotemporal weighted regression model is: Among them, y i Indicates that the explained variable is at the i-th spatiotemporal position (u i , v i , t i ) observations; Indicates that the k1th constant coefficient corresponds to the explanatory variable at the i-th spatiotemporal position (u i , v i , t i ) observations; Indicates that the k2th spatiotemporal coefficient corresponds to the explanatory variable at the ith spatiotemporal position (u i , v i , t i ) observations; is a constant coefficient; is the spatiotemporal coefficient; n is the spatiotemporal sample size; q is the number of constant coefficients; p is the number of explanatory variables; ε i represents the random error term of the model; set z1 = 1 or x q+1 =1 means that the model contains a constant intercept term or an intercept term that varies with time and space; Step 2, obtaining an initial estimation formula of the constant coefficients of the mixed spatiotemporal weighted regression model; Step 3: Obtain the spatial and temporal position (u i , v i , t i )’s initial estimate of the time-varying coefficient; Step 4, calculate the optimal spatiotemporal window width by using the corrected Akaike information criterion, and combine step 2 and step 3 to obtain initial estimates of constant coefficients and initial estimates of spatiotemporal variable coefficients respectively; Step 5, setting the initial estimated value of the constant coefficient obtained in step 4 as the initial estimated value of the scale-adaptive spatiotemporal weighted regression SAGTWR method; Step 6: Set the initial estimate of the spatiotemporal coefficient obtained in step 4 as the initial estimate of the scale-adaptive spatiotemporal weighted regression SAGTWR method: Step 7: For each explanatory variable, repeat step 6 to complete the first iterative update of all spatiotemporal coefficients and the corresponding optimal spatiotemporal window width, and complete the first update of the explained variables of the univariate GTWR model; Step 8, repeat the above steps 6 and 7 until the convergence criterion is met; Step 9, update the constant coefficients.

2. The method for acquiring parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation according to claim 1, characterized in that: The initial estimation formula of the constant coefficient of the mixed spatiotemporal weighted regression model is as follows: in, is the k1th constant coefficient estimate; I is the unit matrix, Y = (y1, y2, ..., y n ) T is the vector of observed values ​​of the explained variable, and Z is the matrix of observed values ​​of the explanatory variables corresponding to the constant coefficients; is the vector of constant coefficient estimates; in, is the k3th row of the matrix X (k3 = 1, 2, ..., n); X represents the matrix of observed values ​​of the explanatory variables corresponding to the spatiotemporal coefficients, W(u i , v i , t i )=Diag(w i1 , w i2 ,…,w in ) is about the space-time position (u i , v i , t i ) is the weight matrix of .

3. The method for acquiring parameters of a hybrid spatiotemporal weighted regression model based on scale adaptation according to claim 1, characterized in that: The spatial and temporal position (u i , v i , t i The initial estimate of the time-varying coefficient of ) is as follows: in, is the time-varying coefficient estimate vector; is the estimated value of the k2th spatiotemporal coefficient; W(u i , v i , t i )=Diag(w i1 , w i2 ,…,w in ) is about the space-time position (u i , v i , t i ) weight matrix; X represents the matrix composed of the observed values ​​of the explanatory variables corresponding to the spatiotemporal coefficients; is a vector of constant coefficient estimates; Y = (y1, y2, ..., y n ) T is a vector of observed values ​​of the explained variable, and Z is a matrix of observed values ​​of the explanatory variables corresponding to the constant coefficients.

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