Estimation and simulation method of spatial point event distribution based on B-spline smoothing

Abstract

A method for estimating and simulating the distribution of spatial point events based on B-spline smoothing, comprising: collecting independently recorded spatial point location data corresponding to a number of spatial location nodes; selecting the least squares estimator or maximum likelihood estimator using B-spline smoothing according to the number of samples, and obtaining the optimal coefficient matrix by solving the corresponding optimization problem; calculating the intensity function using the obtained optimal coefficient matrix estimate, and applying the Thinning algorithm to generate pseudo data for simulating the distribution of spatial point events, which are further used in Monte Carlo simulation to infer the expected value of spatial statistics. The method exhibits significant flexibility, smoothness and robustness when processing spatial data, effectively reduces the impact of noise data, and improves the accuracy and reliability of the model. Through experimental verification, the present invention is superior to the traditional kernel method in multiple evaluation indicators, and provides a new and efficient solution for the estimation and simulation of spatial point event distribution.

Description

Technical Field

[0001] The invention relates to spatial statistics and spatial point event distribution estimation technology, and in particular to a spatial point event distribution estimation and simulation method based on B-spline smoothing. Background Art

[0002] A spatial point process is a point process model that describes the distribution of random events in space. It can be applied to the modeling and analysis of discrete point data distributed in a specific spatial area. For example, the location distribution of trees or bird nests in statistical ecology, the location distribution of stars and galaxies in astrostatistics, the location of silicon wafer point defects in materials science, and the distribution of patients' home addresses in epidemiology can all be analyzed and inferred using spatial point processes, helping various fields to optimize resource allocation and improve decision-making efficiency. In the field of emergency management, spatial point processes are used to describe the demand for automated external defibrillators (AEDs) and quantify their spatial risk factors, which is conducive to accurately evaluating and optimizing the location deployment of AEDs.

[0003] With the widespread application of spatial data in various fields, spatial Poisson process intensity estimation has become an important research hotspot in statistical inference. Spatial Poisson processes are widely used to model and analyze spatial point patterns, such as event distribution in disease epidemic areas, spatial structure of brain neurons, and location of earthquakes. However, accurately estimating the intensity function of spatial Poisson processes remains a challenging and important problem, and is the key to data analysis and simulation. Researchers have proposed many intensity estimators, including parametric methods and non-parametric methods. Classical parametric methods include improved second-order characteristic ratio estimators, spatial point process intensity consistency estimators, and multidimensional point process model residual fitting estimators. Traditional parametric methods do not consider spatial heterogeneity and correlation. The Bayesian semi-parametric mixed model combines covariates such as population density, transportation, and buildings, introduces conditional autoregressive priors to adapt to possible spatial correlations, and improves statistical prediction accuracy and model interpretability. However, the above parametric methods require experts to understand the meaning of parameters, formulate reasonable hyperparameters, and correct deviations. It is difficult for non-professionals to interpret and directly apply them; in addition, improper parameter models and prior assumptions can easily affect model accuracy. Nonparametric estimation does not require the assumption of model architecture and can be directly interpreted and applied by non-professionals in practice. It is more flexible and robust when the actual distribution is unknown. The most classic nonparametric estimation uses piecewise constant functions as the basic form of the intensity function, proving the consistency and effectiveness of the estimation. Nonparametric estimation with piecewise linear functions and piecewise quadratic functions as basis functions effectively improves the problem of continuity of the intensity curve, but the non-smoothness and limited differentiability of the curve still limit the input model to a high uncertainty, and the modeling error is very easy to increase. Kernel methods are then widely used in this type of estimator to ensure the smoothness and differentiability of intensity estimation, such as classic kernel smoothing estimation, kernel estimation based on covariate-dependent Gibbs process, and reproducing kernel estimation in Hilbert space. Kernel methods rely on hyperparameters such as bandwidth and have a strong dependence on data distribution. If the bandwidth is not selected properly, it is easy to cause underfitting or overfitting, resulting in unstable estimation results; in addition, the kernel function maps the data to a high-dimensional space and estimates it by calculating the inner product, which lacks obvious physical or geometric meaning and is difficult to perform statistical reasoning and expansion, making this method less interpretable and difficult to expand.

[0004] It should be noted that the information disclosed in the above background technology section is only used for understanding the background of the present application, and therefore may include information that does not constitute prior art known to ordinary technicians in the field. Summary of the invention

[0005] The main purpose of the present invention is to overcome the defects existing in the above-mentioned background technology and provide a method for estimating and simulating spatial point event distribution based on B-spline smoothing.

[0006] To achieve the above object, the present invention adopts the following technical solutions:

[0007] A method for estimating and simulating spatial point event distribution based on B-spline smoothing comprises the following steps:

[0008] S1: Collecting independently recorded spatial point position data corresponding to a number of spatial position nodes;

[0009] S2: Based on the collected spatial point position data, if the number of independent recorded samples is less than the set sample number threshold, the least squares estimator smoothed by B-spline is used to obtain the optimal coefficient matrix by solving the constrained quadratic programming problem; if the number of independent recorded samples is greater than or equal to the set sample number threshold, the maximum likelihood estimator smoothed by B-spline is used to obtain the optimal coefficient matrix by solving the likelihood function maximization problem through the trust region algorithm; the set sample number threshold is determined based on the results of numerical experimental analysis conducted in advance;

[0010] S3: Based on the estimated value of the optimal coefficient matrix obtained in step 2, calculate the intensity function representing the probability of occurrence of the spatial point event, use the thinning algorithm to generate pseudo data simulating the distribution of the spatial point event according to the estimated value of the intensity function, and apply the pseudo data to the Monte Carlo simulation to infer the expected value of the spatial statistics related to the spatial point event.

[0011] Furthermore, the set sample number threshold is 5.

[0012] Furthermore, the maximum likelihood estimator using B-spline smoothing solves the likelihood function maximization problem through a trust region algorithm to obtain an optimal coefficient matrix, specifically including:

[0013] Constructing a log-likelihood function, including a log-likelihood term of the data, an integral term of the intensity function, and a regular term, wherein the regular term is used to ensure the smoothness and generalization ability of the estimator;

[0014] Convert the log-likelihood function into a matrix form;

[0015] Using the trust region algorithm, the optimal solution of the coefficient matrix C that maximizes the log-likelihood function is found through an iterative process under the premise of ensuring that the intensity function is non-negative; in each iteration, the boundary of the trust region is updated according to the current coefficient matrix estimate to ensure the stability and convergence of the search process;

[0016] When the algorithm converges or reaches the preset number of iterations, the final coefficient matrix estimate is output.

[0017] Furthermore, the least squares estimator using B-spline smoothing obtains the optimal coefficient matrix by solving the constrained quadratic programming problem, specifically including:

[0018] Performing a two-dimensional gridding of the spatial region to create a plurality of grid cells, each grid cell defining a subregion in the space, so as to calculate an expected count of events within each grid cell;

[0019] Construct the loss function W(C), where C is the coefficient matrix, W(C) includes the sum of the squares of the differences between the event counts and the cumulative intensity function in all grids, and the regularization term;

[0020] Convert the loss function W(C) into a quadratic programming form and find the optimal coefficient matrix C that minimizes the loss function W(C);

[0021] When the optimal solution that satisfies all constraints is found, the optimal coefficient matrix C is output as the least squares estimator.

[0022] Furthermore, for the loss function W(C) converted into the quadratic programming form, the constrained quadratic programming method is used to consider the non-negativity of the intensity function to solve the optimal coefficient matrix C that minimizes the loss function W(C).

[0023] Furthermore, the method further comprises the following steps:

[0024] S4: Based on the optimal coefficient matrix estimate obtained in step 2, the weight of each control point in the B-spline is output, the weights of the spatial position nodes are sorted, and the key spatial position nodes that have the greatest impact on the event occurrence pattern are identified.

[0025] Furthermore, the B-spline smoothing operation is to construct a piecewise polynomial curve to flexibly adapt to and approximate the distribution of spatial point events, thereby achieving accurate estimation and simulation of the probability of event occurrence.

[0026] Furthermore, the spatial point events include but are not limited to: the location distribution of organisms in statistical ecology; the location distribution of stars and galaxies in astrostatistics; the location distribution of silicon wafer point defects in materials science; the distribution of patients' home addresses in epidemiology; and the location distribution of AED equipment in the field of emergency management. The spatial point location data include but are not limited to: the geographic coordinate data of organisms collected from ecological studies; the celestial coordinate data of stars and galaxies obtained from astronomical studies; the spatial location data of silicon wafer point defects recorded from material science testing; the geographic information data of patients' home addresses collected from epidemiological surveys; and the geographic location data of AED equipment deployment recorded in emergency management systems.

[0027] A computer-readable storage medium stores a computer program, which, when executed by a processor, implements the spatial point event distribution estimation and simulation method based on B-spline smoothing.

[0028] A computer program product comprises a computer program, wherein when the computer program is executed by a processor, the method for estimating and simulating spatial point event distribution based on B-spline smoothing is implemented.

[0029] The present invention has the following beneficial effects:

[0030] In response to the problems of low stability and interpretability of existing event distribution modeling methods in spatial statistics, the present invention proposes a spatial point event distribution estimation and simulation method based on B-spline smoothing, innovatively proposes a spatial point process intensity function B-spline smoothing estimator, and performs effective statistical analysis on spatial event occurrence patterns. It shows remarkable flexibility, smoothness and robustness when processing spatial data, can approximate complex spatial point event distributions, and effectively reduce the impact of noise data, thereby significantly improving the accuracy and reliability of spatial point event distribution modeling and enhancing the value and significance of the model in practical applications.

[0031] In view of the problems of poor stability, difficulty in expansion, low interpretability, and easy aggravation of analytical errors in actual use of the non-parametric estimation method of intensity function in traditional technology, the estimator based on the B-spline smoothing method and the data simulation algorithm under intensity estimation designed by the present invention provide more reliable estimation and simulation results for the statistical inference and numerical calculation of spatial point processes than traditional models, which helps actual practitioners to analyze and make accurate decisions on the spatial patterns of event occurrence. The algorithm of the present invention effectively expands the B-spline smoothing method to the spatial point process model, making the analytical modeling of discrete point data in the spatial region more accurate and robust, and its physical or geometric meaning is also easier to explain, and it is suitable for sparse data scenarios. It has been verified that in terms of practicality, the accuracy and stability of the present invention are better than those of existing similar algorithms, especially in the case of sparse data often encountered in actual production and life, showing obvious advantages, improving the accuracy and rationality of the analysis conclusions, and can be used as a reference for professionals to optimize facility layout and improve decision-making efficiency.

[0032] The present invention has high application value in multiple professional fields, including but not limited to statistical ecology, astrostatistics, materials science, and epidemiology, and can provide more accurate spatial point event distribution estimation and simulation methods, thereby playing an important role in resource optimization allocation, risk assessment, decision support, etc.

[0033] Other beneficial effects of the embodiments of the present invention will be further described below. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] Figure 1These are the true intensity function image and the intensity function estimation image in an independent experiment of an embodiment of the present invention, including: (a) true intensity function; (b) Spline-CQP intensity function estimation; (c) Spline-MLE intensity function estimation; (d) RHKS intensity function estimation; (e) RBF intensity function estimation.

[0035] Figure 2 1 and 2 are the changes of the intensity estimation accuracy evaluation index under different sample sizes in the embodiment of the present invention, including: (a) rMSE_grid; (b) rMSE_train; (c) MAD; (d) IAD.

[0036] Figure 3 1 is a B-spline smoothing estimate of the intensity function based on the chorley data set according to an embodiment of the present invention, wherein: (a) Spline-CQP estimation; (b) Spline-MLE estimation.

[0037] Figure 4 It is an overall flow chart of the spatial point event distribution estimation and simulation method based on B-spline smoothing of the present invention. DETAILED DESCRIPTION

[0038] The following is a detailed description of the embodiments of the present invention. It should be emphasized that the following description is only exemplary and is not intended to limit the scope and application of the present invention.

[0039] The core of spatial point process modeling lies in the estimation and analysis of intensity functions. Traditional non-parametric estimation methods of intensity functions have problems such as poor stability, difficulty in expansion, and low interpretability, which can easily aggravate analysis errors in actual use. In response to these problems, the present invention proposes a spatial point event distribution estimation and simulation method based on B-spline smoothing, designs an estimator based on the B-spline smoothing method and a data simulation algorithm under intensity estimation, and provides more reliable estimation and simulation results for statistical inference and numerical calculation of spatial point processes than traditional models, which helps actual practitioners to analyze the spatial pattern of event occurrence and make accurate decisions.

[0040] See also Figure 4 The spatial point event distribution estimation and simulation method based on B-spline smoothing in the embodiment of the present invention comprises the following steps:

[0041] Step S1: collecting independently recorded spatial point position data corresponding to a number of spatial position nodes;

[0042] Step S2: Based on the collected spatial point position data, if the number of independent recorded samples is less than the set sample number threshold, use the B-spline smoothed least squares estimator to obtain the optimal coefficient matrix by solving the constrained quadratic programming problem; if the number of independent recorded samples is greater than or equal to the set sample number threshold, use the B-spline smoothed maximum likelihood estimator to solve the likelihood function maximization problem through the trust region algorithm to obtain the optimal coefficient matrix; the set sample number threshold is determined based on the results of numerical experimental analysis conducted in advance. Preferably, the set sample number threshold is 4-6. Experimental results show that the effects of the two B-spline smoothing estimators are comparable between 4-6 and are both better than other kernel estimators. The performance of the B-spline smoothed maximum likelihood estimator has an inflection point between 4-6 (around 5), and the performance after the inflection point is better than all estimators. Optimally, the set sample number threshold is 5.

[0043] Step S3: Based on the estimated value of the optimal coefficient matrix obtained in step 2, calculate the intensity function representing the probability of occurrence of the spatial point event, use the thinning algorithm to generate pseudo data simulating the distribution of the spatial point event according to the estimated value of the intensity function, and apply the pseudo data to the Monte Carlo simulation to infer the expected value of the spatial statistics related to the spatial point event.

[0044] In some embodiments, in step S2, the maximum likelihood estimator smoothed using B-spline is solved by a trust region algorithm to obtain an optimal coefficient matrix by maximizing the likelihood function, specifically including: constructing a log-likelihood function, including a log-likelihood term of the data, an integral term of the intensity function, and a regularization term, wherein the regularization term is used to ensure the smoothness and generalization ability of the estimator; converting the log-likelihood function into a matrix form; using a trust region algorithm, through an iterative process, finding the optimal solution of the coefficient matrix C that maximizes the log-likelihood function while ensuring that the intensity function is non-negative; in each iteration, updating the boundary of the trust region according to the current coefficient matrix estimate to ensure the stability and convergence of the search process; when the algorithm converges or reaches a preset number of iterations, outputting the final coefficient matrix estimate.

[0045] In some embodiments, in step S2, the least squares estimator using B-spline smoothing obtains the optimal coefficient matrix by solving a constrained quadratic programming problem, specifically including: performing two-dimensional grid division on the spatial region, creating multiple grid cells, each grid cell defines a sub-region in the space, so as to calculate the expected count of events in each grid; constructing a loss function W(C), where C is a coefficient matrix, W(C) includes the sum of squares of the difference between the event counts in all grids and the cumulative intensity function, and a regularization term; converting the loss function W(C) into a quadratic programming form, solving the optimal coefficient matrix C that minimizes the loss function W(C); when the optimal solution that satisfies all constraints is found, outputting the optimal coefficient matrix C as the least squares estimator. For the loss function W(C) converted into a quadratic programming form, a constrained quadratic programming method is specifically used, considering the non-negativity of the intensity function, to solve the optimal coefficient matrix C that minimizes the loss function W(C).

[0046] In step S2, the operation of B-spline smoothing is to construct a piecewise polynomial curve to flexibly adapt to and accurately approximate the spatial point event distribution, thereby realizing accurate estimation and simulation of the probability of event occurrence. The present invention introduces B-spline smoothing into the spatial point process model, realizes global fitting through piecewise polynomials, provides higher smoothness and continuity for the intensity function, and effectively reduces the influence of noise data; in terms of dependence on data distribution, the construction of spline function mainly controls smoothness through nodes, is more flexible in use and is not easily affected by local data characteristics, and is more applicable in sparse data scenarios; in terms of interpretability, the spline function clearly defines the estimation model through control points and basis functions, and the contribution of each control point to function estimation can be clearly observed, which has good interpretability.

[0047] In some embodiments, the spatial point event distribution estimation and simulation method further includes the following steps:

[0048] Step S4: Based on the optimal coefficient matrix estimate obtained in step 2, the weight of each control point in the B-spline is output, the weights of the spatial position nodes are sorted, and the key spatial position nodes with the greatest impact on the event occurrence pattern are identified.

[0049] In different embodiments, in the spatial point event distribution estimation and simulation method of the present invention, the spatial point events include but are not limited to: the position distribution of organisms in statistical ecology; the position distribution of stars and galaxies in astronomical statistics; the position distribution of silicon wafer point defects in materials science; the distribution of patients' home addresses in epidemiology; the position distribution of AED equipment in the field of emergency management. The spatial point location data include but are not limited to: the geographic coordinate data of organisms collected from ecological studies; the celestial coordinate data of stars and galaxies obtained from astronomical studies; the spatial location data of silicon wafer point defects recorded from material science testing; the geographic information data of patients' home addresses collected from epidemiological surveys; the geographic location data of AED equipment deployment recorded from emergency management systems.

[0050] The spatial point event distribution estimation and simulation method of the present invention shows remarkable flexibility, smoothness and robustness when processing spatial data. By constructing a piecewise polynomial curve, the present invention can accurately approximate the complex spatial point event distribution, while effectively reducing the influence of noise data. Compared with the traditional kernel method, the present invention innovatively applies the B-spline smoothing method to the estimation and simulation of spatial point event distribution. B-spline smoothing shows higher flexibility in data distribution dependence and is not easily affected by local data characteristics. Especially in the case of sparse data, it can provide more stable estimation results. In addition, the B-spline smoothing method is more interpretable. The estimation model is clearly defined by control points and basis functions, so that the contribution of each control point to the function estimation is clearly visible, which is convenient for practitioners to understand and apply. In terms of performance, the present invention is superior to existing similar algorithms in both accuracy and stability, especially in the case of sparse data, it can significantly improve the accuracy and rationality of the analysis conclusion. Through numerical experimental verification (which will be further described later), the method of the present invention shows superior performance in multiple evaluation indicators, which can provide strong support for professionals in optimizing facility layout and improving decision-making efficiency.

[0051] The following further describes an algorithm example and experimental verification of a specific embodiment of the present invention.

[0052] First consider Given a series of point process data within the range, these point data record the spatial location distribution of events, The independent records are obtained, assuming that Records of Point, The points are recorded as Modeling this spatial point process requires estimating the intensity function based on the data and then running a generation algorithm based on the estimate for simulation.

[0053] B-spline curves as the basic form of intensity function estimators improve the flexibility, smoothness and robustness of traditional estimates and have been proven to reduce the input uncertainty of discrete simulation models. Order B-spline function, assuming that there is an interval of of Control Points And a set of basis functions ,in is the index of the basis function, the curve can be expressed as:

[0054]

[0055] Among them, the basis function , and for of Calculated by the following recursive formula

[0056]

[0057] In a given and In the case of yes subcontinuously differentiable, and

[0058] In fact, the calculation of the basis function can be more easily derived as:

[0059]

[0060] in , and the coefficient matrix It can be calculated as follows:

[0061]

[0062] In this formula, and Indicates a dimensional zero vector, is defined as

[0063]

[0064] Based on the above definition of B-spline, assuming that the space point process The intensity function is as follows

[0065]

[0066] For simplicity of expression, It can be converted into the following matrix form:

[0067]

[0068] in , and C is a The coefficient matrix of the spatial point process. When the intensity function is known, the spatial point process can generate random data through the Thinning algorithm, which is helpful in the field of spatial statistics to calculate the point distribution statistics using Monte Carlo simulation and comprehensively analyze and explain the event occurrence pattern.

[0069] Finally, the point process estimation and simulation problem has two main goals: first, based on the observed point data, the coefficient matrix C is estimated to obtain the intensity function The second step is to apply the intensity function estimate obtained in the previous step, and generate pseudo data based on the Thinning algorithm for simulation. For the estimation target, the intensity function is the core of spatial point process modeling, which can be extended to calculate spatial statistics in multiple dimensions and is also the basis for Monte Carlo simulation. For the simulation target, the expected value of the statistic of the point process model usually has no explicit expression and needs to be calculated using numerical methods. High-accuracy simulation helps to calculate relevant statistics and indicators more accurately and efficiently, and output reasonable analysis conclusions.

[0070] The present invention provides a spatial point event distribution estimation and simulation method based on B-spline smoothing, provides a more accurate and robust intensity function estimator than existing methods, and generates high-quality simulation data for spatial point pattern statistical analysis, comprising the following steps:

[0071] Step S1: Collect and obtain a number of independently recorded spatial point position data.

[0072] Step S2: Based on the collected data, if the number of independent record samples is <5, use the least squares estimator with B-spline smoothing to solve the constrained quadratic programming problem and obtain the optimal coefficient matrix; if the number of independent record samples is ≥5, use the maximum likelihood estimator with B-spline smoothing and use the trust region algorithm to solve the likelihood function maximization problem and obtain the optimal coefficient matrix.

[0073] Step S3: Based on the estimated value of the coefficient matrix, the corresponding intensity function is calculated, and the thinning algorithm is used to generate pseudo data according to the estimated value of the intensity function, which is applied to the Monte Carlo simulation to infer the expected value of the spatial statistic.

[0074] Step S4: Based on the estimated value of the coefficient matrix, the weight of each control point in the B-spline is output, and information is further collected at important geographical location nodes for interpretability analysis.

[0075] The step S1 specifically includes: collecting The spatial point position data recorded independently, Records of Point, The points are recorded as .

[0076] The step S2 specifically includes: calculating the intensity function estimation based on B-spline smoothing. The present invention provides two different B-spline smoothing point process intensity function estimations, including maximum likelihood estimation and least squares estimation.

[0077] The essence of maximum likelihood estimation is to find the optimal solution that can maximize the likelihood function as the estimated value of the parameter. In the estimation problem of the point process intensity function, it is generally considered to add a regular term to the log-likelihood function to ensure the smoothness and generalization ability of the estimator, and the log-likelihood function is derived as follows:

[0078]

[0079] is the regularization term coefficient. For the sake of simplicity, the function can be transformed into the following form:

[0080]

[0081] in , , In fact, it can be approximated as a constant matrix. Considering that the intensity function is non-negative, the above log-likelihood function can use the trust region algorithm to find the optimal solution with all elements in C being bounded to 0, and substitute it into as a maximum likelihood estimator.

[0082] The least squares estimation uses the property that the expected event count in the Poisson process is equal to the cumulative intensity function, and combines the mean square error and the regularization term as the loss function. Divide into grid, assuming the In the record, it fell into Line The number of events in each grid is , then the loss function is:

[0083]

[0084] It can be transformed into a quadratic programming form:

[0085]

[0086] in

[0087]

[0088]

[0089] Note that the estimation problem has been transformed into a quadratic programming form. Combined with the non-negativity of the intensity function, the optimal parameter C can be obtained by minimizing the loss function using a constrained quadratic programming method and used as a least squares estimator.

[0090] In the case of sparse data, the least squares estimation pays more attention to the overall number of events within the regional unit and is less likely to overfit to small sample data than the maximum likelihood estimation. The numerical experiments of the present invention verify (see performance analysis for details) that when the number of data record samples is less than 5, the B-spline smoothed least squares estimation can control the error at a low level.

[0091] The step S3 is specifically as follows: based on the intensity function estimate calculated in step S2, the Thinning algorithm is used to generate pseudo data according to the steps shown in Algorithm 1. Repeating this step to generate multiple sets of pseudo data can be used for Monte Carlo simulation and numerical calculation of spatial statistical expectations, which is convenient for practitioners to analyze and make decisions.

[0092] Algorithm 1: Thinning algorithm for non-homogeneous Poisson spatial point process

[0093] Input: Maximum bounds U, V; Intensity function estimate ; Estimate the maximum value of the intensity function

[0094] Output: Generate data

[0095] 1result←[](initialization)

[0096] 2

[0097] 3for i=1 to do

[0098] 4

[0099] 5

[0100] 6

[0101] 7result.append( )

[0102] 8return result

[0103] The step S4 specifically includes: based on the estimated value of the coefficient matrix C outputted in step S2, obtaining the coefficient matrix C corresponding to each control point The corresponding weights are arranged in descending order to obtain the locations of the key control points that have the greatest impact on the event distribution pattern. Practitioners can further collect additional information on relevant geographic locations for further interpretable analysis and gain a deep understanding of the event permissible regularities.

[0104] Performance Analysis

[0105] Performance Analysis The accuracy and robustness of the proposed B-spline smooth intensity function estimator are verified through numerical experiments, and the practical applicability of the two is discussed. It is carried out according to the following steps: (a) define the true intensity function. (b) generate the The independent data of the group are used as the observation data. (c) The intensity function estimation value is calculated based on the generated data, and the distance between the intensity function estimation and the true intensity function is compared. Among them, the classical smoothing kernel estimation (RBFkernel) and the Hilbert space reproducing kernel estimation (RHKS) are used as benchmark models to compare with the B-spline smoothed least squares estimation (Spline-CQP) and B-spline smoothed maximum likelihood estimation (Spline-MLE) proposed in this invention.

[0106] This numerical experiment selects four evaluation indicators to measure the accuracy of the estimation: rMSE_grid, rMSE_train, maximum absolute distance (MAD), and integral absolute distance (IAD). Among them, rMSE_grid focuses on the rMSE between the true function and the estimated function at uniform points in two-dimensional space; rMSE_train focuses on the rMSE between the two functions at the observed data points; MAD measures the maximum value of the absolute distance between the two; IAD calculates the overall integral of the absolute value of the distance between the true function and the estimated function.

[0107] Figure 1 The gap between the true intensity function and the estimated intensity function in one experiment is visualized. The method of the present invention fits the shape of the true intensity function well, while the benchmark kernel method suffers from underfitting.

[0108] In order to prove the stability and accuracy of the method of the present invention, 15 independent repeated experiments were carried out based on the original numerical experimental steps, and the experimental results are shown in Tables 1 and 2. Two conclusions can be drawn from this: (1) For all evaluation indicators, the two B-spline smoothing estimators proposed in the present invention are superior to other benchmark models in most cases, and the rMSE and the integral absolute value distance between the true intensity function and the intensity function estimate have reached the lowest level, indicating that the B-spline smoothing method can overcome the shortcomings of the kernel method to a certain extent and optimize the accuracy of the estimation. (2) Overall, when the number of data sample records is 20, the performance of Spline-MLE is better than that of Spline-CQP, which may be because Spline-MLE makes more full use of the location information of the event location in the inference of the likelihood function, while Spline-CQP only uses the event count information within the grid. When the sample size is large, a large amount of accurate location information prevents overfitting and enhances the estimation accuracy.

[0109] Table 1 Comparison of average performance of all candidate estimation methods in fifteen independent experiments

[0110]

[0111] Table 2 Frequency of the best performance of the method of the present invention under various evaluation indicators in fifteen independent experiments

[0112]

[0113] Furthermore, numerical experiments with different data sample sizes are designed to verify the robustness of the proposed method and discuss its applicable scenarios. All estimation methods are tested in turn with data record sample sizes increasing from 1 to 30. A set of experiments with a certain sample size will be repeated 20 times independently and the average evaluation index results will be recorded. Figure 2The trend of each evaluation index under different methods with increasing sample size is shown, and the following three conclusions can be summarized from it. (1) In all estimation methods, as the number of independent record samples of spatial point data increases, the gap between the true intensity function and the intensity estimate becomes smaller, and the estimation accuracy increases. The only exception is the RHKS model in rMSE_train, which is speculated to be caused by the greater degree of noise interference of RHKS under large samples. (2) Spline-MLE is suitable for scenarios with a large number of spatial point sample records (>5). Its estimation error has an inflection point when the number of data samples is small. When the number of samples is five, it can quickly improve the accuracy and then maintain a stable low error level that is better than other methods. (3) Spline-CQP is suitable for sparse data scenarios with very small data volumes (<=5). When the data volume is very small, other methods are not stable enough under different evaluation indicators, while Spline-CQP can maintain medium to low errors in all indicators and has high robustness. This may be because this method pays more attention to the number of event counts and pays more attention to the overall situation when the data is sparse, thus preventing overfitting.

[0114] Example test

[0115] In order to make the purpose, technical solutions and advantages of the present invention clearer, the process of the present invention is further described in detail below by taking the public data set chorley:Chorley-Ribble Cancer Data of the spatstat package in R software as an example. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments.

[0116] The Chorley public dataset provides the specific addresses of throat cancer (58 cases) and lung cancer (978 cases) recorded by the Chorley and South Ribble Health Authority in Lancashire, England between 1974 and 1983, and gives the corresponding geographical coordinates. This dataset has been widely analyzed in the field of spatial point processes.

[0117] The steps of the example test are as follows: (a) Calculate the intensity function estimation under different methods based on the observed data. (b) Generate simulation data based on the intensity estimation using the Thinning algorithm. (c) Compare the distribution differences between the simulation data and the observed data. Among them, the evaluation indicators of high-dimensional distribution differences are the maximum mean difference (MMD) and the Wasserstein distance (EMD). MMD quantifies the distance by mapping the data into a high-dimensional feature space and measuring the mean difference between the two distributions in the space. The MMD calculation formula based on sample estimation is: ,in is the number of samples under the two distributions, the kernel function Select a Gaussian kernel. EMD measures the minimum amount of work required to transfer the probability mass of the original distribution P to the target distribution Q. It can be defined by the following formula ,in Indicates from arrive The quality of the transmission, is the set of all possible transport plans, Represents the distance between two samples.

[0118] First, different estimators are applied to the Chorley dataset to calculate the intensity estimate, such as Figure 3 As shown, the method of the present invention visualizes the cancer incidence hotspots, and the trend is consistent with the real data distribution. The inferred intensity estimation can be applied to the statistical inference of spatial point patterns, which is convenient for professionals to analyze. Secondly, based on the simulation effect of intensity estimation, the Thinning generation algorithm based on the B-spline smoothing estimator can quickly and accurately generate pseudo data, which is helpful for the Monte Carlo simulation of spatial point processes, and then infer the corresponding statistical expectations and interval estimates. Table 3 quantifies the data simulation quality under different estimation methods and random simulations based on the high-dimensional distribution distance evaluation index, from which the following two conclusions can be inferred. (1) Data simulation is performed under the B-spline estimation with the lowest estimation error. The more similar the data distribution generated by the simulation is to the original observed data distribution, the higher the simulation quality. (2) The simulation data obtained by Spline-CQP is better than Spline-MLE, which is mutually confirmed by the results of the previous data sample size experiment: the numerical experimental results show that when the number of records is less than five, it is more reasonable to use the Spline-CQP method for estimation. However, the number of point process records in this public data set is only 1, so the Spline-CQP estimation itself is more accurate, so the quality of the generated simulation data is also higher.

[0119] Table 3 Comparison of the differences between the distribution of simulated data and the distribution of observed data under different estimators

[0120]

[0121] In terms of model interpretability, the B-spline smoothing estimator also corresponds to the weight of each control point when it is output. By sorting the weights in descending order and taking the topK control points, we can get the K nodes that have the greatest influence on the overall event distribution. Taking K=20 as an example, it is found that the important locations where cancer cases occur are located in three parts of the region. Practitioners can further collect additional information about these locations, determine the specific factors that affect the distribution of events, and form interpretable analytical conclusions, which will help optimize solutions for problems such as medical facility allocation and staff scheduling.

[0122] An embodiment of the present invention further provides a storage medium for storing a computer program, which at least performs the above method when executed.

[0123] An embodiment of the present invention further provides a control device, comprising a processor and a storage medium for storing a computer program; wherein the processor is configured to execute at least the method described above when executing the computer program.

[0124] An embodiment of the present invention further provides a processor, wherein the processor executes a computer program and at least executes the method described above.

[0125] The storage medium can be implemented by any type of non-volatile storage device, or a combination thereof. Among them, the non-volatile memory can be a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), a magnetic random access memory (FRAM), a ferromagnetic random access memory (Flash Memory), a magnetic surface memory, an optical disc, or a compact disc read-only memory (CD-ROM); the magnetic surface memory can be a disk memory or a tape memory. The storage medium described in the embodiments of the present invention is intended to include, but is not limited to, these and any other suitable types of memory.

[0126] In the several embodiments provided by the present invention, it should be understood that the disclosed systems and methods can be implemented in other ways. The device embodiments described above are only schematic. For example, the division of the units is only a logical function division. There may be other division methods in actual implementation, such as: multiple units or components can be combined, or can be integrated into another system, or some features can be ignored or not executed. In addition, the coupling, direct coupling, or communication connection between the components shown or discussed can be through some interfaces, and the indirect coupling or communication connection of the devices or units can be electrical, mechanical or other forms.

[0127] The units described above as separate components may or may not be physically separated, and the components displayed as units may or may not be physical units, that is, they may be located in one place or distributed on multiple network units; some or all of the units may be selected according to actual needs to achieve the purpose of the present embodiment.

[0128] In addition, all functional units in the embodiments of the present invention may be integrated into one processing unit, or each unit may be separately used as a unit, or two or more units may be integrated into one unit; the above-mentioned integrated units may be implemented in the form of hardware or in the form of hardware plus software functional units.

[0129] A person skilled in the art can understand that: all or part of the steps of implementing the above method embodiment can be completed by hardware related to program instructions, and the aforementioned program can be stored in a computer-readable storage medium. When the program is executed, it executes the steps of the above method embodiment; and the aforementioned storage medium includes: mobile storage devices, read-only memories (ROM, Read-Only Memory), random access memories (RAM, Random Access Memory), disks or optical disks, etc. Various media that can store program codes.

[0130] Alternatively, if the above-mentioned integrated unit of the present invention is implemented in the form of a software function module and sold or used as an independent product, it can also be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the embodiment of the present invention can be essentially or partly reflected in the form of a software product that contributes to the prior art. The computer software product is stored in a storage medium and includes several instructions for a computer device (which can be a personal computer, a server, or a network device, etc.) to execute all or part of the methods described in each embodiment of the present invention. The aforementioned storage medium includes: various media that can store program codes, such as mobile storage devices, ROM, RAM, magnetic disks or optical disks.

[0131] The methods disclosed in the several method embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new method embodiments.

[0132] The features disclosed in several product embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new product embodiments.

[0133] The features disclosed in several method or device embodiments provided by the present invention can be arbitrarily combined without conflict to obtain new method embodiments or device embodiments.

[0134] The above contents are further detailed descriptions of the present invention in combination with specific preferred embodiments, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For those skilled in the art of the present invention, several equivalent substitutions or obvious variations can be made without departing from the concept of the present invention, and the performance or use is the same, which should be regarded as belonging to the protection scope of the present invention.

Claims

1. A method for estimating and simulating spatial point event distribution based on B-spline smoothing, characterized in that: The following steps are involved: S1: Collecting independently recorded spatial point position data corresponding to a number of spatial position nodes; S2: Based on the collected spatial point position data, if the number of independent recorded samples is less than the set sample number threshold, the least squares estimator smoothed by B-spline is used to obtain the optimal coefficient matrix by solving the constrained quadratic programming problem; if the number of independent recorded samples is greater than or equal to the set sample number threshold, the maximum likelihood estimator smoothed by B-spline is used to obtain the optimal coefficient matrix by solving the likelihood function maximization problem through the trust region algorithm; the set sample number threshold is determined based on the results of numerical experimental analysis conducted in advance; S3: Based on the estimated value of the optimal coefficient matrix obtained in step S2, an intensity function representing the probability of occurrence of a spatial point event is calculated, a thinning algorithm is used to generate pseudo data simulating the distribution of spatial point events according to the estimated value of the intensity function, and the pseudo data is applied to a Monte Carlo simulation to infer the expected value of a spatial statistic related to the spatial point event; The spatial point events and the corresponding spatial point location data include: location distribution of organisms in statistical ecology, and geographic coordinate data of organisms collected from ecological research; The maximum likelihood estimator using B-spline smoothing is used to solve the likelihood function maximization problem through a trust region algorithm to obtain an optimal coefficient matrix, which specifically includes: Constructing a log-likelihood function, including a log-likelihood term of the data, an integral term of the intensity function, and a regular term, wherein the regular term is used to ensure the smoothness and generalization ability of the estimator; Convert the log-likelihood function into a matrix form; Using the trust region algorithm, the optimal solution of the coefficient matrix C that maximizes the log-likelihood function is found through an iterative process under the premise of ensuring that the intensity function is non-negative; in each iteration, the boundary of the trust region is updated according to the current coefficient matrix estimate to ensure the stability and convergence of the search process; When the algorithm converges or reaches the preset number of iterations, the final coefficient matrix estimate is output; The least squares estimator using B-spline smoothing obtains the optimal coefficient matrix by solving the constrained quadratic programming problem, specifically including: Performing a two-dimensional gridding of the spatial region to create a plurality of grid cells, each grid cell defining a subregion in the space, so as to calculate an expected count of events within each grid cell; Construct the loss function W(C), where C is the coefficient matrix, W(C) includes the sum of the squares of the differences between the event counts and the cumulative intensity function in all grids, and the regularization term; Convert the loss function W(C) into a quadratic programming form and find the optimal coefficient matrix C that minimizes the loss function W(C); When the optimal solution that satisfies all constraints is found, the optimal coefficient matrix C is output as the least squares estimator.

2. The method for estimating and simulating spatial point event distribution based on B-spline smoothing according to claim 1, characterized in that: The set sample number threshold is 4-6.

3. The method for estimating and simulating spatial point event distribution based on B-spline smoothing according to claim 1, characterized in that: For the loss function W(C) converted into quadratic programming form, the constrained quadratic programming method is used to consider the non-negativity of the intensity function to solve the optimal coefficient matrix C that minimizes the loss function W(C).

4. The method for estimating and simulating spatial point event distribution based on B-spline smoothing according to any one of claims 1 to 3, characterized in that: The following steps are also included: S4: Based on the optimal coefficient matrix estimation value obtained in step S2, the weight of each control point in the B-spline is output, the weights of the spatial position nodes are sorted, and the key spatial position nodes with the greatest impact on the event occurrence pattern are identified.

5. The method for estimating and simulating spatial point event distribution based on B-spline smoothing according to any one of claims 1 to 3, characterized in that: The B-spline smoothing operation is to construct a piecewise polynomial curve to flexibly adapt to and approximate the spatial point event distribution, thereby achieving accurate estimation and simulation of the probability of event occurrence.

6. The method for estimating and simulating spatial point event distribution based on B-spline smoothing according to any one of claims 1 to 3, characterized in that: The spatial point event and the corresponding spatial point position data also include any one or more of the following items: The position distribution of stars and galaxies in astrostatistics, and the celestial coordinate data of stars and galaxies obtained from astronomical research; The location distribution of silicon wafer point defects in materials science, and the spatial location data of silicon wafer point defects recorded from materials science testing; The location distribution of AED devices in the emergency management field, and the geographical location data of AED device deployment recorded from the emergency management system.

7. A computer-readable storage medium storing a computer program, characterized in that: When the computer program is executed by a processor, the method for estimating and simulating spatial point event distribution based on B-spline smoothing as described in any one of claims 1 to 6 is implemented.

8. A computer program product, comprising a computer program, characterized in that When the computer program is executed by a processor, the method for estimating and simulating spatial point event distribution based on B-spline smoothing as described in any one of claims 1 to 6 is implemented.

Images