Physical-data cooperative driving long-span bridge typhoon effect probability prediction method

Abstract

The invention discloses a long-span bridge typhoon effect probability prediction method based on physical-data cooperative driving, and the method comprises the steps: obtaining typhoon and typhoon effect monitoring data recorded by a structure health monitoring system, and calculating typhoon characteristic parameters and bridge vibration response parameters; then, taking typhoon characteristic parameters as model input and bridge vibration response parameters as model output, constructing a sample set, and dividing the sample set into a training set and a test set; a deep integration strategy of physical-data cooperative driving is further adopted to adjust a neural network architecture, and a typhoon effect probability prediction model providing response mean value and variance dynamic estimation is constructed; then, based on the divided data set and the established probability prediction model, carrying out model training and response prediction; and finally, carrying out model interpretation analysis by adopting an improved Shapril additive interpretation method. According to the method, the typhoon effect prediction accuracy and uncertainty quantification performance of the long-span bridge can be effectively improved, and the robustness and interpretability of the model are synchronously enhanced.

Description

Technical Field

[0001] The present invention relates to the field of bridge wind engineering. By combining integrated learning and interpretability methods, real-time intelligent prediction of wind-induced vibration responses of long-span bridges with physical and numerical integration is carried out. Specifically, it relates to a physical-data collaborative-driven probabilistic prediction method for typhoon effects on long-span bridges. Background Art

[0002] Long-span bridges exhibit strong wind sensitivity due to their flexibility and slenderness. Especially for long bridges spanning mountains, valleys, rivers, lakes and seas, strong / typhoon events can endanger the structural stability, thereby increasing the risk of wind-induced disasters and traffic accidents. However, the management and operation and maintenance of bridges under extreme weather conditions still largely depend on the subjective experience of maintenance personnel, and effective early warning technologies have not been fully developed. Accurately predicting wind-induced responses helps to timely take reasonable intervention measures to ensure the wind resistance safety of bridges and the stability of their service states.

[0003] Data-driven methods have opened up a new way for the intelligent prediction of typhoon effects on bridges. Especially for algorithms based on machine learning and deep learning, they only capture the non-linear mapping between inputs and outputs from a data perspective, effectively avoiding complex geometric modeling. Currently, structural health monitoring systems have been deployed on many long-span bridges, and reliable platforms for the intelligent perception of typhoon effects are provided by the collaborative acquisition of measured data through multiple devices. However, some problems still emerge in the process of practice: data-driven models usually only rely on training data sets to learn the inline laws of wind-induced vibrations, so predictions beyond the data sets may be doubtful; most practices only focus on deterministic predictions, that is, generating single-value outputs for each independent sample point, and the uncertainty of predictions is not fully considered; when the training samples are insufficient, single models often have problems such as overfitting, resulting in unstable prediction performance and poor robustness; due to the black-box nature of the models, data-driven methods have limitations in prediction scenarios that emphasize high interpretability.

[0004] In recent years, physics-data co-driven approaches have become increasingly attractive. Among them, physical laws and equations are regarded as prior knowledge and integrated into surrogate models to guide the training and prediction processes. On this basis, the model architecture is further appropriately adjusted to achieve the synchronous output of the response mean and variance, so as to adapt to prediction uncertainties. To further enhance the physics-data fusion performance, an ensemble learning strategy is used to strategically integrate multiple surrogate models with random initializations, which has significant effects in improving the stability and robustness of probabilistic predictions. Explainable artificial intelligence helps to understand why the model produces specific prediction behaviors by quantitatively visualizing feature contributions, thereby cultivating trust in the model. However, the above methods have not yet formed a systematic analysis framework in the field of bridge wind engineering. Therefore, it is urgent to carry out probabilistic predictions of typhoon effects on long-span bridges from the perspective of physics-data co-driven based on ensemble learning and explainable methods to improve the effectiveness and reliability of prediction results. Summary of the Invention

[0005] Technical Problem: Aiming at the problems of limited prediction accuracy, difficulty in quantifying uncertainties, unstable prediction performance, and lack of interpretability in existing typhoon effect prediction methods, the present invention proposes a physics-data co-driven probabilistic prediction method for typhoon effects on long-span bridges, so as to effectively improve the prediction accuracy of the vibration response of long-span bridges under typhoon action and the uncertainty quantification performance, while taking into account the marginal contribution analysis of feature variables to the prediction results and enhancing the interpretability of the model.

[0006] Technical Solution: The present invention is a physics-data co-driven probabilistic prediction method for typhoon effects on long-span bridges, and the method includes the following steps: S1. Obtain the monitoring data of typhoons and their effects recorded by the structural health monitoring system of the long-span bridge, and calculate the typhoon characteristic parameters and the bridge vibration response parameters; S2. Using the typhoon characteristic parameters calculated in step S1 as the model input and the bridge vibration response parameters as the model output, construct a sample set, and divide the sample sets of multiple typhoons and their effects into a training set and a test set; S3. Adopt a physics-data co-driven deep ensemble strategy to adjust the scalable neural network architecture, and construct a probabilistic prediction model for typhoon effects that provides dynamic estimates of the response mean and variance; S4. Based on the training set and test set divided in step S2 and the probabilistic prediction model established in step S3, carry out model training and response prediction; S5. Adopt an improved Shapley additive explanation method to carry out explanation analysis of the probabilistic prediction model, and clarify the marginal contributions of each feature variable and their substantial impacts on the final prediction results.

[0007] Among them, In step S1, the typhoon and its effect monitoring data are obtained based on the anemometer and accelerometer at the mid-span position of the main girder; the typhoon characteristic parameters include average wind speed, wind direction angle, turbulence intensity, turbulence standard deviation, turbulence integral scale, and gust factor; the bridge vibration response parameters include vertical acceleration and torsional acceleration.

[0008] In step S2, the total number of samples of the typhoon and its effects is not less than 10, the test set contains 2 independent samples of the typhoon and its effects, and the remaining samples are used as the training set.

[0009] In step S3, a physics-data co-driven deep integration strategy is used to adjust the scalable neural network architecture. The specific steps are as follows: S31, using the scoring rule as the training criterion, for a model with input X , output Y , and hyperparameters , let its predicted distribution , the true distribution , the ideal scoring rule is written as: (1) In the formula, is the scoring function; represents the performance score of the predicted distribution under the true distribution ; represents the theoretical optimal score of predicting the true distribution itself; when and only when , ; when maximizing the likelihood, the scoring rule satisfying the Gibbs inequality is expressed as: (2) S32, in the physics-data co-driven method, the loss function of the surrogate model needs to consider the physics-informed loss term and the data-driven loss term ; for the physics-informed loss term , taking the bridge wind vibration equation as the control condition: (3) In the formula, M , C , K represent the mass, damping, and stiffness matrices respectively; ü ( x , t ), , u ( x , t ) represent the spatial coordinates x and the time scalet Acceleration, velocity, and displacement vectors under F aero ( x , t ) represents the external wind load vector, which consists of aerodynamic lift F L (t), aerodynamic drag F D (t), and aerodynamic torque F M (t). When the bridge buffets, they can be expressed respectively as follows: (4) (5) (6) In the formula, ρ is the air density; U ( t ) is the incoming flow wind speed; B is the width of the main girder; H is the height of the main girder; α ( t ) is the effective wind attack angle of the main girder section; C L α ( t )], C D α ( t )], C M α ( t )] are the lift coefficient, drag coefficient, and torque coefficient respectively; C L ’ α ( t )], C D ’ α ( t )], C M ’ α ( t )] are the first-order derivatives of the lift coefficient, drag coefficient, and torque coefficient with respect to the wind attack angle respectively; χ L , χ L ’ , χ D , χ D ​​​’ , χ M , χ M ’ is the aerodynamic admittance function; h ( t ) and v ( t ) are the non-stationary fluctuating wind speeds in the along-wind and vertical directions, respectively; On this basis, the bridge acceleration response is further expressed as: (7) (8) (9) In the formula, and respectively represent the vertical acceleration components on the left and right sides of the main girder; and respectively represent the vertical acceleration and torsional acceleration; Accordingly, the physics-informed loss term is expressed as: (10) In the formula, Y v and Y t respectively represent the vertical acceleration and torsional acceleration predicted by the model output; β 1 and β 2 respectively represent the weights of the vertical acceleration residual term and the torsional acceleration residual term; For the data-driven loss term , the Gaussian negative log-likelihood is introduced to simultaneously obtain the mean and variance of the independent model to adapt to the uncertainty of the prediction: (11) The final loss function is composed of the above two parts of loss terms: (12) In the formula, and are respectively the weights of the physics-informed loss term and the data-driven loss term ; the goal of model training is to minimize the loss function ; S33. Apply adversarial training to smooth the prediction distribution. Add adversarial perturbations in the gradient direction that may increase the loss based on the fast gradient sign method to generate pseudo-samples similar to the real samples, so as to enhance the original dataset. This process is written as: (13) In the formula, X p is the generated adversarial sample; ξ is a small value to keep the maximum norm of the perturbation bounded; sign is to take the sign of the gradient; represents the gradient of the loss function in the input space; S34. In the deep ensemble system, introduce a uniform weighted mixing strategy to approximate the Gaussian distribution. On this basis, multiple groups of predictions are integrated as: (14) The final output result based on ensemble learning is expressed as: (15) (16) In the formula, is the uniform weighted distribution function; N and i represent the total number of models and the model serial number of ensemble learning respectively; is the i th model's prediction distribution; is the i th model's hyperparameter; and represent the mean and variance of the i th model respectively; and correspond to the mean and variance of the final prediction respectively.

[0010] In step S4, use annealing search to obtain the local optimal setting of the model hyperparameters, and then apply the Bayesian algorithm to update the prior distribution, so as to identify the global optimal setting.

[0011] In step S5, when calculating the marginal contribution of the feature variables using the improved Shapley additive explanation method, the feature interaction effect is considered. The specific calculation method is as follows: (17) (18) In the formula, r and s represent different feature terms; P and Q are the finite set not including the analyzed feature and the complete set including all features respectively; is a predictor; is the Shapley value considering r and s the interaction effect of features; is the Shapley value of all combinations of the interaction effect of r a feature and other feature variables; is the base value; q is the order of the combination; The substantial impact of each feature variable on the final prediction result is quantified as follows: (19) (20) In the formula, and are the average absolute Shapley value of feature r and its corresponding contribution ratio; p is the number of features.

[0012] Beneficial effects: Compared with the prior art, the present invention has the following advantages: (1) Different from the pure data-driven method, the physics-data co-driven method integrates physical laws and equations as prior knowledge into the surrogate model, thereby guiding model training and response prediction, effectively improving the rationality and accuracy of the prediction result; (2) Different from the deterministic prediction that only provides a single-value output, by introducing the Gaussian negative log-likelihood, the mean and variance are synchronously obtained from the perspective of probabilistic prediction, realizing good uncertainty quantification performance, and providing more comprehensive information for typhoon disaster prevention; (3) An ensemble learning strategy is adopted to strategically integrate multiple independent models to obtain more reliable prediction results, effectively enhancing the stability and robustness of the surrogate model; (4) An improved Shapley additive explanation method is adopted to carry out model interpretation analysis, clarifying the marginal contribution and substantial impact of feature variables on the prediction result on the basis of considering the feature interaction effect, and enhancing the interpretability of the surrogate model. Brief description of the drawings

[0013] Figure 1 is the technical flow chart of the probability prediction method for typhoon effects of long-span bridges driven by physics-data collaboration; Figure 2 is the specific classification schematic diagram of model input and model output; Figure 3 is the schematic diagram of the analysis framework for model training and response prediction; Figure 4 is the schematic diagram of hyperparameter optimization based on annealing search and Bayesian algorithm. Detailed implementation manners

[0014] Combined with the accompanying drawings in the embodiments of the present application, the technical solutions in the embodiments of the present application are clearly and completely described. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative efforts belong to the scope of protection of the present application.

[0015] The present application proposes a physical-data collaborative-driven probabilistic prediction method for typhoon effects on long-span bridges. The main technical process of the implementation solution is as follows (see Figure 1 ): S1. Obtain the typhoon and its effect monitoring data recorded by the long-span bridge structural health monitoring system, and calculate the typhoon characteristic parameters and bridge vibration response parameters.

[0016] Specifically, the typhoon and its effect monitoring data are obtained based on the anemometer and accelerometer at the mid-span position of the main girder; the typhoon characteristic parameters include average wind speed, wind direction angle, turbulence intensity, turbulence standard deviation, turbulence integral scale, and gust factor; the bridge vibration response parameters include vertical acceleration and torsional acceleration.

[0017] S2. Using the typhoon characteristic parameters calculated in step S1 as the model input and the bridge vibration response parameters as the model output, construct a sample set (see Figure 2 ), and divide the sample sets of multiple typhoons and their effects into training sets and test sets.

[0018] Specifically, the total number of sample sets of typhoons and their effects is not less than 10, the test set contains 2 independent sample sets of typhoons and their effects, and the remaining sample sets are used as training sets.

[0019] S3. Adopt a physical-data collaborative-driven deep integration strategy to adjust the scalable neural network architecture, and construct a typhoon effect probability prediction model that provides dynamic estimation of response mean and variance.

[0020] Specifically, the steps to adopt a physical-data collaborative-driven deep integration strategy to adjust the scalable neural network architecture are as follows: S31. Use the scoring rule as the training criterion. For a model with input X , output Y , and hyperparameter , let its prediction distribution , the true distribution , and the ideal scoring rule is written as: (1) In the formula, is the scoring function; represents the true distribution under which the prediction distribution Performance score; Represents the true distribution The theoretically optimal score for prediction by itself; if and only if When ; When maximizing the likelihood, the scoring rule that satisfies Gibbs' inequality is expressed as: (2) S32. In the physics-data co-driven method, the loss function of the surrogate model needs to consider the physics-informed loss term And the data-driven loss term ; For the physics-informed loss term , the bridge wind-induced vibration equation is used as the control condition: (3) In the formula, M , C , K Respectively represent the mass, damping, and stiffness matrices; ü ( x , t ), , u ( x , t ) Respectively represent the acceleration, velocity, and displacement vectors at the spatial coordinate x And the time scale t ; F aero ( x , t ) Represents the external wind load vector, which consists of the aerodynamic lift F L (t), the aerodynamic drag F D (t) and the aerodynamic torque F M (t) and consists of three parts. During bridge buffeting, they can be expressed respectively as follows: (4) (5) (6) In the formula, ρ Is the air density; U ( t ) Is the incoming flow wind speed; B Is the main girder width; H Is the main girder height; α ( t ) Is the effective wind attack angle of the main girder section; C L α (​t )], C D α ( t )], C M α ( t )] are the lift coefficient, drag coefficient, and torque coefficient respectively; C L ’ α ( t )], C D ’ α ( t )], C M ’ α ( t )] are the first-order derivatives of the lift coefficient, drag coefficient, and torque coefficient with respect to the wind attack angle respectively; χ L , χ L ’ , χ D , χ D ’ , χ M , χ M ’ are the aerodynamic admittance functions; h ( t ) and v ( t ) are the crosswind and vertical non-stationary fluctuating wind speeds respectively; On this basis, the bridge acceleration response is further expressed as: (7) (8) (9) In the formula, and respectively represent the vertical acceleration components on the left and right sides of the main girder; and respectively represent the vertical acceleration and torsional acceleration; Accordingly, the physically informed loss term is expressed as: (10) In the formula,​​Y v and Y t respectively represent the vertical acceleration and torsional acceleration of the model prediction output; β 1 and β 2 respectively represent the weights of the vertical acceleration residual term and the torsional acceleration residual term; For the data-driven loss term , the Gaussian negative log-likelihood is introduced to synchronously obtain the mean and variance of the independent model to adapt to the uncertainty of the prediction: (11) The final loss function consists of the above two parts of loss terms: (12) In the formula, and are respectively the physical-informed loss term and the data-driven loss term weights; the goal of model training is to minimize the loss function ; S33. Apply adversarial training to smooth the prediction distribution. Add adversarial perturbations in the gradient direction that may increase the loss based on the fast gradient sign method to generate pseudo-samples similar to the real samples to enhance the original dataset. This process is written as: (13) In the formula, X p is the generated adversarial sample; ξ is a small value to keep the maximum norm of the perturbation bounded; sign is the sign of taking the gradient; represents the gradient of the loss function in the input space; S34. In the deep ensemble system, introduce a uniform weighted mixing strategy to approximate the Gaussian distribution. On this basis, multiple groups of predictions are integrated as: (14) The final output result based on ensemble learning is expressed as: (15) (16) In the formula, is the uniform weighted distribution function; N and i respectively represent the total number of models and the model number of ensemble learning; is the i th model's prediction distribution; For the i hyperparameters of the th model; i and represent the mean and variance of the th

[0021] model respectively; Figure 3 and

[0022] correspond to the mean and variance of the final prediction respectively. Figure 4 S4. Based on the training set and test set divided in step S2 and the probability prediction model established in step S3, carry out model training and response prediction (see

[0023] ).

[0024] Specifically, use annealing search to obtain the local optimal settings of the model hyperparameters, and then apply the Bayesian algorithm to update the prior distribution to identify the global optimal settings (see ). S5. Use the improved Shapley additive explanation method to carry out the explanation analysis of the probability prediction model, and clarify the marginal contributions of each feature variable and its substantial impact on the final prediction result. Specifically, when calculating the marginal contributions of feature variables using the improved Shapley additive explanation method, the feature interaction effect is considered, and the specific calculation method is as follows: r and s represent different feature terms; P and Q are the finite set excluding the analysis feature and the complete set including all features respectively; is the predictor; is the Shapley value considering the interaction effect of features r and s ; is the Shapley value considering all combinations of the interaction effect of feature r with other feature variables; is the base value; q is the order of the combination; The substantial impact of each feature variable on the final prediction result is quantified in the following way: (19) (20) where and are the average absolute Shapley value of feature r and its corresponding contribution ratio respectively; p is the number of features.

[0025] Those of ordinary skill in the art will realize that the embodiments described herein are provided to assist the reader in understanding the principles of the present invention, and it should be understood that the scope of protection of the present invention is not limited to such specific statements and embodiments. Those of ordinary skill in the art can make various other specific deformations and combinations that do not depart from the essence of the present invention based on these technical revelations disclosed in the present invention, and these deformations and combinations are still within the scope of protection of the present invention.

Claims

1. A physical-data collaborative driven probabilistic prediction method for typhoon effects of long-span bridges, characterized in that, The method specifically includes the following steps: S1. Obtain the typhoon and its effect monitoring data recorded by the long-span bridge structural health monitoring system, and calculate the typhoon characteristic parameters and bridge vibration response parameters; S2. Use the typhoon characteristic parameters calculated in step S1 as the model input and the bridge vibration response parameters as the model output to construct a sample set, and divide the sample sets of multiple typhoons and their effects into a training set and a test set; S3. Adopt a physics-data co-driven deep integration strategy to adjust the scalable neural network architecture, and construct a typhoon effect probability prediction model that provides dynamic estimates of response mean and variance; S4. Based on the training set and test set divided in step S2 and the probability prediction model established in step S3, carry out model training and response prediction; S5. Adopt an improved Shapley additive explanation method to carry out explanation analysis of the probability prediction model, and clarify the marginal contributions of each characteristic variable and its substantial impact on the final prediction result.

2. The probability prediction method for typhoon effects of long-span bridges driven by physical-data collaboration according to claim 1, characterized in that In step S1, the typhoon and its effect monitoring data are obtained based on the anemometer and accelerometer at the mid-span position of the main girder; The typhoon characteristic parameters include average wind speed, wind direction angle, turbulence intensity, turbulence standard deviation, turbulence integral scale, and gust factor; the bridge vibration response parameters include vertical acceleration and torsional acceleration.

3. The physical-data collaborative-driven probabilistic prediction method for typhoon effects of long-span bridges according to claim 1, characterized in that In step S2, the total number of sample sets of typhoons and their effects is not less than 10, the test set contains 2 independent sample sets of typhoons and their effects, and the remaining sample sets are used as the training set.

4. The probability prediction method for typhoon effects of long-span bridges driven by physical-data collaboration according to claim 1, wherein In step S3, the scalable neural network architecture is adjusted by adopting a physics-data co-driven deep integration strategy. The specific steps are as follows: S31, using the scoring rule as the training criterion, for the model with the input X , the output is Y , and the hyperparameters are , let its predicted distribution be , the true distribution , the ideal scoring rule is written as: （1） In the formula, is the scoring function; represents the true distribution under which the performance score of the predicted distribution is; represents the theoretical optimal score for making predictions by the true distribution itself; if and only if then ; when maximizing the likelihood, the scoring rule that satisfies Gibbs' inequality is expressed as: （2） S32. In the physical-data collaborative driving method, the loss function of the surrogate model needs to consider the physics-informed loss term and the data-driven loss term ; for the physics-informed loss term , the bridge wind-induced vibration equation is used as the control condition: （3） In the formula, M , C , K represent the mass, damping, and stiffness matrices respectively; ü ( x , t ), , u ( x , t ) represent the acceleration, velocity, and displacement vectors at the spatial coordinate x and the time scale t respectively; F aero ( x , t ) represents the external wind load vector, which consists of three parts: the aerodynamic lift F L (t), the aerodynamic drag F D (t), and the aerodynamic torque F M (t). During the buffeting of the bridge, they can be expressed separately as follows: （4） （5） （6） In the formula, ρ is the air density; U ( t ) is the incoming flow wind speed; B is the width of the main beam; H is the height of the main beam; α ( t ) is the effective wind attack angle of the main beam section; C L α ( t )], C D α ( t )], C M α ( t )] are the lift coefficient, drag coefficient, and torque coefficient respectively; C L ’ α ( t )], C D ’ α ( t )], C M ’ α ( t )] are the first-order derivatives of the lift coefficient, drag coefficient, and torque coefficient with respect to the wind attack angle respectively; χ L , χ L ’ , χ D , χ D ’ , χ M , χ M ’ are the aerodynamic admittance functions; h ( t ) and v ( t ) are the non-stationary fluctuating wind speeds in the along-wind and vertical directions respectively;​​​​ On this basis, the bridge acceleration response is further expressed as: （7） （8） （9） In the formula, and respectively represent the vertical components of acceleration on the left and right sides of the main beam; and respectively represent the vertical acceleration and the torsional acceleration; Accordingly, the physics-informed loss term is expressed as: （10） In the formula, Y v and Y t respectively represent the vertical acceleration and torsional acceleration of the model prediction output; β 1 and β 2 respectively represent the weights of the vertical acceleration residual term and the torsional acceleration residual term; For the data-driven loss term , the Gaussian negative log-likelihood is introduced to synchronously obtain the mean and variance of the independent model to accommodate the uncertainty in prediction: （11） Final loss function From the above two loss terms Composition: （12） wherein, and are the weights of the physics-informed loss term and the data-driven loss term respectively; The goal of model training is to minimize the loss function ; S33. Apply adversarial training to smooth the prediction distribution, add adversarial perturbations in the gradient direction that may increase the loss based on the fast gradient sign method, and generate pseudo-samples similar to the real samples to enhance the original data set. This process is written as: （13） wherein, X p is the generated adversarial sample; ξ is a small value for keeping the maximum norm of the perturbation bounded; sign is the sign of taking the gradient; represents the gradient of the loss function in the input space; S34. In the deep integration system, introduce a uniform weighted mixing strategy to approximate the Gaussian distribution. On this basis, multiple groups of predictions are integrated as: （14） The final output result based on ensemble learning is expressed as: （15） （16） In the formula, is the uniform weighted distribution function; N and i respectively represent the total number of models and the model serial number in the ensemble learning; is the prediction distribution of the i -th model; is the hyperparameter of the i -th model; and respectively represent the mean and variance of the i -th model; and respectively correspond to the mean and variance of the final prediction.

5. The probability prediction method for typhoon effects of long-span bridges driven by physical-data collaboration according to claim 1, characterized in that In step S4, adopt annealing search to obtain the local optimal setting of the model hyperparameters, and then apply the Bayesian algorithm to update the prior distribution to identify the global optimal setting.

6. The probability prediction method for typhoon effects of long-span bridges driven by physical-data collaboration according to claim 1, wherein In step S5, when calculating the marginal contribution of the characteristic variable by adopting the improved Shapley additive explanation method, the characteristic interaction effect is considered. The specific calculation method is as follows: （17） （18） In the formula, r and s represent different characteristic items; P and Q are respectively a finite set excluding analysis features and a complete set including all features; is a predictor; is the Shapley value considering the interaction effect of feature r and s ; is the Shapley value considering all combinations of the interaction effect of feature r and other feature variables; is a base value; q is the order of the combination; The substantial impact of each characteristic variable on the final prediction result is quantified by the following method: （19） （20） In the formula, and are the average absolute Shapley value of feature r and its corresponding contribution ratio, respectively; p is the number of features.

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