CN115495991A - Rainfall interval prediction method based on time convolution network - Google Patents

Abstract

The invention discloses a rainfall interval prediction method based on a time convolution network, which comprises the following steps of collecting rainfall data and meteorological factor data to form an initial time sequence data set, and preprocessing the data; analyzing the characteristics of meteorological factors, constructing a characteristic extraction algorithm, acquiring the incidence relation between the meteorological factors and precipitation, and screening out a time sequence of the meteorological factors which have great influence on precipitation by adopting a maximum information coefficient-asynchronous principal component analysis algorithm; and (3) constructing a time convolution network model for predicting the drainage surface rainfall by combining the drainage surface rainfall, training the model by using data in the training set, then adjusting the model parameter evaluation model to converge the model, and finally predicting by using the test set data. According to the method, the capability of TCN for capturing effective long-time sequence features is utilized, the optimized LUBE coverage width evaluation standard is used as one of training loss objective functions to generate probability interval prediction of future observation results, and the precision of rainfall prediction is effectively improved.

Description

A Precipitation Interval Prediction Method Based on Time Convolution Network

technical field

The invention belongs to the technical field of precipitation forecasting, and in particular relates to a precipitation interval forecasting method based on a temporal convolutional network.

Background technique

Mid- and long-term precipitation forecasting is an important basis for precise management of water resources in river basins. Data-driven models are one of the important ways to carry out medium and long-term precipitation forecasting. The start, duration, and end of the rainy season are determined by monthly precipitation, and monthly precipitation provides more accurate precipitation distribution data within a year than seasonal precipitation, so it is necessary to conduct research on precipitation prediction on a monthly time scale. The sample size of precipitation statistical data is limited, and the available predictors have a high dimension. Under the condition of limited sample size, the key to successful medium- and long-term precipitation prediction is to screen out highly relevant meteorological factors and build a robust data-driven model.

In precipitation prediction, commonly used models are mainly based on physical (process) driven and data-driven. The physical driving model is based on the hydrological conceptual model. This method analyzes the various factors of precipitation formation, and establishes a general physical equation suitable for a certain watershed to simulate the precipitation process. The model parameters need to rely on experience and manual interaction to complete the calibration. The accuracy of the model forecast depends on the knowledge and experience of the modeler and the completeness of the data. The parameters contained in this type of model have physical meaning and have good explanatory properties, but the complex factors affecting medium and long-term precipitation increase the difficulty of parameter calibration. In recent years, data-driven models such as linear regression, neural network, and support vector machines have been used in precipitation forecasting. These models regard the hydrological process as a black box, regardless of the internal physical mechanism of the system, and establish a mapping between input and output samples. relationship to model. Due to the inability to obtain accurate precipitation and its influencing factors (precipitation and related meteorological factors), the intelligent model has certain limitations in forecasting performance. In precipitation forecasting, there are uncertainties in various meteorological factors, and the factors are interrelated, which makes both physical-driven models and data-driven models have their own shortcomings.

Contents of the invention

Purpose of the invention: The purpose of the present invention is to provide a precipitation interval prediction method based on a temporal convolutional network, which is suitable for medium and long-term precipitation interval prediction and has high forecasting accuracy.

Technical solution: The present invention provides a method for predicting intervals of precipitation based on time convolution network, comprising the following steps:

(1) Collect precipitation data and meteorological factor data to form an initial time series data set and preprocess the data;

(2) Analyze the characteristics of meteorological factors, build a feature extraction algorithm, obtain the correlation between meteorological factors and precipitation, and use the maximum information coefficient-asynchronous principal component analysis algorithm to screen out the time series of meteorological factors that have a greater impact on precipitation;

(3) Construct a precipitation interval prediction model based on time convolution network, divide the screened data into training set and test set according to the hold-out method, and then perform sliding window sampling to prepare the input of precipitation interval prediction model based on time convolution network data set;

(4) According to the input historical precipitation and highly correlated historical meteorological factor sequence, the TCN time convolutional network model is selected to construct the precipitation interval prediction model based on the time convolutional network, and the LUBE interval prediction method is introduced to output the interval prediction value to realize The training of the precipitation interval prediction model based on the temporal convolutional network completes the comprehensive modeling of the precipitation process;

(5) For the precipitation interval prediction model based on the time convolutional network that has been trained, input new precipitation to complete the prediction of the precipitation interval.

Further, the preprocessing of the data in step (1) includes data cleaning and data missing completion, wherein the data missing completion adopts a linear interpolation processing method in the time dimension.

Further, the implementation process of the time series of meteorological factors that have a greater impact on precipitation using the maximum information coefficient-asynchronous principal component analysis algorithm in step (2) is as follows:

最佳变形路径的所有元素构成插值时间序列的元素，分别是：X″_i＝{p₁(1),p₂(1),…,p_k(1)}和X″_j＝{p₁(2),p₂(2),…,p_k(2)}所有归一化的时间序列应扩展到反映相应异步相关性的时间序列；获得新的插值时间序列，即X＝X”；在2个新的时间序列的散点图，即集合D上进行X″_i×Y划分，将其元素按X″_i值划分到X″_i个格子中，按Y值划分到Y个格子中；计算集合D的点落在给定的网格G上所得到的频率分布D|G，合理选择网格G的划分上界B(n)，B(n) 是关于样本大小的函数，表示网格G的正方形总数b的约束小于B(n)，B(n)＝n^0.6；计算不同的网格G确定不同的概率分布，即I(D|G)，表示点集基于分布D|G的互信息；在基于X″_i×Y网格G的所有可能分布D|G的互信息中找到最大的互信息 maxI(D|G)记为I”(D,m,n)，得到二维数据集D的特征矩阵I”(D)；Construct interpolation time series and asynchronous correlation time series through DTW: organize data set X _m×n , where m in each column represents the length of univariate time series, and n represents the number of attributes in each row; for each time series in data matrix X Perform normalization; use the Z-score method to normalize the time series Xi, so that the standardized X' _i has zero mean and one standard deviation, that is, X' _ih ~ N(0,1); use DTW to calculate the normalization After X′ _i and X _j ', the best deformation path is obtained

All the elements of the optimal deformation path constitute the elements of the interpolation time series, respectively: X″ _i ={p ₁ (1),p ₂ (1),…,p _k (1)} and X″ _j ={p ₁ (2),p ₂ (2),...,p _k (2)} All normalized time series should be extended to time series reflecting corresponding asynchronous correlations; new interpolated time series are obtained, i.e. X = X"; Carry out X″ _i ×Y division on the 2 new time series scatter plots, that is, set D, divide its elements into X″ _i grids according to X″ _i values, and divide them into Y grids according to Y values ;Calculate the frequency distribution D|G obtained by calculating the points of the set D falling on the given grid G, and choose the upper bound B(n) of the grid G reasonably. B(n) is a function of the sample size, which means The constraint of the total number of squares b of the grid G is less than B(n), B(n)=n ^0.6 ; different grids G are calculated to determine different probability distributions, namely I(D|G), which means that the point set is based on the distribution D| Mutual information of G; find the maximum mutual information maxI(D|G) in the mutual information of all possible distributions D|G based on X″ _i ×Y grid G and denote it as I”(D,m,n), and get The characteristic matrix I"(D) of the two-dimensional data set D;

The maximum information coefficient is used to effectively measure the correlation between the forecast factor and the actual precipitation sequence, and the factors with strong correlation with the actual precipitation are screened out, which are recorded as high-information factors, and the factors with more overlapping information are removed from the high-information factors .

Further, the step (4) selects the TCN time convolution network model to construct the precipitation interval prediction model based on the time convolution network. The realization process is as follows:

In the case of univariate sequences, for a given precipitation input X of a 1D sequence and a filter function ω of size k:{0,...,k-1}, the full dilated causal convolution operation f on successive layers , the definition on the sequence s is as follows:

where, s is the element of the sequence, d is the dilation parameter, which increases exponentially according to the network depth d= ²ⁱ , and d= ²ⁱ is used for the i-level of the network; while sd i describes the past direction; call *d the dilation The convolution operation to distinguish the normal convolution operation.

Further, the implementation process of introducing the LUBE interval prediction method to output the interval prediction value in step (4) is as follows:

The evaluation indicators of the LUBE interval prediction method include PICP, which is evaluated from the probability that the actual observed value is between the upper and lower boundaries of the prediction interval; PINAW, which is evaluated from the width between the upper and lower boundaries of the prediction interval:

Among them, U(x _k ) is the upper boundary value of interval prediction, L(x _k ) is the lower boundary value of interval prediction, n is the sample number of measured values, A is the range of the target variable, that is, the difference between the maximum and minimum values ; then the CWC measuring width and coverage probability can be defined as:

Adding the penalty parameter, the metrics considering width, coverage probability and mean deviation are defined as:

指数放大PICP和e^-η(PICP-μ)之间的差异如果PCWC太小，则使用ε和τ超参数来避免消失。Among them, the parameter τ linearly amplifies PINAW, and the parameter

Exponentially amplify the difference between PICP and e ^{-η (PICP-μ)} If PCWC is too small, ε and τ hyperparameters are used to avoid vanishing.

Beneficial effect: compared with the prior art, the beneficial effect of the present invention: the present invention utilizes the ability of TCN to capture effective long-term sequence features, and uses the optimized LUBE coverage width evaluation standard as one of the training loss objective functions to generate The probability interval prediction of future observation results can effectively improve the accuracy of precipitation prediction; the input data is preprocessed, and the interpolation time series is constructed through DTW, and the MIC matrix is introduced into the APCA method to perform feature screening and further excavate the asynchronous correlation between meteorological factors and precipitation , which is more suitable for feature selection of high-dimensional meteorological factors; the present invention is more suitable for mid- and long-term precipitation interval prediction, has higher prediction accuracy, and is superior to traditional models such as support vector machines, and the prediction accuracy is greatly improved.

Description of drawings

Fig. 1 is a flow chart of the present invention;

Fig. 2 is a schematic diagram of causal dilated convolution in the TCN time convolutional network model in the embodiment of the present invention.

detailed description

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

As shown in Figure 1, the present invention provides a method for predicting intervals of precipitation based on time convolutional networks. First, collect hydrological data in the research basin, and then store the above data in a historical database; Preprocessing such as completeness, data anomaly correction, and data normalization, and then performing feature screening to divide the training set and test set; constructing a precipitation interval prediction model based on a temporal convolutional network, using the data in the training set to train the model, and then adjusting the model parameter evaluation Model, make the model converge, and finally use the test set data to make predictions; specifically include the following steps:

Step 1: Collect precipitation data and meteorological factor data to form an initial time series data set, clean the data, and supplement missing data.

Collect the hydrological data of the target watershed (Qijiang River Basin) from 1979 to 2019 with a data granularity of 1 piece per month. Each piece of data includes precipitation in the watershed, atmospheric circulation factors, sea temperature factors and other meteorological factors, and then store the above data in Historical database; take out data from the historical database and perform data preprocessing, including data missing completion. The present invention uses the linear interpolation method in the time dimension to complete the missing data, that is, the missing value is determined according to the time values before and after the missing part.

Step 2: Analyze the characteristics of meteorological factors, build a feature extraction algorithm for input data, use the maximum information coefficient-asynchronous principal component analysis to filter out the time series of meteorological factors that have a greater impact on precipitation, and obtain the correlation between meteorological factors and precipitation.

Interpolation time series and asynchronous correlation time series are constructed by DTW; the correlation relationship between forecast factors and actual precipitation series is effectively measured by the maximum information coefficient, and the factors with strong correlation with actual precipitation are screened out. These factors are usually high information factors, which are important for forecasting Variables have a significant impact; factors with more overlapping information are removed from high-information factors, and the specific process is as follows:

最佳变形路径的所有元素构成插值时间序列的元素，分别是：X″_i＝{p₁(1),p₂(1),…,p_k(1)}和X″_j＝{p₁(2),p₂(2),…,p_k(2)}所有归一化的时间序列应扩展到反映相应异步相关性的时间序列。获得新的插值时间序列，即X＝X”。在2个新的时间序列的散点图(集合D)上进行X″_i×Y划分，将其元素按X″_i值划分到X″_i个格子中，按Y值划分到Y个格子中。计算集合D的点落在给定的网格G上所得到的频率分布D|G，合理选择网格G的划分上界B(n)，B(n) 是关于样本大小的函数，表示网格G的正方形总数b的约束小于B(n)，B(n)＝n^0.6计算不同的网格G确定不同的概率分布，即I(D|G)，表示点集基于分布D|G的互信息。在基于X″_i×Y网格G的所有可能分布D|G的互信息中找到最大的互信息 maxI(D|G)记为I”(D,m,n)，得到二维数据集D的特征矩阵I”(D)；从上述结果中得到的结果中找出最大者，即为最大信息系数。

分别计算各气象预报因子与实测降水间的最大互信息系数MIC(X″_i,Y)，并按照从大到小的顺序依次排列，筛选出排名靠前的若干因子组成新的因子集X'＝ {X1，X2，…，Xr}，r≤n。分别计算因子集X'中各因子之间的MIC值 MIC(X″_i,X″_j)(i，j≤n)，组成MIC特征矩阵。计算MIC特征矩阵的特征值并使其按从大到小顺序排列λ₁≥λ₂≥…≥λ_m≥0。分别求出对应于特征值λ_i的特征向量e_i(i＝1,2,…,m),其中||e_i||＝1。Organize the data set (or data matrix) X _m×n , where each column m represents the length of the univariate time series, and n represents the number of attributes (or variables) in each row; normalize each time series in the data matrix X change. Use the Z-score method to normalize the time series Xi, so that the standardized X' _i has zero mean and one standard deviation, that is, X' _ih ~N(0,1). Obtain an interpolated time series. Calculate the normalized X' _i and X _j ' with DTW to obtain the best deformation path

All the elements of the optimal deformation path constitute the elements of the interpolation time series, respectively: X″ _i ={p ₁ (1),p ₂ (1),…,p _k (1)} and X″ _j ={p ₁ (2),p ₂ (2),...,p _k (2)} All normalized time series should be extended to time series reflecting the corresponding asynchronous correlations. Obtain a new interpolation time series, that is, X=X". Carry out X" _i ×Y division on the scatter diagram (set D) of the two new time series, and divide its elements into X" _i according to the value of X" _i In the grid, it is divided into Y grids according to the Y value. Calculate the frequency distribution D|G obtained by calculating the points of the set D falling on the given grid G, and choose the upper bound B(n) of the grid G reasonably. B(n) is a function of the sample size, which means that the network The constraint of the total number of squares b of the grid G is less than B(n), B(n)=n ^0.6 Calculate different grid G to determine different probability distributions, that is, I(D|G), which means that the point set is based on the distribution D|G Mutual information. Find the maximum mutual information maxI(D|G) in the mutual information of all possible distributions D|G based on X″ _i ×Y grid G, and record it as I”(D,m,n), and obtain a two-dimensional data set D The characteristic matrix I"(D); find the largest one from the results obtained above, which is the largest information coefficient.

Calculate the maximum mutual information coefficient MIC(X″ _i ,Y) between each meteorological forecast factor and the measured precipitation, and arrange them in order from large to small, and select the top-ranked factors to form a new factor set X' ={X1, X2,...,Xr}, r≤n. Calculate the MIC value MIC(X″ _i , X″ _j )(i, j≤n) between each factor in the factor set X’ respectively, and form the MIC feature Matrix. Calculate the eigenvalues of the MIC eigenmatrix and arrange them in descending order λ ₁ ≥ λ ₂ ≥... ≥ λ _m ≥ 0. Find the eigenvector e _{i corresponding to the eigenvalue λ i (} i=1 ,2,...,m), where ||e _i ||=1.

Calculate the principal component contribution rate and cumulative contribution rate of the eigenvalues, the calculation formula is as follows:

Among them, L ₁ is the contribution rate of the main component, L ₂ is the cumulative contribution rate, i=1, 2,...,m, generally take the q principal components corresponding to the q eigenvalues with a cumulative contribution rate of 80% to 95% , that is to get the final predictor set X”={X″ ₁ , X″ ₂ ,…,X″ _q } Since q<m, dimension reduction can be achieved.

Step 3: Build a precipitation interval prediction model based on time convolutional network, divide the screened data into training set and test set according to the hold-out method, and then perform sliding window sampling to prepare the input of precipitation interval prediction model based on time convolutional network data group.

The feature-screened data is divided into two mutually exclusive sets using the set-out method: the training set and the test set. The ratio of the training set and the test set is 8:2. Data for 2011-2019. Using a sliding window sampling technique, the input data for the model is prepared.

Step 4: According to the input historical precipitation and highly correlated historical meteorological factor sequence, select the TCN temporal convolutional network model to construct the precipitation interval prediction model based on the temporal convolutional network, and introduce the LUBE interval prediction method to output the interval prediction value to realize The training of the precipitation interval prediction model based on the temporal convolutional network completes the comprehensive modeling of the precipitation process.

Construct a TCN temporal convolutional network model, which consists of causal dilated convolutions and residual connections. Causal dilated convolution is shown in Figure 2: Causal convolution means that the output at time t can only be convolved from the input no later than t, which acts as a filter in time. Unlike standard convolution, t The output of the moment has no effect on the future value, which can better capture the causal relationship between historical data and future predicted values. TCN uses 1DFCN (one-dimensional full convolutional network) architecture, and the length of each hidden layer is the same as the length of the input layer. same, and use zero padding to ensure that subsequent layers have the same length. The longer prediction period increases the difficulty of predicting the target value. Dilated causal convolution allows input values by skipping specific steps. The difference between dilated convolution and traditional convolution is that it allows the input of convolution to be sampled at intervals. Such convolutions increase the input range of the network, accessing longer input subsequences. The specific structure is as follows:

In the case of univariate sequences, the full dilated causal convolution operation on the hidden layer f , the definition on the sequence s is as follows:

where s is the element of the sequence, d is the dilation parameter, and sd i describes the past direction. Call *d a dilated convolution operation to distinguish it from a normal convolution operation. The normal convolution operator (*) is a specific version of dilated convolution (when d=1). The receptive field of the causal network is increased in two ways: one is to increase the filter size k; the other is to increase the expansion factor d. In this study, d is exponentially increased according to the network depth d = ²ⁱ to ensure that the long-term history can be effectively covered; specifically, d = ²ⁱ is used for level i of the network.

The LUBE (lower and upper bound estimation) interval prediction method is introduced in the output layer to quantify the uncertainty of the prediction results to generate probability interval predictions for future observations. The LUBE coverage width evaluation standard (PCWC) is optimized to effectively improve the accuracy of precipitation prediction, as follows:

There are two main evaluation indicators for the LUBE interval prediction method. PICP is evaluated from the probability that the actual observed value is between the upper and lower bounds of the prediction interval; PINAW is evaluated from the width between the upper and lower bounds of the prediction interval:

Among them, U(x _k ) is the upper boundary value of interval prediction, L(x _k ) is the lower boundary value of interval prediction, n is the sample number of measured values, A is the range of the target variable, that is, the difference between the maximum and minimum values . Then CWC, which measures width and coverage probability, can be defined as:

In precipitation prediction, under the influence of high-dimensional meteorological factors, the accuracy of prediction results is often greatly affected by features, and it is difficult to adjust the balance between the interval coverage and interval width of precipitation prediction through φ. On this basis, the penalty parameter is added. In summary, the indicators considering width, coverage probability and average deviation can be defined as:

指数放大PICP和e^-η(PICP-μ)之间的差异如果PCWC太小，则使用8和τ超参数来避免消失。Among them, the parameter τ linearly amplifies PINAW, and the parameter

Exponentially amplify the difference between PICP and e ^{-η (PICP-μ)} 8 and τ hyperparameters are used to avoid vanishing if PCWC is too small.

Step 5: For the precipitation interval prediction model based on the time convolutional network that has been trained, input new precipitation to complete the prediction of the precipitation interval.

The optimized PCWC index is introduced to train the model for interval prediction. In addition, two evaluation standards are used to evaluate the performance of the model, namely the root mean square error RMSE and the Nash efficiency coefficient NSE. The model evaluation standard formula is as follows:

root mean square error

Nash efficiency coefficient:

是第i时刻的预测值，

是观测值的平均值，N代表测试期间的观测次数。Among them, y _i is the observed value at the i-th moment,

is the predicted value at time i,

is the average value of observations, and N represents the number of observations during the test period.

The above is only a preferred embodiment of the present invention, and it should be pointed out that for those of ordinary skill in the art, some improvements and modifications can be made without departing from the principle of the present invention. It should be regarded as the protection scope of the present invention.

Claims (5)

1. A method for forecasting intervals of precipitation based on time convolution network, is characterized in that, comprises the following steps: (1) Collect precipitation data and meteorological factor data to form an initial time series data set and preprocess the data; (2) Analyze the characteristics of meteorological factors, build a feature extraction algorithm, obtain the correlation between meteorological factors and precipitation, and use the maximum information coefficient-asynchronous principal component analysis algorithm to screen out the time series of meteorological factors that have a greater impact on precipitation; (3) Construct a precipitation interval prediction model based on time convolution network, divide the screened data into training set and test set according to the hold-out method, and then perform sliding window sampling to prepare the input of precipitation interval prediction model based on time convolution network data set; (4) According to the input historical precipitation and highly correlated historical meteorological factor sequence, the TCN time convolutional network model is selected to construct the precipitation interval prediction model based on the time convolutional network, and the LUBE interval prediction method is introduced to output the interval prediction value to realize The training of the precipitation interval prediction model based on the temporal convolutional network completes the comprehensive modeling of the precipitation process; (5) For the precipitation interval prediction model based on the time convolutional network that has been trained, input new precipitation to complete the prediction of the precipitation interval. 2. A kind of precipitation interval forecasting method based on time convolutional network according to claim 1, it is characterized in that, step (1) described data preprocessing comprises data cleaning and missing completion of data, wherein data Missing completion is processed by linear interpolation in the time dimension. 3. a kind of precipitation interval forecasting method based on time convolutional network according to claim 1, it is characterized in that, step (2) described adopts maximum information coefficient-asynchronous principal component analysis algorithm to screen out the larger impact on precipitation The time series realization process of meteorological factors is as follows: Construct interpolation time series and asynchronous correlation time series through DTW: organize data set X _m×n , where m in each column represents the length of univariate time series, and n represents the number of attributes in each row; for each time series in data matrix X Perform normalization; use the Z-score method to normalize the time series Xi, so that the standardized X' _i has zero mean and one standard deviation, that is, X' _ih ~ N(0,1); use DTW to calculate the normalization After X' _i and X' _j , the best deformation path is obtained

All the elements of the optimal deformation path constitute the elements of the interpolation time series, respectively: X” _i ={p ₁ (1),p ₂ (1),…,p _k (1)} and X” _j ={p ₁ (2),p ₂ (2),...,p _k (2)} All normalized time series should be extended to time series reflecting corresponding asynchronous correlations; new interpolated time series are obtained, i.e. X = X"; Carry out X” _i ×Y division on the 2 new time series scatter plots, that is, set D, divide its elements into X” _i grids according to X” _i values, and divide them into Y grids according to Y values ;Calculate the frequency distribution D|G obtained by calculating the points of the set D falling on the given grid G, and reasonably select the division upper bound B(n) of the grid G, B(n) is a function of the sample size, expressing The constraint of the total number of squares b of the grid G is less than B(n), B(n)=n ^0.6 ; different grids G are calculated to determine different probability distributions, namely I(D|G), which means that the point set is based on the distribution D| Mutual information of G; find the maximum mutual information maxI(D|G) in the mutual information of all possible distributions D|G based on X” _i ×Y grid G and denote it as I”(D,m,n), get The characteristic matrix I"(D) of the two-dimensional data set D; The maximum information coefficient is used to effectively measure the correlation between the forecast factor and the actual precipitation sequence, and the factors with strong correlation with the actual precipitation are screened out, which are recorded as high-information factors, and the factors with more overlapping information are removed from the high-information factors . 4. a kind of precipitation interval prediction method based on time convolution network according to claim 1, it is characterized in that, the described step (4) selects TCN time convolution network model to construct the precipitation interval prediction model based on time convolution network The implementation process is as follows: In the case of univariate sequences, for a given precipitation input X of a 1D sequence and a filter function ω of size k:{0,...,k-1}, the full dilated causal convolution operation f on successive layers , the definition on the sequence s is as follows:

where, s is the element of the sequence, d is the dilation parameter, which increases exponentially according to the network depth d= ²ⁱ , and d= ²ⁱ is used for the i-level of the network; while sd i describes the past direction; call *d the dilation The convolution operation to distinguish the normal convolution operation. 5. a kind of precipitation interval prediction method based on time convolutional network according to claim 1, it is characterized in that, described in step (4) introduces LUBE interval prediction method output interval prediction value realization process is as follows: The evaluation indicators of the LUBE interval prediction method include PICP, which is evaluated from the probability that the actual observed value is between the upper and lower boundaries of the prediction interval; PINAW, which is evaluated from the width between the upper and lower boundaries of the prediction interval:

Among them, U(x _k ) is the upper boundary value of interval prediction, L(x _k ) is the lower boundary value of interval prediction, n is the sample number of measured values, A is the range of the target variable, that is, the difference between the maximum and minimum values ; then the CWC measuring width and coverage probability can be defined as:

Adding the penalty parameter, the metrics considering width, coverage probability and mean deviation are defined as:

Among them, the parameter τ linearly amplifies PINAW, and the parameter

Exponentially amplify the difference between PICP and e ^{-η (PICP-μ)} If PCWC is too small, ε and τ hyperparameters are used to avoid vanishing.

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