CN111831966B - Combined river channel water level forecasting method based on high-dimensional probability distribution function - Google Patents

Abstract

The present invention relates to the technical field of hydrological forecasting, and more specifically, relates to a combined river water level forecasting method based on a high-dimensional probability distribution function, including: selecting a time series of river water levels in a certain period; The marginal distribution functions F(X _t‑1 ) and F(X _t ) of the data series construct the joint distribution probability function of the water level variable, and select the three Copula models with the smallest AIC value; use the joint distribution probability function to input the known t‑ From the water level variable data series X _t‑1 at time 1, the conditional distribution probability _of the water level variable data series at time t is obtained; the conditional distribution probability function is converted into its inverse function form, and the pseudo Combine water level data series X _t ; obtain the three best Copula model prediction values, set the weight of each Copula function prediction value according to the AIC value, and calculate the weighted average of the three Copula function prediction values, which is the final prediction value . The invention can accurately forecast the river water level variable and other hydrological variables, and has important application value.

Description

A Combined River Water Level Forecasting Method Based on High Dimensional Probability Distribution Function

technical field

The invention relates to the technical field of hydrological forecasting, and more specifically, relates to a combined river water level forecasting method based on a high-dimensional probability distribution function.

Background technique

Hydrological forecasting is based on the laws of the formation and movement of various hydrological processes in nature, combined with currently acquired hydrometeorological data, to predict the hydrological situation in a certain period of time in the future. The results of hydrological forecasting provide services for the rational utilization and protection of water resources, flood control and rescue, water conservancy project construction and operation management, and industrial and agricultural safety production. The hydrological theory is based on the physical hydrological process, because the physical process can scientifically and reasonably reflect the actual hydrological process. Therefore, many physical hydrological models have been established to achieve the purpose of hydrological forecasting. However, due to too many uncertain factors in nature, it is difficult for the physical model to fully reflect the influence relationship between variables, resulting in unsatisfactory effects of the physical model. In addition, physical models usually require a large number of underlying surface and climate data types, and in practice, these data are often difficult to obtain, thus limiting the application of physical models.

Contents of the invention

The purpose of the present invention is to overcome the deficiencies in the prior art, to provide a combined river water level forecasting method based on high-dimensional probability distribution function, through the method of dynamic weight, realize the combined forecasting of multiple models, and improve the accuracy of forecasting.

In order to solve the problems of the technologies described above, the technical solution adopted in the present invention is:

A combined river water level prediction method based on a high-dimensional probability distribution function is provided, including the following steps:

S1. For a certain hydrological station, select the time series of water level variables in a certain period, let X _t-1 be the water level data series at t-1 time, and X _t be the water level data series at t time;

S2. Using the marginal distribution functions F(X _t-1 ) and F(X _{t ) of the water level data series at time t-1 and time t} , based on the binary Copula function, construct the joint of water level variables at time t-1 and time t Distribution probability function, select the three Copula models with the smallest AIC (AKAIKE INFORMATION CRITERION) value;

S3. Utilize the joint distribution probability function constructed in step S2, input the water level variable data series X _t-1 of known t-1 moment, obtain the conditional distribution probability function of the water level variable data series at t time;

S4. Further convert the conditional distribution probability function into its inverse function form, so as to obtain the fitting water level data series X _t at time t with X _t-1 as an input variable;

S5. After obtaining the predicted values of the three Copula models, set the weight of each Copula function predicted value according to the AIC value, calculate the weighted average of the predicted values of the three Copula functions, and obtain the final predicted value.

The combined river water level prediction method based on the high-dimensional probability distribution function of the present invention introduces a variety of Coupla function types, assigns dynamic weights according to the predicted performance, and realizes combined forecasts based on dynamic weights; the present invention can accurately predict the river water level variables , to provide scientific and effective methods and means for the prevention and control of drought and flood disasters, which has important application value.

Preferably, in step S1, the predictor variable is the water level of the river channel, but the predictor variable of the present invention is not limited to the water level of the river channel, and can also be extended to other hydrological variables such as water flow, precipitation and soil moisture, and the time scale of the time series includes Hour, day, month, year.

Preferably, in step S2, the type of the binary copula function is selected from: Gaussian copula function, BB1 copula function, BB6 copula function, BB7 copula function, BB8 copula function.

Preferably, in step S2, calculate the AIC value of Copula model by the following formula:

In the formula, k is the number of parameters of the proposed Copula function, N is the number of variable observations, c(U _{t, i} , U _{t-1, i} |θ) means U _{t, i} , U _{t-1, i} The joint density function c of and its parameter θ, i is the ith observed value.

Preferably, in step S3, the conditional distribution probability F(X _t ) of the water level variable data series at time t is calculated according to the following steps:

S31. Let U _t-1 = F(X _t-1 ), F(X _t-1 ) is the marginal distribution function of X _t-1 ;

S32. For each value of F(Xt _-1 ) in step S31, all utilize three Copula model functions that have been selected to obtain the corresponding conditional distribution probability:

F(U _t |U _t-1 )＝h(U _t |U _t-1 ; θ) (2)

Where θ is the parameter of the joint probability function of U _t and U _t-1 , which can be calculated by the above copula model, and the h function is a standard conditional distribution function.

In addition, the present invention further provides a 3-dimensional conditional forecasting model, and its formula can be expressed as:

F(U _t |U _t-1 ，U _t-2 )＝h[h(U _t |U _t-2 )|h(U _t-1 |U _t-2 )] (3)

Among them, U _t-2 is the marginal distribution function of X _t-2 time series, and X _t-2 is two moments ahead;

S33. By calculating the inverse function of formula (2), the probability distribution value U _t of the variable to be predicted can be calculated:

U _t ＝h ^-1 (γ|U _t-1 ; θ) (4)

For a 3-dimensional conditional prediction model, it can be written as:

U _t ＝h ^-1 (h ^-1 (γ|h(U _t-1 |U _t-2 )|U _t-1 ) (5)

Among them, h ^-1 is the inverse function of the h conditional function, γ is the probability value that needs to be set, and the range is [0, 1]. Multiple γ values can be generated through the sampling algorithm, and according to formula (4) for each specific The water level data values that need to be predicted generate multiple U _t .

In step S4, the fitting water level data series X _t at time t is obtained according to the following steps:

S41. Assuming that the water level variable data series at time t and the water level variable data series at t-1 time obey the same distribution, the inverse function of the marginal distribution function can be obtained according to the water level variable data series at t-1 time;

S42. Input the fitted conditional distribution probability into the inverse function to obtain simulated data. Right now:

X _t ＝F ^-1 (U _t ) (6)

In the formula, X _t represents the predicted value at time t, and F ^-1 represents the inverse function of the marginal distribution function U _t ;

S43. According to step S33, n gamma values are generated by sampling algorithm, then n probability distribution values U _t at time t can be obtained according to formula (4), and n water level variables at time t can be calculated by formula (6) Predicted value X _t . At this time, by taking the average of n predicted values definition:

Then use the data value at time t-1 to predict the water level data series at time t, which is the value to be predicted

Preferably, in step S5, the weight of each Copula function prediction value is calculated as follows:

In the formula, w _i represents the weight of the predicted value of the i-th copula function, and AIC ₁ , AIC ₂ , and AIC ₃ represent the AIC values of the first, second, and third copula functions, respectively.

Preferably, when the predicted values of the three Copula functions are Y ₁ , Y ₂ , and Y ₃ , the final predicted value Y is calculated as follows:

Y=Y ₁ ×w ₁ +Y ₂ ×w ₂ +Y ₃ ×w ₃ (9).

Preferably, after step S5 , the accuracy of the final predicted value is evaluated by using the fitness index, which includes root mean square error RMSE, Nash efficiency coefficient NSE and coefficient of determination R ² .

Compared with prior art, the beneficial effect of the present invention is:

The present invention utilizes the conventional Gaussian copula and the new copula function of BB for forecasting, and more comprehensively fits the water level data and the joint relationship; provides a three-dimensional conditional forecasting model to obtain higher forecasting sensitivity; introduces various copula types, and according to The performance of its prediction is endowed with dynamic weights, so as to realize the combined forecast based on dynamic weights; the invention can accurately forecast the water level variables of the river, and provide scientific and effective methods and means for the prevention and control of drought and flood disasters, which has important applications value.

Description of drawings

Fig. 1 is the structure schematic diagram of the combined river channel water level prediction method based on the high-dimensional probability distribution function of the present invention;

Fig. 2 is three optimal Copula probability density figures of water level under the daily scale of fitting of the present invention;

Fig. 3 is three optimal Copula probability density figures of water level under the monthly scale fitted by the present invention;

Fig. 4 is the comparison chart of the measured value and the simulated value of the water level under the daily scale predicted by the method of the present invention;

Fig. 5 is a comparison chart of the actual measured value and the simulated value of the water level on a monthly scale predicted by the method of the present invention.

Detailed ways

The present invention will be further described below in combination with specific embodiments. Wherein, the accompanying drawings are only for illustrative purposes, showing only schematic diagrams, rather than physical drawings, and should not be construed as limitations on this patent; in order to better illustrate the embodiments of the present invention, some parts of the accompanying drawings will be omitted, Enlargement or reduction does not represent the size of the actual product; for those skilled in the art, it is understandable that certain known structures and their descriptions in the drawings may be omitted.

Example

As shown in Figure 1, be the embodiment of the combined river course water level forecasting method based on high-dimensional probability distribution function of the present invention, comprise the following steps:

S1. For a certain hydrological station, select the time series of water level variables in a certain period, let X _t-1 be the water level data series at t-1 time, and X _t be the water level data series at t time; Wherein, the present embodiment uses t -1 moment is taken as an example to illustrate the steps but is not limited thereto. The method of the present invention can be further expanded to include more time-delay series as input variables;

S2. Using the marginal distribution functions F(X _t-1 ) and F(X _{t ) of the water level data series at time t-1 and time t} , based on the binary Copula function, construct the joint of water level variables at time t-1 and time t Distribution probability function, select the three Copula models with the smallest AIC value of the Akaike information content criterion;

S3. Utilize the joint distribution probability function constructed in step S2, input the water level variable data series X _t-1 of known t-1 moment, obtain the conditional distribution probability function of the water level variable data series at t time;

S4. Further convert the conditional distribution probability function into its inverse function form, so as to obtain the fitting water level data series X _t at time t with X _t-1 as an input variable;

S5. Select three models with the best performance from multiple copula models to further form a combined forecast model, wherein the copula models selected in this embodiment are not limited to three, and other numbers of copula models can also be set according to actual needs. . That is, after obtaining the predicted values of the three Copula models, the weight of the predicted values of each Copula function is set according to the AIC value, and the weighted average of the predicted values of the three Copula functions is calculated to obtain the final predicted value.

In step S1, the predictor variable is the river water level, but the predictor variable in this embodiment is not limited to the river water level, and can also be extended to other water level variables, such as water flow, precipitation and soil moisture. The time scale of the time series includes hourly, daily , month, year.

In step S2, the type of the binary copula function is selected from: Gaussian copula function, BB1 copula function, BB6 copula function, BB7 copula function, BB8 copula function.

In step S2, the AIC value of the Copula model is calculated according to the following formula:

In the formula, k is the number of parameters of the proposed Copula function, N is the number of variable observations, c(U _t , U _{t-1, i} |θ) represents the joint of U _{t, i} , U _{t-1, i} The density function c and its parameters θ, i are the i-th observed value; the AIC values of the Copula function calculated in this embodiment are all less than 0, and the smaller the AIC value, the higher the fitting degree of the Copula model.

In step S3, the conditional distribution probability F(X _t ) of the water level variable data series at time t is calculated according to the following steps:

S31. Let U _t-1 = F(X _t-1 ), F(X _t-1 ) is the marginal distribution function of X _t- 1 (i.e. lag 1 one time before);

S32. For each value of F(Xt _-1 ) in step S31, all utilize three Copula model functions that have been selected to obtain the corresponding conditional distribution probability:

F(U _t |U _t-1 )＝h(U _t |U _t-1 ; θ) (2)

Where θ is the parameter of the joint probability function of U _t and U _t-1 , which can be calculated by the above copula model, and the h function is a standard conditional distribution function.

In addition, the present invention further provides a 3-dimensional conditional forecasting model, and its formula can be expressed as:

F(U _t |U _t-1 ，U _t-2 )＝h[h(U _t |U _t-2 )|h(U _t-1 |U _t-2 )] (3)

Wherein, U _t-2 is the marginal distribution function of the time series of X _t-2 (that is, 2 moments before: lag 2); this embodiment can be implemented under actual demand according to the above-mentioned three-dimensional conditional forecast model;

S33. By calculating the inverse function of formula (2), the probability distribution value U _t of the variable to be predicted can be calculated. For the 2-dimensional conditional prediction model, it can be written as:

U _t ＝h ^-1 (γ|U _t-1 ) (4)

For a 3-dimensional conditional prediction model, it can be written as:

U _t ＝h ^-1 (h ^-1 (γ|h(U _t-1 |U _t-2 )|U _t-1 ) (5)

Among them, h ^-1 is the inverse function of the h conditional function, γ is the probability value that needs to be set, the range is [0, 1], multiple γ values can be generated through the sampling algorithm, and the number of γ values can be taken as 200; then according to Formula (4), for each data value (that is, each specific water level data value that needs to be predicted), multiple U _t can be generated, and the number of U _t is equal to the γ value.

In step S4, the fitting water level data series X _t at time t is obtained according to the following steps:

S41. Assuming that the water level variable data series at time t and the water level variable data series at t-1 time obey the same distribution, the inverse function of the marginal distribution function can be obtained according to the water level variable data series at t-1 time, and the data is restored using the inverse function;

S42. Input the fitted conditional distribution probability into the inverse function to obtain simulated data. Right now:

X _t ＝F ^-1 (U _t ) (6)

In the formula, X _t represents the predicted value at time t, and F ^-1 represents the inverse function of the marginal distribution function U _t ;

S43. According to step S33, generate n gamma values through the sampling algorithm, and the possible value of n is 200; according to the formula (4), the probability distribution value U _t of n time t can be obtained, and then can be calculated by the formula (6). The predicted value X _t of the water level variable at n times t. At this time, by taking the average of n predicted values definition:

Then use the data at time t-1 to predict the average value at time t is the final predicted value />

In step S5, three models with the best performance are selected from multiple copula models (there may also be multiple models, set according to actual needs) to further form a combined forecast model. The weight of the predicted value of each Copula function is calculated as follows:

In the formula, w _i represents the weight of the predicted value of the i-th copula function, and AIC ₁ , AIC ₂ , and AIC ₃ represent the AIC values of the first, second, and third copula functions, respectively.

When the predicted values of the three Copula functions are Y ₁ , Y ₂ , and Y ₃ , the final predicted value Y is calculated as follows:

Y=Y ₁ ×w ₁ +Y ₂ ×w ₂ +Y ₃ ×w ₃ (9).

After step S5, the accuracy of the final predicted value is evaluated by using the fitness index, which includes root mean square error RMSE, Nash efficiency coefficient NSE and coefficient of determination R ² .

In order to verify the forecasting effect of the method of this embodiment, the present embodiment selects the water level data of a certain hydrological station from 1989 to 2011 to test (mainly taking the 2-dimensional forecasting model as an example, and the 3-dimensional forecasting model can be based on the 3-dimensional forecasting model provided by the present invention. The forecast formula is implemented under actual demand), including the following steps:

Convert the data at time t and time t-1 into marginal distribution functions respectively, using the marginal distributions F(X _t-1 ) and F(X t ) of the data series at time t-1 and time _t , based on the binary Copula function, Construct the joint distribution probability function of the water level variable at time t-1 and time t; the type of copula function is selected from Gaussian copula function, BB1 copula function, BB6 copula function, BB7 copula function, BB8 copula function, and the calculated copula types The parameters and AIC values are shown in Table 1. The copula types selected on the daily scale are BB1 copula function, BB6 copula function, and BB7 copula function. The schematic diagrams of the three functions are shown in Figure 2. Copula function, BB8 copula function, the schematic diagrams of the three functions are shown in Figure 3.

Table 1 Copuia function fitting

Use the selected Copula model to construct the conditional distribution function of the water level variable data series at time t-1 and the water level variable data series at time t;

Based on the three optimal Copula models screened, the conditional distribution function of the present invention is utilized to set up a conditional prediction model to obtain the water level prediction values of the three models respectively;

The weights of the predicted values of the three models are calculated according to the AIC to obtain the weighted average of the predicted water level values, and the weighted average is used as the final predicted value of this embodiment.

Through the above steps, the evaluation analysis of the final predicted value obtained in this embodiment is shown in Table 2:

Table 2 The prediction effect of the model of the present invention

It can be seen from Table 2 that the simulation effect of the daily-scale water level data is good, and the simulation effect of the monthly-scale water level data is relatively poor. It can be seen from Figures 4 and 5 that the measured values of the daily-scale water level data are relatively close to the simulated values, and the simulation effect is good; however, the measured values of the monthly-scale water level data deviate from the simulated values, and the simulation effect is relatively poor. This is because the amount of water level data on the daily scale is large, and the more data input when calibrating the model, the more representative the obtained model is and the closer to the actual situation. Therefore, the RMSE value is smaller, and the NSE and ^R2 are larger. That is to say, the prediction effect of the model is good; while the amount of data referenced when establishing the monthly scale model is relatively limited, the representativeness is poor, and the forecast effect is not as good as that of the daily scale model, but the RMSE value is small, indicating that the prediction effect of the present invention is acceptable. It can be seen that when the combined water level forecasting method of this embodiment performs water level forecasting, the amount of data selected should be as large as possible, so as to improve the accuracy of model forecasting and achieve a good forecasting effect. In addition, the selection of different copula types may also have a certain impact on the prediction results. Therefore, the present invention proposes to introduce multiple copula types and assign certain weights according to their specific performances, thereby realizing the combined forecast based on weights. Though in the example of the present invention only adopted the 2-dimensional conditional combination forecasting model, the present invention has provided the specific calculation formula of 3-dimensional conditional combinational forecasting model simultaneously, if more input variables are arranged as predictors, can further improve The accuracy of the forecast.

Apparently, the above-mentioned embodiments of the present invention are only examples for clearly illustrating the present invention, rather than limiting the implementation of the present invention. For those of ordinary skill in the art, on the basis of the above description, other changes or changes in different forms can also be made. It is not necessary and impossible to exhaustively list all the implementation manners here. All modifications, equivalent replacements and improvements made within the spirit and principles of the present invention shall be included within the protection scope of the claims of the present invention.

Claims (5)

1. A combined river channel water level forecasting method based on high-dimensional probability distribution function, is characterized in that, comprises the following steps: Step S1. For a certain hydrological station, select the time series of water level variables in a certain period, let X _t-1 be the water level data series at t-1 time, and Xt be the water level data series at t time; Step S2. Utilize the marginal distribution functions F(X _t-1 ) and F(X _t ) of the water level data series at the time t-1 and the time t, and based on the binary Copula function, construct the water level variable at the time t-1 and the time t Combined distribution probability function to select three Copula models with the smallest AIC value of Akaike information content criterion; Step S3. Utilize the joint distribution probability function constructed in step S2, input the water level variable data series X _t-1 of known t-1 moment, obtain the conditional distribution probability function of the water level variable data series at t time; Step S4. Convert the conditional distribution probability function to its inverse function form, so as to obtain the fitting water level data series X _t at time t with X _t-1 as the input variable; Step S5. After obtaining the predicted values of the three Copula models, set the weight of each Copula function predicted value according to the AIC value, calculate the weighted average of the predicted values of the three Copula functions, and obtain the final predicted value; In step S2, the type of the binary copula function is selected from: Gaussian copula function, BB1 copula function, BB6 copula function, BB7 copula function, BB8 copula function; In step S2, the AIC value of the Copula model is calculated according to the following formula: In the formula, k is the number of parameters of the proposed Copula function, N is the number of variable observations, c(U _{t, i} , U _{t-1, i} |θ) means U _{t, i} , U _{t-1, i} The joint density function c and its parameter θ; In step S3, the conditional distribution probability of the water level variable data series at time t is calculated according to the following steps: S31. Let U _t-1 = F(X _t-1 ), F(X _t-1 ) is the marginal distribution function of X _t-1 ; S32. For each value of F(Xt _-1 ) in step S31, all utilize three Copula model functions that have been selected to obtain the corresponding conditional distribution probability: F(U _t |U _t-1 )＝h(U _t |U _t-1 ; θ) (2) Where θ is the parameter of the joint probability function of U _t and U _t-1 , which is calculated by the copula model function, and the h function is the standard conditional distribution function; Provide a 3-dimensional conditional forecast model, expressed as: F(U _t |U _t-1 ，U _t-2 )＝h[h(U _t |U _t-2 )|h(U _t-1 |U _t-2 )](3) Among them, U _t-2 is the marginal distribution function of X _t-2 time series, and X _t-2 is two moments ahead; S33. By calculating the inverse function of formula (2), calculate the marginal distribution function U _t of the water level variable that needs to be predicted: U _t ＝h ^-1 (γ|U _t-1 ; θ) (4) For a 3-dimensional conditional forecast model, it is expressed as: U _t ＝h ^-1 (h ^-1 (γ|h(U _t-1 |U _t-2 )|U _t-1 )) (5) Among them, h ^-1 is the inverse function of the h conditional function, γ is the probability value that needs to be set, and the range is [0, 1]. Multiple γ values are generated through the sampling algorithm, and according to formula (4) for each specific need The predicted water level data values generate multiple U _t ; In step S4, the fitting water level data series X _t at time t is obtained according to the following steps: S41. Assume that the water level variable data series at time t and the water level variable data series at t-1 time obey the same distribution, and obtain the inverse function of the marginal distribution function according to the water level variable data series at t-1 time; S42. Input the fitted conditional distribution probability into the inverse function of the predicted variable distribution function to obtain simulated data; namely: X _t ＝F ^-1 (U _t ) (6) In the formula, X _t represents the predicted value at time t, and F ^-1 represents the inverse function of the marginal distribution function U _t of the water level variable at time t; S43. According to step S33, generate n γ values through sampling algorithm, obtain n probability distribution values U _t at time t according to formula (4), and then calculate n water level variable prediction values X at time t through formula (6) _t ; at this time, by taking the average of n predicted values definition: Then use the data value at time t-1 to predict the water level data series at time t, which is the value to be predicted 2. The combined river water level forecast method based on high-dimensional probability distribution function according to claim 1, characterized in that, in step S1, the time scale of the time series includes hours, days, months, and years. 3. the combined river course water level prediction method based on high-dimensional probability distribution function according to claim 1, is characterized in that, in step S5, the weight of each Copula function prediction value is calculated as follows: In the formula, w _i represents the weight of the predicted value of the i-th copula function, and AIC ₁ , AIC ₂ , and AIC ₃ represent the AIC values of the first, second, and third copula functions, respectively. 4. the combined river channel water level forecasting method based on high-dimensional probability distribution function according to claim 3, is characterized in that, when the predicted value of three Copula functions Y ₁ , Y ₂ , Y ₃ , then final predicted value Y presses The following formula is calculated: Y=Y ₁ ×w ₁ +Y ₂ ×w ₂ +Y ₃ ×w ₃ (9). 5. the combined river channel water level forecasting method based on high-dimensional probability distribution function according to claim 3, is characterized in that, after step S5, adopts the degree of fit index to assess the accuracy of final predicted value, described degree of fit index Including root mean square error RMSE, Nash efficiency coefficient NSE and coefficient of determination R ² .