CN111831966A - Combined river water level forecasting method based on high-dimensional probability distribution function - Google Patents
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Abstract
The invention relates to the technical field of hydrologic prediction, in particular to a combined river channel water level prediction method based on a high-dimensional probability distribution function, which comprises the following steps of: selecting a time sequence of river water levels in a certain period; edge distribution function F (X) using water level data series at time t-1 and time tt‑1) And F (X)t) Constructing a joint distribution probability function of the water level variable, and screening three Copula models with the minimum AIC value; inputting water level variable data series X at known t-1 moment by using joint distribution probability functiont‑1Solving the conditional distribution probability of the water level variable data series at the time t; converting conditional distribution probability function into its inverse function form to realize Xt‑1Obtaining a series of t-time fitting water level data X for an input variablet(ii) a Obtaining three optimal Copula model predicted values, setting the weight of each Copula function predicted value according to the AIC value, and calculating three Copula functionsAnd the weighted average value of the predicted values is the final predicted value. The method can accurately forecast the river water level variable and other hydrological variables, and has important application value.
Description
Technical Field
The invention relates to the technical field of hydrologic prediction, in particular to a combined river channel water level prediction method based on a high-dimensional probability distribution function.
Background
The hydrologic forecast is used for forecasting the hydrologic conditions in a certain period of time in the future according to the formation and motion rules of various hydrologic processes in the nature and by combining the currently acquired hydrologic and meteorological data. The hydrologic forecast result provides services for reasonable utilization and protection of water resources, flood prevention and emergency rescue, hydraulic engineering construction, dispatching, application and 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, a plurality of physical hydrological models are established to achieve the aim of hydrological prediction. However, because there are too many uncertain factors in nature, the physical model is difficult to completely reflect the influence relationship between variables, and the effect of the physical model is not satisfactory. In addition, physical models often require a large amount of underlying surface and climate data types, which are often difficult to obtain in full in practice, limiting the utility of physical models.
Disclosure of Invention
The invention aims to overcome the defects in the prior art, provides a combined river water level forecasting method based on a high-dimensional probability distribution function, realizes the combined forecasting of multiple models through a dynamic weight method, and improves the forecasting precision.
In order to solve the technical problems, the invention adopts the technical scheme that:
the combined river water level forecasting method based on the high-dimensional probability distribution function comprises the following steps:
S1.selecting a time sequence of water level variables of a certain period aiming at a certain hydrological site, and enabling Xt-1Water level data series at time t-1, XtIs a water level data series at the time t;
s2, utilizing an edge distribution function F (X) of the water level data series at the t-1 moment and the t momentt-1) And F (X)t) Based on a binary Copula function, constructing a joint distribution probability function of water level variables at the t-1 moment and the t moment, and screening out three Copula models with the minimum red pool INFORMATION quantity criterion AIC (AKAIKE INFORMATION criterion) value;
s3, inputting the water level variable data series X at the known t-1 moment by using the combined distribution probability function constructed in the step S2t-1Solving a conditional distribution probability function of the water level variable data series at the time t;
s4, further converting the conditional distribution probability function into an inverse function form thereof, thereby realizing the purpose of using Xt-1Obtaining a fitted water level data series X at time t for an input variablet；
And S5, after three Copula model predicted values are obtained, setting the weight of each Copula function predicted value according to the size of the AIC value, and calculating the weighted average value of the three Copula function predicted values to obtain a final predicted value.
According to the combined river water level forecasting method based on the high-dimensional probability distribution function, multiple Coipla function types are introduced, dynamic weight is given according to the predicted performance, and combined forecasting based on the dynamic weight is achieved; the method can accurately forecast the river water level variable, provides a scientific and effective method and means for the control of drought and flood disasters, and has important application value.
Preferably, in step S1, the forecast variable is river level, but the forecast variable of the present invention is not limited to river level, and can be extended to other hydrological variables such as water flow, precipitation and soil humidity, and the time scale of the time series includes hour, day, month and year.
Preferably, in step S2, the type of the binary Copula function is selected from: gaussian copula function, BB1copula function, BB6copula function, BB7 copula function, BB8 copula function.
Preferably, in step S2, the AIC value of the Copula model is calculated according to the following formula:
where k is the number of parameters of the proposed Copula function, N is the number of observed values of the variables, and c (U)t，i，Ut-1，iTheta) represents Ut，i，Ut-1，iAnd the joint density function c and the parameter theta, i thereof are the ith observation value.
Preferably, in step S3, the conditional distribution probability F (X) of the water level variable data series at time tt) The method comprises the following steps:
s31, order Ut-1＝F(Xt-1)，F(Xt-1) Is Xt-1The edge distribution function of (1);
s32. for F (X) in step S31t-1) The corresponding conditional distribution probability is solved by utilizing the selected three Copula model functions:
F(Ut|Ut-1)＝h(Ut|Ut-1；θ) (2)
wherein theta is Ut、Ut-1The parameters of the joint probability function can be calculated by the copula model, and the h function is a standard conditional distribution function.
In addition, the invention further provides a 3-dimensional condition forecasting model, and the formula can be expressed as:
F(Ut|Ut-1，Ut-2)＝h[h(Ut|Ut-2)|h(Ut-1|Ut-2)](3)
wherein, Ut-2Is Xt-2Edge distribution function of time series, Xt-22 moments ahead;
s33, calculating the probability distribution value U of the variables needing to be predicted through the inverse function of the calculation formula (2)t：
Ut＝h-1(γ|Ut-1；θ) (4)
For a 3-dimensional conditional prediction model, it can be written as:
Ut＝h-1(h-1(γ|h(Ut-1|Ut-2)|Ut-1) (5)
wherein h is-1Is the inverse function of the h condition function, gamma is the probability value to be set, and the range is [0, 1%]Multiple gamma values may be generated by a sampling algorithm and multiple U's generated for each particular water level data value to be predicted according to equation (4)t。
In step S4, fitting water level data series X at time ttThe preparation method comprises the following steps:
s41, assuming that the water level variable data series at the time t and the water level variable data series at the time t-1 obey the same distribution, and obtaining an inverse function of an edge distribution function according to the water level variable data series at the time t-1;
and S42, inputting the fitted conditional distribution probability into an inverse function to obtain simulation data. Namely:
Xt＝F-1(Ut) (6)
in the formula, XtIndicates the predicted value at time t, F-1Representing the edge distribution function UtThe inverse function of (c);
s43, according to the step S33, n gamma values are generated through a sampling algorithm, and then the probability distribution values U of n t moments can be obtained according to the formula (4)tAnd the predicted value X of the water level variable at n times t can be calculated by the formula (6)t. At this time, the average value of the n predicted values is taken
Defining:
then the water level data series at the time t is obtained by utilizing the data value prediction at the time t-1, namely the water level data series at the time t is the value to be predicted
Preferably, in step S5, the weight of each Copula function prediction value is calculated according to the following formula:
in the formula, wiWeight, AIC, representing the predicted value of the ith Copula function1、AIC2、AIC3Respectively, the AIC values of the 1 st, 2 nd and 3 rd Copula functions are shown.
Preferably, when the predicted value of the three Copula functions is Y1、Y2、Y3Then the final predicted value Y is calculated as:
Y＝Y1×w1+Y2×w2+Y3×w3(9)。
preferably, after step S5, the accuracy of the final predicted value is evaluated using a fitness index including the root mean square error RMSE, the nash efficiency coefficient NSE, and the decision coefficient R2。
Compared with the prior art, the invention has the beneficial effects that:
the method utilizes the conventional Gaussian copula and BB novel copula functions to forecast, and more comprehensively fits the water level data and the joint relation; providing a three-dimensional condition forecasting model to obtain higher forecasting sensitivity; introducing a plurality of copula types, and endowing dynamic weight according to the predicted performance of the copula types, thereby realizing combined forecast based on the dynamic weight; the method can accurately forecast the river water level variable, provides a scientific and effective method and means for the control of drought and flood disasters, and has important application value.
Drawings
FIG. 1 is a schematic structural diagram of a combined river water level forecasting method based on a high-dimensional probability distribution function according to the present invention;
FIG. 2 is a graph of the probability density of three best Copula's for water level at daily scale fitted by the present invention;
FIG. 3 is three best Copula probability density maps of water level at monthly scale fitted by the present invention;
FIG. 4 is a comparison graph of the measured water level and the simulated water level at the daily scale predicted by the method of the present invention;
FIG. 5 is a comparison graph of measured and simulated water levels at the monthly scale predicted by the method of the present invention.
Detailed Description
The present invention will be further described with reference to the following embodiments. Wherein the showings are for the purpose of illustration only and are shown by way of illustration only and not in actual form, and are not to be construed as limiting the present patent; to better illustrate the embodiments of the present invention, some parts of the drawings may be omitted, enlarged or reduced, and do not represent the size of an actual product; it will be understood by those skilled in the art that certain well-known structures in the drawings and descriptions thereof may be omitted.
Examples
Fig. 1 shows an embodiment of the combined river water level forecasting method based on the high-dimensional probability distribution function according to the present invention, which includes the following steps:
s1, aiming at a hydrological station, selecting a time sequence of water level variables in a certain period, and enabling X to bet-1Water level data series at time t-1, XtIs a water level data series at the time t; in the embodiment, the step is described by taking the time t-1 as an example, but not limited to this, and the method of the present invention can be further expanded to more time-lag series as input variables;
s2, utilizing an edge distribution function F (X) of the water level data series at the t-1 moment and the t momentt-1) And F (X)t) Based on a binary Copula function, constructing a joint distribution probability function of water level variables at the t-1 moment and the t moment, and screening out three Copula models with the minimum value of the information content criterion AIC of the Chichi pool;
s3, inputting the water level variable data series X at the known t-1 moment by using the combined distribution probability function constructed in the step S2t-1Solving a conditional distribution probability function of the water level variable data series at the time t;
s4, further converting the conditional distribution probability function into an inverse function form thereof, thereby realizing the purpose of using Xt-1As input variablesObtaining a fitting water level data series X at the moment tt；
And S5, selecting three models with the best expression effect from the plurality of copula models to further form a combined forecasting model, wherein the number of the selected copula models is not limited to three, and other numbers of copula models can be set according to actual requirements. Namely, after three Copula model predicted values are obtained, the weight of each Copula function predicted value is set according to the size of the AIC value, and the weighted average value of the three Copula function predicted values is calculated to obtain the final predicted value.
In step S1, the forecast variable is the river level, but the forecast variable of this embodiment is not limited to the river level, and may also be extended to other water level variables, such as water flow, precipitation and soil humidity, and the time scale of the time series includes hour, day, month and year.
In step S2, the type of the binary Copula function is selected from: gaussian copula function, BB1copula function, BB6copula function, BB7 copula function, BB8 copula function.
In step S2, the AIC value of the Copula model is calculated according to the following formula:
where k is the number of parameters of the proposed Copula function, N is the number of observed values of the variables, and c (U)t，Ut-1，iTheta) represents Ut，i，Ut-1，iThe joint density function c and the parameter theta, i thereof are the ith observation value; in this embodiment, all the AIC values of the Copula functions are calculated to be less than 0, and the smaller the AIC value is, the higher the fitting degree of the Copula model is.
In step S3, the conditional distribution probability F (X) of the water level variable data series at time tt) The method comprises the following steps:
s31, order Ut-1＝F(Xt-1)，F(Xt-1) Is Xt-1(i.e., 1 prior time lag 1) edge distribution function;
s32. for F (X) in step S31t-1) Each value of (a) is obtained by using three Copula modes which are already selectedThe type function finds the corresponding conditional distribution probability:
F(Ut|Ut-1)＝h(Ut|Ut-1；θ) (2)
wherein theta is Ut、Ut-1The parameters of the joint probability function can be calculated by the copula model, and the h function is a standard conditional distribution function.
In addition, the invention further provides a 3-dimensional condition forecasting model, and the formula can be expressed as:
F(Ut|Ut-1，Ut-2)＝h[h(Ut|Ut-2)|h(Ut-1|Ut-2)](3)
wherein, Ut-2Is Xt-2(i.e., 2 previous time instants: lag 2) edge distribution function of the time series; the embodiment can be implemented under the actual requirement according to the three-dimensional condition forecasting model;
s33, calculating the probability distribution value U of the variables needing to be predicted through the inverse function of the calculation formula (2)tFor a 2-dimensional conditional prediction model, it can be written as:
Ut＝h-1(γ|Ut-1) (4)
for a 3-dimensional conditional prediction model, it can be written as:
Ut＝h-1(h-1(γ|h(Ut-1|Ut-2)|Ut-1) (5)
wherein h is-1Is the inverse function of the h condition function, gamma is the probability value to be set, and the range is [0, 1%]A plurality of gamma values may be generated by a sampling algorithm, and the number of gamma values may be 200; then, according to equation (4), for each data value (i.e., each specific water level data value that needs to be predicted), a plurality of U's may be generatedt，UtThe number of (d) is equal to the value of gamma.
In step S4, fitting water level data series X at time ttThe preparation method comprises the following steps:
s41, assuming that the water level variable data series at the time t and the water level variable data series at the time t-1 obey the same distribution, obtaining an inverse function of an edge distribution function according to the water level variable data series at the time t-1, and restoring data by using the inverse function;
and S42, inputting the fitted conditional distribution probability into an inverse function to obtain simulation data. Namely:
Xt＝F-1(Ut) (6)
in the formula, XtIndicates the predicted value at time t, F-1Representing the edge distribution function UtThe inverse function of (c);
s43, according to the step S33, generating n gamma values through a sampling algorithm, wherein n can be 200; then, the probability distribution values U at n times t can be obtained according to equation (4)tAnd the predicted value X of the water level variable at n times t can be calculated by the formula (6)t. At this time, the average value of the n predicted values is taken
Defining:
then the average value at time t is predicted by using the data at time t-1
For the predicted values finally obtained
In step S5, three models (or a plurality of models, which may be set according to actual needs) having the best expression effect are selected from the plurality of copula models, and a combined prediction model is further composed. The weight of each Copula function prediction value is calculated according to the following formula:
in the formula, wiWeight, AIC, representing the predicted value of the ith Copula function1、AIC2、AIC3Respectively represent the 1 st,AIC values of 2 nd and 3 rd Copula functions.
When the predicted value of the three Copula functions is Y1、Y2、Y3Then the final predicted value Y is calculated as:
Y＝Y1×w1+Y2×w2+Y3×w3(9)。
after step S5, the accuracy of the final predicted value is evaluated using a fitness index that includes the root mean square error RMSE, the Nash efficiency coefficient NSE, and the decision coefficient R2。
In order to verify the prediction effect of the method of the present embodiment, the present embodiment selects water level data of a certain hydrological station from 1989 to 2011 for testing (mainly taking a 2-dimensional forecasting model as an example, and a 3-dimensional forecasting model can be implemented according to a 3-dimensional forecasting formula provided by the present invention under actual requirements), and includes the following steps:
converting the data at the time t and the time t-1 into edge distribution functions respectively, and utilizing the edge distribution F (X) of the data series at the time t-1 and the time tt-1) And F (X)t) Constructing a joint distribution probability function of the water level variables at the t-1 moment and the t moment based on a binary Copula function; the types of the Copula functions are selected from a Gaussian Copula function, a BB1Copula function, a BB6Copula function, a BB7 Copula function and a BB8 Copula function, the calculated parameters and AIC values of the Copula types are shown in table 1, the Copula types selected by daily scale are a BB1Copula function, a BB6Copula function and a BB7 Copula function, the schematic diagrams of the three functions are shown in fig. 2, the Copula models selected by monthly scale are a BB1Copula function, a BB6Copula function and a BB8 Copula function, and the schematic diagrams of the three functions are shown in fig. 3.
TABLE 1 copuisa function fitting cases
Constructing a conditional distribution function of the water level variable data series at the t-1 moment and the water level variable data series at the t moment by using the selected Copula model;
based on the three screened optimal Copula models, establishing a condition prediction model by using the condition distribution function of the invention to respectively obtain the water level prediction values of the three models;
and calculating the weights of the three model predicted values according to the AIC to obtain a weighted average value of the predicted water level values, and taking the weighted average value as a final predicted value of the embodiment.
Through the above steps, the evaluation analysis of the final predicted value obtained in this example is as shown in table 2:
TABLE 2 model prediction effect of the invention
As can be seen from Table 2, the water level data of the daily scale has a good simulation effect, and the water level data of the monthly scale has a relatively poor simulation effect. As can be seen from fig. 4 and 5, the measured value of the daily scale water level data is closer to the analog value, and the simulation effect is good; the measured value of the monthly-scale water level data deviates from the analog value, and the simulation effect is relatively poor. This is because the water level data amount on the daily scale is large, and the more data amount input at the time of rating the model, the more representative the acquired model is, and the closer the model is to the actual condition, and therefore, the RMSE value is small, and NSE and R are small2The model is large, namely the model has good prediction effect; and the data volume of reference is relatively limited when the monthly scale model is established, the representativeness is poor, the forecasting effect is not as good as that of the daily scale model, but the RMSE value is small, which shows that the forecasting effect of the invention is still good. Therefore, when the combined water level forecasting method of the embodiment is used for forecasting the water level, the selected data quantity should be as large as possible, so that the model forecasting precision is improved, and a good forecasting effect is achieved. In addition, different copula types are selected possibly to have certain influence on the prediction result, so that the invention provides that a certain weight is given according to the specific performance of the copula types by introducing the copula types, thereby realizing the weight-based combined prediction. Although only the 2-dimensional condition combination forecasting model is adopted in the embodiment of the invention, the invention also provides a specific calculation formula of the 3-dimensional condition combination forecasting model, and if more input variables are used as the forecasting factors, the forecasting precision can be further improved.
It should be understood that the above-described embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. Other variations and modifications will be apparent to persons skilled in the art in light of the above description. And are neither required nor exhaustive of all embodiments. Any modification, equivalent replacement, and improvement made within the spirit and principle of the present invention should be included in the protection scope of the claims of the present invention.
Claims (9)
1. A combined river channel water level forecasting method based on a high-dimensional probability distribution function is characterized by comprising the following steps:
s1, aiming at a hydrological station, selecting a time sequence of water level variables in a certain period, and enabling X to bet-1Water level data series at time t-1, XtIs a water level data series at the time t;
s2, utilizing an edge distribution function F (X) of the water level data series at the t-1 moment and the t momentt-1) And F (X)t) Based on a binary Copula function, constructing a joint distribution probability function of water level variables at the t-1 moment and the t moment, and screening out three Copula models with the minimum value of the information content criterion AIC of the Chichi pool;
s3, inputting the water level variable data series X at the known t-1 moment by using the combined distribution probability function constructed in the step S2t-1Solving a conditional distribution probability function of the water level variable data series at the time t;
s4, further converting the conditional distribution probability function into an inverse function form thereof, thereby realizing the purpose of using Xt-1Obtaining a fitted water level data series X at time t for an input variablet；
And S5, after three Copula model predicted values are obtained, setting the weight of each Copula function predicted value according to the size of the AIC value, and calculating the weighted average value of the three Copula function predicted values to obtain a final predicted value.
2. The method for forecasting the combined river water level according to claim 1, wherein the time scales of the time series include hour, day, month and year in step S1.
3. The method for forecasting the river water level in combination according to claim 1, wherein in step S2, the binary Copula function is selected from the following types: gaussian copula function, BB1copula function, BB6copula function, BB7 copula function, BB8 copula function.
4. The method for forecasting the river water level in combination according to claim 3, wherein in step S2, the AIC value of the Copula model is calculated according to the following formula:
where k is the number of parameters of the proposed Copula function, N is the number of observed values of the variables, and c (U)t,Ut-1,iTheta) represents Ut,i,Ut-1,iAnd the joint density function c and the parameter theta, i thereof are the ith observation value.
5. The method for forecasting the water level of a combined river channel based on high-dimensional probability distribution function of claim 1, wherein 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, order Ut-1＝F(Xt-1)，F(Xt-1) Is Xt-1The edge distribution function of (1);
s32. for F (X) in step S31t-1) The corresponding conditional distribution probability is solved by utilizing the selected three Copula model functions:
F(Ut|Ut-1)＝h(Ut|Ut-1；θ) (2)
wherein theta is Ut、Ut-1The parameters of the joint probability function can be calculated by copula model function, and the h function is standardA conditional distribution function;
providing a 3-dimensional conditional prediction model represented as:
F(Ut|Ut-1，Ut-2)＝h[h(Ut|Ut-2)|h(Ut-1|Ut-2)](3)
wherein, Ut-2Is Xt-2Edge distribution function of time series, Xt-22 moments ahead;
s33, calculating the probability distribution value U of the variables needing to be predicted through the inverse function of the calculation formula (2)t：
Ut＝h-1(γ|Ut-1；θ) (4)
For a 3-dimensional conditional prediction model, it is then expressed as:
Ut＝h-1(h-1(γ|h(Ut-1|Ut-2)|Ut-1) (5)
wherein h is-1Is the inverse function of the h condition function, gamma is the probability value to be set, and the range is [0, 1%]Multiple gamma values may be generated by a sampling algorithm and multiple U's generated for each particular water level data value to be predicted according to equation (4)t。
6. The method for forecasting the river water level in combination according to claim 5, wherein the fitting water level data series X at time t is obtained in step S4tThe preparation method comprises the following steps:
s41, assuming that the water level variable data series at the time t and the water level variable data series at the time t-1 obey the same distribution, and obtaining an inverse function of an edge distribution function according to the water level variable data series at the time t-1;
s42, inputting the fitted conditional distribution probability into an inverse function of a predicted variable distribution function to obtain simulation data; namely:
Xt＝F-1(Ut) (6)
in the formula, XtIndicates the predicted value at time t, F-1Representing the edge distribution function UtThe inverse function of (c);
s43, according to the step S33, generating n gamma values through a sampling algorithm, and obtaining n probability distribution values U at t moments according to the formula (4)tAnd then the predicted value X of the water level variable at n times t is obtained by calculation according to the formula (6)t(ii) a At this time, the average value of the n predicted values is taken
Defining:
then the water level data series at the time t is obtained by utilizing the data value prediction at the time t-1, namely the water level data series at the time t is the value to be predicted
7. The method for forecasting the river water level in combination according to claim 1, wherein the weight of each Copula function prediction value is calculated according to the following formula in step S5:
in the formula, wiWeight, AIC, representing the predicted value of the ith Copula function1、AIC2、AIC3Respectively, the AIC values of the 1 st, 2 nd and 3 rd Copula functions are shown.
8. The method according to claim 7, wherein the prediction value of each Copula function is Y1、Y2、Y3Then the final predicted value Y is calculated as:
Y＝Y1×w1+Y2×w2+Y3×w3(9)。
9. the method for forecasting the river water level in combination based on the high-dimensional probability distribution function of claim 7, wherein after step S5, the accuracy of the final predicted value is evaluated by using the fitness index, which includes the Root Mean Square Error (RMSE), the Nash efficiency coefficient (NSE) and the decision coefficient (R)2。
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