CN110598181A - Extreme hydrological event risk analysis method and system based on maximum entropy - Google Patents

Extreme hydrological event risk analysis method and system based on maximum entropy Download PDF

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CN110598181A
CN110598181A CN201910821570.XA CN201910821570A CN110598181A CN 110598181 A CN110598181 A CN 110598181A CN 201910821570 A CN201910821570 A CN 201910821570A CN 110598181 A CN110598181 A CN 110598181A
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陈璐
张寒松
仇红亚
钟文杰
易彬
吴常运
齐尧乐
林橙
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Abstract

The invention discloses an extreme hydrological event risk analysis method and system based on maximum entropy, belonging to the field of extreme hydrological event risk analysis in hydrology, and comprising the following steps: sampling acquired measured flow values of a drainage basin for years at maximum year of use to obtain long-sequence hydrological data; taking various generalized extreme value distribution functions as a model to be selected, fitting long-sequence hydrological data, performing parameter estimation on the model to be selected to obtain a parameter estimation value of the model to be selected, and determining a probability density distribution function of the model to be selected according to the parameter estimation value of the model to be selected; calculating the entropy value of the probability density distribution function of the model to be selected, and taking the model to be selected with the maximum entropy value as an optimal model; and (4) carrying out extreme hydrological event risk analysis by using the optimal model to obtain the designed flood value of the drainage basin in one hundred years and one thousand years. The invention is not limited by the number of the probability density function parameters of the selected model, and can give consistent estimation and always correctly select the optimal distribution for the model with more parameters.

Description

Extreme hydrological event risk analysis method and system based on maximum entropy
Technical Field
The invention belongs to the field of extreme hydrological event risk analysis in hydrology, and particularly relates to an extreme hydrological event risk analysis method and system based on maximum entropy.
Background
The extreme hydrological event risk analysis is to establish a probability distribution statistical model according to historical extreme hydrological information, reveal the space-time distribution rule and the evolution trend of the extreme hydrological event, estimate the recurrence period of the extreme hydrological event, and invert the design rainstorm and the design flood value of a certain design standard. The core of the extreme hydrologic event risk analysis is the determination of the distribution function form of the hydrologic extreme events. The main problems existing in the current extreme hydrological event risk analysis and design flood calculation are as follows: the probability risk calculated by different distribution functions and the inverted design flood value have obvious difference under the limitation of the form, the bias characteristic and the tail characteristic of the distribution function. Therefore, the invention introduces a generalized distribution function cluster to participate in calculation. How to select the optimal distribution function is a difficult and hot problem. The existing model selection method mainly depends on the root mean square error RMSE, the AIC criterion and the BIC criterion. However, the AIC criterion and the BIC criterion are limited by the number of the parameters of the probability density function of the selected model, and for the model with more parameters, a consistent estimation cannot be given, and the optimal distribution cannot be always correctly selected.
Therefore, the prior art has the technical problems that the number of the parameters of the probability density function of the selected model is limited, the models with more parameters cannot give consistent estimation, and the optimal distribution cannot be always correctly selected.
Disclosure of Invention
In view of the above defects or improvement needs of the prior art, the present invention provides a maximum entropy-based extreme hydrological event risk analysis method and system, thereby solving the technical problems of the prior art that the number of parameters of the probability density function of the selected model is limited, the model with more parameters cannot give consistent estimation, and the optimal distribution cannot be always correctly selected.
To achieve the above object, according to one aspect of the present invention, there is provided a maximum entropy-based extreme hydrologic event risk analysis method, comprising the steps of:
(1) sampling acquired measured flow values of a drainage basin for years at maximum year of use to obtain long-sequence hydrological data;
(2) taking various generalized extreme value distribution functions as a model to be selected, fitting long-sequence hydrological data, performing parameter estimation on the model to be selected to obtain a parameter estimation value of the model to be selected, and determining a probability density distribution function of the model to be selected according to the parameter estimation value of the model to be selected;
(3) calculating the entropy value of the probability density distribution function of the model to be selected, and taking the model to be selected with the maximum entropy value as an optimal model;
(4) and (4) carrying out extreme hydrological event risk analysis by using the optimal model to obtain the designed flood value of the drainage basin in one hundred years and one thousand years.
Further, the plurality of generalized extremum distribution functions includes: generalized Gamma distribution, generalized Beta second-type distribution, Halphen A distribution, Halphen B distribution and Halphen IB distribution.
Further, the specific implementation manner of performing parameter estimation in step (2) is as follows:
and fitting the long-sequence hydrological data, constructing a function mapping relation between a maximum entropy principle Lagrange multiplier and various generalized extreme value distribution function parameters, establishing a parameter estimation equation set of the model to be selected according to the function mapping relation, and calculating the parameter estimation value of the model to be selected by using the parameter estimation equation set of the model to be selected.
Further, the entropy of the probability density distribution function of the candidate model is:
h (x) is an entropy function of continuous information entropy, namely an entropy value of a probability density distribution function of the model to be selected, f (x) is the probability density distribution function of the model to be selected, and a and b are a lower limit and an upper limit in entropy value calculation respectively.
Further, the step (4) comprises:
and drawing a probability flow curve by using a probability density distribution function of the optimal model and the fitted long-sequence hydrological data, performing extreme hydrological event risk analysis by using the probability flow curve, taking the flow corresponding to the probability flow curve with the probability of 1% as a designed flood value of the basin in one hundred years, and taking the flow corresponding to the probability flow curve with the probability of 0.1% as a designed flood value of the basin in one thousand years.
According to another aspect of the present invention, there is provided an extreme hydrologic event risk analysis system based on maximum entropy, comprising:
the first module is used for sampling acquired measured flow values of a drainage basin for years at maximum year of use to obtain long-sequence hydrological data;
the second module is used for taking various generalized extreme value distribution functions as a model to be selected, fitting the long-sequence hydrological data, performing parameter estimation on the model to be selected to obtain a parameter estimation value of the model to be selected, and determining a probability density distribution function of the model to be selected according to the parameter estimation value of the model to be selected;
the third module is used for calculating the entropy value of the probability density distribution function of the model to be selected, and taking the model to be selected with the maximum entropy value as the optimal model;
and the fourth module is used for carrying out extreme hydrological event risk analysis by utilizing the optimal model to obtain the designed flood value of the drainage basin in one hundred years and one thousand years.
Further, the plurality of generalized extremum distribution functions includes: generalized Gamma distribution, generalized Beta second-type distribution, Halphen A distribution, Halphen B distribution and Halphen IB distribution.
Further, the specific implementation manner of the parameter estimation in the second module is as follows:
and fitting the long-sequence hydrological data, constructing a function mapping relation between a maximum entropy principle Lagrange multiplier and various generalized extreme value distribution function parameters, establishing a parameter estimation equation set of the model to be selected according to the function mapping relation, and calculating the parameter estimation value of the model to be selected by using the parameter estimation equation set of the model to be selected.
Further, the entropy of the probability density distribution function of the candidate model is:
h (x) is an entropy function of continuous information entropy, namely an entropy value of a probability density distribution function of the model to be selected, f (x) is the probability density distribution function of the model to be selected, and a and b are a lower limit and an upper limit in entropy value calculation respectively.
Further, the fourth module includes:
and drawing a probability flow curve by using a probability density distribution function of the optimal model and the fitted long-sequence hydrological data, performing extreme hydrological event risk analysis by using the probability flow curve, taking the flow corresponding to the probability flow curve with the probability of 1% as a designed flood value of the basin in one hundred years, and taking the flow corresponding to the probability flow curve with the probability of 0.1% as a designed flood value of the basin in one thousand years.
In general, compared with the prior art, the above technical solution contemplated by the present invention can achieve the following beneficial effects:
(1) the method breaks through the theoretical bottleneck of extreme hydrological event risk analysis from three aspects of distribution form-parameter estimation method-model selection by introducing various generalized extreme value distributions and utilizing the maximum entropy principle to carry out parameter estimation and model selection. The accuracy and the rationality of risk analysis and design flood calculation are improved, and the resource waste caused by overhigh design value and the destructive risk caused by overlow design value are effectively reduced. The invention is not limited by the number of the selected model probability density function parameters, compatible estimation can be given for the model with more parameters, and the optimal distribution can be always correctly selected.
(2) The method is used as a new model selection method, and mutually verifies with the existing method to verify the rationality of the proposed model. The theoretical bottlenecks that the extreme value distribution probability density function of the existing method is single in shape, the deviation of the design result after the frequency curve is extended is large, and the deviation and the heavy tail characteristic of the hydrological extreme value sequence cannot be fully reflected are broken through. Economic losses caused by excessive safety due to the fact that the design value is once met in the process of overestimating T years in engineering design and dam break and dam overflow risks of the dyke reservoir caused by underestimating the design value are avoided, the operation benefits of the reservoir in the flood season are brought into play, and the reasonability and feasibility of flood calculation are improved.
Drawings
FIG. 1 is a flowchart of a maximum entropy-based extreme hydrologic event risk analysis method according to an embodiment of the present invention;
fig. 2 is a probability flow graph of five distributions provided by the embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the present invention is described in further detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative of the invention and are not intended to limit the invention. In addition, the technical features involved in the embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.
As shown in fig. 1, a maximum entropy-based extreme hydrologic event risk analysis method includes the following steps:
(1) sampling acquired measured flow values of a drainage basin for years at maximum year of use to obtain long-sequence hydrological data;
(2) taking various generalized extreme value distribution functions as a model to be selected, fitting long-sequence hydrological data, performing parameter estimation on the model to be selected to obtain a parameter estimation value of the model to be selected, and determining a probability density distribution function of the model to be selected according to the parameter estimation value of the model to be selected;
(3) calculating the entropy value of the probability density distribution function of the model to be selected, and taking the model to be selected with the maximum entropy value as an optimal model;
(4) and (4) carrying out extreme hydrological event risk analysis by using the optimal model to obtain the designed flood value of the drainage basin in one hundred years and one thousand years.
Further, the plurality of generalized extremum distribution functions includes: generalized Gamma distribution, generalized Beta second-type distribution, Halphen A distribution, Halphen B distribution and Halphen IB distribution. The 5 generalized extreme value distribution functions have strong adaptability and are suitable for risk analysis of extreme hydrological events.
Further, the specific implementation manner of performing parameter estimation in step (2) is as follows:
and fitting the long-sequence hydrological data, constructing a function mapping relation between a maximum entropy principle Lagrange multiplier and various generalized extreme value distribution function parameters, establishing a parameter estimation equation set of the model to be selected according to the function mapping relation, and calculating the parameter estimation value of the model to be selected by using the parameter estimation equation set of the model to be selected.
Further, the entropy of the probability density distribution function of the candidate model is:
h (x) is an entropy function of continuous information entropy, namely an entropy value of a probability density distribution function of the model to be selected, f (x) is the probability density distribution function of the model to be selected, and a and b are a lower limit and an upper limit in entropy value calculation respectively.
Further, the step (4) comprises:
and drawing a probability flow curve by using a probability density distribution function of the optimal model and the fitted long-sequence hydrological data, performing extreme hydrological event risk analysis by using the probability flow curve, taking the flow corresponding to the probability flow curve with the probability of 1% as a designed flood value of the basin in one hundred years, and taking the flow corresponding to the probability flow curve with the probability of 0.1% as a designed flood value of the basin in one thousand years.
The basis for selecting the model with the maximum entropy value as the optimal model is as follows:
according to the maximum entropy principle, under the condition of no constraint condition, the probability of occurrence of each data in the data set is equal, at the moment, the entropy value of the function is maximum, and the entropy value is continuously reduced along with the increase of the constraint condition. Verification is performed by discrete information entropy and continuous information entropy as follows:
using the expression for discrete information entropy:
wherein H (p) is an entropy function of discrete information entropy; p is a radical ofiIs the probability of each event occurring. The unit of entropy is related to the base of the logarithm in the formula.
Without other constraints, the method comprises:
using Lagrange multiplier method to obtain conditional extremum to construct Lagrange multiplier L (p) of H (p)i) Solving for the maximum value of H (p) as follows:
in the formula, L (p)i) Lagrange multiplier type;is a constraint; the undetermined coefficient λ is the lagrange multiplier.
Are respectively to piCarrying out derivation, making the derived value be 0, and solving for piAnd λ, the following equation is obtained:
obtaining by solution:
pi＝eλ-1
consists of:
the following can be obtained:
it is stated that without constraints, the probability of each data in the data set occurring is equal, when the entropy of the function is at a maximum.
Using the expression of continuous information entropy:
wherein H (x) is an entropy function of continuous information entropy; f (x) is a probability density function of the variable x. Assuming that the probability distribution f (x) satisfies the condition:
in the formula, Ar(x) Is an arbitrary function with respect to x; crIs Ar(x) Is calculated from the expected value of (c).
As can be seen from the above discrete entropy equation, for the continuous entropy function, the lagrangian multiplier method is also used to solve the conditional extremum, construct the lagrangian multiplier formula of f (x), and can solve the expression of f (x) as follows:
then:
substituting the expression f (x) into Hm(f) Can obtain the following components:
in the formula, λrA lagrange multiplier determined for the constraint; r represents a number.
Let q (x) be another probability distribution that satisfies n constraints, where: n is more than or equal to m
Order:
and (3) constructing a function:
g(x)＝xlnx-x+1
wherein x is greater than 0.
Derivation of g (x) can give:
g' (x) ═ lnx g (x) monotonically decreases at (0, 1) and monotonically increases at (1, ∞).
Then:
g (x) g (1) ═ 0:
xlnx≥x-1
the above formula can be rewritten as:
wherein h (x) > 0.
Order:the following can be obtained:
multiplying q (x) on both sides of the equation gives:
namely:
then:
substituting the expressions q (x) and f (x) into the above equation:
wherein:
since n is more than or equal to m, then: lambda [ alpha ]0、λ1、...λmIf present and meaningful, the above formula can be written:
the following can be obtained:
Hn(q)≤Hm(f)，n≥m
as explained above, as the constraint increases, the entropy value decreases.
In the verification process, the model with the largest entropy value among the 5 distribution models is selected as the most optimal model, because the model is the model with the least increase of the constraint conditions under the condition of containing known information, and is also the model with the least unknown assumption. In accordance with the principle of insufficient inference of the Laplace, and in accordance with the principle of maximum entropy. Namely: in the case of containing known information, unknown events should be treated as equal probability events without making any unknown assumptions. Therefore, the rationale should be a best fit model.
In order to more clearly show the purpose, structure and technical scheme of the invention, the invention is further described in detail by using the Qingjiang river basin and the attached drawings, and the specific implementation steps comprise:
step 1, acquiring annual runoff data of a Qingjiang river basin for 43 years, and calculating annual representative annual runoff by using an annual maximum value method to obtain long-sequence hydrological data serving as calculation fitting data.
And 2, selecting 5 generalized distribution models of generalized Gamma distribution, generalized second-class Beta distribution (GB2), Halphen A distribution, Halphen B distribution and Halphen IB distribution according to the step 1, and fitting.
And 3, obtaining estimation parameters of each model by utilizing a maximum entropy principle according to the step 2, and further deducing probability density expressions of the 5 models.
And 4, according to the step 3, utilizing an expression of the continuous information entropy:
and respectively substituting the probability density functions of the 5 distribution models into a continuous information entropy formula to obtain entropy values of the 5 distribution models, and selecting the model with the maximum entropy value from the 5 distribution models as an optimal model according to a maximum entropy principle. The results obtained are shown in Table 1.
TABLE 1 maximum entropy principle calculation results Table
Distribution function Entropy value H (x)
GG 3.435
GB2 3.668
Hal-A 2.827
Hal-B 2.442
Hal-IB 2.212
And 5, calculating 5 distributed AIC values, BIC values and RMSE values according to the step 4, selecting an optimal model, and comparing and analyzing the optimal model with the model selected by the maximum entropy principle to obtain the optimal model. The results are shown in Table 2 and FIG. 2.
TABLE 2 AIC, BIC, RMSE value calculation result table
Distribution function RMSE AIC BIC
GG 0.0259 -256.8 -248.4
GB2 0.0196 -263.1 -254.7
Hal-A 0.0269 -255.5 -247.1
Hal-B 0.0288 -252.9 -244.5
Hal-IB 0.0379 -211.9 -203.5
And 6, drawing a probability-flow curve of 5 kinds of distribution according to the step 5, and combining the optimal model distribution to calculate the design flood value of the Qingjiang river basin with one hundred years and one thousand years of distribution prediction. The results are shown in Table 3.
TABLE 3 flood value table for Qingjiang river basin
Distribution function One hundred year meeting All the year round
GB2 12630.92 19606.77
GG 11635.92 14959.42
Hal-A 11448.49 14480.81
Hal-B 11174.42 13945.87
Hal-IB 11322.21 14532.21
According to the table 1, the model with the largest entropy value is GB2 distribution, the optimal distribution is selected as the maximum entropy principle, according to the table 2, the model selected according to the AIC value, BIC value and RMSE value is GB2 distribution, and the maximum entropy principle conclusion is met. The optimal distribution of the annual runoff of the Qingjiang river basin can be obtained as GB2 distribution. According to fig. 2, the probability-flow curve of the GB2 distribution is significantly more consistent with the event situation. As shown in Table 3, the designed flood values of the Qingjiang river basin are 12630.92m for one hundred years and one thousand years respectively3/s、19606.77m3And s. It can be seen that the GB2 predicted value has obvious difference with other distribution predicted values, and the real situation can be better fitted.
In summary, the invention introduces various generalized extremum distribution functions as the model to be selected; performing parameter estimation by using a maximum entropy principle, and determining a specific probability density function expression of a distribution function; according to a discrete entropy and continuous entropy formula, deducing a principle of selecting a distribution function by using the maximum entropy, calculating an entropy value of the distribution function, and selecting an optimal distribution function form by using the size of the entropy value; and further, the designed flood value of the drainage basin in one hundred years and one thousand years is calculated. The method improves the accuracy and rationality of risk analysis and design flood calculation, effectively reduces resource waste caused by overhigh design value and destructive risk caused by overlow design value, and breaks through the theoretical bottlenecks that the extreme value distribution probability density function of the existing method has single shape, the design result has larger deviation after the frequency curve is delayed, and the deviation and the heavy tail characteristic of the hydrological extreme value sequence cannot be fully reflected.
It will be understood by those skilled in the art that the foregoing is only a preferred embodiment of the present invention, and is not intended to limit the invention, and that any modification, equivalent replacement, or improvement made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (10)

1. The extreme hydrological event risk analysis method based on the maximum entropy is characterized by comprising the following steps of:
(1) sampling acquired measured flow values of a drainage basin for years at maximum year of use to obtain long-sequence hydrological data;
(2) taking various generalized extreme value distribution functions as a model to be selected, fitting long-sequence hydrological data, performing parameter estimation on the model to be selected to obtain a parameter estimation value of the model to be selected, and determining a probability density distribution function of the model to be selected according to the parameter estimation value of the model to be selected;
(3) calculating the entropy value of the probability density distribution function of the model to be selected, and taking the model to be selected with the maximum entropy value as an optimal model;
(4) and (4) carrying out extreme hydrological event risk analysis by using the optimal model to obtain the designed flood value of the drainage basin in one hundred years and one thousand years.
2. The extreme hydrologic event risk analysis method of maximum entropy based on claim 1, wherein the plurality of generalized extreme distribution functions comprises: generalized Gamma distribution, generalized Beta second-type distribution, Halphen A distribution, Halphen B distribution and Halphen IB distribution.
3. The extreme hydrologic event risk analysis method based on maximum entropy as claimed in claim 1 or 2, characterized in that the specific implementation manner of parameter estimation in step (2) is as follows:
and establishing a function mapping relation between the maximum entropy principle Lagrange multiplier and various generalized extreme value distribution function parameters, establishing a parameter estimation equation set of the model to be selected according to the function mapping relation, and calculating the parameter estimation value of the model to be selected by using the parameter estimation equation set of the model to be selected.
4. The extreme hydrological event risk analysis method based on maximum entropy according to claim 1 or 2, characterized in that the entropy of the probability density distribution function of the candidate model is:
h (x) is an entropy function of continuous information entropy, namely an entropy value of a probability density distribution function of the model to be selected, f (x) is the probability density distribution function of the model to be selected, and a and b are a lower limit and an upper limit in entropy value calculation respectively.
5. A maximum entropy based extreme hydrologic event risk analysis method as claimed in claim 1 or 2, characterized in that said step (4) comprises:
and drawing a probability flow curve by using a probability density distribution function of the optimal model and the fitted long-sequence hydrological data, performing extreme hydrological event risk analysis by using the probability flow curve, taking the flow corresponding to the probability flow curve with the probability of 1% as a designed flood value of the basin in one hundred years, and taking the flow corresponding to the probability flow curve with the probability of 0.1% as a designed flood value of the basin in one thousand years.
6. An extreme hydrologic event risk analysis system based on maximum entropy, comprising:
the first module is used for sampling acquired measured flow values of a drainage basin for years at maximum year of use to obtain long-sequence hydrological data;
the second module is used for taking various generalized extreme value distribution functions as a model to be selected, fitting the long-sequence hydrological data, performing parameter estimation on the model to be selected to obtain a parameter estimation value of the model to be selected, and determining a probability density distribution function of the model to be selected according to the parameter estimation value of the model to be selected;
the third module is used for calculating the entropy value of the probability density distribution function of the model to be selected, and taking the model to be selected with the maximum entropy value as the optimal model;
and the fourth module is used for carrying out extreme hydrological event risk analysis by utilizing the optimal model to obtain the designed flood value of the drainage basin in one hundred years and one thousand years.
7. The extreme hydrological event risk analysis system of claim 6, wherein the plurality of generalized extremum distribution functions comprises: generalized Gamma distribution, generalized Beta second-type distribution, Halphen A distribution, Halphen B distribution and Halphen IB distribution.
8. The extreme hydrological event risk analysis system based on maximum entropy of claim 6 or 7, wherein the specific implementation manner of parameter estimation in the second module is as follows:
and establishing a function mapping relation between the maximum entropy principle Lagrange multiplier and various generalized extreme value distribution function parameters, establishing a parameter estimation equation set of the model to be selected according to the function mapping relation, and calculating the parameter estimation value of the model to be selected by using the parameter estimation equation set of the model to be selected.
9. The extreme hydrological event risk analysis system based on maximum entropy according to claim 6 or 7, wherein the entropy of the probability density distribution function of the candidate model is:
h (x) is an entropy function of continuous information entropy, namely an entropy value of a probability density distribution function of the model to be selected, f (x) is the probability density distribution function of the model to be selected, and a and b are a lower limit and an upper limit in entropy value calculation respectively.
10. The maximum entropy-based extreme hydrologic event risk analysis system of claim 6 or 7, wherein the fourth module comprises:
and drawing a probability flow curve by using a probability density distribution function of the optimal model and the fitted long-sequence hydrological data, performing extreme hydrological event risk analysis by using the probability flow curve, taking the flow corresponding to the probability flow curve with the probability of 1% as a designed flood value of the basin in one hundred years, and taking the flow corresponding to the probability flow curve with the probability of 0.1% as a designed flood value of the basin in one thousand years.
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