CN107622162A - A kind of rating curve calculation method based on Copula functions - Google Patents
A kind of rating curve calculation method based on Copula functions Download PDFInfo
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Abstract
The invention discloses a kind of rating curve calculation method based on Copula functions, by collecting section water level and data on flows data, it is determined that on the basis of marginal probability distribution function, the joint probability distribution function of water level and flow is built using Copula functions, and then the conditional probability distribution function of flow when giving water level is solved, rating curve and analysis of uncertainty are inquired into according to statistical principle on this basis.The present invention has stronger statistical theory basis, it is allowed to which water level and flow have any type of edge distribution, can describe the non-linear and Singular variance correlation structure between water level and flow exactly.In addition, can not only obtain the point estimate of flow, and the uncertainty of model parameter and model structure can be considered simultaneously more fully hereinafter, obtain flow and integrate uncertain section.
Description
Technical field
The invention belongs to hydraulic engineering field, more particularly to a kind of rating curve based on Copula functions pushes away
Seek method.
Background technology
Stage discharge relation refers to the relation between the water level of river cross-section and the corresponding discharge.Due to discharge measurement technology
More complicated, cost is costly, it is difficult to is carried out continuously, generally passes through continuous water level prediction in hydrological data compilation
It is converted to continuous flow data, while also commonly uses it in hydrologic forecast, the hydrology calculate and water conservancy management works to do water
Conversion between position, flow.In addition, highest, the corresponding discharge of lowest water level can not be measured because of the limitation of conditions, need to be according to stage-discharge
Relation curve makees the extension of height water, and whether this extension is appropriate, can directly influence the scale and size of engineering design project.Cause
This, rating curve has important Practical significance[1]。
Rating curve corresponds to data on flows to determine according to the multiple measured water level of section with it, conventional method one
As presuppose stage discharge relation and obey a certain mathematics line style, then solved under selected Optimality Criteria using optimized algorithm
Parameter, so that it is determined that the specific mathematical equation of stage discharge relation[2].At present using more line style be include power function type,
Polynomial type and logarithmic function type, Optimality Criteria include residual sum of squares (RSS) minimum criteria, absolute residuals absolute value and minimum criteria
And relative residual absolute value and minimum criteria, optimized algorithm mainly have least square method, genetic algorithm, ant group algorithm, particle
It is group's algorithm, chaos algorithm, mixing tabu search algorithm, artificial fish-swarm algorithm, artificial bee colony algorithm, gregarious spider algorithm, immune
Evolution algorithm, differential evolution algorithm etc.[3].For the deficiency of conventional method, there is scholar to propose to apply artificial neural network[4], branch
Hold vector machine[5]And genetic program[6]The methods of be fitted stage discharge relation, although avoiding the office for presupposing specific functional expression
Limit, but all remain some problem and shortage.The structure of artificial neural network can only be selected by experience, lack unified reason
By guidance, SVMs is sensitive to missing data, how to select suitable kernel function dispute, genetic program convergence efficiency to be present
Low, obtained regression formula is excessively complicated and unstable.In addition, some scholars use the Markov Chain based on bayesian theory
Parameter and its uncertainty in Monte Carlo (MCMC) method estimation power function type rating curve, and give water
The confidential interval of bit traffic relation curve[7].But MCMC methodology still needs the mathematics line style for presetting stage discharge relation,
And estimation is only capable of due to rating curve confidential interval caused by model parameter uncertainty, and model knot can not be considered
The uncertainty of structure.
Copula functions can construct the Joint Distribution of multiple stochastic variables with any edge distribution, can catch well
The abnormal feature between variable and non-linear, the Singular variance dependency relation between them are caught, in obtaining for hydrographic water resource field
It is widely applied[8].At present, Copula functions are introduced into during rating curve inquires into by no document.
The content of the invention
In view of the deficienciess of the prior art, the invention provides a kind of stage discharge relation based on Copula functions is bent
Line calculation method.
In order to solve the above technical problems, the present invention adopts the following technical scheme that：
A kind of rating curve calculation method based on Copula functions, including step：
Step 1, section water level and data on flows data are collected；
Step 2, according to the water level in step 1 and data on flows data, appropriate edge distribution line style is chosen, and estimate it
Parameter；
Step 3, using Copula construction of function water level and the joint probability distribution function of flow, and Copula functions are estimated
Parameter；
Step 4, the joint distribution function that the marginal distribution function and step 3 estimated according to step 2 are built solves given water
The analytical expression of flow condition probability-distribution function during position；
Step 5, the analytical expression according to the conditional probability distribution function obtained by step 4, according to statistical principle, is pushed away
Ask rating curve and analysis of uncertainty.
In the step 2, the marginal probability distribution function line style using the distribution of P-III types as water level and flow.
In the step 2, using the parameter of linear Moment method estimators marginal probability distribution function.
In the step 3, using the joint probability of Gumbel-Hougaard Copula construction of function water levels and flow point
Cloth function, the parameter of Gumbel-Hougaard Copula functions is estimated using Kendall rank correlations Y-factor method Y.
The present invention is by collecting section water level and data on flows data, it is determined that on the basis of marginal probability distribution function,
The condition of flow is general when the joint probability distribution function of water level and flow is built using Copula functions, and then solving given water level
Rate distribution function, rating curve and analysis of uncertainty are inquired into according to statistical principle on this basis.
Compared with prior art, the beneficial effects of the present invention are：
(1) present invention has stronger statistical theory basis, it is allowed to and water level and flow have any type of edge distribution,
Non-linear and Singular variance correlation structure between water level and flow can be described exactly.
(2) compared with normal water level discharge relation curve calculation method, the present invention can not only obtain the point estimation of flow
Value, and the uncertainty of model parameter and model structure can be considered simultaneously more fully hereinafter, it is uncertain to obtain flow synthesis
Property section.
Brief description of the drawings
Fig. 1 is the flow chart of the inventive method.
Fig. 2 is the rating curve schematic diagram inquired into based on Copula functions.
Embodiment
Below by embodiment, and with reference to accompanying drawing, the invention will be further described.
As Figure 1-Figure 2, a kind of rating curve calculation method based on Copula functions, section water is collected
Position and data on flows data, it is determined that on the basis of marginal probability distribution function, water level and flow are built using Copula functions
Joint probability distribution function, and then when solving given water level flow conditional probability distribution function, on this basis according to number
Reason Statistics inquires into rating curve and analysis of uncertainty.Fig. 1 is the calculation flow chart of the present embodiment, according to
Lower step is carried out：
1. collect section water level and data on flows data.
The time scale of measured water level and data on flows data is day in this specific implementation.Section day water level and daily flow money
Expect to obtain from the Water Year Book at hydrometric station.
2. determine the marginal probability distribution function of water level and flow.
According to the water level in step 1 and data on flows data, appropriate edge distribution line style is chosen, and estimates its parameter,
This step includes two sub-steps：
2.1 selection edge distribution line styles
Because the overall distribution frequency curves of water level and flow are unknown, generally from energy good fit majority hydrology sample
The line style of this data system.China finds water level of the P-III types distribution for China major part river by com-parison and analysis for many years
It is preferable with flow data fitting, recommend to use in engineering practice.
The edge distribution line style as water level and flow is distributed using P-III types in this specific implementation.
The parameter of 2.2 estimation edge distribution line styles
After curve type of frequency distribution is selected, next need to carry out the parameter for estimating frequency distribution.Currently used method
Mainly there are moments method, maximum-likelihood method, suitable collimation method, probability-weighted moment, weight-function method and linear moments method (L- moments methods) etc..Wherein,
L- moments methods are the actual parameter methods of estimation generally acknowledged both at home and abroad at present, and maximum feature is that do not have to the maximum and minimum of sequence
Conventional square is so sensitive, and the estimates of parameters tried to achieve is more sane.
The parameter of L- Moment method estimators edge distribution line styles is used in this specific implementation.
3. utilize Copula functions structure water level and the joint probability distribution function of flow.
According to the marginal probability distribution function estimated in the water level in step 1 and data on flows data and step 2, choose
Appropriate Copula construction of function water level and the joint probability distribution function of flow, and estimate its parameter, this step includes two
Sub-step：
3.1 selection Copula functions
Assuming that H, Q represent water level and flow respectively, h, q are respectively corresponding implementation value.FH(h)、FQ(q) it is marginal probability
Distribution function, corresponding probability density function are fH(h)、fQ(q).From Sklar theorems, H, Q joint distribution function can be with
Represented with a dimensional Co pula functions C：
FH,Q(h, q)=Cθ(FH(h),FQ(q))=Cθ(u,v) (1)
Wherein, θ is the parameter of Copula functions；U=FH(h), v=FQ(q) it is marginal distribution function.
In this specific implementation, using the joint probability of Gumbel-Hougaard Copula construction of function water levels and flow point
Cloth function, its expression formula are as follows：
The parameter of 3.2 estimation Copula functions
In this specific implementation, Gumbel-Hougaard Copula functions are estimated using Kendall rank correlations Y-factor method Y
Parameter.The relation of Kendall coefficient correlations τ and parameter θ is：
Make { (x1,y1),…,(xn,yn) represent from the middle n observation extracted of continuous random variable (X, Y) with press proof
This, then have in the sampleThe different observation combination (x of kindi,yi) and (xj,yj).Sample Kendall rank correlation coefficients τ passes through
Following formula calculates
Wherein, sign () is sign function.
4. the conditional probability distribution function of flow when solving given water level.
During given water level H value h, corresponding flow Q value is simultaneously not exclusive, but changeable, simply occurs not
Probability with value is different, and there is a conditional probability distribution function
FQ|H(q)=P (Q≤q | H=h) (5)
By Copula functions, conditional probability distribution function FQ|H(q) can be expressed as：
Formula (2) is substituted into：
5. inquire into rating curve and analysis of uncertainty.
Obtain flow Q conditional probability distribution function FQ|H(q) after, according to statistical principle, middle position can be calculated
Number is used as flow Q point estimate, and the flow Q median functions obtained accordingly are stage discharge relation function.Meanwhile obtain to
The interval estimation determined under confidence level carries out analysis of uncertainty.
Flow Q median qmSolved by following formula：
FQ|H(qm)=0.5 (8)
Formula (8) is solved using dichotomy tentative calculation in this specific implementation and obtains numerical solution.
Flow Q median q during by solving any given H=hm, it is possible to obtain what is inquired into based on Copula functions
Stage discharge relation function, is shown below：
Q=qm(h) (9)
Select certain confidence level (1- ξ), make flow Q values appear in distribution both ends probability be ξ, it is possible to define
R interval estimation, the upper and lower limit of confidence are provided by following two formula respectively：
FQ|H(ql)=ξ1 (10)
FQ|H(qu)=1- ξ2 (11)
Wherein, ξ1+ξ2=ξ, represent significance；ξ is taken in this specific implementation1=ξ2=ξ/2.
Formula (10) is solved using dichotomy tentative calculation in this specific implementation, (11) obtain numerical solution.Therefore
P(ql≤Q≤qu)=1- ξ (12)
That is [ql,qu] for flow Q confidence level (1- ξ) interval estimation, flow Q can be estimated according to confidential interval
The comprehensive uncertain progress quantitative assessment of value.
As shown in Fig. 2 give the rating curve schematic diagram inquired into based on Copula functions.Wherein, black
Round dot is measured value, and solid line is median result, and dash area represents the 90% uncertain section of flow of estimation.
To sum up, the present invention is by collecting section water level and data on flows data, it is determined that the base of marginal probability distribution function
On plinth, the flow when joint probability distribution function of water level and flow is built using Copula functions, and then solving given water level
Conditional probability distribution function, rating curve and uncertain point are inquired into according to statistical principle on this basis
Analysis.The present invention has stronger statistical theory basis, it is allowed to which water level and flow have any type of edge distribution, can be accurate
Ground describes the non-linear and Singular variance correlation structure between water level and flow.In addition, it can not only obtain the point estimation of flow
Value, and the uncertainty of model parameter and model structure can be considered simultaneously more fully hereinafter, it is uncertain to obtain flow synthesis
Property section.
It is obvious to a person skilled in the art that the invention is not restricted to the details of above-mentioned one exemplary embodiment, Er Qie
In the case of without departing substantially from spirit or essential attributes of the invention, the present invention can be realized in other specific forms.Therefore, no matter
From the point of view of which point, embodiment all should be regarded as exemplary, and be nonrestrictive, the scope of the present invention is by appended power
Profit requires rather than described above limits, it is intended that all in the implication and scope of the equivalency of claim by falling
Change is included in the present invention.Any reference in claim should not be considered as to the involved claim of limitation.
Moreover, it will be appreciated that although the present specification is described in terms of embodiments, not each embodiment is only wrapped
Containing an independent technical scheme, this narrating mode of specification is only that those skilled in the art should for clarity
Using specification as an entirety, the technical solutions in the various embodiments may also be suitably combined, forms those skilled in the art
It is appreciated that other embodiment.
Claims (4)
1. a kind of rating curve calculation method based on Copula functions, it is characterised in that comprise the following steps：
Step 1, section water level and data on flows data are collected；
Step 2, according to the water level in step 1 and data on flows data, appropriate edge distribution line style is chosen, and estimate its ginseng
Number；
Step 3, using Copula construction of function water level and the joint probability distribution function of flow, and the ginseng of Copula functions is estimated
Number；
Step 4, when the joint distribution function that the marginal distribution function and step 3 estimated according to step 2 are built solves given water level
The analytical expression of flow condition probability-distribution function；
Step 5, the analytical expression according to the conditional probability distribution function obtained by step 4, according to statistical principle, inquires into water
Bit traffic relation curve and analysis of uncertainty.
2. a kind of rating curve calculation method based on Copula functions as claimed in claim 1, its feature exists
In：In the step 2, the marginal probability distribution function line style using the distribution of P-III types as water level and flow.
3. a kind of rating curve calculation method based on Copula functions as claimed in claim 1, its feature exists
In：In the step 2, using the parameter of linear Moment method estimators marginal probability distribution function.
4. a kind of rating curve calculation method based on Copula functions as claimed in claim 1, its feature exists
In：In the step 3, using Gumbel-Hougaard Copula construction of function water levels and the joint probability distribution letter of flow
Number, the parameter of Gumbel-Hougaard Copula functions is estimated using Kendall rank correlations Y-factor method Y.
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| CN108596998A (en) * | 2018-04-24 | 2018-09-28 | 江西省水利科学研究院 | A kind of rainfall runoff correlation drawing drawing method based on Copula functions |
| CN109214588A (en) * | 2018-09-28 | 2019-01-15 | 郑州大学 | Mountain flood probability rainfall pattern calculation method based on copula function |
| CN109344993A (en) * | 2018-08-23 | 2019-02-15 | 江西省水利科学研究院 | A kind of river flood-peak stage forecasting procedure based on conditional probability distribution |
| CN109902266A (en) * | 2019-03-07 | 2019-06-18 | 江西省水利科学研究院 | A kind of streamflow routing method based on Copula function |
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| CN111708980A (en) * | 2020-06-22 | 2020-09-25 | 江西省水利科学研究院 | Staged design flood calculation method considering historical flood information |
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| CN109214588A (en) * | 2018-09-28 | 2019-01-15 | 郑州大学 | Mountain flood probability rainfall pattern calculation method based on copula function |
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| CN109902267A (en) * | 2019-03-07 | 2019-06-18 | 江西省水利科学研究院 | A kind of river safety discharge calculation method influenced by downstream lake jacking |
| CN109902267B (en) * | 2019-03-07 | 2022-09-23 | 江西省水利科学研究院 | River channel safety discharge amount calculation method influenced by downstream lake jacking |
| CN109902266B (en) * | 2019-03-07 | 2022-10-28 | 江西省水利科学研究院 | Riverway flow calculation method based on Copula function |
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| CN111831966B (en) * | 2020-05-21 | 2023-08-15 | 中山大学 | Combined river channel water level forecasting method based on high-dimensional probability distribution function |
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Effective date of registration: 20210811 Address after: No.1038, Beijing East Road, Nanchang, Jiangxi 330000 Patentee after: Jiangxi Academy of water resources Address before: No.1038, Beijing East Road, Nanchang, Jiangxi 330000 Patentee before: JIANGXI PROVINCE WATER CONSERVANCY SCIENCE Research Institute |