CN107622162B - Copula function-based water level flow relation curve calculation method - Google Patents
Copula function-based water level flow relation curve calculation method Download PDFInfo
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Abstract
The invention discloses a Copula function-based water level flow relation curve deduction method, which comprises the steps of collecting data of cross section water level and flow, constructing a joint probability distribution function of water level and flow by using a Copula function on the basis of determining an edge probability distribution function, further solving a conditional probability distribution function of flow at a given water level, and deducting a water level flow relation curve and uncertainty analysis on the basis according to a mathematical statistics principle. The invention has stronger statistical theory basis, allows the water level and the flow to have any form of edge distribution, and can accurately describe the nonlinear and heteroscedastic correlation structure between the water level and the flow. In addition, the point estimation value of the flow can be obtained, and the uncertainty of the model parameters and the model structure can be considered more comprehensively at the same time to obtain the flow comprehensive uncertainty interval.
Description
Technical Field
The invention belongs to the field of hydraulic engineering, and particularly relates to a Copula function-based water level flow relation curve calculation method.
Background
The water level flow relation refers to the relation between the water level of the river cross section and the corresponding flow. Because the flow rate testing technology is complex, expensive and difficult to be continuously performed, the continuous water level data is usually converted into the continuous flow rate data when the hydrologic data is compiled, and the conversion between the water level and the flow rate is also commonly used in hydrologic forecasting, hydrologic calculation and water conservancy management work. In addition, the corresponding flow of the highest and lowest water levels cannot be measured due to condition restrictions, and the high and low water needs to be prolonged according to a water level flow relation curve, and whether the extension is proper or not directly influences the scale and size of an engineering design project. Therefore, the water level and flow relation curve has important practical significance.
The water level flow relation curve is determined according to the water level measured for many times on the section and the corresponding flow data, the traditional method generally presupposes that the water level flow relation obeys a certain mathematical line type, and then the parameters are solved by using an optimization algorithm under the selected optimization criterion, so that a specific mathematical equation of the water level flow relation is determined. At present, the more used line types comprise power function types, polynomial types and logarithmic function types, the optimization criteria comprise a residual square sum minimum criterion, an absolute residual absolute value and minimum criterion and a relative residual absolute value and minimum criterion, and the optimization algorithms mainly comprise a least square method, a genetic algorithm, an ant colony algorithm, a particle swarm algorithm, a chaotic algorithm, a mixed tabu search algorithm, an artificial fish colony algorithm, an artificial bee colony algorithm, a colony spider algorithm, an immune evolution algorithm, a differential evolution algorithm and the like. Aiming at the defects of the traditional method, some scholars propose to apply methods such as an artificial neural network, a support vector machine and a genetic program to fit a water level flow relation, and although the limitation of a pre-assumed specific functional formula is avoided, some problems and defects still exist. The structure of the artificial neural network can only be selected by experience, unified theoretical guidance is lacked, a support vector machine is sensitive to missing data, a dispute exists on how to select a proper kernel function, the convergence efficiency of a genetic program is low, and an obtained regression formula is too complex and unstable. In addition, a learner also estimates parameters and uncertainty of the power function type water level flow rate relation curve by using a Markov Chain Monte Carlo (MCMC) method based on Bayesian theory, and a confidence interval of the water level flow rate relation curve is given. However, the MCMC method still needs to preset a mathematical line of the water level flow relationship, and can only estimate a confidence interval of the water level flow relationship curve due to uncertainty of model parameters, but cannot consider uncertainty of the model structure.
The Copula function can construct the joint distribution of a plurality of random variables with any edge distribution, can well capture the non-normal characteristics among the variables and the non-linear and heteroscedastic correlation relations among the variables, and is widely applied to the field of hydrology and water resources. At present, no literature introduces Copula function into the water level flow rate relation curve calculation.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides a Copula function-based water level flow relation curve calculation method.
In order to solve the technical problems, the invention adopts the following technical scheme:
a Copula function-based water level flow relation curve calculation method comprises the following steps:
step 1, collecting data of section water level and flow;
step 2, selecting a proper edge distribution line type according to the water level and flow data information in the step 1, and estimating parameters of the edge distribution line type;
step 3, constructing a joint probability distribution function of water level and flow by using a Copula function, and estimating parameters of the Copula function;
step 4, solving an analytical expression of a flow conditional probability distribution function at a given water level according to the edge distribution function estimated in the step 2 and the combined distribution function constructed in the step 3;
and 5, according to the analytical expression of the conditional probability distribution function obtained in the step 4 and according to a mathematical statistic principle, calculating a water level flow relation curve and analyzing uncertainty.
In the step 2, the P-III type distribution is used as the edge probability distribution function line type of the water level and the flow.
In the step 2, parameters of the marginal probability distribution function are estimated by adopting a linear moment method.
In the step 3, a Gumbel-Hougaard Copula function is adopted to construct a joint probability distribution function of water level and flow, and a Kendall rank correlation coefficient method is adopted to estimate parameters of the Gumbel-Hougaard Copula function.
In the step 5, a bisection method is adopted to try out and solve the median Q of the flow Q when any given H is HmAnd interval estimate of confidence level (1- ξ) [ ql,qu]。
According to the invention, by collecting data of the water level and the flow of the cross section, on the basis of determining the marginal probability distribution function, a Copula function is utilized to construct a joint probability distribution function of the water level and the flow, and further a conditional probability distribution function of the flow at a given water level is solved, and on the basis, a water level flow relation curve and uncertainty analysis are obtained according to a mathematical statistics principle.
Compared with the prior art, the invention has the beneficial effects that:
(1) the invention has stronger statistical theory basis, allows the water level and the flow to have any form of edge distribution, and can accurately describe the nonlinear and heteroscedastic correlation structure between the water level and the flow.
(2) Compared with the conventional water level flow relation curve calculation method, the method can obtain the point estimation value of the flow, and can comprehensively consider the uncertainty of the model parameters and the model structure at the same time to obtain the comprehensive uncertainty interval of the flow.
Drawings
FIG. 1 is a flow chart of the method of the present invention.
FIG. 2 is a schematic diagram of a relationship curve of water level and flow rate based on a Copula function.
Detailed Description
The invention is further illustrated by the following examples in conjunction with the accompanying drawings.
As shown in fig. 1-2, a Copula function-based water level flow relation curve calculation method collects data of cross-section water level and flow, constructs a joint probability distribution function of water level and flow by using Copula function on the basis of determining edge probability distribution function, further calculates a conditional probability distribution function of flow at a given water level, and calculates a water level flow relation curve and uncertainty analysis on the basis of mathematical statistics principle. Fig. 1 is a calculation flowchart of the present embodiment, which is performed according to the following steps:
1. collecting data of water level and flow rate of cross section.
The time scale of actually measuring the water level and flow data in the embodiment is day. The section daily water level and daily flow data are obtained from a hydrological yearbook of a hydrological station.
2. Determining marginal probability distribution functions of water level and flow.
Selecting a proper edge distribution line type according to the water level and flow data information in the step 1 and estimating parameters thereof, wherein the step comprises two substeps:
2.1 Selective edge Profile
Since the overall frequency profile of water level and flow is unknown, a profile is usually chosen that fits well to most hydrological sample data series. Through years of analysis and comparison, the P-III type distribution is found to be good in fitting to water level and flow data of most rivers in China, and is recommended to be adopted in engineering practice.
In this embodiment, a P-III type distribution is used as the edge profile of the water level and flow.
2.2 estimating parameters of the edge profile
The method mainly comprises a moment method, a maximum likelihood method, an adaptive line method, a probability weight moment method, a weight function method, a linear moment method (L-moment method) and the like, wherein the L-moment method is an effective parameter estimation method recognized at home and abroad at present, and has the biggest characteristic that the maximum value and the minimum value of a sequence are not as sensitive as the conventional moment, and the obtained parameter estimation value is more stable.
In this embodiment, L-moment method is used to estimate the parameters of the edge profile.
3. And constructing a joint probability distribution function of the water level and the flow by using a Copula function.
Selecting a proper Copula function to construct a joint probability distribution function of the water level and the flow according to the data of the water level and the flow in the step 1 and the edge probability distribution function estimated in the step 2, and estimating parameters of the joint probability distribution function, wherein the step comprises two substeps:
3.1 selecting Copula function
Let H, Q denote water level and flow rate, respectively, and h, q are corresponding implementations, respectivelyThe value is obtained. FH(h)、FQ(q) is the edge probability distribution function, corresponding to a probability density function of fH(h)、fQ(q) is carried out. According to the Sklar theorem, the joint distribution function of H and Q can be represented by a two-dimensional Copula function C:
FH,Q(h,q)＝Cθ(FH(h),FQ(q))＝Cθ(u,v) (1)
wherein, θ is a parameter of the Copula function; u ═ FH(h),v＝FQ(q) is an edge distribution function.
In the specific implementation, a Gumbel-Hougaard Copula function is adopted to construct a joint probability distribution function of water level and flow, and the expression is as follows:
3.2 estimating the parameters of the Copula function
In the specific implementation, a Kendall rank correlation coefficient method is adopted to estimate parameters of a Gumbel-Hougaard Copula function. The Kendall correlation coefficient tau is related to the parameter theta by:
let { (x)1,y1),…,(xn,yn) Denotes random samples of n observations taken from successive random variables (X, Y), among which are samples
Different combinations of observations (x)i,yi) And (x)j,yj). The Kendall rank correlation coefficient tau of the sample is calculated by the following formula
Where sign (·) is a sign function.
4. And solving the conditional probability distribution function of the flow at the given water level.
When the water level H is given a value H, the corresponding flow Q is not unique but can be larger or smaller, only the probability of different values is different, and a conditional probability distribution function exists
FQ|H(q)＝P(Q≤q|H＝h) (5)
By means of Copula function, conditional probability distribution function FQ|H(q) may be expressed as:
substituting formula (2) into:
5. and (5) calculating a water level flow relation curve and analyzing uncertainty.
Obtaining a conditional probability distribution function F of the flow QQ|HAnd (Q) calculating to obtain a median serving as an estimated value of a flow Q point according to a mathematical statistic principle, wherein the obtained flow Q median function is a water level flow relation function. Meanwhile, interval estimation under a given confidence level is obtained for uncertainty analysis.
Median Q of flow QmSolving by:
FQ|H(qm)＝0.5 (8)
in the present embodiment, a numerical solution is obtained by trial calculation of a solution formula (8) by a dichotomy.
By solving for the median Q of the flow Q at any given H-HmThen, a water level and flow rate relation function based on Copula function can be obtained, as shown in the following formula:
Q＝qm(h) (9)
selecting a certain confidence level (1- ξ), and making the probability of the flow Q value appearing at both ends of the distribution be ξ, the interval estimation of R can be defined, and the lower confidence limit and the upper limit are respectively given by the following two formulas:
FQ|H(ql)＝ξ1(10)
FQ|H(qu)＝1-ξ2(11)
wherein, ξ1+ξ2ξ, signifying the significance level, ξ in this particular implementation1＝ξ2＝ξ/2。
In the present embodiment, the numerical solutions are obtained by solving equations (10) and (11) through a binary trial calculation. Thus, it is possible to provide
P(ql≤Q≤qu)＝1-ξ (12)
I.e. [ q ]l,qu]For interval estimation of confidence levels (1- ξ) of flow rate Q, the aggregate uncertainty of the flow rate Q estimates can be quantitatively evaluated based on the confidence intervals.
As shown in fig. 2, a schematic diagram of a water level flow rate relation curve based on Copula function estimation is given. Wherein, the black dots are measured values, the solid lines are median results, and the shaded parts represent the uncertainty interval of the estimated flow rate of 90%.
In summary, the invention collects data of cross-section water level and flow rate, on the basis of determining the marginal probability distribution function, the Copula function is used for constructing a joint probability distribution function of the water level and the flow rate, and further the conditional probability distribution function of the flow rate at the given water level is solved, and on the basis, a water level flow rate relation curve and uncertainty analysis are obtained according to the mathematical statistics principle. The invention has stronger statistical theory basis, allows the water level and the flow to have any form of edge distribution, and can accurately describe the nonlinear and heteroscedastic correlation structure between the water level and the flow. In addition, the point estimation value of the flow can be obtained, and the uncertainty of the model parameters and the model structure can be considered more comprehensively at the same time to obtain the flow comprehensive uncertainty interval.
It will be evident to those skilled in the art that the invention is not limited to the details of the foregoing illustrative embodiments, and that the present invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof. The present embodiments are therefore to be considered in all respects as illustrative and not restrictive, the scope of the invention being indicated by the appended claims rather than by the foregoing description, and all changes which come within the meaning and range of equivalency of the claims are therefore intended to be embraced therein. Any reference sign in a claim should not be construed as limiting the claim concerned.
Furthermore, it should be understood that although the present description refers to embodiments, not every embodiment may contain only a single embodiment, and such description is for clarity only, and those skilled in the art should integrate the description, and the embodiments may be combined as appropriate to form other embodiments understood by those skilled in the art.
Claims (1)
1. A Copula function-based water level flow relation curve calculation method is characterized by comprising the following steps:
step 1, collecting data of section water level and flow;
step 2, selecting a proper edge distribution line type according to the water level and flow data information in the step 1, and estimating parameters of the edge distribution line type;
step 3, constructing a joint probability distribution function of water level and flow by using a Copula function, and estimating parameters of the Copula function;
step 4, solving an analytical expression of a flow conditional probability distribution function at a given water level according to the edge distribution function estimated in the step 2 and the combined distribution function constructed in the step 3;
step 5, according to the analytical expression of the conditional probability distribution function obtained in the step 4, according to a mathematical statistic principle, a water level flow relation curve and uncertainty analysis are obtained;
in the step 2, the P-III type distribution is taken as an edge probability distribution function line type of the water level and the flow;
in the step 2, parameters of the marginal probability distribution function are estimated by adopting a linear moment method;
in the step 3, a Gumbel-Hougaard Copula function is adopted to construct a joint probability distribution function of water level and flow, and a Kendall rank correlation coefficient method is adopted to estimate parameters of the Gumbel-Hougaard Copula function;
in the step 5, a dichotomy trial calculation is adopted to solve the median Q of the flow Q when any given water level H is equal to the value HmAnd a lower confidence limit q for the interval corresponding to the significance level ξlAnd upper confidence limit qu。
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