CN109344993A - A kind of river flood-peak stage forecasting procedure based on conditional probability distribution - Google Patents
A kind of river flood-peak stage forecasting procedure based on conditional probability distribution Download PDFInfo
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Abstract
The river flood-peak stage forecasting procedure based on conditional probability distribution that the invention discloses a kind of, pass through statistics upstream and downstream flood-peak stage and propagation time series, on the basis of determining marginal probability distribution function, upstream flood-peak stage and downstream flood-peak stage, the two-dimentional joint probability distribution function in propagation time are constructed respectively using Copula function, and then downstream flood-peak stage, the conditional probability distribution function in propagation time when solving given upstream flood-peak stage, and then forecast downstream flood-peak stage and its time of occurrence.The present invention is based on probability theory and mathematical statistics methods quantitatively to give upstream section flood-peak stage and downstream section flood-peak stage, the regression curve of flood transmission time, compared with conventional freehand empirical relation curve method, theoretical basis is stronger, arbitrariness brought by freehand is effectively reduced, it is more objective reasonable.
Description
Technical field
The invention belongs to the forecasting and warning field that controls flood, in particular to a kind of river flood-peak stage based on conditional probability distribution
Forecasting procedure.
Background technique
Flood-peak stage forecast in river is one of important content of nonstructural measures of flood control, can be emergency flood fighting and industrial or agricultural
Safety in production provide technical support.Flood in natural river course, with flood wave morphology, road is moved from upstream toward downstream along the river, flood
Peak water level first occurs in river upstream section, then successively occurs in downstream section.Therefore, using the fortune of flood wave in river
Dynamic rule, by the flood-peak stage at upstream section current time, to forecast the flood-peak stage of downstream section future time instance.
Most popular river flood-peak stage forecasting procedure is equivalent water level method to engineering in practice at present, i.e., according to previous
Peb process data, point draw the corresponding flood-peak stage correlation figure of upstream and downstream section and upstream section flood-peak stage and flood transmission
Time correlation figure, by the empirical relation in figure, forecast by the current flood-peak stage of upstream section downstream section flood-peak stage and its
Time of occurrence.Equivalent water level method is although simple and practical, but in view of upstream section flood-peak stage and downstream section flood-peak stage, flood
There is apparent non-linear, abnormal feature between the water propagation time, traditional linear regression is caused to be difficult to be applicable in.Thus, it should
Method generally presses point group distribution trend, and think in terms of the majority an evidence, and freehand empirical relation curve, subjective, precision is also difficult
Exempt to be affected.
In fact, inquiring into upstream section flood-peak stage and downstream section flood-peak stage, the regression curve of flood transmission time
When essence is to solve for given upstream section flood-peak stage, downstream section flood-peak stage, the conditional probability distribution of flood transmission time.
Copula function can construct the Joint Distribution of multiple stochastic variables with any edge distribution, and then solving condition probability
Analytical expression can describe the abnormal feature and nonlinear correlation structure of stochastic variable well.Currently, without document by base
It is introduced into river flood-peak stage prediction research in the conditional probability distribution construction method of Copula function.
Summary of the invention
In view of the deficiencies of the prior art, the present invention provides a kind of river flood-peak stage based on conditional probability distribution
Forecasting procedure.
In order to solve the above technical problems, the present invention adopts the following technical scheme that:
A kind of river flood-peak stage forecasting procedure based on conditional probability distribution, comprising steps of
Step 1, upstream and downstream flood-peak stage and propagation time series are counted；
Step 2, according to the upstream and downstream flood-peak stage and propagation time series materials in step 1, edge appropriate point is chosen
Cloth line style, and estimate its parameter；
Step 3, upstream flood-peak stage and downstream flood-peak stage, the two dimension in propagation time are constructed using Copula function respectively
Joint probability distribution function, and estimate the parameter of Copula function；
Step 4, according to step 2 estimate marginal distribution function and step 3 construct two-dimentional joint distribution function solve to
Determine the analytical expression of downstream flood-peak stage when the flood-peak stage of upstream, the conditional probability distribution function in propagation time；
Step 5, according to the resulting conditional probability distribution function of step 4 analytical expression, forecast downstream flood-peak stage and
Its time of occurrence.
In the step 2, by normal distribution, logarithm normal distribution, Gumbel distribution, Gamma distribution and Pearson I II type
It is distributed alternately marginal probability distribution function line style, and using the ginseng of linear Moment method estimators candidate edge probability-distribution function
Number.
In the step 2, by the smallest candidate edge probability of the root-mean-square error of one-dimensional theory frequency and empirical Frequency point
Cloth function is as optimal marginal probability distribution function.
In the step 3, Gumbel-Hougaard and Frank Copula construction of function upstream flood-peak stage is respectively adopted
With the two-dimentional joint probability distribution function of downstream flood-peak stage, propagation time.
In the step 3, using the ginseng of maximum-likelihood method estimation Gumbel-Hougaard and Frank Copula function
Number.
The present invention is by statistics upstream and downstream flood-peak stage and propagation time series, in the base for determining marginal probability distribution function
On plinth, upstream flood-peak stage and downstream flood-peak stage, the two-dimentional joint probability in propagation time are constructed respectively using Copula function
Distribution function, and then downstream flood-peak stage, the conditional probability distribution function in propagation time when giving upstream flood-peak stage are solved, into
And forecast downstream flood-peak stage and its time of occurrence.
Compared with prior art, the beneficial effects of the present invention are:
The present invention is based on probability theory and mathematical statistics methods quantitatively to give upstream section flood-peak stage and downstream section
Flood-peak stage, the regression curve of flood transmission time, compared with conventional freehand empirical relation curve method, theoretical basis
It is stronger, arbitrariness brought by freehand is effectively reduced, it is more objective reasonable.
Detailed description of the invention
Fig. 1 is the flow chart of the method for the present invention.
Fig. 2 is the river flood-peak stage forecast regression curve schematic diagram based on conditional probability distribution.
Specific embodiment
The invention will be further described below by way of examples and with reference to the accompanying drawings.
As Figure 1-Figure 2, a kind of river flood-peak stage forecasting procedure based on conditional probability distribution, statistics upstream and downstream flood
Peak water level and propagation time series, on the basis of determining marginal probability distribution function, structurally using Copula function difference
Flood-peak stage and downstream flood-peak stage, the two-dimentional joint probability distribution function in propagation time are swum, and then solves given upstream flood peak
Downstream flood-peak stage, the conditional probability distribution function in propagation time when water level, and then forecast downstream flood-peak stage and its when occurring
Between.Fig. 1 is the calculation flow chart of the present embodiment, is followed the steps below:
1. counting upstream and downstream flood-peak stage and propagation time series.
The corresponding flood-peak stage of section upstream and downstream and its time of occurrence, upstream and downstream flood-peak stage point are taken passages from Water Year Book
Z is not denoted as itsAnd Zx, time of occurrence is denoted as T respectivelysAnd Tx, the corresponding propagation time is Tc=Tx-Ts.Flood peak in this specific implementation
The unit of water level is rice, and the unit in propagation time is hour.
2. determining the marginal probability distribution function of upstream and downstream flood-peak stage and propagation time.
According to the upstream and downstream flood-peak stage and propagation time series materials in step 1, edge distribution line style appropriate is chosen,
And estimate its parameter, this step includes three sub-steps:
2.1 candidate edge probability-distribution function line styles
Due to the overall distribution frequency curves of upstream and downstream flood-peak stage and propagation time be it is unknown, usually select can preferably
It is fitted the line style of most hydrology sample data series.It uses in this specific implementation by normal distribution, logarithm normal distribution, Gumbel
Distribution, Gamma distribution and Peason III distribution alternately marginal probability distribution function line style.
The parameter of 2.2 estimation edge distribution line styles
After curve type of frequency distribution is selected, the parameter of estimation frequency distribution is next carried out.Currently used method
Mainly there are moments method, maximum-likelihood method, suitable collimation method, probability-weighted moment, weight-function method and linear moments method etc..Wherein, linear moments method
It is the actual parameter estimation method generally acknowledged both at home and abroad at present, maximum feature is to the maximum and minimum of sequence without conventional square
So sensitive, the estimates of parameters acquired is more steady.
The parameter of L- Moment method estimators candidate edge probability-distribution function is used in this specific implementation.
2.3 optimal edge probability-distribution functions determine
The one-dimensional theory being distributed using root-mean-square error (Root Mean Square Error, RMSE) criterion evaluation edge
Frequency and empirical Frequency fit solution, RMSE value is smaller, illustrates that fitting effect is better.
In formula: F (xi) it is observation xiTheoretic frequency；M (i) is to meet x≤x in actual measurement seriesiObservation number, n
For sample length.
In this specific implementation, using the smallest candidate edge probability-distribution function of RMSE value as optimal marginal probability point
Cloth function.
3. establishing upstream flood-peak stage and downstream flood-peak stage, the two-dimentional joint probability distribution function in propagation time respectively.
According in step 1 upstream and downstream flood-peak stage and propagation time data information and step 2 in the edge estimated it is general
Rate distribution function is joined using the two dimension that Copula function constructs upstream flood-peak stage and downstream flood-peak stage, propagation time respectively
Probability-distribution function is closed, and estimates the parameter of Copula function；This step includes two sub-steps:
3.1 selection Copula functions
Enable upstream flood-peak stage Zs, downstream flood-peak stage ZxWith propagation time TcImplementation value be respectively zs、zxAnd tc, edge
Probability-distribution function is respectively u=Fs(zs), v=Fx(zx) and w=Fc(tc), corresponding probability density function is fs(zs)、fx
(zx) and fc(tc)。
By Copula function, Zs、ZxAnd Zs、TcTwo-dimentional joint probability distribution function can indicate are as follows:
Wherein, θ is the parameter of Copula function.
In this specific implementation, using Gumbel-Hougaard Copula construction of function Zs、ZxTwo-dimentional joint probability distribution
Function, expression formula are as follows:
Using Frank Copula construction of function Zs、TcTwo-dimentional joint probability distribution function, mathematic(al) representation is as follows:
The parameter of 3.2 estimation Copula functions
The common method of the parameter of estimation Copula function has Kendall correlation coefficient process, maximum-likelihood method, limit at present
Deduction method and kernel density estimation method etc..Wherein, the thought of maximum-likelihood method is to maximize likelihood function about parameter θ, is obtained
The estimated value of parameter θ.
In this specific implementation, using the ginseng of maximum-likelihood method estimation Gumbel-Hougaard and Frank Copula function
Number.
4. downstream flood-peak stage, the conditional probability distribution function in propagation time when calculating given upstream flood-peak stage.
Given upstream flood-peak stage Zs=zs, corresponding downstream flood-peak stage Zx, propagation time TcValue and not exclusive,
The probability for different values occur is different, is respectively present conditional probability distribution function
Fx|s(zx)=P (Zx≤zx|Zs=zs) (6)
Fc|s(tc)=P (Tc≤tc|Zs=zs) (7)
Utilize Copula function, conditional probability distribution function Fx|s(zx)、Fc|s(tc) respectively indicate are as follows:
5. forecasting downstream flood-peak stage and its time of occurrence.
Obtain conditional probability distribution function Fx|s(zx)、Fc|s(tc) after, using conditional medial as given upstream flood peak water
Position Zs=zsWhen, downstream flood-peak stage Zx, propagation time TcRepresentative predicted value, to obtain downstream flood-peak stage ZxAnd it propagates
Time TcRegression curve.
Downstream flood-peak stage ZxMedian zxm, propagation time TcMedian tcmIt is solved by following formula:
Fx|s(zxm)=0.5, Fc|s(tcm)=0.5 (10)
Formula (10) are solved using dichotomy tentative calculation in this specific implementation and obtain numerical solution.
By solving any given Zs=zsWhen downstream flood-peak stage ZxMedian zxm, propagation time TcMedian tcm,
It can be obtained by the downstream flood-peak stage Z based on conditional probability distributionxAnd propagation time TcRegression curve is shown below:
zx=zxm(zs), tc=tcm(zs) (11)
As shown in Fig. 2, giving the river flood-peak stage forecast regression curve schematic diagram based on conditional probability distribution.Its
In, black circles are measured value, and solid line is median regression result.
TsAnd TxImplementation value be respectively tsAnd tx, according to upstream flood-peak stage time of occurrence and the propagation time of estimation, under
The time of occurrence of trip flood-peak stage is calculate by the following formula:
tx=ts+tc=ts+tcm(zs) (12)
In actual job forecast, according to given upstream flood-peak stage zsWith time of occurrence ts, so that it may forecast downstream section
Flood-peak stage is zxm(zs), time of occurrence ts+tcm(zs)。
To sum up, the present invention is determining marginal probability distribution letter by statistics upstream and downstream flood-peak stage and propagation time series
On the basis of number, joined using the two dimension that Copula function constructs upstream flood-peak stage and downstream flood-peak stage, propagation time respectively
Downstream flood-peak stage, the conditional probability distribution in propagation time when closing probability-distribution function, and then solving given upstream flood-peak stage
Function, and then forecast downstream flood-peak stage and its time of occurrence.The present invention is based on probability theory and mathematical statistics methods quantitatively to give
Upstream section flood-peak stage and downstream section flood-peak stage, the regression curve of flood transmission time are gone out, have manually been drawn with conventional
Empirical relation curve method processed is compared, and theoretical basis is stronger, effectively reduces arbitrariness brought by freehand, more objective conjunction
Reason.
Claims (5)
1. a kind of river flood-peak stage forecasting procedure based on conditional probability distribution, it is characterised in that the following steps are included:
Step 1, upstream and downstream flood-peak stage and propagation time series are counted；
Step 2, according to the upstream and downstream flood-peak stage and propagation time series materials in step 1, edge distribution line appropriate is chosen
Type, and estimate its parameter；
Step 3, upstream flood-peak stage is constructed respectively using Copula function to combine with the two dimension in downstream flood-peak stage, propagation time
Probability-distribution function, and estimate the parameter of Copula function；
Step 4, the two-dimentional joint distribution function that the marginal distribution function and step 3 estimated according to step 2 construct solves on given
The analytical expression of downstream flood-peak stage, the conditional probability distribution function in propagation time when swimming flood-peak stage；
Step 5, it according to the analytical expression of the resulting conditional probability distribution function of step 4, forecasts downstream flood-peak stage and its goes out
Between current.
2. a kind of river flood-peak stage forecasting procedure based on conditional probability distribution as described in claim 1, it is characterised in that:
In the step 2, using normal distribution, logarithm normal distribution, Gumbel distribution, Gamma distribution and Peason III distribution as
Candidate edge probability-distribution function line style, and using the parameter of linear Moment method estimators candidate edge probability-distribution function.
3. a kind of river flood-peak stage forecasting procedure based on conditional probability distribution as described in claim 1, it is characterised in that:
In the step 2, the smallest candidate edge probability-distribution function of the root-mean-square error of one-dimensional theory frequency and empirical Frequency is made
For optimal marginal probability distribution function.
4. a kind of river flood-peak stage forecasting procedure based on conditional probability distribution as described in claim 1, it is characterised in that:
In the step 3, Gumbel-Hougaard and Frank Copula construction of function upstream flood-peak stage and downstream flood is respectively adopted
Peak water level, the two-dimentional joint probability distribution function in propagation time.
5. a kind of river flood-peak stage forecasting procedure based on conditional probability distribution as described in claim 1, it is characterised in that:
In the step 3, using the parameter of maximum-likelihood method estimation Gumbel-Hougaard and Frank Copula function.
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