CN109583030A - A kind of superposition of wind wave and swell Distribution of wave height and period construction method based on mixed model - Google Patents
A kind of superposition of wind wave and swell Distribution of wave height and period construction method based on mixed model Download PDFInfo
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Abstract
The present invention proposes a kind of superposition of wind wave and swell Distribution of wave height and period construction method based on mixed model, comprising: A, selection original distribution function；B, two-dimentional mixed model is established；C, the Q function of EM algorithm is constructed；D, the initial value of iterative calculation is chosen；E, Mixture Distribution Model parameter is calculated；F, the Mixture Distribution Model of wave height and period is constructed.The present invention uses the probabilistic model of parameter, avoids the restrictive condition that the narrow spectrum of theoretical model is assumed, the scope of application is wider.Meanwhile compared with common parameter model, Mixture Distribution Model can simulate the bimodal situation that Distribution of wave height and period in the case of superposition of wind wave and swell may be presented, and fitting effect is more preferable.
Description
Technical field
The invention belongs to wave stock assessment technical fields, and in particular to a kind of superposition of wind wave and swell wave height based on mixed model and
Period Joint Distribution construction method.
Background technique
In ocean engineering field, accurate description wave feature is particularly significant, and constructs the short-term of sea wave height and period
Closing distribution is exactly a kind of common method.Since the geometry that the short-term Joint Distribution of wave height and period contains a large amount of wave is special
Reference breath, therefore play an important role in terms of designing marine structure, evaluation.About building wave height and week
The problem of joint probability density function of phase, is sufficiently complex, and many scholars propose the theoretical Two dimensional Distribution in many wave height and period
Model, wherein most representative with the model that Longuet-Higgins is proposed in nineteen eighty-three.By many and measured data and mould
Quasi- data comparison shows that this model fitting degree under narrow spectral condition is preferable.
Existing theoretical distribution be all based on narrow spectrum and Gaussian process it is assumed that the therefore spectrum width and shape feature of ocean wave spectrum
The fitting degree of theoretical Joint Distribution model can be had a huge impact.Numerous studies comparison shows in the biggish feelings of spectrum width
Under condition, the Joint Distribution in wave height and period and narrow spectrum situation difference are larger, in some instances it may even be possible to Double-peak Phenomenon, existing theory mould occur
Type fitting degree is poor.It modifies although many scholars are based on this to existing theoretical model, bigger spectrum can be adapted to
Wide scope, but it is still not applicable in the case of spectrum width is very big.
Since practical sea situation is the mixing of multiple wave systems, ocean wave spectrum may show two spectral peaks.Wherein there is frequency
Highest rate is exactly stormy waves, the case where mixing of surging.The short-term Joint Distribution of wave height and period in this case, with theory
The hypothesis deviation of the narrow spectrum of Joint Distribution is larger, therefore fit solution is also and bad.There is scholar studies have shown that surging as wind of advocating peace
Wave is surged under comparable mixing sea situation, when stormy waves spectrum peak frequency and larger spectral peak frequency phase-difference of surging, the connection of wave height and period
Bi-modal case, the theory that existing theoretical distribution only has Lindgren and Rychlik to propose in nineteen eighty-two can be presented by closing distribution
Distribution can show this feature.
Wave Data deviation under theoretical distribution model and the larger situation of spectrum width is larger, is fitted by parameter distribution model
The distribution in wave height and period becomes another selectable method.There is scholar to be fitted wave height respectively using Weibull distribution
The conditional probability distribution of edge distribution and period passes through the comparison with existing theoretical model, conditional probability model fitting degree
More preferably, the scope of application is wider.However, it is unimodal that this parameter distribution model, which still can only be fitted Distribution of wave height and period,
Situation.
Summary of the invention
In order to solve mixing sea situation under Distribution of wave height and period fitting problems, the present invention using Mixture Distribution Model come
Its two-dimentional Joint Distribution is constructed, so as to show the bimodal feature of wave height, period Joint Distribution.
The present invention, which is that the following technical solution is employed, to be realized:
A kind of superposition of wind wave and swell Distribution of wave height and period construction method based on mixed model, comprising:
A, original distribution function is selected；
B, two-dimentional mixed model is established；
V=(X, Y) is enabled to indicate that sample is the observation of n, X and Y respectively represent across zero point wave height and corresponding period, then
The Mixture Distribution Model of X and Y can be expressed as
Wherein, ζ={ ω, θ }, ωjIt indicates the relative frequency of each ingredient, meetsθjIt indicates
The parameter of j-th of Mixture Distribution Model ingredient；
C, the Q function of EM algorithm is constructed；
D, the initial value of iterative calculation is chosen；
E, Mixture Distribution Model parameter is calculated；
F, the Mixture Distribution Model of wave height and period is constructed.
Further, it in the step A, is distributed using two dimensional logarithmic normal distribution and Gumbel logistic, two dimension is right
Number normal distribution is expressed as:
In formula, (ξx,εx) and (ξy,εy) be respectively ln (X) and ln (Y) mathematic expectaion and standard deviation, ρ be X and Y phase
Relationship number；
The probability density function of Gumbel logistic distribution indicates are as follows:
In formula, (ux,αx) and (uy,αy) be respectively X and Y mathematic expectaion and standard deviation, η be describe X and Y relationship phase
According to property coefficient.
Further, in the step D, the initial value of EM algorithm is chosen using randomized, observation is randomly divided into two groups,
Two ingredients are indicated, for each observation viIf it is assigned to jth group, then it is assumed that this observation is only by j-th
The model of ingredient generates, the model parameter θ of j-th of ingredientjIt can estimate to obtain according to the subsample of jth group.
Further, in the step E, Mixture Distribution Model parameter is calculated using EM algorithm.EM algorithm contains E step
With M step.E step is exactly the Q (ζ) calculated in step C according to previous ζ (r), and M step is to solve for the maximum of Q (ζ) and obtains
To ζ (r+1)；
When carrying out M step, the Q (ζ) in step C is decomposed into two, it may be assumed that
First item is maximized, weight is estimated with following formula:
The maximum of Section 2 is obtained according to its partial derivative for 0:
Further, in the step E, when | | ζ(r+1)-ζ(r)| | when < ε (ε > 0), iteration stopping.
Further, in the solution generation that mixed model is obtained by EM algorithm, is obtained into two dimension into the distributed model in step B
Mixture Distribution Model.
Compared with prior art, the advantages and positive effects of the present invention are:
The present invention uses the probabilistic model of parameter, avoids the restrictive condition that the narrow spectrum of theoretical model is assumed, the scope of application is more
Extensively.Meanwhile compared with common parameter model, Mixture Distribution Model can simulate wave height and period joint in the case of superposition of wind wave and swell
It is distributed the bimodal situation that may be presented, fitting effect is more preferable.
Detailed description of the invention
Fig. 1 is that the present invention is based on the superposition of wind wave and swell Distribution of wave height and period construction method flow diagrams of mixed model；
Fig. 2 is that the embodiment of the present invention mixes two dimensional logarithmic normal distribution joint probability density figure；
Fig. 3 is two dimensional logarithmic of embodiment of the present invention normal distribution joint probability density figure；
Fig. 4 is that the embodiment of the present invention mixes Gumbel logistic distribution joint probability density figure；
Fig. 5 is that Gumbel of embodiment of the present invention logistic is distributed joint probability density figure.
Specific embodiment
It include as follows the present invention is based on the superposition of wind wave and swell Distribution of wave height and period construction method of mixed model with reference to Fig. 1
Step:
Step 1: selection original distribution function, we are using two dimensional logarithmic normal distribution and Gumbellogistic points here
Cloth.Assuming that X and Y respectively represent across zero point wave height and corresponding period, then two dimensional logarithmic normal distribution is represented by
In formula, (ξx,εx) and (ξy,εy) be respectively ln (X) and ln (Y) mathematic expectaion and standard deviation, ρ be X and Y phase
Relationship number.
The probability function of Gumbel logistic distribution can be written as
F (x, y)=exp {-[(- lnF (x))η+(-lnF(y))η]1/η}(η≥1) (2)
In formula, η is the interdependent property coefficient for indicating relationship between X and Y, is defined as:
F (x) and F (y) is respectively the edge Gumbel distribution of X and Y, can be write as:
In formula, (ux,αx) and (uy,αy) be respectively X and Y mathematic expectaion and standard deviation.Then Gumbel logistic points
The probability density function of cloth can indicate are as follows:
Step 2: establishing two-dimentional mixed model.
It is traditional to be fitted the distributed model generally used in step A when the distribution of unimodal situation, and to be more preferably fitted point
There is bimodal situation in cloth, then needs step 2 construction mixed model through the invention.Two ingredient mixing two dimensional model parameters are
Twice of archetype can preferably be fitted original distribution, to bi-modal case so that mixed model has greater flexibility
Under fitting effect it is naturally also just more preferable.
V=(X, Y) is enabled to indicate that sample is the observation of n, then the mixed distribution of stochastic variable X and Y comprising two ingredients
Model can be expressed as
Wherein, ζ={ ω, θ }, ωjIt indicates the relative frequency of each ingredient, meetsθjIndicate the
The parameter of j Mixture Distribution Model ingredient.
Step 3: the Q function of construction EM (expectation maximization) algorithm.Introduce a variable H={ HiI=1,2 ..., n, it uses
It indicates to generate the ingredient of corresponding observation, can indicate are as follows:
f(Vi=vi,Hi=j | ζ)=ωjfj(vi|θj) (7)
Then H can be written as dependent on the conditional probability function of V:
Based on formula (8), our the iteration expression formulas of available EM algorithm are as follows:
Step 4: choosing the initial value of iterative calculation.Here using randomized choose EM algorithm initial value, i.e., by observation with
Machine is divided into two groups, indicates two ingredients.I.e. for each observation viIf it is assigned to jth group, then it is assumed that this sight
Measured value is only generated by the model of j-th of ingredient.In this way, the model parameter θ of j-th of ingredientjAccording to the increment of jth group
This estimation obtains.And the weight of subsample is expressed as the initial value of ω (0), i.e.,
Wherein, njIndicate the sample size of jth group.
Step 5: Mixture Distribution Model parameter is calculated using EM algorithm.
Mixture Distribution Model parameter is more, solve it is more difficult, it is most of at present in ocean engineering field using less
The case where using one-dimensional distributed model is also only limitted to, therefore research achievement in this respect is also less.The present invention uses EM algorithm
It then effectively avoids being likely encountered the case where not restraining.
EM algorithm of the present invention contains E step and M step.E step is exactly according to previous ζ (r) calculating formula (9), M step
It is then to solve for the maximum of Q (ζ) and obtains ζ (r+1):
ζ(r+1)=argmaxQ (ζ | ζ(r)) (11)
When carrying out M step, it is noted that Q (ζ) can be decomposed into two in formula (9), and respectively seek its maximum.Q (ζ) writes
Are as follows:
First item is maximized, then weight is estimated with following formula:
The maximum of Section 2 can be obtained according to its partial derivative for 0:
In order to ensure the precision of obtained estimated value, it would be desirable to define the standard of iteration stopping.Here pass through limitation two
The difference of secondary subsequent iteration result formulates standard:
|ζ(r+1)-ζ(r)| | < ε (ε > 0) (15)
Step 6: the Mixture Distribution Model of building wave height and period.In the solution generation that mixed model is obtained by EM algorithm, is arrived into formula
(6), two-dimentional Mixture Distribution Model is obtained.
The present invention is described in detail below with reference to specific example.
The Wave Data in the case of Bimodal Spectra is now obtained using the method for numerical simulation, Bimodal Spectra uses Ochi and Hubble
In six Parameter Spectrums that 1976 propose:
Parameter setting uses the superposition of wind wave and swell situation based on surging in Rodr í guez and Guedes Soares (1999), such as table
Shown in 1.
1 six Parameter Spectrum parameter setting of table
For each simulated series, the sampling interval is set as 0.5s, and sequence total length is 4096s, simulates 20 sequences altogether
Column.Wave height (H) and period (T) sequence are obtained across balance method on, and nondimensionalization is carried out to it in the following ways:
Wherein, mkFor the k rank square of ocean wave spectrum:
Superposition of wind wave and swell Distribution of wave height and period building process is as follows:
1, original distribution function is selected.
Here we are distributed using two dimensional logarithmic normal distribution and Gumbel logistic.
2, two-dimentional mixed model is established.
Assuming that X and Y respectively represent nondimensionalization on across zero point wave height and corresponding period, enable V=(X, Y) indicate sample
For the observation of n, then the Mixture Distribution Model of stochastic variable X and Y comprising two ingredients can be expressed as
3, the Q function of EM (expectation maximization) algorithm is constructed.
4, the initial value of iterative calculation is chosen.
The initial value that EM algorithm is chosen using randomized then mixes two dimensional logarithmic normal distribution and mixing Gumbellogistic
The initial value of distribution is as shown in table 2.
2 mixed distribution initial parameter values of table
5, Mixture Distribution Model parameter is calculated using EM algorithm.
Mixture Distribution Model parameter is calculated using EM algorithm, then mixes two dimensional logarithmic normal distribution and mixing Gumbel
The parameter of logistic distribution is as shown in table 3.
3 mixed distribution estimates of parameters of table
Then mix the expression formula of two dimensional logarithmic normal distribution are as follows:
Mix the expression formula of Gumbel logistic distribution are as follows:
6, the Mixture Distribution Model of wave height and period is constructed.
By the joint probability density figure of our available Mixture Distribution Models of formula (06) and (07), as a result such as Fig. 2 and Fig. 4
It is shown.In order to embody the dominance of mixed model, we also give two dimensional logarithmic normal distribution model (Fig. 3) and Gumbel
The result of logistic model (Fig. 5).Using sum of squares of deviations come the error of descriptive model, it the results are shown in Table 4.It by comparing can be with
It was found that compared with archetype, mixed model can show experience well and be distributed bimodal characteristic, fitting effect obviously compared with
It is good.
4 distributed model error of table
The above described is only a preferred embodiment of the present invention, being not that the invention has other forms of limitations, appoint
What those skilled in the art changed or be modified as possibly also with the technology contents of the disclosure above equivalent variations etc.
It imitates embodiment and is applied to other fields, but without departing from the technical solutions of the present invention, according to the technical essence of the invention
Any simple modification, equivalent variations and remodeling to the above embodiments, still fall within the protection scope of technical solution of the present invention.
Claims (6)
1. a kind of superposition of wind wave and swell Distribution of wave height and period construction method based on mixed model, characterized by comprising:
A, original distribution function is selected；
B, two-dimentional mixed model is established；
V=(X, Y) is enabled to indicate that sample is the observation of n, X and Y respectively represent across zero point wave height and corresponding period, then X and Y
Mixture Distribution Model can be expressed as
Wherein, ζ={ ω, θ } indicates model variable, ωjIt indicates the relative frequency of each ingredient, meetsωj∈[0,
1],θjIndicate the parameter of j-th of Mixture Distribution Model ingredient；
C, the Q function of EM algorithm is constructed；
Wherein, variable H={ Hi}I=1,2 ..., nFor indicating to generate the ingredient of corresponding observation.
D, the initial value of iterative calculation is chosen；
E, Mixture Distribution Model parameter is calculated；
F, the Mixture Distribution Model of wave height and period is constructed.
2. the superposition of wind wave and swell Distribution of wave height and period construction method according to claim 1 or 2 based on mixed model,
It is characterized in that: in the step A, being distributed using two dimensional logarithmic normal distribution and Gumbel logistic,
Two dimensional logarithmic normal distribution is expressed as:
In formula, (ξx,εx) and (ξy,εy) be respectively ln (X) and ln (Y) mathematic expectaion and standard deviation, ρ be X and Y phase relation
Number；
The probability density function of Gumbel logistic distribution indicates are as follows:
In formula, (ux,αx) and (uy,αy) be respectively X and Y mathematic expectaion and standard deviation, η be describe X and Y relationship dependence system
Number.
3. the superposition of wind wave and swell Distribution of wave height and period construction method according to claim 2 based on mixed model, special
Sign is, in the step D, the initial value of EM algorithm is chosen using randomized, and observation is randomly divided into two groups, indicate two at
Point, for each observation viIf it is assigned to jth group, then it is assumed that this observation is only by the model of j-th of ingredient
It generates, the model parameter θ of j-th of ingredientjIt can estimate to obtain according to the subsample of jth group.
4. the superposition of wind wave and swell Distribution of wave height and period construction method according to claim 1 based on mixed model, special
Sign is, in the step E,
Mixture Distribution Model parameter is calculated using EM algorithm.EM algorithm contains E step and M step.E step is exactly according to previous
A ζ (r) calculates the Q (ζ) in step C, and M step is to solve for the maximum of Q (ζ) and obtains ζ (r+1)；
When carrying out M step, the Q (ζ) in step C is decomposed into two, it may be assumed that
First item is maximized, weight is estimated with following formula:
The maximum of Section 2 is obtained according to its partial derivative for 0:
5. the superposition of wind wave and swell Distribution of wave height and period construction method according to claim 4 based on mixed model, special
Sign is, in the step E, when | | ζ(r+1)-ζ(r)| | when < ε (ε > 0), iteration stopping.
6. the superposition of wind wave and swell Distribution of wave height and period construction method according to claim 1 based on mixed model, special
Sign is: by the solution generation for obtaining mixed model by EM algorithm into the distributed model in step B, obtaining two-dimentional mixed distribution mould
Type.
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