A coastal tidal extreme value prediction method based on extreme value mixed distribution

Abstract

The present invention discloses a method for predicting extreme values ​​of coastal tide levels, comprising the following steps: obtaining several years of historical data measured by a tide gauge at a location, determining the maximum sample value every X hours, and forming independent tide level analysis data. The selected tide level analysis data is subjected to extreme value mixed distribution family parameter fitting, and the extreme value mixed distribution with the highest negative logarithmic fit is calculated and parameter values ​​are determined. The coastal tide level value under a specific return period is calculated based on the obtained optimal extreme value mixed distribution form. The method selects an extreme value mixed distribution, calculates extreme values ​​under a specific return period, uses a generalized Pareto distribution as a tail and splices it with other distributions, uses a threshold as part of a distribution parameter, and automatically selects the optimal threshold. This method avoids the empirical and subjective nature of threshold selection in traditional super-threshold extreme value analysis, and fully considers the integrity of the analysis data, providing a more reliable numerical reference for structural design and disaster prevention.

Description

Technical Field

[0001] The present invention relates to a method for predicting ocean data, and in particular to a method for predicting coastal tide extreme values ​​based on extreme value mixed distribution. Background Art

[0002] When designing coastal structures and preventing coastal disasters, it is often necessary to consider tidal levels that occur only once in many years. These rare extreme values, while infrequent, can cause significant damage when they do occur. Statistical methods are currently widely used to estimate design parameters for marine and coastal structures, both domestically and internationally. Extreme value theory is a statistical theory specifically designed to study rare extreme events. It extrapolates extreme values ​​corresponding to a return period of once in N years. Two common interpretations of the recurrence level for a return period of T years are: 1. Waiting time: The average waiting time before the next event is T years. 2. Number of events: The average number of events occurring within a T-year period is 1.

[0003] Currently, the block extreme value method and the super-threshold method are widely used methods for selecting extreme value data. The block extreme value method selects the maximum data value in a block as the analysis data and fits a generalized extreme value distribution family. This method has been well-established in various structural designs that require consideration of extreme values ​​of environmental design parameters, but it suffers from low data utilization. The super-threshold method (POT) sets a threshold and fits a generalized Pareto distribution to data exceeding the threshold. This method has been widely used and studied in recent years. However, the selection of the threshold in the POT method often requires drawing and empirical methods, and there is currently no universally applicable automatic threshold selection method. Summary of the Invention

[0004] Purpose of the invention: In response to the above problems, the present invention proposes a method for predicting coastal tide extremes based on extreme value mixed distribution, which can overcome the problem of low computational efficiency caused by the subjectivity and experience of threshold selection in the POT method, automatically select the threshold, and further improve the prediction accuracy.

[0005] Technical solution: The technical solution adopted by the present invention is a method for predicting coastal tidal extremes based on extreme value mixed distribution, comprising the following steps:

[0006] (1) Obtain the historical tide data measured at a location, select the maximum tide data value every X hours, and form independent tide analysis data;

[0007] (2) Select an extreme value mixture distribution form based on the tail fraction of the bulk model or an extreme value mixture distribution form based on the parameterized tail fraction, use the threshold as the variable of the distribution function, consider the number of thresholds and the continuity at the threshold, design several different forms of extreme value mixture distribution models, and fit the extreme value mixture distribution family parameters to the tide analysis data; calculate the negative log-likelihood value of the fitted model, determine the threshold value corresponding to the minimum negative log-likelihood value, and thus establish a certain form of extreme value mixture distribution model, and select the model with the minimum negative log-likelihood value among various forms of extreme value mixture distribution models as the prediction model; for the extreme value mixture distribution form, the main model adopts normal distribution, Weibull distribution or the tail model adopts generalized Pareto distribution.

[0008] The described design of several different forms of extreme value mixed distribution models specifically includes the following forms: a single-threshold extreme value mixed distribution form based on the body model tail fraction without considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the body model tail fraction without considering the continuity at the threshold; a single-threshold extreme value mixed distribution form based on the parameterized tail fraction without considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the parameterized tail fraction without considering the continuity at the threshold; a single-threshold extreme value mixed distribution form based on the body model tail fraction with considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the body model tail fraction with considering the continuity at the threshold; a single-threshold extreme value mixed distribution form based on the parameterized tail fraction with considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the parameterized tail fraction with considering the continuity at the threshold.

[0009] The single-threshold extreme value mixed distribution form F(x|θ,u,σ) based on the tail fraction of the volume model u ,ξ) is:

[0010]

[0011] Where H represents the main distribution, G represents the generalized Pareto distribution, θ represents the parameter vector of the main distribution, u represents the threshold, σ u is the scale parameter of the generalized Pareto distribution, and ξ is the shape parameter of the generalized Pareto distribution.

[0012] The single-threshold extreme value mixture distribution based on the parameterized tail fraction is:

[0013]

[0014] Where H represents the main distribution, G represents the generalized Pareto distribution, θ represents the parameter vector of the main distribution, u represents the threshold, σ u is the scale parameter of the generalized Pareto distribution, ξ is the shape parameter of the generalized Pareto distribution, φ u Indicates the tail fraction.

[0015] The double-threshold extreme value mixed distribution form based on the tail fraction of the body model:

[0016]

[0017] where Θ=(u l ,σ ul ,ξ l ,μ,β,μ r ,σ ur ,ξ r ) represents all parameter vectors, u l represents a smaller threshold, σ ul represents the scale parameter of the GPD at the lower tail, ξ l represents the shape parameter of the GPD at the lower tail, μ represents the expectation of the main normal distribution, β represents the standard deviation of the main normal distribution, and u r represents a larger threshold, σ ur represents the scale parameter of the GPD at the upper and lower tails, ξ r represents the shape parameter of the GPD at the upper tail, and H(·|μ,β) indicates that it is a normal distribution.

[0018] A dual-threshold extreme value mixture distribution based on parameterized tail fractions:

[0019]

[0020] where Θ'=(φ ul ,μl l ,σ ul ,ξ l ,μ,β,μ r ,σ ur ,ξ r ,φ ur ) is the parameter vector, u l represents a smaller threshold, σ ul represents the scale parameter of the GPD at the lower tail, ξ l represents the shape parameter of the GPD at the lower tail, μ represents the expectation of the main normal distribution, β represents the standard deviation of the main normal distribution, and u r represents a larger threshold, σ ur represents the scale parameter of the GPD at the upper and lower tails, ξ r represents the shape parameter of the GPD at the upper tail, H(·|μ,β) indicates a normal distribution, φ ul =P(X<u l ) represents the lower tail fraction, the upper tail fraction φ ur =P(X>u r ).

[0021] The continuity constraint is implemented by equating the bulk distribution to the generalized Pareto distribution at lower and upper thresholds.

[0022] Continuity constraint σ of extreme value mixture distribution form based on tail fraction of volume model u for:

[0023]

[0024] Continuity Constraints σ on the Extreme Value Mixture Distribution Form Based on Parameterized Tail Fraction u for:

[0025]

[0026] H(u|θ) represents the distribution function of the subject model, h(u|θ) represents the subject distribution density function, φ u Indicates the tail fraction.

[0027] The calculation formula of the negative log-likelihood value -l(θ) is:

[0028]

[0029] Where m is the sample size, F G is the selected extreme value mixture distribution form.

[0030] (3) Using the obtained prediction model, randomly generate Y tide level data, take the maximum value of these Y data, repeat the above process N times to generate N data values, and take the average of these N data values ​​to obtain the extreme tide level of the area. The extreme value mixed distribution form is adopted by the main model as normal distribution, Weibull distribution or generalized Pareto distribution, and the tail model adopts generalized Pareto distribution. Gradually increase the number of randomly generated tide level values ​​(Y) and the number of repeated calculations (N) so that the error between the calculated tide level extreme values ​​is less than the threshold.

[0031] Beneficial effects: Compared with the existing technology, the present invention has the following advantages: the present invention uses the extreme value mixed distribution model to fit the ocean tide data as a whole, automatically selects the optimal threshold, overcomes the difficulty of threshold selection in the extreme value analysis process of the POT method, and improves the efficiency of extreme value analysis; and considers the integrity of the data, thereby improving the accuracy of tide extreme value prediction. BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Figure 1 It is a negative log-likelihood value radar chart of the coastal tide extreme value prediction method based on extreme value mixed distribution described in the present invention;

[0033] Figure 2 It is a quantile test diagram of the coastal tide extreme value prediction method based on extreme value mixed distribution described in the present invention;

[0034] Figure 3 It is a density function test diagram of the coastal tide extreme value prediction method based on extreme value mixed distribution described in the present invention;

[0035] Figure 4 It is a recurrence level prediction map of the coastal tide extreme value prediction method based on extreme value mixed distribution described in the present invention. DETAILED DESCRIPTION

[0036] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0037] The method for predicting coastal tidal extremes based on extreme value mixed distribution of the present invention comprises the following steps:

[0038] Step 1: Data Collection: Using RTUDIO, we collected hourly tide data from 1846 to 2016 for a coastal tidal station in a certain area. Other tools, such as Python and Matlab, can also be used to obtain historical tide data for a selected location.

[0039] Step 2: Analysis data selection: Determine the maximum tide level value within every X hours, form independent tide level analysis data, see Table 1, and determine the maximum tide level value within every 72 hours.

[0040] Table 1 Tidal level analysis data of a certain area

[0041]

[0042] Step 3. Fitting the extreme value mixed distribution family data: Fit the obtained tidal analysis data to the extreme value mixed distribution family, including the mixed distribution forms of Weibull+GPD, Normal+GPD, and GPD+Normal+GPD. Considering the continuity at the threshold and the tail fraction estimation method, the negative logarithm is calculated, and the threshold corresponding to the minimum negative logarithm is selected to fit the extreme value mixed distribution form with the best effect. The extreme value mixed distribution family can also use other mixed distribution forms. For the mixed distribution form of GPD+Normal+GPD, its main distribution is Normal. In nature, extremely large and extremely small values ​​do not occur frequently. The tail of the normal distribution is not sufficient to represent the extreme values ​​in reality, so GPD is used to fit the two tails of the data.

[0043] The single-threshold extreme value mixture distribution form based on the tail fraction of the body model is:

[0044]

[0045] A single-threshold extreme value mixture distribution based on a parameterized tail fraction:

[0046]

[0047] Where H(x|η) can be a normal distribution or a Weibull distribution or other distribution forms, φ u =P(X>u), the model before the threshold u is called the "main model", and the model after the threshold u is called the "tail model". u =1-H(u|θ), it is transformed into a single-threshold extreme value mixed distribution based on the tail fraction of the body model.

[0048] The double-threshold extreme value mixed distribution form based on the tail fraction of the body model:

[0049]

[0050] where Θ=(μ l ,σ ul ,ξ l ,μ,β,μ r ,σ ur ,ξ r ) is the parameter vector, and H(·|μ,β) is the normal distribution.

[0051] A dual-threshold extreme value mixture distribution based on parameterized tail fractions:

[0052]

[0053] where Θ'=(φ ul ,μ l ,σ ul ,ξ l ,μ,β,μ r ,σ ur ,ξ r ,φ ur ) is the parameter vector, φ ul =P(X<u l ),φ ur =P(X>u r ).

[0054] Continuity constraint σ based on the tail fraction of the body model u :

[0055]

[0056] Continuity constraint σ based on parameterized tail fraction u :

[0057]

[0058] H(u|θ) represents the distribution function of the subject model, h(u|θ) represents the subject distribution density function, φ uIt represents the probability that the selected analysis data exceeds the set threshold u, also known as the tail fraction.

[0059] Use maximum likelihood estimation to estimate the parameters of the extreme value mixture distribution in (3) and (4)

[0060]

[0061] Where L is the likelihood function, F G is the defined distribution (known as an extreme value mixture distribution).

[0062] The negative log-likelihood formula is expressed as:

[0063]

[0064] For different thresholds u seq =(u1,u2,...,u x ) can get different negative log-likelihood values, using the threshold sequence u seq The obtained -l(θ) value is searched and the u corresponding to the minimum -l(θ) value is selected -l(θ)min as a threshold.

[0065] Similarly, set u lseq =(u 1seq ,u 2seq ,...,u xseq ) and u rseq =(u rseq ,u rseq ,...,u yseq ) where x∈(1,n), y∈(1,n), u rseq >u lseq , using u lseq and u rseq As the double threshold, fit the double threshold extreme value mixed distribution, calculate the negative log likelihood value, and select the u corresponding to the minimum -l(θ) value l and u r .

[0066] Taking the analysis data obtained in step 2 as an example, different mixed distribution models are combined to fit the historical data. A total of 16 extreme value mixed distribution models are considered, including 8 single-threshold extreme value mixed distribution models and 8 double-threshold extreme value mixed distribution models. The detailed description of the distribution models is shown in Table 2 and Table 3. Comparing the above 16 extreme value mixed distribution models, since the extreme value mixed distribution model uses the minimum negative log-likelihood value to determine the optimal parameter value. The 16 extreme value mixed distributions are used to fit the tide analysis data in Table 1. The minimum negative log-likelihood value obtained by each distribution model is used to draw a radar chart ( Figure 1 ), select the extreme value mixed distribution model N with the minimum negative log-likelihood d1. Calculated N d The specific parameters of 1 are, μ=6836.42,β=976.16,u l =5400,σ ul =195.29,ξ l =-0.23,φ ul =0.034,u r =6900,σ ur =504.39,ξ r =0.40,φ ur =0.27.

[0067] Figure 2 Given the mixed distribution model N d 1 Fit the quantile plot of the data in Table 1, Figure 3 The density test of the distribution model fitting the data in Table 1 is given. Figure 4 A recurrence level prediction graph is given.

[0068] Table 2 Single threshold extreme value mixed distribution model

[0069]

[0070] Table 3 Double threshold extreme value mixed distribution model

[0071] name Subject distribution Tail fraction (lower tail) Tail fraction (upper tail) Continuity at threshold <![CDATA[N d 1]]> Normal Parameterization Parameterization discontinuous <![CDATA[N d 2]]> Normal Volume-based model Parameterization discontinuous <![CDATA[N d 3]]> Normal Parameterization Volume-based model discontinuous <![CDATA[N d 4]]> Normal Volume-based model Volume-based model discontinuous <![CDATA[N d 5]]> Normal Parameterization Parameterization continuous <![CDATA[N d 6]]> Normal Volume-based model Parameterization continuous <![CDATA[N d 7]]> Normal Parameterization Volume-based model continuous <![CDATA[N d 8]]> Normal Volume-based model Volume-based model continuous

[0072] Step 4: Extreme value prediction:

[0073] Using the obtained N d Specific parameters of the distribution model: randomly generate Y tidal level data points, take the maximum value among these Y data points, repeat this process 1000 times to generate 1000 data points, and take the average of these 1000 data points to obtain the tidal level value corresponding to the Y / 122-year return period. Therefore, Y corresponding to a 20-year return period is 2440, Y corresponding to a 50-year return period is 6100, and Y corresponding to a 100-year return period is 12200. Table 4 shows the estimated extreme tidal levels for the corresponding return periods.

[0074] Table 4 Estimation results of extreme tidal values ​​corresponding to the return period

[0075] Return period 20 years 50 years 100 years Estimated value 8066.48 8089.46 8104.57

[0076] The return period can also be calculated using the prediction model. The formula for calculating the specific return period based on the final determined extreme value mixture distribution form is:

[0077]

[0078] Where T is the return period, x is the corresponding extreme tidal value, and F(x) is the upper tail of the generalized Pareto distribution (GPD) of the extreme value mixture model. In statistics, the area of ​​the distribution function is 1. The tail that makes the distribution function area close to 1 is the upper tail, or right tail, and the tail that is close to 0 is the lower tail, or left tail. Because only the maximum value is predicted, only the upper tail is used. For example, if the prediction model calculates the probability of a 500mm tidal level exceedance to be 0.001, and if one data point is analyzed each year, T is 1000, meaning the 1000-year tidal level is 500mm.

Claims

1. A method for predicting coastal tidal extremes based on extreme value mixed distribution, characterized in that: The following steps are involved: (1) Obtain the historical tide data measured at a location, select the maximum tide data value every X hours, and form independent tide analysis data; (2) Select an extreme value mixture distribution form based on the tail fraction of the volume model or an extreme value mixture distribution form based on the parameterized tail fraction, use the threshold as the variable of the distribution function, consider the number of thresholds and the continuity at the threshold, design several different forms of extreme value mixture distribution models, and fit the extreme value mixture distribution family parameters to the tide analysis data; calculate the negative log-likelihood value of the fitted model, determine the threshold value corresponding to the minimum negative log-likelihood value, and thus establish a certain form of extreme value mixture distribution model. Among the various forms of extreme value mixture distribution models, the model with the minimum negative log-likelihood value is selected as the prediction model; (3) Using the obtained prediction model, randomly generate Y tide level data, take the maximum value of these Y data, repeat the above process N times to generate N data values, and take the average of these N data values ​​to obtain the extreme tide level of the area; The described design of several different forms of extreme value mixed distribution models specifically includes the following forms: a single-threshold extreme value mixed distribution form based on the body model tail fraction without considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the body model tail fraction without considering the continuity at the threshold; a single-threshold extreme value mixed distribution form based on the parameterized tail fraction without considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the parameterized tail fraction without considering the continuity at the threshold; a single-threshold extreme value mixed distribution form based on the body model tail fraction with considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the body model tail fraction with considering the continuity at the threshold; a single-threshold extreme value mixed distribution form based on the parameterized tail fraction with considering the continuity at the threshold, a double-threshold extreme value mixed distribution form based on the parameterized tail fraction with considering the continuity at the threshold.

2. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The single-threshold extreme value mixed distribution form F(x|θ,u,σ) based on the tail fraction of the volume model u ,ξ) is: Where H represents the main distribution, G represents the generalized Pareto distribution, θ represents the parameter vector of the main distribution, u represents the threshold, σ u is the scale parameter of the generalized Pareto distribution, and ξ is the shape parameter of the generalized Pareto distribution.

3. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The single-threshold extreme value mixture distribution form F(x|θ,u,σ) based on the parameterized tail fraction u ,ξ,φ u )for: Where H represents the main distribution, G represents the generalized Pareto distribution, θ represents the parameter vector of the main distribution, u represents the threshold, σ u is the scale parameter of the generalized Pareto distribution, ξ is the shape parameter of the generalized Pareto distribution, φ u Indicates the tail fraction.

4. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The double-threshold extreme value mixed distribution form F(x|Θ) based on the tail fraction of the body model is: where Θ=(u l ,σ ul ,ξ l ,μ,β,μ r ,σ ur ,ξ r ) represents all parameter vectors in the distribution function, u l represents the smaller threshold value in the dual threshold, σ ul represents the scale parameter of the GPD at the lower tail, ξ l represents the shape parameter of the GPD at the lower tail, μ represents the expectation of the main normal distribution, β represents the standard deviation of the main normal distribution, and u r represents the larger threshold value in the dual threshold, σ ur represents the scale parameter of the GPD at the upper and lower tails, ξ r represents the shape parameter of the GPD at the upper tail, and H(·|μ,β) indicates that it is a normal distribution.

5. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The double-threshold extreme value mixture distribution form F(x|Θ') based on the parameterized tail fraction is: where Θ'=(φ ul ,μ l ,σ ul ,ξ l ,μ,β,μ r ,σ ur ,ξ r ,φ ur ) is the parameter vector of all distribution functions, u l represents the smaller threshold value in the dual threshold, σ ul represents the scale parameter of the GPD at the lower tail, ξ l represents the shape parameter of the GPD at the lower tail, μ represents the expectation of the main normal distribution, β represents the standard deviation of the main normal distribution, and u r represents the larger threshold value in the dual threshold, σ ur represents the scale parameter of the GPD at the upper and lower tails, ξ r represents the shape parameter of the GPD at the upper tail, H(·|μ,β) indicates a normal distribution, φ ul =P(X<u l ) represents the lower tail fraction, the upper tail fraction φ ur =P(X>u r ).

6. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The continuity constraint is implemented by equating the bulk distribution to the generalized Pareto distribution at lower and upper thresholds. Continuity constraint σ of extreme value mixture distribution form based on tail fraction of volume model u for: Continuity Constraints σ on the Extreme Value Mixture Distribution Form Based on Parameterized Tail Fraction u for: H(u|θ) represents the distribution function of the subject model, h(u|θ) represents the subject distribution density function, φ u Indicates the tail fraction.

7. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The calculation formula of the negative log-likelihood value -l(θ) is: Where m is the sample size, F G is the selected extreme value mixture distribution form.

8. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The extreme value mixed distribution form adopts normal distribution or Weibull distribution as the main model and generalized Pareto distribution as the tail model.

9. The method for predicting coastal tidal extreme values ​​based on extreme value mixed distribution according to claim 1, characterized in that: The number of randomly generated tide values ​​Y and the number of repeated calculations N are gradually increased so that the error between the calculated tide extremes is less than the threshold.

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