Specifications for climatic feasibility demonstration—Probability and statistic analysis of extremum
Some standard content:
ICS07.060
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Meteorological Industry Standard of the People's Republic of China
QX/T529—2019
Specifications for climatic feasibility demonstration-Probability and extreme value probability statistical analysis
Specifications for climatic feasibility demonstration-Probability and extreme value probability statistical analysis Statistical analysis of extreme value
Released on 2019-12-26
China Meteorological Administration
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Implemented on 2020-04-01
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Normative references
Terms and definitions
Data collection and processing
Data collection
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Homogeneity processing of extreme value sequences
Selection of extreme value distribution type
6 Estimation of extreme value distribution parameters||tt| |Empirical Probability Distribution Plots
Parameter Estimation
Gumbel Distribution
Pearson III Distribution
Weibull Distribution
Generalized Extreme Value (GEV) Distribution
Poisson-Gumbel Composite Distribution6.2.5
Other Distributions
7 Comprehensive Goodness of Fit Analysis
7.1 Distribution Function Conformity Test
7.2 Goodness of Fit Analysis
7.3 Comprehensive Judgment
7.4 Other Methods
8 Extreme Value Recurrence Period
8.1 Cumulative probability
8.2 Calculation of return period
Appendix A (Informative Appendix)
Extreme value sequence
Extreme value sequence and its empirical probability distribution
Empirical probability distribution of extreme value sequence
Appendix B (Informative Appendix) Extreme value distribution and its parameter estimation method B.1 Gumbel distribution and its parameter estimation method.
Pearson III distribution and its parameter estimation method B.2
Weibull distribution and its parameter estimation method Generalized extreme value (GEV) distribution and its parameter estimation method Poisson-Gumbel composite distribution Cloth and its parameter estimation method B.5
Appendix C (Informative Appendix) Goodness of fit test method C.1
Kolmogorov-Smirnov test method Residual variance
Relative deviation of fit
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QX/T529—2019
References
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This standard was drafted in accordance with the rules given in GB/T1.1-2009. This standard was proposed and managed by the National Technical Committee for Climate and Climate Change Standardization (SAC/TC540). QX/T529—2019
The drafting units of this standard are: Shenyang Institute of Atmospheric Environment, China Meteorological Administration, Shenyang Regional Climate Center, Guangdong Meteorological Disaster Prevention Technical Service Center.
The main drafters of this standard are: Wang Hongyu, Gong Qiang, Huang Haohui. m
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1Scope
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Specification for Climate Feasibility Demonstration
Extreme Value Probability Statistical Analysis
QX/T529—2019
This standard specifies the probability statistical analysis methods and requirements for data collection and processing, distribution type selection, distribution parameter estimation, goodness of fit comprehensive analysis, and return period calculation involving extreme values of meteorological elements in climate feasibility demonstration. This standard is applicable to climate feasibility demonstration of engineering project planning, design, and construction. 2 Normative references
The following documents are indispensable for the application of this document. For dated references, only the dated version applies to this document. For undated references, the latest version (including all amendments) applies to this document QX/T457-2018 Climate Feasibility Demonstration Specification Meteorological Data Processing QX/T469-2018 Climate Feasibility Demonstration Specification General Principles
3 Terms and definitions
The following terms and definitions apply to this document. 3.1
Extremum
The maximum or minimum value of an element within a certain period of time Note: Rewrite GB/T34293-2017. Definition 2.4. 3.2
probabilitydistribution
Probability distribution
Used to express the probability law of random variable values. Note: Depending on the type of random variable, the probability distribution takes different forms of expression (function). 3.3
populationdistribution
Population distribution
Probability distribution of a random variable population.
Note: The entirety of the research objects (random variables) is the population3.4
Parameter estimation
parameterestimation
The process of estimating the value of an unknown parameter in a probability distribution function using a measured finite sample and a probability distribution function3.5
recurrenceinterval
Recurrence period
The statistical time interval at which a specific value of a statistic recursNote 1: Also known as the average reproduction interval, the unit is generally years (a). That is, the average interval between two possible extreme events greater than or equal to a certain intensity value. This intensity event is often called a "××-year" event. Note 2: Rewrite GB/T34293-2017, definition 2.9. 1
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4 Data collection and processing
4.1 Data collection
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According to the requirements of the climate feasibility demonstration project, it is advisable to collect the extreme value sequence of meteorological elements according to the annual extreme value method, such as maximum wind speed, maximum wind speed, maximum temperature, minimum temperature, maximum precipitation of different durations, maximum snow depth, maximum frozen soil depth, and minimum pressure in the center of tropical cyclones. The extreme value sequence should adopt the longest historical record of local observations, and the time series should be no less than 30 years. Data collection should meet the requirements of Chapter 6 of QXT469-2018.
4.2 Uniformity processing of extreme value sequence
When the observation site, observation instrument model, observation method, etc. of the meteorological station have changed in each historical stage, necessary consistency test, correction and interpolation should be carried out before the extreme value statistical analysis. Including 10-minute and 2-minute maximum wind speed conversion, observation height standardization correction, maximum wind speed and maximum wind speed conversion, and correction of the influence of observation field environment change. The data should be processed according to the requirements and methods of Chapter 9 of QX/T469-2018 and Chapter 5 of QX/T457-2018 5 Selection of extreme value distribution type
First, the recommended extreme value distribution type should be selected according to the engineering standards and specifications related to the demonstration project. 5.2 If there are no relevant standards and specifications, the Gumbel distribution is generally used as the primary choice; for elements with strong randomness (such as extreme wind speed, extreme precipitation, etc.), the Pearson III distribution should be used as the primary choice; for extreme value sequences with discontinuities (such as regional typhoon extreme wind speed, etc.), the Poisson-Gumbel composite distribution should be used as the primary choice. 5.3 The extreme value distributions recommended by standards and specifications and those selected as the primary choice should all be subjected to a comprehensive goodness of fit analysis in accordance with the requirements of Chapter 7. 5.4 When the extreme value distributions in 5.1 and 5.2 are not suitable for the extreme value sequence of the analyzed meteorological elements or there are other more suitable extreme value distributions, other extreme value distributions such as the Weibull distribution and the generalized extreme value (GEV) distribution may be used, but the results should be compared and explained on the basis of a comprehensive goodness of fit analysis.
6 Extreme value distribution parameter estimation
6.1 Plotting points of empirical probability distribution
The plotting points of the empirical probability distribution curve can generally be determined by the Weibull probability mean method, see A.2 of Appendix A.6.2 Parameter Estimation
6.2.1 Gumbel Distribution
Common parameter estimation methods in the Gumbel distribution function include the moment method, Gumbel method, maximum likelihood method, least squares method, etc. It is advisable to use the Gumbel method or least squares method for parameter estimation, see B.1 of Appendix B. Note: The Gumbel distribution (also known as the extreme value type I probability distribution) is a relatively complete extreme value theory distribution and is the limiting distribution when the sample size is very large. 6.2.2 Pearson III distribution Common parameter estimation methods in the Pearson III distribution function include the moment method, linear moment method, maximum likelihood method, numerical integration single weight function method, numerical integration double weight function method, line fitting method, maximum value adjustment line fitting method, etc. It is advisable to use the linear moment method, numerical integration single (or double) weight function method or fitting method for parameter estimation, see B.2 of Appendix B.2
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Note: There are quite a few natural phenomena that conform to the Pearson III distribution, which are widely used in the fields of precipitation and hydrology. 6.2.3 Weibull distribution
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The commonly used parameter estimation methods in the Weibull distribution function include the moment method, maximum likelihood method, least squares method, etc. It is advisable to use the least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Fréchet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B.6.2.5 Poisson-Gumbel Composite Distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to Poisson distribution, and the extreme value of a certain factor under the influence of this type of event conforms to the Weibull distribution, the Poisson-Gumbel composite distribution can be applied to analyze it. The parameter estimation method in its distribution function can use the Weibull method, etc., see B.5 of Appendix B.6.2.6 Other distributions
When it is necessary to use methods other than the distributions and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distributions and parameter estimation methods based on a comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see C.1 of Appendix C.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variance of the last 15% samples (daily value samples) of the extreme value sequence (arranged in order from small to large), and the smaller value is better: if the residual variances are consistent, you can compare the relative deviation of fit of the 15% large value samples, and the smaller value is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the fitting curve of the extreme value distribution and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, a variety of extreme value distribution functions can be used for fitting tests, and the best function result can be selected. In combination with the safety and economic requirements of the engineering project, a comprehensive analysis should be conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than those in 7.1 and 7.2, the inspection results should be compared and analyzed and explained. 3
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Extreme value return period
Cumulative probability
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The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
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A.1 Extreme value sequence
A.1.1 Sequence
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Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the position of the plot point of the sequence on the probability graph. There are multiple formulas to choose from. The choice of formula has a great influence on the parameter estimation using the fitting method. In general, the Weibull probability mean method (i.e. the common empirical distribution algorithm) is used to draw the points of the empirical probability curve, see formula (A.4): Fw(a)=
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(i =1,2,,n)
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Appendix B
(Informative Appendix)
Extreme value distribution and its parameter estimation method
Gumbel distribution and its parameter estimation method B.1.1 Gumbel distribution
The probability density function f(α) and cumulative distribution function F(α) of the Gumbel distribution are shown in formula (B.1) and formula (B.2), respectively: f()=ae-4(-h)-e
Where:
Scale parameter;
Location parameter.
F(r) = e(-)
The conversion variable y=a(r-b) can be set to facilitate calculation. Given the cumulative probability P(Tp), the extreme value of the extreme event can be calculated according to formula (B.3): -ln(-ln(P(rp)))/a+b
Common parameter estimation method of Gumbel distribution
The mathematical expectation and variance of the conversion variable y can be obtained by integration, see formula (B.4): (E(y)=0.5772
[a(y)2
From the relationship between y and work, we can get:
a(αx)
6=E(α)
E(y)o(x)
=E(r)-0.45004g(α)
....(B.1)
. .(B.2)
Substitute the estimated values of the sequence mean and standard deviation of A.1 in Appendix A (see formula (A.1) and formula (A.2)) to obtain the estimated values of parameters α and b.
Gumbel method
Gumbel method also belongs to the moment estimation method.
Construct a new sequence yG. according to formula (B.6), where: yG.=-In(—In(Fw(r,))
Where:
It can be obtained:
Empirical probability distribution function, see formula (A.4).ika-cJouakA
(i=1,2,...,n)
..(B.6)1 Gumbel distribution
Common parameter estimation methods in the Gumbel distribution function include the moment method, Gumbel method, maximum likelihood method, least squares method, etc. It is advisable to use the Gumbel method or least squares method for parameter estimation, see B.1 of Appendix B. Note: The Gumbel distribution (also known as the extreme value type I probability distribution) is a relatively complete extreme value theory distribution, which is the limiting distribution when the sample size is very large. 6.2.2 Pearson III distribution Common parameter estimation methods in the Pearson III distribution function include the moment method, linear moment method, maximum likelihood method, numerical integration single weight function method, numerical integration double weight function method, line fitting method, maximum value adjustment line fitting method, etc. It is advisable to use the linear moment method, numerical integration single (or double) weight function method or fitting method for parameter estimation, see B.2 of Appendix B.2
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Note: There are quite a few natural phenomena that conform to the Pearson III distribution, which are widely used in the fields of precipitation and hydrology. 6.2.3 Weibull distribution
QX/T529—2019
The commonly used parameter estimation methods in the Weibull distribution function include the moment method, maximum likelihood method, least squares method, etc. It is advisable to use the least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Frechet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B.6.2.5 Poisson-Gumbel Composite Distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to Poisson distribution, and the extreme value of a certain factor under the influence of this type of event conforms to Weibull distribution, the Poisson-Gumbel composite distribution can be applied to analyze it. The parameter estimation method in its distribution function can use Weibull method, etc., see B.5 of Appendix B.6.2.6 Other distributions
When it is necessary to use methods other than the distributions and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distributions and parameter estimation methods based on a comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see C.1 of Appendix C.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variance of the last 15% samples (daily value samples) of the extreme value sequence (arranged in order from small to large), and the smaller value is better: if the residual variances are consistent, you can compare the relative deviation of fit of the 15% large value samples, and the smaller value is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the fitting curve of the extreme value distribution and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, a variety of extreme value distribution functions can be used for fitting tests, and the best function result can be selected. In combination with the safety and economic requirements of the engineering project, a comprehensive analysis should be conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than those in 7.1 and 7.2, the inspection results should be compared and analyzed and explained. 3
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Extreme value return period
Cumulative probability
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The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
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A.1 Extreme value sequence
A.1.1 Sequence
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Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the plotting point position of the sequence on the probability graph. There are multiple formulas to choose from. The choice of formula has a great influence on the parameter estimation using the fitting method. In general, the Weibull probability mean method (i.e. the common empirical distribution algorithm) is used to draw the points of the empirical probability curve, see formula (A.4): Fw(a)=
ikAa-cJouakAa
(i =1,2,,n)
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Appendix B
(Informative Appendix)
Extreme value distribution and its parameter estimation method
Gumbel distribution and its parameter estimation method B.1.1 Gumbel distribution
The probability density function f(α) and cumulative distribution function F(α) of the Gumbel distribution are shown in formula (B.1) and formula (B.2), respectively: f()=ae-4(-h)-e
Where:
Scale parameter;
Location parameter.
F(r) = e(-)
The conversion variable y=a(r-b) can be set to facilitate calculation. Given the cumulative probability P(Tp), the extreme value of the extreme event can be calculated according to formula (B.3): -ln(-ln(P(rp)))/a+b
Common parameter estimation method of Gumbel distribution
The mathematical expectation and variance of the conversion variable y can be obtained by integration, see formula (B.4): (E(y)=0.5772
[a(y)2
From the relationship between y and work, we can get:
a(αx)
6=E(α)
E(y)o(x)
=E(r)-0.45004g(α)
....(B.1)
. .(B.2)
Substitute the estimated values of the sequence mean and standard deviation of A.1 in Appendix A (see formula (A.1) and formula (A.2)) to obtain the estimated values of parameters α and b.
Gumbel method
Gumbel method also belongs to the moment estimation method. Www.bzxZ.net
Construct a new sequence yG. according to formula (B.6), where: yG.=-In(—In(Fw(r,))
Where:
It can be obtained:
Empirical probability distribution function, see formula (A.4).ika-cJouakA
(i=1,2,...,n)
..(B.6)1 Gumbel distribution
Common parameter estimation methods in the Gumbel distribution function include the moment method, Gumbel method, maximum likelihood method, least squares method, etc. It is advisable to use the Gumbel method or least squares method for parameter estimation, see B.1 of Appendix B. Note: The Gumbel distribution (also known as the extreme value type I probability distribution) is a relatively complete extreme value theory distribution, which is the limiting distribution when the sample size is very large. 6.2.2 Pearson III distribution Common parameter estimation methods in the Pearson III distribution function include the moment method, linear moment method, maximum likelihood method, numerical integration single weight function method, numerical integration double weight function method, line fitting method, maximum value adjustment line fitting method, etc. It is advisable to use the linear moment method, numerical integration single (or double) weight function method or fitting method for parameter estimation, see B.2 of Appendix B.2
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Note: There are quite a few natural phenomena that conform to the Pearson III distribution, which are widely used in the fields of precipitation and hydrology. 6.2.3 Weibull distribution
QX/T529—2019
The commonly used parameter estimation methods in the Weibull distribution function include the moment method, maximum likelihood method, least squares method, etc. It is advisable to use the least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Fréchet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B.6.2.5 Poisson-Gumbel Composite Distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to Poisson distribution, and the extreme value of a certain factor under the influence of this type of event conforms to the Weibull distribution, the Poisson-Gumbel composite distribution can be applied to analyze it. The parameter estimation method in its distribution function can use the Weibull method, etc., see B.5 of Appendix B.6.2.6 Other distributions
When it is necessary to use methods other than the distributions and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distributions and parameter estimation methods based on a comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see C.1 of Appendix C.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variance of the last 15% samples (daily value samples) of the extreme value sequence (arranged in order from small to large), and the smaller value is better: if the residual variances are consistent, you can compare the relative deviation of fit of the 15% large value samples, and the smaller value is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the fitting curve of the extreme value distribution and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, a variety of extreme value distribution functions can be used for fitting tests, and the best function result can be selected. In combination with the safety and economic requirements of the engineering project, a comprehensive analysis should be conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than those in 7.1 and 7.2, the inspection results should be compared and analyzed and explained. 3
YTkAa-cJouaki
QX/T529—2019
Extreme value return period
Cumulative probability
Yika-JouakA
The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
ika-cJouak
A.1 Extreme value sequence
A.1.1 Sequence
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Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the position of the plot point of the sequence on the probability graph. There are multiple formulas to choose from. The choice of formula has a great influence on the parameter estimation using the fitting method. In general, the Weibull probability mean method (i.e. the common empirical distribution algorithm) is used to draw the points of the empirical probability curve, see formula (A.4): Fw(a)=
ikAa-cJouakAa
(i =1,2,,n)
QX/T529—2019
HiKAa-cJouakA
Appendix B
(Informative Appendix)
Extreme value distribution and its parameter estimation method
Gumbel distribution and its parameter estimation method B.1.1 Gumbel distribution
The probability density function f(α) and cumulative distribution function F(α) of the Gumbel distribution are shown in formula (B.1) and formula (B.2), respectively: f()=ae-4(-h)-e
Where:
Scale parameter;
Location parameter.
F(r) = e(-)
The conversion variable y=a(r-b) can be set to facilitate calculation. Given the cumulative probability P(Tp), the extreme value of the extreme event can be calculated according to formula (B.3): -ln(-ln(P(rp)))/a+b
Common parameter estimation method of Gumbel distribution
The mathematical expectation and variance of the conversion variable y can be obtained by integration, see formula (B.4): (E(y)=0.5772
[a(y)2
From the relationship between y and work, we can get:
a(αx)
6=E(α)
E(y)o(x)
=E(r)-0.45004g(α)
....(B.1)
. .(B.2)
Substitute the estimated values of the sequence mean and standard deviation of A.1 in Appendix A (see formula (A.1) and formula (A.2)) to obtain the estimated values of parameters α and b.
Gumbel method
Gumbel method also belongs to the moment estimation method.
Construct a new sequence yG. according to formula (B.6), where: yG.=-In(—In(Fw(r,))
Where:
It can be obtained:
Empirical probability distribution function, see formula (A.4).ika-cJouakA
(i=1,2,...,n)
..(B.6)2 Pearson III distribution Common parameter estimation methods in Pearson III distribution function include moment method, linear moment method, maximum likelihood method, numerical integration single weight function method, numerical integration double weight function method, line fitting method, maximum value adjustment line fitting method, etc. It is advisable to use linear moment method, numerical integration single (or double) weight function method or line fitting method for parameter estimation, see B.2 of Appendix B.2
YkAa-cJouaki
YTKA-JouaKAa
Note: There are quite a few natural phenomena that conform to Pearson III distribution, which are widely used in precipitation and hydrology. 6.2.3 Weibull distribution
QX/T529—2019
Common parameter estimation methods in Weibull distribution function include moment method, maximum likelihood method, least squares method, etc. It is advisable to use least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Frechet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B. 6.2.5 Poisson-Gumbel Composite Distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to Poisson distribution, and the extreme value of a certain factor under the influence of this type of event conforms to the Union distribution, the Poisson-Gumbel composite distribution can be applied to analyze it. The parameter estimation method in its distribution function can adopt the joint Bell method, etc., see B.5 of Appendix B.6.2.6 Other distributions
When it is necessary to use methods other than the distribution and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distribution and parameter estimation methods based on the comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see C.1 of Appendix C.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, the goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variances of the last 15% samples (daily samples) of the extreme value sequence (arranged in ascending order), and the smaller one is better: if the residual variances are consistent, you can compare the relative deviations of the fitting of the 15% large value samples, and the smaller one is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the extreme value distribution fitting curve and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, multiple extreme value distribution functions can be used for fitting tests, and the best function result can be selected. Combined with the safety and economic requirements of the engineering project, a comprehensive analysis is conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than 7.1 and 7.2, the inspection results should be compared and explained. 3
YTkAa-cJouaki
QX/T529—2019
Extreme value return period
Cumulative probability
Yika-JouakA
The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
ika-cJouak
A.1 Extreme value sequence
A.1.1 Sequence
ikAa-Jouaka
Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the position of the plot point of the sequence on the probability graph. There are multiple formulas to choose from. The choice of formula has a great influence on the parameter estimation using the fitting method. In general, the Weibull probability mean method (i.e. the common empirical distribution algorithm) is used to draw the points of the empirical probability curve, see formula (A.4): Fw(a)=
ikAa-cJouakAa
(i =1,2,,n)
QX/T529—2019
HiKAa-cJouakA
Appendix B
(Informative Appendix)
Extreme value distribution and its parameter estimation method
Gumbel distribution and its parameter estimation method B.1.1 Gumbel distribution
The probability density function f(α) and cumulative distribution function F(α) of the Gumbel distribution are shown in formula (B.1) and formula (B.2), respectively: f()=ae-4(-h)-e
Where:
Scale parameter;
Location parameter.
F(r) = e(-)
The conversion variable y=a(r-b) can be set to facilitate calculation. Given the cumulative probability P(Tp), the extreme value of the extreme event can be calculated according to formula (B.3): -ln(-ln(P(rp)))/a+b
Common parameter estimation method of Gumbel distribution
The mathematical expectation and variance of the conversion variable y can be obtained by integration, see formula (B.4): (E(y)=0.5772
[a(y)2
From the relationship between y and work, we can get:
a(αx)
6=E(α)
E(y)o(x)
=E(r)-0.45004g(α)
....(B.1)
. .(B.2)
Substitute the estimated values of the sequence mean and standard deviation of A.1 in Appendix A (see formula (A.1) and formula (A.2)) to obtain the estimated values of parameters α and b.
Gumbel method
Gumbel method also belongs to the moment estimation method.
Construct a new sequence yG. according to formula (B.6), where: yG.=-In(—In(Fw(r,))
Where:
It can be obtained:
Empirical probability distribution function, see formula (A.4).ika-cJouakA
(i=1,2,...,n)
..(B.6)2 Pearson III distribution Common parameter estimation methods in Pearson III distribution function include moment method, linear moment method, maximum likelihood method, numerical integration single weight function method, numerical integration double weight function method, line fitting method, maximum value adjustment line fitting method, etc. It is advisable to use linear moment method, numerical integration single (or double) weight function method or line fitting method for parameter estimation, see B.2 of Appendix B.2
YkAa-cJouaki
YTKA-JouaKAa
Note: There are quite a few natural phenomena that conform to Pearson III distribution, which are widely used in precipitation and hydrology. 6.2.3 Weibull distribution
QX/T529—2019
Common parameter estimation methods in Weibull distribution function include moment method, maximum likelihood method, least squares method, etc. It is advisable to use least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Frechet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B. 6.2.5 Poisson-Gumbel Composite Distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to Poisson distribution, and the extreme value of a certain factor under the influence of this type of event conforms to the Union distribution, the Poisson-Gumbel composite distribution can be applied to analyze it. The parameter estimation method in its distribution function can adopt the joint Bell method, etc., see B.5 of Appendix B.6.2.6 Other distributions
When it is necessary to use methods other than the distribution and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distribution and parameter estimation methods based on the comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see C.1 of Appendix C.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, the goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variances of the last 15% samples (daily samples) of the extreme value sequence (arranged in ascending order), and the smaller one is better: if the residual variances are consistent, you can compare the relative deviations of the fitting of the 15% large value samples, and the smaller one is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the extreme value distribution fitting curve and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, multiple extreme value distribution functions can be used for fitting tests, and the best function result can be selected. Combined with the safety and economic requirements of the engineering project, a comprehensive analysis is conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than 7.1 and 7.2, the inspection results should be compared and explained. 3
YTkAa-cJouaki
QX/T529—2019
Extreme value return period
Cumulative probability
Yika-JouakA
The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
ika-cJouak
A.1 Extreme value sequence
A.1.1 Sequence
ikAa-Jouaka
Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the position of the plot point of the sequence on the probability graph. There are multiple formulas to choose from. The choice of formula has a great influence on the parameter estimation using the fitting method. In general, the Weibull probability mean method (i.e. the common empirical distribution algorithm) is used to draw the points of the empirical probability curve, see formula (A.4): Fw(a)=
ikAa-cJouakAa
(i =1,2,,n)
QX/T529—2019
HiKAa-cJouakA
Appendix B
(Informative Appendix)
Extreme value distribution and its parameter estimation method
Gumbel distribution and its parameter estimation method B.1.1 Gumbel distribution
The probability density function f(α) and cumulative distribution function F(α) of the Gumbel distribution are shown in formula (B.1) and formula (B.2), respectively: f()=ae-4(-h)-e
Where:
Scale parameter;
Location parameter.
F(r) = e(-)
The conversion variable y=a(r-b) can be set to facilitate calculation. Given the cumulative probability P(Tp), the extreme value of the extreme event can be calculated according to formula (B.3): -ln(-ln(P(rp)))/a+b
Common parameter estimation method of Gumbel distribution
The mathematical expectation and variance of the conversion variable y can be obtained by integration, see formula (B.4): (E(y)=0.5772
[a(y)2
From the relationship between y and work, we can get:
a(αx)
6=E(α)
E(y)o(x)
=E(r)-0.45004g(α)
....(B.1)
. .(B.2)
Substitute the estimated values of the sequence mean and standard deviation of A.1 in Appendix A (see formula (A.1) and formula (A.2)) to obtain the estimated values of parameters α and b.
Gumbel method
Gumbel method also belongs to the moment estimation method.
Construct a new sequence yG. according to formula (B.6), where: yG.=-In(—In(Fw(r,))
Where:
It can be obtained:
Empirical probability distribution function, see formula (A.4).ika-cJouakA
(i=1,2,...,n)
..(B.6)3 Weibull distribution
QX/T529—2019
Common parameter estimation methods in Weibull distribution function include moment method, maximum likelihood method, least squares method, etc. It is advisable to use least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Fréchet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B. 6.2.5 Poisson-Gumbel composite distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to the Poisson distribution, and the extreme value of a factor under the influence of such an event conforms to the Poisson-Gumbel distribution, the Poisson-Gumbel composite distribution can be used to analyze it. The parameter estimation method in its distribution function can use the Poisson-Gumbel method, etc., see Appendix B B.56.2.6 Other distributions
When it is necessary to use methods other than the distributions and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distributions and parameter estimation methods based on the comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see Appendix C C.1.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variances of the last 15% samples (daily value samples) of the extreme value sequence (arranged in ascending order), and the smaller value is better: if the residual variances are consistent, you can compare the relative deviation of fit of the 15% large value samples, and the smaller value is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the fitting curve of the extreme value distribution and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, a variety of extreme value distribution functions can be used for fitting tests, and the best function result can be selected. In combination with the safety and economic requirements of the engineering project, a comprehensive analysis should be conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than those in 7.1 and 7.2, the inspection results should be compared and analyzed and explained. 3
YTkAa-cJouaki
QX/T529—2019
Extreme value return period
Cumulative probability
Yika-JouakA
The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
ika-cJouak
A.1 Extreme value sequence
A.1.1 Sequence
ikAa-Jouaka
Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the position of the plot point of the sequence on the probability graph. There are multiple formulas to choose from. The choice of formula has a great influence on the parameter estimation using the fitting method. In general, the Weibull probability mean method (i.e. the common empirical distribution algorithm) is used to draw the points of the empirical probability curve, see formula (A.4): Fw(a)=
ikAa-cJouakAa
(i =1,2,,n)
QX/T529—2019
HiKAa-cJouakA
Appendix B
(Informative Appendix)
Extreme value distribution and its parameter estimation method
Gumbel distribution and its parameter estimation method B.1.1 Gumbel distribution
The probability density function f(α) and cumulative distribution function F(α) of the Gumbel distribution are shown in formula (B.1) and formula (B.2), respectively: f()=ae-4(-h)-e
Where:
Scale parameter;
Location parameter.
F(r) = e(-)
The conversion variable y=a(r-b) can be set to facilitate calculation. Given the cumulative probability P(Tp), the extreme value of the extreme event can be calculated according to formula (B.3): -ln(-ln(P(rp)))/a+b
Common parameter estimation method of Gumbel distribution
The mathematical expectation and variance of the conversion variable y can be obtained by integration, see formula (B.4): (E(y)=0.5772
[a(y)2
From the relationship between y and work, we can get:
a(αx)
6=E(α)
E(y)o(x)
=E(r)-0.45004g(α)
....(B.1)
. .(B.2)
Substitute the estimated values of the sequence mean and standard deviation of A.1 in Appendix A (see formula (A.1) and formula (A.2)) to obtain the estimated values of parameters α and b.
Gumbel method
Gumbel method also belongs to the moment estimation method.
Construct a new sequence yG. according to formula (B.6), where: yG.=-In(—In(Fw(r,))
Where:
It can be obtained:
Empirical probability distribution function, see formula (A.4).ika-cJouakA
(i=1,2,...,n)
..(B.6)3 Weibull distribution
QX/T529—2019
Common parameter estimation methods in Weibull distribution function include moment method, maximum likelihood method, least squares method, etc. It is advisable to use least squares method or moment method for parameter estimation, see B.3 of Appendix B. Note: Weibull distribution is an extreme value theory distribution, which is often used in the probability distribution of wind speed and the study of wind energy resources. 6.2.4 Generalized Extreme Value (GEV) Distribution
Integrating Gumbel distribution, Fréchet distribution and Weibull distribution into two distribution functions is called generalized extreme value distribution. Common parameter estimation methods in its distribution function include linear moment method, maximum likelihood method, etc., see B.4 of Appendix B. 6.2.5 Poisson-Gumbel composite distribution A random event belongs to discrete distribution, and the extreme value of a certain factor under the influence of this type of event can constitute a continuous distribution. If the random event conforms to the Poisson distribution, and the extreme value of a factor under the influence of such an event conforms to the Poisson-Gumbel distribution, the Poisson-Gumbel composite distribution can be used to analyze it. The parameter estimation method in its distribution function can use the Poisson-Gumbel method, etc., see Appendix B B.56.2.6 Other distributions
When it is necessary to use methods other than the distributions and parameter estimation methods in 6.2.1 to 6.2.5, they should be compared with the above distributions and parameter estimation methods based on the comprehensive analysis of goodness of fit. 7 Comprehensive analysis of goodness of fit
7.1 Distribution function conformity test
The Kolmogorov-Smirnov test method can be used to determine whether the extreme value sequence obeys a certain type of extreme value distribution, see Appendix C C.1.
7.2 Goodness of fit analysis
For more than two alternative distribution functions that pass the conformity test, goodness of fit analysis can be performed using methods such as residual variance and relative deviation of fit, see C.2 and C.3 of Appendix C.
You can also compare the residual variances of the last 15% samples (daily value samples) of the extreme value sequence (arranged in ascending order), and the smaller value is better: if the residual variances are consistent, you can compare the relative deviation of fit of the 15% large value samples, and the smaller value is better, see B.2.2.6 and B.2.2.7 of Appendix B.
7.3 Comprehensive judgment
A comparison chart of the fitting curve of the extreme value distribution and the empirical probability distribution of the measured data should be drawn. If the fitting effect of the extreme value distribution function specified in the relevant engineering specifications is not good, a variety of extreme value distribution functions can be used for fitting tests, and the best function result can be selected. In combination with the safety and economic requirements of the engineering project, a comprehensive analysis should be conducted to confirm the extreme value distribution type and corresponding parameters suitable for the extreme value sequence of the meteorological element. 7.4 Other methods
When it is necessary to use inspection and analysis methods other than those in 7.1 and 7.2, the inspection results should be compared and analyzed and explained. 3
YTkAa-cJouaki
QX/T529—2019
Extreme value return period
Cumulative probability
Yika-JouakA
The probability that a random variable X does not exceed a certain fixed value p is called the cumulative probability (left-hand probability), see formula (1): P(X≤ap)=F(ap)
Where:
P(X≤p)
Return period calculation
The probability that X does not exceed a certain fixed value Tp. Cumulative probability (left-hand probability).
The calculation of the return period T(p) of the extreme value is shown in formula (2): 1
T(ap)=:
1-P(X≤rp)
=1-F(rp)
ika-cJouak
A.1 Extreme value sequence
A.1.1 Sequence
ikAa-Jouaka
Appendix A
(Informative Appendix)
Extreme value sequence and its empirical probability distribution
Let the sequence (α) be the value of the sample in the sequence. E = E(1) is the mathematical expectation of the sequence (x), that is, the mean = (α) is the standard deviation of the sequence ().
QX/T529—2019
In actual calculations, the mean and standard deviation of a finite sample size can be used as the estimated values of E(x) and. (). After sorting the extreme value sequence to be analyzed, we get: x≤≤…≤α, n is the total number of samples in the sequence, and it is a finite and ordered sample sequence of the sequence "x".
A.1.2 Estimated value of the mean of the sample sequence
The estimated value of the mean of the sample sequence is shown in formula (A.1) 1
A.1.3 Estimated value of the standard deviation of the sample sequence
The estimated value of the standard deviation of the sample sequence. See formula (A.2): (-)
A.1.4 Estimated value of the skewness coefficient of the sample sequence The estimated value of the skewness coefficient r of the sample sequence is shown in formula (A, 3): A.2 Empirical probability distribution of extreme value sequence
..(A.2)
.（A.3)
F（1）is the empirical probability value of the extreme value ordered sequence at x, which determines the position of the plot point of the sequence on the probability graph. There are multiple formulas to choose from. The choice of f
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