Detection of Urbanization signals in Extreme Winter
Minimum Temperatures change over Northern China
Qingxiang Li, Jiayou Huang, Zhihong Jiang, Liming Zhou, Peng Chu, Kaixi Hu
GEV
The GEV distribution is a family of continuous probability distributions
developed within extreme value theory to combine the Gumbel, Fréchet and Weibull
families, with a theoretical distribution function as follows:
exp{ [1   ( x   ) / a]1 / k }, k  0, x    a / k

F ( x)  
exp{  exp[ ( x   )]},   0
exp{ [1   ( x   ) / a]1 / k }, k  0, x    a / k

(1)
Where ξ represents the location parameter (referred to as CM) determining the
position of the distribution; α is the scale parameter (referred to as CF) determining
the range of extension of measurement distribution curve; k is the shape parameter
(referred to as CK) determining the tail behavior of the distribution (i.e., the pattern of
extreme distribution): when k=0, it is the Gumbel distribution; when k>0, it is the
Weibull distribution; when k<0, it is the Fréchet distribution. Therefore, the GEV
probability distribution is described by the 3 parameters (characteristic values).
Estimating the GEV distribution parameters generally consists of two major
methods, maximum likelihood methods and L-moments methods (Hosking, 1990;
Coles, 2001). The calculation procedures are as follows:
First, the 30 extreme minimum temperature samples for each 10-year window
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are sorted to form a new sequence in the ascending order:
xi (i  1,2,  n)
Where n is the sample size, and the linear matrix is:
1  b0
(2)
2  2b1  b0
(3)
3  6b2  6b1  b0
(4)
where,
b0 
1 n
 xi
n i 1
b1 
1 n (i  1)
xi

n i 2 (n  1)
b2 
1 n (i  1) (i  2)

xi

n i 3 (n  1) (n  2)
are the sample probability weight matrix. The shape parameter in the GEV
distribution can be estimated as follows:
k  7.8590C + 2.9554C 2
(5)
Where,
C

2
ln2
 (3  3 ) 3
2 ln3
The scale parameter can be estimated by,

k2
(1  2 )(1  k )
k
(6)
and the position parameter can be calculated by:
  1  [1  (1  k )] / k
(7)
Where (1  k ) is the function of  .
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References
Coles, S. G. (2001), An Introduction to Statistical Modeling of Extreme Values.
Springer.
Hosking J. R. M. (1990), L-moments: Analysis and Estimation of Distributions using
Linear Combinations of Order Statistics. J. R. Statist. Soc. B, 52, 105-124
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