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
1
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,
k2
(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 .
2
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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