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Phase synchronization clustering method is used to detect the process of extreme weather events rather than extreme values events mathematically. The applicability is discussed from the aspects of noise intensity and sequence length and the observed data are applied practically. The detection process shows that clustering measure difference can detect the temporal process objectively to a certain degree and it has certain application to detect the temporal process of extreme weather events.

With global warming, extreme weather events have become increasingly common. The third and fourth assessment reports by the IPCC both gave a definite definition about extreme weather events: for a particular place at a particular time, extreme weather events are small probability ones, whose occurring probability is about 10% or even more lower [

A. Hutt and co-workers proposed a method to detect mutual phase synchronization [

In physics, phase reflects the state of a signal. Phase synchronization analysis is to separate the information about amplitude and phase from signal and only phase information and phase relativity are considered. The phase of a real signal s(t) can be defined via its corresponding analytical signal.

is Hilbert transform about,

where, the integral in Equation (2) refers to the Cauchy principal value. Then

is the phase of signal.

In order to detect temporal process objectively, temporal phase sequences are clustered by K-means cluster algorithm and cluster quality measure is used to give the proper number of clusters [

where is the normalized factor. and

denote the nearest and the second-nearest cluster center of data point i, respectively. represents a subset of members of the cluster to which data point i is associated. The dataset is partitioned into distinct subsets reflecting consecutive time segments each. For every number of clusters K the subsets represent consecutive time segments. Usually the optimal number of clusters is unknown resulting in an uncertainty about a proper choice of K. To minimize this uncertainty a statistical approach and average different cluster measures with increasing K are used and yields the so so-called cluster quality measure,

where,. R is the maximum number of clusters. An increasing number of clusters K yields an increasing number of subsets and subsequently, it diminishes the cluster measures. In general, an optimal value of the upper bound R depends on the real number of clusters in the data but R is usually in the range of tens.

In order to compare cluster qualities across different datasets, a reference system is introduced by randomizing the examined dataset with respect to its temporal order. Because the surrogates do not contain any temporal structure they can be used to normalize the original values [

The difference

reveals significant peaks at segment borders between different clusters [

Based on the above analysis, the detection of phase synchronization can classify the temporal phase sequences. A cluster means a temporal phase window, i.e., a state of some event, so the method provides a kind of way to give the process of event.

In order to compare with the result of A. Hutt and coworkers’, the following stochastic dynamical system is also discussed,

where, ,

. The values represent phases that evolve along the gradient of a potential,

Considering the complexity of practically observed meteorological elements data, N = 1 is chosen. Equation (8) is simulated solution as a trial being obtained by decreasing β from −1 to 1 for Q = 0.001 in 500 equidistant steps. At each step the system relaxes for 1000 integrations and the final one is stored. The initial phase angles were.

It can be known from Equation (8) that this system shows various forms of phase locking and/or bifurcation patterns depending on parameters β and Q for N = 1. Here how noise intensity Q affects the result of detection is considering.

Because small noise intensity is better for the detection, Q = 0.001 is chosen. Then the influence of sequence length is considered. For the phase system, it is assumed that phase changes as the same with the time changing.

For the sake of the adaptability of practically observed data to phase clustering, MSPI (Multi-scales Standardized Precipitation Index) [

In view of extreme values events mathematically rather than the process about extreme weather events, phase synchronization clustering method is introduced and the applicability of the method is discussed from the aspects of noise intensity and sequence length. At last the observed data are applied. The results show that clustering measure difference can detect the temporal process objectively to a certain degree and it has certain application to detect the temporal process of extreme weather events. For simplicity, we only study one-dimension data and multi-dimensions data deserve further research which our future research would focus on.

Funding was obtained from National Science and Technology Support program under Grant No. 2012CB955901 and National Natural Science Foundation of China under Grant No.41105033.