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 [12]. The following is mainly about the method of detection of phase synchronization approached by A. Hutt and co-workers. Because the phase data represent time series and all the data are well ordered in time. Therefore, clusters can be considered as temporal segments as the K-means algorithm maps data points to their nearest cluster centers. For every number of clusters K, each data point i is associated with a cluster measure,

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 [13]. An effective clustering measure is defined by means of

The difference

reveals significant peaks at segment borders between different clusters [13].

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. Figure 1 shows the detection result. The phase changes show that this system switches at about β = 0.5 and about β = −0.5 which is in accord with potential change of the system [14,15]. values reveal significant peaks at corresponding β value, which indicates can distinguish different state objectively.

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. Figure 2 is phase changing for Q = 0.001, 0.01, 0.05, 1 respectively and Figure 3 is the corresponding detection result. It indicates that enough strong noise intensity makes phase distortion which leads not to detect different clusters with phase synchronization.

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. Figure 4 is the clustering result for different sequence length. It shows that with sequence length n increasing, reveals significant peaks at the borders of different states and value becomes smaller. Because of unchanged dynamic mechanisms of phase system, time points of segment borders are unchanged too, while value means the ratio of some cluster to all possible clusters, hence its value should get smaller over time which is in accord with physical meanings of.