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J ournal o f A pp Published Onli n http://dx.doi.or g Open Access Dete c ABSTRA C In view of ex t nization clust e tensity and s e can detect the ess of extrem e Keywords: P r 1. Introdu c With global w come increas sessment rep o about extrem e p articular ti m ability ones, w even more lo w tion curve of events were strictly extre m what damage cause to soc i affected area focus on the p the fall of e x searchers hav these method s artificial judg m A. Hutt an d mutual phas e kind of thou g The data rec o to be split in t the one han d scales on the considered to p resenting te m Here the ap p detect tempo r detail. p lied Mathemat i n e November 2 0 g /10.4236/jamp . c tion o f Colle g C T t reme values e e ring method e quence lengt h temporal pro c e weather eve n r ocess; Phase c tion w arming extr e ingly commo n o rts by the IP C e weathe r eve n m e, extreme w e w hose occurri n w e r [1]. Then meteorologic a defined and m e values ev e every extrem e i ety and econ o and duration . p rocess, i.e., t x treme weath e e concerned a s are not obj e m ent. d co-workers e synchroniza t g ht to detect t h o rded in certai n t o temporal s e d and time w i other. The pa r be a state or a m poral partiti o p licability ab o r al process of i cs and Ph y sics , 0 13 (http://ww w .2013.16002 f the Pr Zhongh u g e of Physical S c e vents mathe m is introduced h . At last the o c ess objective l n ts. Synchronizati o e me weather e n . The third C C both gave a n ts: for a part i e ather events n g probability based on pro b a l elements, e researche d [ 2 e nts mathem a e weather eve n o my depend . So it is ver y t he rise, the d e e r events. Re c a bout this issu e e ctive enough proposed a m t ion [12], w h h e process of n open syste m e quences of f a i ndows of na r r t of phase sy n a cluster abou t o n, i.e., the te m o ut phase sy n weather eve n , 2013, 1, 6-10 w .scirp.org/jour n ocess a b u a Qian, Ze n c ience & Techn Email: q i Recei v m atically rathe r a nd the appli c o bserved data l y to a certain o n; Clustering e vents have b and fourth a a definite defi n i cular place at are small pro b is about 10% o b ability distrib u e xtreme weath 2 -9], which a r a tically. In fa c n t amount cou l on its streng t y significant t e velopment a n c ently some r e e [10,11], wh i and need so m m ethod to det e h ich provides weather even t m s is consider e a st transients o r row- b and ti m n chronization t time windo w m poral proce s n chronization t n ts is studied i n al / jamp ) b out E x n gping Zha n n ology, Yangzh o i anzh@yzu.edu v e d August 201 3 r than the pro c c ability of the is applied. T h degree and it h e- a s- n e a b - o r u - er r e c t, ld t h, t o n d e- i le m e e ct a t s. e d o n m e is w s, s s. t o in 2. Me 2.1. D e In phy synchr o about a inform a phase corres p ( ) H t where the Ca u is the p 2.2. P h M In ord e p hase s rithm a p roper about t approa c p hase d ordere d x treme n g, Guolin F o u University, Y .cn 3 c ess about ext r method is dis c h e results sho w h as certain ap p thod of De t e finition of P sics, phase r e o nization ana l a mplitude an d a tion and ph a ()t of a re a p onding analyt i () s t ) is Hilbert t r () 1 Ht () 1/ht t , t u chy principal () a t p hase of signal h ase Cluster i M easure e r to detect te m s equences are a nd cluster q u number of cl u t he method o f c hed by A. H d ata represent d in time. The r Weath e eng Y angzhou, Chin a r eme weather e c ussed from t h w that clusteri n p lication to de t t ectin g the P P hase e flects the st a l ysis is to se p d phase from s a se relativity a l signal s(t) c i cal signal s˜( t () ()st iHt r ansform abou t () () () stht d s HV d t t he integral in value. Then a rctan() /Ht s . i ng and Clu s m poral proces s clustered by K u ality measur e u s t ers [12]. T h f detection of p H utt and co- w time series a n r efore, cluster s e r Eve n a e vents, phase s h e aspects of n n g measure d i t ect the tempo P rocess a te of a sign a p arate the inf o s ignal and on l are conside r c an be define d t ). t () s t, d d Equation (2) () s t s ter Quality s objectively, t K -means clus t e is used to h e following i s p hase synchr o w orkers. Bec a n d all the data s can be cons i JAMP n ts s ynchro- n oise in- i fference ral proc- a l. Phase o rmation l y phase r ed. The d via its (1) (2) refers to (3) t emporal t er algo- give the s mainly o nization a use the are well i dered as Z.-H. QIAN ET AL. Open Access JAMP 7 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 clus- ter measure () K A i, () [() ()]/ K s jnjK ji Ai dC xdCxN (4) where 1 () T KK i NAi is the normalized factor. n C and s C denote the nearest and the second-nearest cluster center of data point i, respectively. i represents a subset of members of the cluster to which data point i is associated. The dataset is partitioned into distinct sub- sets i reflecting consecutive time segments each. For every number of clusters K the subsets i 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 statis- tical approach and average different cluster measures with increasing K are used and yields the so so-called cluster quality measure, () ()/piAiA (5) where 2 1 () () 1 R K K A iAi R , 1 () T i A Ai . R is the maximum number of clusters. An increasing number of clusters K yields an increasing number of subsets i and subsequently, it diminishes the cluster measures () K A i. 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 randomiz- ing the examined dataset with respect to its temporal or- der. Because the surrogates () () s pi do not contain any temporal structure they can be used to normalize the original values ()pi [13]. An effective clustering mea- sure eff p is defined by means of () ()max{0,()max[( )]} s eff pipip j (6) The difference () () ()max{0, (1)() max[(1)()]} ss peffip ip i p jpj (7) 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 se- quences. 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. 3. Numerical Simulation of Phase Synchronization 3.1. Detection of One-Dimensional Data In order to compare with the result of A.Hutt and co- workers’, the following stochastic dynamical system is also discussed, sinsin 22 k kkk dQ dt (8) where 1, 2,,kN , () 0 kt , ()( ')2(') kl kl tt tt . The values () kk t represent phases that evolve along the gradient of a potential, () cos/2cos2 kk k V (9) Considering the complexity of practically observed meteorological elements data, N = 1 is chosen. Equation (8) is simulated solution as a trial being obtained by de- creasing from −1 to 1 for Q = 0.001 in 500 equidis- tant steps. At each step the system relaxes for 1000 inte- grations and the final one is stored. The initial phase an- gles were (0) . Figure 1 shows the detection result. The phase changes show that this system switches at about = 0.5 and about = −0.5 which is in accord with potential change of the system [14,15]. eff p val- ues reveal significant peaks at corresponding value, which indicates eff p can distinguish different state objectively. 3.2. Sensitive Numerical Simulation of Noise Intensity Q It can be known from Equation (8) that this system shows various forms of phase locking and/or bifurcation patterns depending on parameters and Q for N = 1. Here how noise intensity Q affects the result of detec- tion is considering. Figure 2 is phase changing for Q = 0.001, 0.01, 0.05, 1 respectively and Figure 3 is the cor- responding detection result. It indicates that enough strong noise intensity makes phase distortion which leads not to detect different clusters with phase synchroniza- tion. 3.3. Sensitive Numerical Simulation of Sequence Length 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, eff p reveals significant peaks at the borders of different Open Access 8 -1.0 -1 0 1 2 3 4 (a) -0.5 0.0 0 - -1 0 1 2 3 4 - -2 -1 0 1 2 3 4 F Figure 2 Figure 3. 0 .5 1.0 p - 1.0 -0.5 - 1.0 -0.5 Z.-H. Q F igure 1. Phase 2 . Phase chang e Phase clusteri n -1.0 - 0 0.0 0.1 0.2 0.3 0.4 p i eff (b) 0.0 0.5 Q=0.001 0.0 0.5 Q=0.05 Q IAN ET A L clustering for e s for different n g for differen t 0 .5 0.00.5 1.0 -1.0 -2 -1 0 1 2 3 4 1.0 -1.0 -6 -4 -2 0 2 4 L . Q = 0.001. noise intensity t noise intensit y 1.0 0 . 0 . 0 . 0 . pi e f -0.5 0.0 -0.5 0.0 Q. y Q. -1.0 -0.5 . 0 . 1 . 2 . 3 f f (c) 0.5 1.0 Q=0.01 0.5 1.0 Q=1 0.0 0.5 JAMP 1.0 Open Access states and unchanged d y p oints of se g eff p value m b le clusters, h which is in ac 4. Applica t For the sake data to phas e dized Precipi t Figure 4 . eff p value b e y namic mech a g ment border s m eans the rati o h ence its valu e cord with phy s t ion of Pra c of the adapta b e clustering, M t ation Index) . Phase cluster i e comes small e a nisms of pha s s are unchan g o of some clu s e should ge t s m s ical meaning s c ticall y Obs b ility of prac t M SPI (Multi- [16] January 2 i ng for sequen c Z.-H. Q e r. Because o s e system, ti m g ed too, wh i s ter to all pos s m aller over ti m s of eff p. erved Data t ically observ e scales Stand a 2 009-Decemb c e length n. Q IAN ET A L o f m e i le s i- m e e d ar - e r 2012 o analyz e ure 5 Guizh o tion e x drough t getting tempor drough t getting 5. Su m In vie w than t h synchr o applic a of noi s served measu r p roces s applic a weathe r studie d search w 6. Ac k Fundin nology Fi g MSPI i L . o f 31 observa t e d on Southw e shows the c l o u province a s x perienced to t t , oscillating moist. Phas e al process obj t event, som e drought from m mar y an d w of extreme v h e process ab o o nization clus t a bility of the m s e intensity a n data is appli e r e difference s objectively t o a tion to detec t r events. Fo r d only, multi- d w hich is our f u k nowled g e m g was obtain e Support p rog r g ure 5. Phase c l -4.2 -2.1 0.0 2.1 6 6 0 2 4 6 8 MSPI p i eff t ion stations i e st Drought E v l ustering res u s an example. t al three pro c continuous d e clustering m ectively to a c e station in s o January 2009 . d Discussio n v alues events o ut extreme w t ering method m ethod is disc u n d sequence l e e d. The result s eff p can o a certain de g t the tempor a r simplicity o d imensions d a u ture research m ents e d from Natio n r am under Gr a l ustering detec t 12 18 2 12 18 2 5770 7 i n southwest C v ent in fall 2 0 u lt of Bijie s t It shows that c esses, i.e., i n d rought and g m ethod can d e c ertain degree. o uthwest Chi n . n mathematical l w eather even t is introduce d u ssed from th e e ngth. At las t s show that c l detect the t g ree and it ha a l process of o ne-dimension a ta deserve fu r would focus o n al Science a n a nt No. 2012C B t ion in Bijie sta 2 43036 2 43036 7 JAMP 9 C hina is 0 09. Fig- t ation in this sta- n creasing g radually e tect the For this n a began l y rather t s, phase d and the e aspects t the ob- l ustering t emporal s certain extreme data is rther re- o n. n d Tech- B 955901 a tion. Z.-H. QIAN ET AL. Open Access JAMP 10 and National Natural Science Foundation of China under Grant No.41105033. REFERENCES [1] IPCC, “Summary for Policymakers of the Synthesis Re- port of the IPCC Fourth Assessment Report,” Cambridge University Press, Cambridge, 2007, pp. 1-15. [2] P. Frich, L. V. Alexander and P. M. Della-Marta, “Obse- vered Coherent Changes in Climatic Extremes during the Second Half of the 20th Century,” Climate Researc h, Vol. 19, No. 3, 1993, pp. 193-212. http://dx.doi.org/10.3354/cr019193 [3] M. J. Manton, P. M. Della-Martaand M. R. 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