1. Abstract Most of the classical time series analyses require the time series to be stationary and/or linear. However, financial time series are usually nonlinear and nonstationary. In this study, we use the Hilbert-Huang Transformation (HHT) to investigate the dairy return of several international stock markets from 2005 to 2010. The HHT approach mainly consists of application of the empirical mode decomposition (EMD) and the instantaneous frequency analysis. We find that there is an obvious change in the behavior of the trading activity among these stock markets since the U.S. sub-prime mortgage credit crunch in 2008. Keywords: instantaneous frequency, empirical mode decomposition, HHT. INTRODUCTION Stock markets are complex dynamical systems whose elements are investors and traders with varying capital sizes using different investment and trading strategies. For the last two decades the international finance has higher correlation because of the internationalization. For example: Exchange rate and interest rate, economic policy adjustment, international main stock market's change, petroleum price fluctuation, financial crisis, war or other big events. This study tries to address the question: “Is there a high similarity in the trading activities of different markets when encounter a financial crisis?” To resolve these problems, we use the instantaneous angle and instantaneous frequency, developed by Huang et al. [2], to examine whether there is any similarity in time evolution of prices in the two segments and any change in similarity between two indexes before and after a financial crisis. This method was proposed based on nonlinear or nonstationary concern and has been designed to extract dynamic information from time series. HILBERT-HUANG TRANSFOMTION METHOD The algorithm to create IMFs in the EMD has two main steps. After repeated the sifting process up to k times until some acceptable tolerance is reached: If the resulting time series is the first IMF, it is designated as. Finally, we get For the k-th IMF, this can be done by first calculating the conjugate pair of, i.e., where P indicates the Cauchy principal value. We define. Then, we can calculate the instantaneous phase according to the above equations. STOCK INDEXES In this study, we use several major world stock indexes from 2005 to 2010 for analysis. These indexes include the TWI, HSI, FTSE, CAC, DAX, S&P, STI and NASDAQ. The data was retrieved from the Yahoo database [3]. The international main stock prices in the last decade were changing enormously. In the years 1999-2000, the NASDAQ vigorously exploded and jumped up around 50% and bring the positively impact to the other international stock markets. This rally was based on expected earnings of brand new internet and tech companies with no track record. Promptly, the stock price declined even more powerful since the internet bubble. The years 2007-2008 were another period of extremes. Financial companies found a way to make money even easier - trading and selling financial derivatives. This finincial crises of sub- prime mortgage credit, causes the American housing market murky dispirited. The association attacks the American economy, and has the negative chain effect to the global economic and stocks. ANALYSIS OF STOCK INDEXES In the stock time series analysis, the researchers usually use the logarithmic return scale for application. The logarithmic return is simply defined by,where stands for the stock price at time t. Fig. 2 gives the logarithmic return of NASDAQ index and the first 3 IMFs obtained through the EMD method result in decomposition of 9 components and 1 residue. Therefore, the further application of the phase statistics approach is based on analysis on the first IMF. Consequently, we take an illustration with the NASDAQ index, the probability density functions of the amplitude for its first-IMF is roughly Boltzmann distributed. Beside this, we find from Fig. 3 that except for the first IMF, the phases of the other IMFs have almost equal probability for To investigate the correlative behaviors between any two different stock markets, we pair-wisely calculate the instantaneous phases between two indexes. Here, the phase differences is defined as Fig. 4 gives the probability distributions of phase differences between the first-IMFs of returns for two indexes for 2005–2010. We summarize some finding based various market regions and certain periods. (1) The third row plots provide the angle difference distribution of S&P against the other indices. It can be seen, when subtracting against those of TWI, STI, or HSI, each distribution all have positive mod. Surprisingly, similar results are displayed in the fourth row pots. That indicates that the stock activity of American market have the leading impact to the Asia stock indexes. (2) The phase difference distribution between the S&P and the NASDAQ has high kurtosis and narrow range. Hence these two markets have high correlated activities in the whole period. More specifically, the (3,4)-th and the (4,3)-th subplots show that the more correlative activities occurred after the sub-prime mortgage credit crunch in 2008. (3) Such correlative activities can also be found among the FTSE, the CAC, and the DAX. (4) From the first, the second and the fifth rows, one can see that each of the Asia indexes verse the European indexes, all have low Kurtosis, and hence a relatively less correlated activities between these two Continents. From above observation, the universal existence of regional correlation among international stock markets, hence we take only TWI and NASDAQ as benchmarks for the following event investigation. We summarize the main result as follows: (1) The first row of Fig. 4 shows the TWI and the American stock markets have higher correlative after the sub-prime mortgage credit crunch in 2008. However, the TWI verse the other Asia stock markets or the TWI verse the European markets have higher correlative during 2007b-2008a rather than the period 2008b-2009a. (2) For the NASDAQ index verse the other international indexes, the correlations have remarkable change during 2007b-2008a and elevate continually after 2010. FIGURES REFERENCE 1.M.-C. Wu and C.-K. Hu, Phys. Rev. E 73, 051917 (2006). 2.N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C. Yen, C.-C. Tung and H. H. Liu, Proc. R. Soc. Lond. A 454, 903 (1998). 3.Data sets are available from http://finance.yahoo.com/ Instantaneous Frequency to Stock Returns in a Financial Crisis Tsair-chuan Lin Department of Statistics, National Taipei University Amanda S. M. Chang Center of General Education, Chang Gung University Fig. 4. The array plot of the probability distribution for. The indexes i and j are labeled by the order: 1(TWI), 2(STI), 3(SP), 4(NASDQ), 5(HSI), 6(FTSE), 7(CAC), and 8(DAX). Fig. 1. The dairy series of the TWI, STI, SP, NASDQ, HSI, FTSE, CAC, and DAX indexes from 2005 to 2010. Fig. 2 The application of EMD for the NASDAQ index. The return series is decomposed into 9 IMFs and 1 residue; here, only the first 3 components are displayed. Figure 3. The distribution of (a) phases and (b) amplitudes of the first three IMFs of the NASDAQ index return.