以經驗模態分解方法來分析臺灣股票加權指數與世界經濟大國股票指數之相關性及趨勢研究

經驗模態分解(Empirical Mode Decomposition)是非線性及非定態時間序列資料分析的新方法，可將複雜的資料分解成有限且數量少的簡單振盪模態，在自然科學有很好的應用，在社會科學領域應用不多。艾略特波浪理論(Elliott Wave Principle)預測趨勢的準確度毀譽參半，很少有客觀的科學事實來證明特定波的存在。本文的目的是使用EMD方法來解釋股市之艾略特波浪理論，將複雜的股票指數分解成數量少的本質模態函數(Intrinsic Mode Function， IMF)，以研究臺灣股票加權指數(TWI)與美國道瓊工業指數(DJI)、英國金融時報指數(FTSE)、德國法蘭克福指數(GDAX)、日本日經指數(N225)及香港恆生指數(HIS)之相關性及趨勢。以各國240筆的月資料(1989年1月至2008年12月)收盤價做為相關性分析對象，每一國的月資料可分解出5個IMFs(c1到c5)，各國IMFs的平均週期只有c1大致相同(3.0個月左右)，其他的平均週期都不一樣，表示每一個波都有自己波動的頻率。相關性以交叉相關函數(CCF)來做比較。本研究結果顯示，只看個別波IMFs之間的相關性所能得到的訊息不多，符合每一個波都有自己波動頻率的特性。與TWI趨勢最相關的指數排序依次為DJI (CCF 0.990)、HSI (CCF 0.989)、FTSE(CCF 0.877)、GDAX(CCF 0.764)、N225(CCF -0.887)，顯示除了日本外，台灣與其他國的趨勢是一致的。本研究結果發現，將EMD趨勢的定義再擴充，也就是由低頻率IMF向高頻率IMF的波值與殘餘值相加後就形成特定時間尺度的趨勢，可得到不同平均週期的趨勢波。這些趨勢波可明顯看出是艾略特波浪理論的不同層級的波。擴充EMD趨勢c5-residue(c5+EMD趨勢)就是艾略特波浪理論c5-residue層級的波(以平均週期來定義)。以c4-residue層級(即c4+c5+EMD趨勢)來說，在2009年4月TWI正處在第四波的C波，DJI正處在第四波的A波，FTSE正處在第四波的C波，GDAX正處在第四波的C波，N225正處在第四波的C波，HSI正處在第四波的A波，各國都是處在下降波。擴充EMD趨勢的定義使艾略特波浪理論的波浪變得很乾凈，波浪計數當中沒有干擾，讓波浪計數更有依據更簡單。

關鍵字

經驗模態分解；本質模態函數；交叉相關係數；趨勢；艾略特波浪

並列摘要

The empirical mode decomposition (EMD) method is a novel powerful procedure for analyzing non-stationary and non-linear time series. By EMD any complicated data set can be decomposed into a finite and small number of simple oscillatory modes, defined as an intrinsic mode function (IMF), with significantly different frequencies. EMD was successfully applied in many area of nature science and engineering, but little applied in social science so far. The position of Elliott wave forecasts is controversial. There were no consensus and no objective methods to prove the existence of a specific Elliott wave. The purpose of this paper is using the sequential combination of the residue and IMFs to help to explain and count the Elliott wave and study the correlations and trends between Taiwan Capitalization Weighted Stock Index (TWI) and Dow Jones Industrial Average (DJI), TWI and Hang Seng Index (HIS), TWI and Nikkei Stock Average 225 (N225), TWI and German stock index (GDAX), and TWI and Financial Times Stock Exchange (FTSE). The 20-year (Jan. 1989 – Dec. 2008) monthly closed data of above indices were used in cross correlation analysis. Five IMFs (c1-c5) and a residue (named as EMD trend) were derived from sifting processes of each data set. Mean periods of each IMF of indices were different except the one of first IMF (c1), which was about 3.0 month for all of indices. This indicates each oscillation has significantly different frequencies. Correlation between each IMF of TWI and other indices was evaluated by cross-correlation function (CCF). One of our results showed that little information could be derived from the point of view of each IMF so far. EMD method can extract various trends from a data set. EMD trend is defined as the overall adaptive trend or the sum of a residue and the latest IMFs, which may already satisfy the definition of a trend. Results of the study showed that the correlations of trend with TWI trend in descending order were DJI (CCF 0.990), HSI (CCF 0.989), FTSE(CCF 0.877), GDAX(CCF 0.764) and N225(CCF -0.887). We find that an extended EMD trend of a specific time scale is a different degree of Elliott wave of a specific time scale. We define the extended EMD trends as EMD trend plus the remaining IMF sequentially from low frequency to high frequency. For example, c5-residue (c5+EMD trend) is the c5-residue degree of Elliott wave. If we counted the c4-residue degree of Elliott wave, the present position of TWI, FTSE, GDAX, and N225 were in the minor wave C of intermediate wave (4), and DJI and HSI in the minor wave A of intermediate wave (4). EMD-assisted Elliott wave counting makes the counting of Elliott wave simple, easy and straightforward.

並列關鍵字

Empirical mode decomposition ； Intrinsic mode function ； Cross-correlationfunction ； Trend ； Elliott wave principle

參考文獻

Box, G. E., Jenkins, G. M., & Reinsel, G. C. (2008). Time series analysis: Forecasting and control. Hoboken, New Jersey, U.S.A: John Wiley & Sons, Inc.

Huang, N. E., Shen, Z., Long, S. R., Wu, M. G., Hish, H. H., et. al. (1998). The empirical mode decomposition and the Hilbert spectrum for nonlinear & nonstationary time series analysis. Proceedings of the Royal Society A, vol. 454, 903-995.

Huang, N. E., Wu, M. L., Qu, W., Long, S. R., & Shen, S. S. P. (2003). Application of Hilbert-Hunag transform to non-stationary financial time series analysis. Applied Stochastic Models in Business and Industry, vol. 19, 245-268.

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Wei, W. W. S. (2006). Time series analysis: univariate and multivariate methods. Bonston, U.S.A.: Pearson education, Inc.

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以經驗模態分解方法來分析臺灣股票加權指數與世界經濟大國股票指數之相關性及趨勢研究

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