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   -> Volume 5, Issue 8

Preprint: Wavelet detection of jump points
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PostPosted: Wed Oct 02, 1996 10:52 am    
Subject: Preprint: Wavelet detection of jump points
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#5 Preprint: Wavelet detection of jump points


Heung Wong and Wai-cheung Ip, Department of Applied Mathematics,
The Hong Kong Polytechinc University, Hong Kong
Yihui Luan and Zhongjie Xie, Department of Probability and Statistics



In recent years, as a mathematical tool, wavelet analysis has been
introduced to the statistical field and has attracted great attention
(see Oppenheim (1995)). Several authors discussed the jump point
problem in economic data, such as Wang (1995) and Nason (1995). Nason
discussed the jump point detection for the simulation data of
Australian versus US dollars exchange rate by universal thresholding,
Globalsure thresholding and cross- validation methods. Wang showed
interesting results in his paper on the jump points of the monthly
stock market return from 1953 to 1991. In the work of Wang (1995), he
suggested a detecting procedure for the jump point under the condition
of uncorrelated white noise. His detection for the jump points of the
US stock market return was relatively successful, but the great
influence of the Gulf crisis during 1989-1991 did not reported in his
paper. His theo- retical result was based on the uncorrelated white
noise, but actually as we know that the innovation series in financial
area are usually not white noises, namely they have correlated
structures. May be this is somewhat a limitation for the applications
of the Wang's method in practice.


In this paper, we firstly introduce some new statistical results on
detecting the jump points of a discontinuous function in the presence
of noise by wavelet analysis, in which the noise process is assumed to
be a sta- tionary noise and the wavelets are assumed to satisfy mild
mathematical con- ditions. Estimators for the numbers and locations of
jump points are proposed and shown to be strongly consistent.
Then we demonstrate the detection of jump points for the exchange rate of
US dollar versus Deutsche Mark during 1-Aug.-1989 to 31-July-1991 (515
data). The detecting procedures are carried out in two statges,
original observation records and the residuals of the data after
extracting the trend component. The trend decomposition procedure is
also realized by the wavelet analysis.


Different jump points have been detected in each stage and they all
contain very evident political and social meaning. We then introduce
the score-testing for detecting the jump points as outlier
interventions, and the results are compared with those obtained by the
wavelet detection suggested in this paper. Interesting results are
shown. Outlier analysis essentially agrees with our method but it
misses an important jump in the data. Finally, we discuss the
advantages and limitations of our theoretical results.

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