文档介绍：An Introduction to the Wavelet Analysis
of Time Series
Don Percival
Applied Physics Lab, University of Washington, Seattle
Dept. of Statistics, University of Washington, Seattle
MathSoft, Inc., Seattle
Overview
• wavelets are analysis tools mainly for
– time series analysis (focus of this tutorial)
– image analysis (will not cover)
• as a subject, wavelets are
– relatively new (1983 to present)
– synthesis of many new/old ideas
– keyword in 10, 558+ articles & books since 1989
(2000+ in the last year alone)
• broadly speaking, have been two waves of wavelets
– continuous wavelet transform(1983 and on)
– discrete wavelet transform(1988 and on)
1
Game Plan
• introduce subject via CWT
• describe DWT and its main ‘products’
– multiresolution analysis (additive position)
– analysis of variance (‘power’ position)
• describe selected uses for DWT
– wavelet variance (related to Allan variance)
– decorrelation of fractionally differenced processes
(closely related to power law processes)
– signal extraction (denoising)
2
What is a Wavelet?
• wavelet is a ‘small wave’(sinusoids are ‘big waves’)
• real-valued ψ(t) is a wavelet if
∞
1. integral of ψ(t) is zero: −∞ψ(t) dt =0
2 ∞ 2
2. integral of ψ(t) is unity: −∞ψ(t) dt =1
(called ‘unit energy’ property)
• wavelets so defined deserve their name because
–#2 says we have, for every small >0,
T
ψ2(t) dt < 1 −,
−T
for some finite T (might be quite large!)
– length of [−T,T] pare to [−∞, ∞]
–#2 says ψ(t) must be nonzero somewhere
–#1 says ψ(t) balances itself above/below 0
• Fig. 1: three wavelets
• Fig. 2: examples plex-valued wavelets
3
Basics of Wavelet Analysis: I
• wavelets tell us about variations in local averages
• to quantify this description, let x(t) be a ‘signal’
– real-valued function of t
– will refer to t as time (but can be, ., depth)
• consider average value of x(t)over[a, b]:
1 b
x(u) du ≡α(a, b)
b − a a
•