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Understanding the Fourier Transform and Its Application in Time Series Analysis

This article delves into the fundamentals of the Fourier Transform, highlighting its significance in time series analysis. It provides insight into regularity conditions, convolution, discrete Fourier Transforms, and the use of power spectral density in analyzing stationary processes. The discussion extends to cross-covariance, coherency, and coherence among different processes, showcasing how these concepts can be employed to understand associations and dependencies in bivariate and trivariate time series data. Practical examples demonstrate the application and relevance of these mathematical concepts.

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Understanding the Fourier Transform and Its Application in Time Series Analysis

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  1. FT makes the New Yorker, October 4, 2010 page 71

  2. The Fourier transform.  regularity conditions Functions, A(), - <  <   |A()|d finite FT. a(t) =  exp{it)A()d - <  <  Inverse A() =(2)-1 exp{-it} a(t) dt unique C()=  A() +  B() c(t) =  c(t) +  b(t) 2  1

  3. Convolution (filtering). d(t) =  b(t-s) c( s)ds D() = B()C() Discrete FT. a(t) = T-1 exp{i2ts/T} A(2s/T) s, t = 0,1,...,T-1 A(2s/T) =  exp {-i2st/T) a(t) FFTs exist

  4. Dirac delta.  g() () d = g(0)  exp {it}() d  = 1 inverse () = (2)-1 exp {-it}dt Heavyside function H() = signum () () = dH()/d

  5. Mixing. Stationary case unless otherwise indicated cov{dN(t+u),dN(t)} small for large |u| |pNN(u) - pNpN| small for large |u| hNN(u) = pNN(u)/pN ~ pN for large |u| qNN(u) = pNN(u) - pNpN u  0  |qNN(u)|du <  cov{dN(t+u),dN(t)}= [(u)pN + qNN(u)]dtdu

  6. Power spectral density. frequency-side, , vs. time-side, t /2 : frequency (cycles/unit time) fNN() = (2)-1 exp{-iu}cov{dN(t+u),dN(t)}/dt = (2)-1 exp{-iu}[(u)pN+qNN(u)]du = (2)-1pN + (2)-1  exp{-iu}qNN(u)]du Non-negative, symmetric Approach unifies analyses of processes of widely varying types

  7. Examples.

  8. Spectral representation. stationary increments - Kolmogorov

  9. Filtering. dN(t)/dt =  a(t-v)dM(v) =  a(t-j ) =  exp{it}A()dZM() with a(t) = (2)-1  exp{it}A()d dZN() = A() dZM() fNN() = |A()|2 fMM()

  10. Association. Measuring? Due to chance? Are two processes associated? Eg. t.s. and p.p. How strongly? Can one predict one from the other? Some characteristics of dependence: E(XY)  E(X) E(Y) E(Y|X) = g(X) X = g (), Y = h(),  r.v. f (x,y)  f (x) f(y) corr(X,Y)  0

  11. Bivariate point process case. Two types of points (j ,k) Crossintensity. a rate Prob{dN(t)=1|dM(s)=1} =(pMN(t,s)/pM(s))dt Cross-covariance density. cov{dM(s),dN(t)} = qMN(s,t)dsdt no () often

  12. Spectral representation approach.

  13. Frequency domain approach. Coherency, coherence Cross-spectrum. Coherency. R MN() = f MN()/{f MM() f NN()} complex-valued, 0 if denominator 0 Coherence |R MN()|2 = |f MN()| 2 /{f MM() f NN()| |R MN()|2 1, c.p. multiple R2

  14. Proof. Filtering. M = {j }  a(t-v)dM(v) =  a(t-j ) Consider dO(t) = dN(t) -  a(t-v)dM(v)dt, (stationary increments) where A() =  exp{-iu}a(u)du fOO () is a minimum at A() = fNM()fMM()-1 Minimum: (1 - |RMN()|2 )fNN() 0  |R MN()|2 1

  15. Proof. Coherence, measure of the linear time invariant association of the components of a stationary bivariate process.

  16. Regression analysis/system identification. dZN() = A() dZM() + error() A() =  exp{-iu}a(u)du

  17. Empirical examples. sea hare

  18. Mississippi river flow

  19. Partial coherency. Trivariate process {M,N,O} “Removes” the linear time invariant effects of O from M and N

  20. Time series variants. details later continuous time case Mixing. cov{Y(t+u),Y(t)} = cYY(u) small for large |u|  |cYY(u)|du < 

  21. Power spectral density. frequency-side, , vs. time-side, t /2 : frequency (cycles/unit time) fYY() = (2)-1 exp{-iu}cov{Y(t+u),Y(t)} = (2)-1 exp{-iu}cYY(u)du -<< Non-negative, symmetric Approach unifies analyses of processes of widely varying types Things in the frequency domain look the same

  22. Spectral representation. Y(t) = exp{it}dZY() - < t <  ZY() random, complex-valued conj{ZY()} = ZY(-) E{dZY()} = ()cNd cov{dZY(),dZY()}=(-)f NN()dd cum{dZY(1),...,dZY(K)} = ...

  23. Filtering. Yt) =  a(t-v)X(v)dv =  exp{it}A()dZX() with a(t) = (2)-1  exp{it}A()d dZY() = A() dZX() fYY() = |A()|2 fXX()

  24. Bivariate time series case. (X(t),Y(t)) - < t <  Cross-covariance function. general case cov{X(s),Y(t)} = cXY(s,t)

  25. Spectral representation approach. FXY(.): cross-spectral measure

  26. Frequency domain approach. Coherency, coherence Cross-spectrum. f XY()= (2)-1 exp{-iu)c XY(u)du -< < complex-valued Coherency. R XY() = f XY()/{f XX() f YY()} 0 if denominator 0 Coherence. |RXY()|2 = |f XY()| 2 /{fXX() fYY()| |RXY()|2 1, c.p. multiple R2

  27. Regression analysis/system identification. dZY() = A() dZX() + error() A() =  exp{-iu}a(u)du

  28. Partial coherency. Trivariate process {X,Y,O} “Removes” the linear time invariant effects of O from X and Y

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