D.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS STATISTICAL FOURIER ANALYSIS

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1 STATISTICAL FOURIER ANALYSIS The Furier Represetati f a Sequece Accrdig t the basic result f Furier aalysis, it is always pssible t apprximate a arbitrary aalytic fucti defied ver a fiite iterval f the real lie, t ay desired degree f accuracy, by a weighted sum f sie ad csie fuctis f harmically icreasig frequecies. The accuracy f apprximati icreases with the umber f fuctis withi the sum. Similar results apply i the case f sequeces, which may be regarded as fuctis mappig frm the set f itegers t the real lie. Fr a sample f T bservatis y,..., y T 1, it is pssible t devise a expressi i the frm (1) y t X j α j cs( j t) + β j si( j t), wherei j πj/t is a multiple f the fudametal frequecy 1 π/t. Thus, the elemets f a fiite sequece ca be expressed exactly i terms f sies ad csies. This expressi is called the Furier decmpsiti f y t ad the set f cefficiets {α j, β j ; j, 1,..., } are called the Furier cefficiets. Whe T is eve, we have T/; ad it fllws that () si( t) si(), cs( t) cs() 1, si( t) si(πt), cs( t) cs(πt) ( 1) t. Therefre, equati (1) becmes 1 X (3) y t α + α j cs( j t) + β j si( j t) + α ( 1) t. j1 Whe T is dd, we have (T 1)/; ad the equati (1) becmes (4) y t α + X j1 α j cs( j t) + β j si( j t). I bth cases, there are T zer cefficiets amgst the set {α j, β j ; j, 1,..., }; ad the mappig frm the sample values t the cefficiets cstitutes a e-t-e ivertible trasfrmati. I equati (3), the frequecies f the trigmetric fuctis rage frm 1 π/t t π; whereas, i equati (4), they rage frm 1 π/t t π(t 1)/T. The frequecy π is the s-called Nyquist frequecy. 1

2 STATISTICAL FOURIER ANALYSIS Althugh the prcess geeratig the data may ctai cmpets f frequecies higher tha the Nyquist frequecy, these will t be detected whe it is sampled regularly at uit itervals f time. I fact, the effects the prcess f cmpets with frequecies i excess f the Nyquist value will be cfuded with thse whse frequecies fall belw it. T demstrate this, csider the case where the prcess ctais a cmpet which is a pure csie wave f uit amplitude ad zer phase whse frequecy lies i the iterval π < < π. Let π. The (5) cs(t) cs (π )t cs(π) cs( t) + si(π) si( t) cs( t); which idicates that ad are bservatially idistiguishable. Here, < π is described as the alias f > π. The Spectral Represetati f a Statiary Prcess By allwig the value f i the expressi (1) t ted t ifiity, it is pssible t express a sequece f idefiite legth i terms f a sum f sie ad csie fuctis. Hwever, i the limit as, the cefficiets α j, β j ted t vaish. Therefre, a alterative represetati i terms f differetials is called fr. By writig α j da( j ), β j db( j ), where A(), B() are step fuctis with disctiuities at the pits { j ; j,..., }, the expressi (1) ca be redered as (6) y t X cs( j t)da( j ) + si( j t)db( j ). j I the limit, as, the summati is replaced by a itegral t give (7) y(t) cs(t)da() + si(t)db(). Here, cs(t) ad si(t), ad therefre y(t), may be regarded as ifiite sequeces defied ver the etire set f psitive ad egative itegers. Sice A() ad B() are disctiuus fuctis fr which derivatives exist, e must avid usig α()d ad β()d i place f da() ad db(). Mrever, the itegral i equati (7) is a Furier Stieltjes itegral. I rder t derive a statistical thery fr the prcess that geerates y(t), e must make sme assumptis ccerig the fuctis A() ad B(). S far, the sequece y(t) has bee iterpreted as a realisati f a stchastic prcess. If y(t) is regarded as the stchastic prcess itself, the the fuctis A(), B() must, likewise, be regarded as stchastic prcesses defied ver

3 the iterval [, π]. A sigle realisati f these prcesses w crrespds t a sigle realisati f the prcess y(t). The first assumpti t be made is that the fuctis da() ad db() represet a pair f stchastic prcesses f zer mea which are idexed the ctiuus parameter. Thus (8) E da() E db(). The secd ad third assumptis are that the tw prcesses are mutually ucrrelated ad that -verlappig icremets withi each prcess are ucrrelated. Thus (9) E da()db(λ) fr all, λ, E da()da(λ) if 6 λ, E db()db(λ) if 6 λ. The fial assumpti is that the variace f the icremets is give by (1) V da() V db() df () f()d. We ca see that, ulike A() ad B(), F () is a ctiuus differetiable fucti. The fucti F () ad its derivative f() are the spectral distributi fucti ad the spectral desity fucti, respectively. I rder t express equati (7) i terms f cmplex expetials, we may defie a pair f cjugate cmplex stchastic prcesses: (11) d() 1 da() idb(), d () 1 da() + idb(). Als, we may exted the dmai f the fuctis A(), B() frm [, π] t [, π] by regardig A() as a eve fucti such that A( ) A() ad by regardig B() as a dd fucti such that B( ) B(). The, there is (1) d () d( ). Frm cditis uder (9), it fllws that (13) E d()d (λ) if 6 λ, E{d()d ()} f()d. 3

4 STATISTICAL FOURIER ANALYSIS These results may be used t reexpress equati (7) as (14) y(t) Ω (e it + e it ) da() i (eit e it ) db() Ω it {da() idb()} it {da() + idb()} e + e Ω e it d() + e it d (). Whe the itegral is exteded ver the rage [, π], this becmes (15) y(t) e it d(). This is cmmly described as the spectral represetati f the prcess y(t). The Autcvariaces ad the Spectral Desity Fucti The sequece f the autcvariaces f the prcess y(t) may be expressed i terms f the spectrum f the prcess. Frm equati (15), it fllws that the autcvariace y t at lag τ t k is give by (16) Ω γ τ C(y t, y k ) E λ e it d() e iλk d( λ) λ e it e iλk E{d()d (λ)} e iτ E{d()d ()} e iτ f()d. Here the fial equalities are derived by usig the results (1) ad (13). This equati idicates that the Furier trasfrm f the spectrum is the autcvariace fucti. The iverse mappig frm the autcvariaces t the spectrum is give by (17) f() 1 π τ 1 γ + π γ τ e iτ τ1 γ τ cs(τ). This fucti is directly cmparable t the peridgram f a data sequece. Hwever, the peridgram has T empirical autcvariaces c,..., c T 1 i place 4

5 f a idefiite umber f theretical autcvariaces. Als, it differs frm the spectrum by a scalar factr f 4π. I may texts, equati (17) serves as the primary defiiti f the spectrum. T demstrate the relatiship which exists betwee equatis (16) ad (17), we may substitute the latter it the frmer t give (18) γ τ 1 π e iτ 1 π κ γ κ τ γ τ e iτ d e i(τ κ) d. Frm the fact that Ω π, if κ τ; (19) e i(τ κ) d, if κ 6 τ, it ca be see that the RHS f the equati reduces t γ τ. This serves t shw that equatis (16) ad (17) d ideed represet a Furier trasfrm ad its iverse. The essetial iterpretati f the spectral desity fucti is idicated by the equati () γ f()d, which cmes frm settig τ i equati (16). This equati shws hw the variace r pwer f y(t), which is γ, is attributed t the cyclical cmpets f which the prcess is cmpsed. It is easy t see that a flat spectrum crrespds t the autcvariace fucti which characterises a white-ise prcess ε(t). Let f ε f ε () be the flat spectrum. The, frm equati (16), it fllws that (1) γ πf ε, f ε ()d ad, frm equati (16), it als fllws that () γ τ f ε. f ε ()e iτ d e iτ d These are the same as the cditis that serve t defie a white-ise prcess. Whe the variace is deted by σ ε, the expressi fr the spectrum f the white-ise prcess becmes (3) f ε () σ ε π. 5

The Excel FFT Function v1.1 P. T. Debevec February 12, The discrete Fourier transform may be used to identify periodic structures in time ht.

The Excel FFT Function v1.1 P. T. Debevec February 12, The discrete Fourier transform may be used to identify periodic structures in time ht. The Excel FFT Fucti v P T Debevec February 2, 26 The discrete Furier trasfrm may be used t idetify peridic structures i time ht series data Suppse that a physical prcess is represeted by the fucti f time,