Imagiary uit: i ( i Complx umbr: z x+ i y Cartsia coordiats: x (ral part y (imagiary part Complx cougat: z x i y Absolut valu: r z x + y Polar coordiats: r (absolut valu or modulus ω (argumt or phas x cos(ω r x r cos(ω, si(ω y r si(ω si( ω y y x ta( ω ω ata( cos( ω x r y z z ( x+ i y( x i y x i y x + y z aalogous if z ot i st quadrat,.g., x<0, y>0 ω ata( y/x
Exrcis: Show that th st C of -dimsioal complx vctors togthr with vctor additio dfid by z w z + w z+w M + M M z w z + w is a commutativ group. FG Exrcis: Show that th st C togthr with vctor additio dfid as abov ad scalar multiplicatio dfid by z λz λ zλ M M z λz is a complx vctor spac, i.., λ(µ z(λ µz, zz, λ(z+w(λ z+(λ w, (λ+µz(λ z+(µ z. FV Exrcis: Show that th ir product of two complx vctors z ad w dfid by z z,w w*z ( w,..., w M z t w t, z satisfis (i v + w, z v, z + w, z, (ii (iii (iv (v (vi (vii z, v+ w z, v + z, w, λ z, w λ z, w, v, λw λ v, w, z, w w, z, z, z 0, z, z 0 z0. FI Two complx vctors z ad w ar said to b orthogoal, if z, w 0.
Exrcis: Show that th orm of a complx vctor z dfid by satisfis z (i z, z zt z t (ii λ z λ z, z t z+ w z + w. FN Hit: Us th Cauchy-Schwarz Iquality z, w z w. Exrcis: Prov th Pythagora thorm z, w 0 z+ w z + w. FP Siusoid: g ( t R si( ωt + φ Paramtrs: R (amplitud ω (frqucy φ (phas A siusoid is priodic with priod p bcaus g ω ( t+ R si( ω( t+ + φ ω ω R si( ωt+φ + R si( ωt+ φ g(t. For fixd, th frqucis ω, 0,, ] ar calld Fourir frqucis. [ ω implis a priod of p. ω p. ω ω implis a priod of M 3
Exrcis: Us th Eulr rlatio FE to show that i ω + i ( ω+ cos( ω i si( ω iω, iω iω i(ω ω,. Th vctors FB iω M, 0,, iω costitut a orthoormal basis for C bcaus,, iω t iω t i( ω ω 0 if. iωt iωt - 0 ( iω t iωt iω t iωt 0, ( i(ω ω t i( ω ω i(ω ω t i(ω -ω i(ω -ω Thus, ay x C has a rprstatio of th form FS 0 x λ. Taig ir products of ach sid w obtai x,, 0λ 0λ, λ, λ, λ x, x x x * x, 0 0 λλ t iωt x,, 0 λ λ λ. 0 Exrcis: Show that ( x t x λ. FF 0 Not: For a giv tim sris x,,x obsrvd at tims,, or tim itrvals (0,,,(,, th siz of λ thrfor idicats how much of th sampl variac ca b attributd to frqucy ω. 4
Suppos that x R. If <, th AZ iω-t ( i t i ( t i t iωt iωt iω - t iω t iω t xt xt x t λ, λ iω t λ + λ iω t λ iω t + λ iω t ( a + ib (cos( ω t i si( ω t + + ( a ib (cos( ω t i si( ω t a cos( ω t b si( ω t R si( φ cos( ω t + R R si( ω t+ φ, cos( φ si( ω t whr R ad φ ar th polar coordiats of b +a i, i.., a si( φ, b cos( φ. R si(α+βsi(αcos(β+cos(αsi(β R, If, th ω iω t it, cos( t + si( t si( t+, 3 t λ xt cos(t xt (- λ iω t R si( t R t+ R, 0 cos( R si( ωt+ φ. If 0, th 0 iω ω 0, t i 0 t, λ λ iω t x. xt xt x R, Thus, [ / ] iωt xt λ0 + λ x+ R si( ωt + φ. 5
Th priodogram of x,,x is dfid by For <, I(ω λ I(ω iωt x t. ( a + b ((a + ( b R 8 8. It will b show latr that for ay ozro Fourir frqucy ω, I(ω ca b writt as whr I(ω ˆ γ ( -iω ( γ( ˆ, ( xt x( xt+ x is th sampl autocovariac at lag. If th obsrvatios x,,x com from a statioary procss x, th priodogram I(ω may b rgardd as a sampl aalogu of th fuctio f(ω γ( -iω which is calld th spctral dsity of th procss x. Th statioarity of th procss x implis that all x t hav th sam ma ad th sam variac ad th autocovariacs γ(cov(x t,x t- dpd oly o but ot o t., 6
Exrcis: Show that - iω dω 0 if 0. A0 Assumig that th itrchag of summatio ad itgratio is ustifid w ca driv th spctral rprstatio of th autocovariac fuctio γ of a statioary procss x with spctral dsity f as follows: iω f(ω dω - - iω γ( γ( γ( -iω γ( - - iω( iω( dω dω dω Rmar: Lt If th f (ω γ( cos(ω. g(ω γ( <, - g(ω dω < ad, by th domiat covrgc thorm, bcaus - lim - lim f (ω dω f (ω dω, (ω f γ(cos(ω γ( g(ω. AR 7
Rmar: It follows from f(ω ad γ( - γ( -iω iω f(ω dω that th spctral dsity f ad th autocovariac fuctio γ cotai th idtical iformatio. Giv obsrvatios x,,x, th priodogram I(ω ˆ ( γ( -iω is a vry rratic stimator for th spctral dsity, bcaus th sampl autocovariacs ˆ γ ( (xt x(xt+ x cotai vry fw products ( x x( x + x if is larg. t t A obvious improvmt is to giv lss wight to th mor variabl sampl autocovariacs. A stimator of th typ f ˆ ( ω ˆ ( w γ( -iω is calld a wightd covariac stimator. A widly usd stimator is th Bartltt stimator which uss th triagl wights M if < M, w 0 ls. Th trucatio poit M is a importat paramtr for cotrollig th smoothss of th stimator. A altrativ mthod of smoothig th priodogram is to ta wightd avrags ovr ighborig frqucis. A widly usd smoothd priodogram stimator is th modifid Daill smoothr, which diffrs from a simpl movig avrag of th priodogram oly i that th first ad th last wight ar oly half as larg as th othrs. 8
Exrcis: Spctral aalysis of th postwar US GDP Crat a worig dirctory, say C:\GDPq, for th aalysis of th quartrly US GDP. Dowload th ral Gross Domstic Product (quartrly, sasoally adustd as a csv fil from th wbsit of th Fdral Rsrv Ba of St. Louis ito your worig dirctory. Th dowloadd fil GDPC.csv cosists of two colums (dats ad GDP valus. Import th data ito R ad plot th GDP, th log GDP, th diffrcd log GDP, ad th priodogram of th diffrcs. stwd("c:/gdpq" # commt: st worig dirctory D <- rad.csv("gdpc.csv" # import data d <- D[,] # st colum of D: dats v <-D[,] # d colum of D: GDP valus d <- as.dat(d # covrt charactr strigs to dats N <- lgth(v # N o. of quartrs lgth of vctor v par(mfrowc(, # subsqut plots i x array par(marc(,,, # st arrow margis for plots plot(d,v,pch0 # plot GDP valus agaist dats y <- log(v; plot(d,y,pch0 # plot charactr: solid circl r <- y[:n]-y[:(n-]; <- N- # o. of diffrcs plot(d[:n],r,pch0,typ"o" # ovrplot poits&lis h <- spc.pgram(r,tapr0,dtrdf,fasf,plof c <- *pi; f <- c*h$frq # Fourir fr. btw 0 ad pi pg <- h$spc/c; plot(f,pg,typ"o",pch0 # priodogr. 9
Smooth th priodogram with th modifid Daill smoothr. par(mfrowc(, # sigl plot plot(f,pg,typ"l" # oly lis, o poits h <- spc.pgram(r,tapr0,dtrdf,fasf,plof, spas3 lis(c*h$frq,h$spc/c,col"gr",lwd # add li to xistig plot with li width twic as wid h <- spc.pgram(r,tapr0,dtrdf,fasf,plof, spas0 lis(c*h$frq,h$spc/c,col"rd",lwd Th highr th spa (th total umbr of trms i th movig avrag, th smoothr th stimat. 0