Time Series Analysis and Its Applications: With R Examples

Second Edition

HOME CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 CHAPTER 5

R Code Used in the Text  (Chapters 1-5)

Everything you see in a box below is R code. You can copy-and-paste any line (or multiple lines) into R. Make sure the data files are in the mydata directory (or change the code accordingly). R code for Chapter 6 has its own page and we also have some code for Chapter 7. We have improved, corrected or streamlined some of the code below, so there are a few cases where the code here differs slightly from what is in the text.

Disclaimer: Sometimes I use t for time ... e.g., t = 1984:2001. This is "bad R practice" because t is used for matrix transpose, i.e., t(A) is the transpose of matrix A. If you don't want to be bad, you can replace my t in the code below with something else. I like to be bad sometimes, but you've been warned ... so don't blame me if you turn to a life of crime.

CHAPTER 1

Example 1.1

jj = scan(“c:/data/jj.dat") # yes forward slash jj = ts(jj, start=1960, frequency=4) plot(jj, ylab="Quarterly Earnings per Share")

Example 1.6

fmri = read.table(“c:/data/fmri.dat") par(mfrow=c(2,1)) # sets up the graphics - ?par for info on mfrow()

ts.plot(fmri[,2:5], lty=c(1,4), ylab="BOLD") ts.plot(fmri[,6:9], lty=c(1,4), ylab="BOLD")

Example 1.7

x = matrix(scan(“c:/data/eq5exp6.dat"), ncol=2) par(mfrow=c(2,1)) plot.ts(x[,1], main="Earthquake", ylab="EQ5") plot.ts(x[,2], main="Explosion", ylab="EXP6")

Example 1.9

w = rnorm(500,0,1) # 500 N(0,1) variates v = filter(w, sides=2, rep(1,3)/3) # moving average par(mfrow=c(2,1)) plot.ts(w, main="white noise") plot.ts(v, main="moving average")# now try this: ts.plot(w, v, lty=2:1, col=1:2, lwd=1:2)

Example 1.10

w = rnorm(550,0,1) # 50 extra to avoid startup problems - the [-(1:50)] on the next line... x = filter(w, filter=c(1,-.9), method="recursive")[-(1:50)] #... removes the first 50 values plot.ts(x)

Example 1.11

set.seed(154) w = rnorm(200,0,1) x = cumsum(w) wd = w +.2 xd = cumsum(wd) plot.ts(xd, ylim=c(-5,55)) lines(x) lines(.2*(1:200), lty="dashed")

Example 1.12

t = 1:500 c = 2*cos(2*pi*t/50 + .6*pi) w = rnorm(500,0,1) par(mfrow=c(3,1)) plot.ts(c) plot.ts(c + w) plot.ts(c + 5*w)

Example 1.24

speech = scan(“c:/data/speech.dat") acf(speech,250)

Example 1.25

soi = ts(scan(“c:/data/soi.dat"), start=1950, frequency=12) rec = ts(scan(“c:/data/recruit.dat"), start=1950, frequency=12) par(mfrow=c(3,1)) acf(soi, 50) acf(rec, 50) ccf(soi, rec, 50)

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CHAPTER 2

Example 2.1

globtemp = ts(scan(“c:/data/globtemp.dat"), start=1856) gtemp = window(globtemp, start=1900) # use years 1990 to 1997 fit = lm(gtemp~time(gtemp), na.action=NULL) # regress gtemp on time#-- note: na.action=NULL is used to retain time series attributes - ?lm for info summary(fit) # view the results

plot(gtemp, type="o", ylab="Global Temperature Deviation") # plot the series abline(fit) # add regression line to the plot

Example 2.2

mort = ts(scan(“c:/data/cmort.dat"),start=1970, frequency=52) temp = ts(scan(“c:/data/temp.dat"), start=1970, frequency=52) part = ts(scan(“c:/data/part.dat"), start=1970, frequency=52) par(mfrow=c(3,1), mar=c(3,4,3,2)) # plot data - ?par for info on mar() plot(mort, main="Cardiovascular Mortality", xlab="", ylab="") plot(temp, main="Temperature", xlab="", ylab="") plot(part, main="Particulates", xlab="", ylab="") x11() # open another graphic device pairs(cbind(mort, temp, part)) # scatterplot matrix temp = temp-mean(temp) temp2 = temp^2 trend = time(mort) # time fit = lm(mort~ trend + temp + temp2 + part, na.action=NULL) summary(fit) # regression results anova(lm(mort~cbind(trend, temp, temp2, part))) # pretty ANOVA table num = length(mort) # sample size AIC(fit)/num - log(2*pi) # AIC (corresponds to def 2.1) AIC(fit, k=log(num))/num - log(2*pi) # BIC (corresponds to def 2.3) (AICc = log(sum(resid(fit)^2)/num)+(num+5)/(num-5-2)) # AICc (as in def 2.2)

Examples 2.3 and 2.4

globtemp = ts(scan(“c:/data/globtemp.dat"), start=1856) gtemp = window(globtemp, start=1900) # use years 1990 to 1997 fit = lm(gtemp~time(gtemp), na.action=NULL) # regress gtemp on time par(mfrow=c(2,1)) plot(resid(fit), type="o", main="detrended") plot(diff(gtemp), type="o", main="first difference") # this and below will do example 2.4 x11() # open a new graphic device par(mfrow=c(3,1)) # plot ACFs acf(gtemp, 48) acf(resid(fit), 48) acf(diff(gtemp), 48)

Example 2.6You can download a function called lag.plot2 that will do the second part of this example over here.

soi = ts(scan(“c:/data/soi.dat"), start=1950, frequency=12) rec = ts(scan(“c:/data/recruit.dat"), start=1950, frequency=12) lag.plot(soi, lags=12, layout=c(3,4), diag=F)

par(mfrow=c(3,3), mar=c(2.5, 4, 4, 1)) # set up plot area for(h in 0:8){ # loop through lags 0-8 plot(lag(soi,-h),rec, main=paste("soi(t-",h,")",sep=""), ylab="rec(t)",xlab="") }

Example 2.8

t = 1:500 x = 2*cos(2*pi*t/50 + .6*pi) + rnorm(500,0,5) I = abs(fft(x)/sqrt(500))^2 # the periodogram P = (4/500)*I # the scaled periodogram f = 0:250/500

plot(f, P[1:251], type="l", xlab="frequency", ylab=" ") abline(v=seq(0,.5,.02), lty="dotted")

Example 2.10

mort = ts(scan(“c:/data/cmort.dat"),start=1970, frequency=52) ma5 = filter(mort, sides=2, rep(1,5)/5) ma53 = filter(mort, sides=2, rep(1,53)/53) plot(mort, type="p", xlab="week", ylab="mortality") lines(ma5) lines(ma53)

Example 2.11

wk = time(mort) - mean(time(mort)) # week (centered) - it's basically t/52 ... wk2 = wk^2 # ... because frequency=52 for mort wk3 = wk^3 c = cos(2*pi*wk) s = sin(2*pi*wk) reg1 = lm(mort~wk + wk2 + wk3, na.action=NULL) reg2 = lm(mort~wk + wk2 + wk3 + c + s, na.action=NULL) plot(mort, type="p", ylab="mortality") lines(fitted(reg1)) lines(fitted(reg2)) # here's the original code where mort isn't a ts object mort = scan(“c:/data/cmort.dat") t = 1:length(mort) t2 = t^2 t3 = t^3 c = cos(2*pi*t/52) s = sin(2*pi*t/52) fit1 = lm(mort~t + t2 + t3) fit2 = lm(mort~t + t2 + t3 + c + s) plot(t, mort)

lines(fit1$fit) lines(fit2$fit)

Example 2.12

plot(mort, type="p", ylab="mortality") lines(ksmooth(time(mort), mort, "normal", bandwidth=10/52)) # or try bandwidth=5/52 lines(ksmooth(time(mort), mort, "normal", bandwidth=2))

# - original code when mort wasn't a time series with frequency=52 plot(t, mort) lines(ksmooth(t, mort, "normal", bandwidth=10)) lines(ksmooth(t, mort, "normal", bandwidth=104)) # Red indicates a change; Fig. 2.15 shows a bandwidth of 10.

Example 2.13

par(mfrow=c(2,1)) plot(mort,type="p", ylab="mortality", main="nearest neighbor") lines(supsmu(time(mort), mort, span=.5)) lines(supsmu(time(mort), mort, span=.01)) plot(mort, type="p", ylab="mortality", main="lowess") lines(lowess(mort, f=.02)) lines(lowess(mort, f=2/3))

Example 2.14

plot(mort, type="p", ylab="mortality") lines(smooth.spline(time(mort), mort)) lines(smooth.spline(time(mort), mort, spar=1))

# - original code when mort wasn't a time series with frequency=52 plot(t, mort)

lines(smooth.spline(t, mort, spar=.0000001)) lines(smooth.spline(t, mort, spar=.1))

Example 2.15

par(mfrow=c(2,1)) plot(temp, mort, main="lowess") lines(lowess(temp,mort)) plot(temp, mort, main="smoothing splines") lines(smooth.spline(temp,mort))

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CHAPTER 3

Example 3.1

par(mfrow=c(2,1)) plot(arima.sim(list(order=c(1,0,0), ar=.9), n=100), ylab="x",main=(expression(AR(1)~~~phi==+.9))) # ~ is a space and == is equal plot(arima.sim(list(order=c(1,0,0), ar=-.9), n=100), ylab="x",main=(expression(AR(1)~~~phi==-.9)))

Example 3.3

par(mfrow=c(2,1)) plot(arima.sim(list(order=c(0,0,1), ma=.5), n=100), ylab="x",main=(expression(MA(1)~~~theta==+.5))) plot(arima.sim(list(order=c(0,0,1), ma=-.5), n=100), ylab="x",main=(expression(MA(1)~~~theta==-.5)))

Example 3.9

set.seed(5) ar2 = arima.sim(list(order=c(2,0,0), ar=c(1.5,-.75)), n = 144) t = (1:144)/12 plot(t, ar2, type="l", xlab="Time (one unit = 12 points)")

abline(v=0:12, lty="dotted")

Example 3.10

ARMAtoMA(ar=.9, ma=.5, 50) # for a list plot(ARMAtoMA(ar=.9, ma=.5, 50)) # for a graph

Example 3.14

ar2.acf = ARMAacf(ar=c(1.5,-.75), ma=0, 24) ar2.pacf = ARMAacf(ar=c(1.5,-.75), ma=0, 24, pacf=T) par(mfrow=c(1,2)) plot(ar2.acf, type="h", xlab="lag") abline(h=0) plot(ar2.pacf, type="h", xlab="lag") abline(h=0)

Example 3.16

rec = ts(scan("c:/data/recruit.dat"), start=1950, frequency=12) par(mfrow=c(2,1)); acf(rec, 48); pacf(rec, 48) (fit = ar.ols(rec,order=2,demean=F,intercept=T)) # estimates fit$asy.se # standard errors

Example 3.26

rec.yw = ar.yw(rec, order=2) rec.yw$x.mean rec.yw$ar sqrt(diag(rec.yw$asy.var.coef)) rec.yw$var.pred rec.pr = predict(rec.yw, n.ahead=24) U = rec.pr$pred + rec.pr$se L = rec.pr$pred - rec.pr$se minx = min(rec,L) maxx = max(rec,U)

ts.plot(rec, fore$pred, xlim=c(1980,1990), ylim=c(minx,maxx)) lines(rec.pr$pred, col="red", type="o") lines(U, col="blue", lty="dashed") lines(L, col="blue", lty="dashed")

Example 3.29

rec.mle = ar.mle(rec, order=2) rec.mle$x.mean rec.mle$ar sqrt(diag(rec.mle$asy.var.coef)) rec.mle$var.pred

Example 3.33

x = scan(“c:/data/ar1boot.dat") # see note below m = mean(x) # estimate of mu fit = ar.yw(x, order=1) phi = fit$ar # estimate of phi nboot = 200 # number of bootstrap replicates resids = fit$resid resids = resids[2:100] # the first resid is NA x.star = x # initialize x.star phi.star = matrix(0, nboot, 1) for (i in 1:nboot) { resid.star = sample(resids, replace=TRUE) for (t in 1:99){ x.star[t+1] = m + phi*(x.star[t]-m) + resid.star[t] } phi.star[i] = ar.yw(x.star, order=1)$ar }

# Red indicates a change; see the errata. # If you want to simulate your own data, use the following commands: e = rexp(150, rate = .5)

u = runif(150,-1,1) de = e*sign(u) x = 50 + arima.sim(n = 100, list(ar =.95), innov = de, n.start = 50)

Example 3.35

gnp96 = read.table(“c:/data/gnp96.dat") gnp = ts(gnp96[,2], start=1947, frequency=4) plot(gnp) acf(gnp, 50) gnpgr = diff(log(gnp)) # growth rate plot.ts(gnpgr) par(mfrow=c(2,1)) acf(gnpgr, 24) pacf(gnpgr, 24)# Fit ARIMA and view results: (gnpgr.ar = arima(gnpgr, order = c(1, 0, 0))) # potential problem here (see R Issue 1) (gnpgr.ma = arima(gnpgr, order = c(0, 0, 2))) ARMAtoMA(ar=.35, ma=0, 10) # prints psi-weights

Example 3.36

tsdiag(gnpgr.ma, gof.lag=20)

Example 3.37

varve = scan(“c:/data/varve.dat") (varve.ma = arima(log(varve), order = c(0, 1, 1))) tsdiag(varve.ma) (varve.arma = arima(log(varve), order = c(1, 1, 1))) tsdiag(varve.arma, gof.lag=20)

Example 3.39

n = length(gnpgr) # sample size

kma = length(gnpgr.ma$coef) # number of parameters in ma model sma=gnpgr.ma$sigma2 # mle of sigma^2 kar = length(gnpgr.ar$coef) # number of parameters in ar model sar=gnpgr.ar$sigma2 # mle of sigma^2 # AIC Returned Value log(sma) + (n+2*kma)/n # MA2 -8.298 log(sar) + (n+2*kar)/n # AR1 -8.294 # AICc log(sma) + (n+kma)/(n-kma-2) # MA2 -8.288 log(sar) + (n+kar)/(n-kar-2) # AR1 -8.285 # BIC log(sma) + kma*log(n)/n # MA2 -9.252 log(sar) + kar*log(n)/n # AR1 -9.264

## Note: You can use sarima.R to get the above ## (remember to source the code first): sarima(gnpgr,0,0,2) # for the MA(2) sarima(gnpgr,1,0,0) # for the AR(1)

Example 3.41

phi = c(rep(0,11),.8) acf = ARMAacf(ar=phi, ma=-.5, 50) pacf = ARMAacf(ar=phi, ma=-.5, 50, pacf=T) par(mfrow=c(1,2)) plot(acf, type="h", xlab="lag") abline(h=0) plot(pacf, type="h", xlab="lag") abline(h=0)

Example 3.43

prod=ts(scan(“c:/data/prod.dat"), start=1948, frequency=12) par(mfrow=c(2,1)) # (P)ACF of data acf(prod, 48)

pacf(prod, 48) par(mfrow=c(2,1)) # (P)ACF of d1 data acf(diff(prod), 48) pacf(diff(prod), 48) par(mfrow=c(2,1)) # (P)ACF of d1-d12 data acf(diff(diff(prod),12), 48) pacf(diff(diff(prod),12), 48)### fit model (iii) prod.fit3 = arima(prod, order=c(1,1,1), seasonal=list(order=c(2,1,1), period=12)) prod.fit3 # to view the results tsdiag(prod.fit3, gof.lag=48) # diagnostics ### forecasts for the final model prod.pr = predict(prod.fit3, n.ahead=12) U = prod.pr$pred + 2*prod.pr$se L = prod.pr$pred - 2*prod.pr$se ts.plot(prod,prodpr$pred, col=1:2, type="o", ylim=c(105,175), xlim=c(1975,1980)) lines(U, col="blue", lty="dashed") lines(L, col="blue", lty="dashed")

## Note: This can be done using acf2.R, sarima.R ## and sarima.for.R (over here) as follows ## (remember to source the code first): prod=scan(“c:/data/prod.dat") acf2(prod,48) acf2(diff(prod), 48) acf2(diff(diff(prod),12), 48)### fit model (iii) sarima(prod,1,1,1,2,1,1,12)### forecasts for the final model sarima.for(prod,12,1,1,1,2,1,1,12)#### -- See the note for this example in the errata.

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CHAPTER 4

Example 4.1

t = 1:100 x1 = 2*cos(2*pi*t*6/100) + 3*sin(2*pi*t*6/100) x2 = 4*cos(2*pi*t*10/100) + 5*sin(2*pi*t*10/100) x3 = 6*cos(2*pi*t*40/100) + 7*sin(2*pi*t*40/100) x = x1 + x2 + x3 par(mfrow=c(2,2)) plot.ts(x1, ylim=c(-10,10), main = expression(omega==6/100~~~A^2==13)) plot.ts(x2, ylim=c(-10,10), main = expression(omega==10/100~~~A^2==41)) plot.ts(x3, ylim=c(-10,10), main = expression(omega==40/100~~~A^2==85)) plot.ts(x, ylim=c(-16,16),main="sum")

Example 4.2

P = abs(2*fft(x)/100)^2 f = 0:50/100 plot(f, P[1:51], type="o", xlab="frequency", ylab="periodogram")

Example 4.7

x = c(1,2,3,2,1) c1 = cos(2*pi*1:5*1/5) s1 = sin(2*pi*1:5*1/5) c2 = cos(2*pi*1:5*2/5) s2 = sin(2*pi*1:5*2/5) omega1 = cbind(c1, s1) omega2 = cbind(c2, s2) anova(lm(x~omega1+omega2)) # ANOVA Table abs(fft(x))^2/5 # the periodogram (as a check)

Example 4.9

soi = scan(“c:/data/soi.dat") rec = scan(“c:/data/recruit.dat") par(mfrow=c(2,1)) soi.per = spec.pgram(soi, taper=0, log="no") abline(v=1/12, lty="dotted") abline(v=1/48, lty="dotted") rec.per = spec.pgram(rec, taper=0, log="no") abline(v=1/12, lty="dotted") abline(v=1/48, lty="dotted") soi.per$spec[40] # soi pgram at freq 1/12 = 40/480 soi.per$spec[10] # soi pgram at freq 1/48 = 10/480# -- conf intervals -- # returned value: U = qchisq(.025,2) # 0.05063562 L = qchisq(.975,2) # 7.377759 2*soi.per$spec[10]/L # 0.1747835 2*soi.per$spec[10]/U # 25.46648 2*soi.per$spec[40]/L # 3.162688 2*soi.per$spec[40]/U # 460.813# -- replace soi with rec above to get recruit values

##########################################################################################- Repeat example using "frequency=12" for monthly data and note the differences -######- basically, "everything" is in multiples of Δ = 1/12 -########################################################################################## soi=ts(scan(“c:/data/soi.dat"), start=1950, frequency=12) rec=ts(scan(“c:/data/recruit.dat"), start=1950, frequency=12) par(mfrow=c(2,1)) soi.per = spec.pgram(soi, taper=0, log="no") abline(v=1, lty="dotted") abline(v=1/4, lty="dotted") rec.per = spec.pgram(rec, taper=0, log="no") abline(v=1, lty="dotted")

abline(v=1/4, lty="dotted") soi.per$spec[40] # soi pgram at freq 1/12 = 40/480 soi.per$spec[10] # soi pgram at freq 1/48 = 10/480 # -- conf intervals -- # returned value: U = qchisq(.025,2) # 0.05063562 L = qchisq(.975,2) # 7.377759 2*soi.per$spec[10]/L # 0.01456530 2*soi.per$spec[10]/U # 2.122207 2*soi.per$spec[40]/L # 0.2635573 2*soi.per$spec[40]/U # 38.40108

Example 4.10

par(mfrow=c(2,1)) k = kernel("daniell",4) soi.ave = spec.pgram(soi, k, taper=0, log="no") abline(v=1/12, lty="dotted") abline(v=2/12, lty="dotted") abline(v=3/12, lty="dotted") abline(v=1/48, lty="dotted")# -- Repeat 5 lines above using rec in place of soi soi.ave$bandwidth # reported bandwidth soi.ave$bandwidth*sqrt(12) # Bw df = soi.ave$df # df = 16.9875 (returned values) U = qchisq(.025,df) # U = 7.555916 L = qchisq(.975,df) # L = 30.17425 soi.ave$spec[10] # 0.5942431 soi.ave$spec[40] # 1.428959# -- intervals -- df*soi.ave$spec[10]/L # 0.334547 df*soi.ave$spec[10]/U # 1.336000 df*soi.ave$spec[40]/L # 0.8044755 df*soi.ave$spec[40]/U # 3.212641# -- repeat above commands with soi replaced by rec

Example 4.11

kernel("modified.daniell", c(3,3)) # for a list plot(kernel("modified.daniell", c(3,3))) # for a graph par(mfrow=c(2,1)) k = kernel("modified.daniell", c(3,3)) soi.smo = spec.pgram(soi, k, taper=0, log="no") abline(v=1/12, lty="dotted") abline(v=1/48, lty="dotted")# -- Repeat above 3 lines with rec replacing soi df = soi.smo$df # df=17.42618 Lh = 1/sum(k[-k$m:k$m]^2) # Lh=9.232413 Bw = Lh/480 # Bw=0.01923419

# An easier way to obtain soi.smo (you can add log="no"): soi.smo = spectrum(soi, spans=c(7,7), taper=0)

Example 4.12

par(mfrow=c(3,1)) spectrum(soi, spans=c(7,7), taper=0, main="No Taper") abline(v=1/12,lty="dashed") abline(v=1/48,lty="dashed") spectrum(soi, spans=c(7,7), main="10% Taper") abline(v=1/12,lty="dashed") abline(v=1/48,lty="dashed") spectrum(soi, spans=c(7,7), taper=.5, main="50% Taper") abline(v=1/12,lty="dashed") abline(v=1/48,lty="dashed")

Example 4.13

x = matrix(scan(“c:/data/eq5exp6.dat"), ncol=2) eqP = x[1:1024, 1]; eqS = x[1025:2048, 1] exP = x[1:1024, 2]; exS = x[1025:2048, 2] par(mfrow=c(2,2)) eqPs=spectrum(eqP, spans=c(21,21), log="no", xlim=c(0,.25), ylim=c(0,.04)) eqSs=spectrum(eqS, spans=c(21,21), log="no", xlim=c(0,.25), ylim=c(0,.4))

exPs=spectrum(exP, spans=c(21,21), log="no", xlim=c(0,.25), ylim=c(0,.04)) exSs=spectrum(exS, spans=c(21,21), log="no", xlim=c(0,.25), ylim=c(0,.4)) exSs$df

Example 4.16

x = ts(cbind(soi,rec)) s = spec.pgram(x, kernel("daniell",9), taper=0) s$df # df = 35.8625 f = qf(.999, 2, s$df-2) # f = 8.529792 c = f/(18+f) # c = 0.3188779 plot(s, plot.type = "coh", ci.lty = 2) abline(h = c)

Example 4.17

par(mfrow=c(3,1)) plot.ts(soi) # the data plot.ts(diff(soi)) # first difference k = kernel("modified.daniell", 6) #-- 12 month filter soif = kernapply(soi,k) plot.ts(soif) x11() # open new graphics device spectrum(soif, spans=9, log="no") #-- spectral analysis abline(v=1/52, lty="dotted") windows() w = seq(0,.5, length=1000) #-- frequency response FR = abs(1-exp(2i*pi*w))^2 plot(w, FR, type="l")

Example 4.19

spec.ar(soi, log="no") # plot min AIC spectrum abline(v=1/52, lty="dotted") # locate El Niño period abline(v=1/12, lty="dotted") # locate yearly period soi.ar = ar(soi, order.max=30) # obtain AICs

plot(0:30, soi.ar$aic, type="l") # plot AICs soi.ar # view results

Example 4.20

eqexp = matrix(scan(“c:/data/eq5exp6.dat"), ncol=2) ex = eqexp[,2] # the explosion series ## -- dynamic spectral analysis -- ## nobs = length(ex) # number of observations wsize = 256 # window size overlap = 128 # overlap ovr = wsize-overlap nseg = floor(nobs/ovr)-1; # number of segments krnl = kernel("daniell", c(1,1)) # kernel ex.spec = matrix(0,wsize/2,nseg) for (k in 1:nseg) { a = ovr*(k-1)+1 b = wsize+ovr*(k-1) ex.spec[,k]=spectrum(ex[a:b],krnl,taper=.5,plot=F)$spec } ## --plot results -- ## x = seq(0, .5, len = nrow(ex.spec)) y = seq(0, ovr*nseg, len = ncol(ex.spec)) persp(x, y, ex.spec, zlab="Power", xlab="frequency", ylab="time", ticktype = "detailed", theta=25, d=2)

Example 4.21

# -- uses the waveslim package --# library(waveslim) eqexp = matrix(scan(“c:/data/eq5exp6.dat"), ncol=2) eq = eqexp[,1] # the earthquake series eq.dwt = dwt(eq, n.levels=6) # -- plot the dwt and calculate TP -- # TP = matrix(0,7,1) par(mfcol=c(7,1), pty="m", mar=c(3,4,2,2)) for(i in 1:6){ plot.ts(up.sample(eq.dwt[[i]], 2^i), type="h", axes=F,

ylab=names(eq.dwt)[i]) abline(h=0) axis(side=2) TP[i]=sum(eq.dwt[[i]]^2) } plot.ts(up.sample(eq.dwt[[7]], 2^6), type="h", axes=F, ylab=names(eq.dwt)[7]) abline(h=0) axis(side=2) axis(side=1) TP[7]=sum(eq.dwt[[7]]^2) TP/sum(eq^2) # the energy distribution# To plot the ordinates of the wavelet transforms on the # same scale, include a command like ylim=c(-1.5,1.5) in # each plot.ts() command.

Example 4.22

# -- uses the waveslim package --## -- assumes eq is retained from previous example --# library(waveslim) eq.dwt = dwt(eq, n.levels=6) eq.trsh = universal.thresh(eq.dwt, hard=F) eq.smo = idwt(eq.trsh) par(mfrow=c(3,1)) plot.ts(eq, ylab="Earthquake", ylim=c(-.5,.5)) plot.ts(eq.smo,ylab="Smoothed Earthquake",ylim=c(-.5,.5)) plot.ts(eq-eq.smo, ylab="Noise", ylim=c(-.5,.5))

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CHAPTER 5

Example 5.1

# -- uses the fracdiff package --# library(fracdiff) lvarve = log(varve)-mean(log(varve))

varve.fd = fracdiff(lvarve, nar=0, nma=0, M=30) varve.fd$d varve.fd$stderror.dpq

Example 5.3

# -- uses the tseries package --# library(tseries) gnp96 = read.table(“c:/data/gnp96.dat") gnpr = diff(log(gnp96[,2])) gnpr.ar = ar.mle(gnpr, order.max=1) # recall phi1 =.347 y = gnpr.ar$resid[2:length(gnpr)] # first resid is NA arch.y = garch(y,order=c(0,1)) summary.garch(arch.y)

NB: You can fit simultaneous [ARMA mean and GARCH variance] models using garchFit() from the package fSeries. Unfortunately, I couldn't get it to work with this example.

Example 5.4

# -- uses the tseries package --# library(tseries) nyse = scan(“c:/data/nyse.dat") nyse.g = garch(nyse, order=c(1,1))# old:# summary.garch(nyse.g)# u = predict.garch(nyse.g)# new: summary(nyse.g) u = predict(nyse.g) plot(800:1000, nyse[800:1000], type="l", xlab="Time", ylab="NYSE Returns") lines(u[,1], col="blue", lty="dashed") lines(u[,2], col="blue", lty="dashed")

Example 5.7This code does not appear in the text where the results were obtained using ASTSA. Here we use gls() from the nlme package.

library(nlme) mort = scan(“c:/data/cmort.dat") temp = scan(“c:/data/temp.dat") part = scan(“c:/data/part.dat") temp = temp - mean(temp) temp2 = temp^2 trend = 1:length(mort)# -- lm() fit in Example 2.2 indicates AR(2) for the residuals fit.gls = gls(mort~trend + temp + temp2 + part, correlation=corARMA(p=2), method="ML")# -- the results (abbreviated) summary(fit.gls) Correlation Structure: ARMA(2,0) Parameter estimate(s): Phi1 Phi2 0.3848530 0.4326282 Coefficients: Value Std.Error t-value p-value(Intercept) 87.63747 2.7327956 32.06880 0.0000trend -0.02915 0.0081705 -3.56781 0.0004temp -0.01881 0.0430264 -0.43715 0.6622temp2 0.01542 0.0020289 7.60244 0.0000part 0.15437 0.0248071 6.22297 0.0000 Residual standard error: 7.699336 Degrees of freedom: 508 total; 503 residual# -- check noise for whiteness (* see note below) w = filter(residuals(fit.gls), filter=c(1,-.3848530, -.4326282), sides=1) w = w[3:508] # first two are NA Box.test(w, 20, type="Ljung") # Ljung-Box-Pierce Statistic X-squared = 26.793, df = 20, p-value = 0.1412 pchisq(26.793, 18, lower=F) # the p-value (they are resids from an ar2 fit) 0.08296072

* The errors from the gls fit, say e(t), are estimated to be of the form e(t) = .3848530 e(t-1) + .4326282 e(t-2) + w(t). The filter statement gets at the w(t), which should be white. top

© Copyright 2006, R.H. Shumway & D.S. Stoffer