Description Details Author(s) References See Also Examples
Linear time series modelling. Methods are given for loglikelihood computation, forecasting and simulation.
Package: | ltsa |
Type: | Package |
Version: | 1.4.5 |
Date: | 2015-08-22 |
License: | GPL (>= 2) |
FUNCTION | SUMMARY |
DHSimulate | Davies and Harte algorithm for time series simulation |
DLAcfToAR | from Acf to AR using Durbin-Levinson recursion |
DLLoglikelihood | exact loglikelihood using Durbin-Levinson algorithm |
DLResiduals | exact one-step residuals, Durbin-Levision algorithm |
DLSimulate | exact simulation of Gaussian time series using DL |
is.toeplitz | test for Toeplitz matrix |
PredictionVariance | two methods provided |
tacvfARMA | theoretical autocovariances |
ToeplitzInverseUpdate | update inverse |
TrenchForecast | general algorithm for forecasting |
TrenchInverse | efficient algorithm for inverse of Toeplitz matrix |
TrenchLogLikelihood | exact loglikelihood |
TrenchMean | exact MLE for mean |
A. I. McLeod, Hao Yu and Zinovi Krougly.
Maintainer: aimcleod@uwo.ca
Hipel, K.W. and McLeod, A.I., (2005). Time Series Modelling of Water Resources and Environmental Systems. Electronic reprint of our book orginally published in 1994. http://www.stats.uwo.ca/faculty/aim/1994Book/.
McLeod, A.I., Yu, Hao, Krougly, Zinovi L. (2007). Algorithms for Linear Time Series Analysis, Journal of Statistical Software.
DHSimulate
,
DLAcfToAR
,
DLLoglikelihood
,
DLResiduals
,
DLSimulate
,
exactLoglikelihood
,
is.toeplitz
,
PredictionVariance
,
tacvfARMA
,
ToeplitzInverseUpdate
,
TrenchForecast
,
TrenchInverse
,
TrenchLoglikelihood
,
TrenchMean
,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 | #Example 1: DHSimulate
#First define acf for fractionally-differenced white noise and then simulate using DHSimulate
`tacvfFdwn` <-
function(d, maxlag)
{
x <- numeric(maxlag + 1)
x[1] <- gamma(1 - 2 * d)/gamma(1 - d)^2
for(i in 1:maxlag)
x[i + 1] <- ((i - 1 + d)/(i - d)) * x[i]
x
}
n<-1000
rZ<-tacvfFdwn(0.25, n-1) #length 1000
Z<-DHSimulate(n, rZ)
acf(Z)
#Example 2: DLAcfToAR
#
n<-10
d<-0.4
r<-tacvfFdwn(d, n)
r<-(r/r[1])[-1]
HoskingPacf<-d/(-d+(1:n))
cbind(DLAcfToAR(r),HoskingPacf)
#Example 3: DLLoglikelihood
#Using Z and rZ in Example 1.
DLLoglikelihood(rZ, Z)
#Example 4: DLResiduals
#Using Z and rZ in Example 1.
DLResiduals(rZ, Z)
#Example 5: DLSimulate
#Using Z in Example 1.
z<-DLSimulate(n, rZ)
plot.ts(z)
#Example 6: is.toeplitz
is.toeplitz(toeplitz(1:5))
#Example 7: PredictionVariance
#Compare with predict.Arima
#general script, just change z, p, q, ML
z<-sqrt(sunspot.year)
n<-length(z)
p<-9
q<-0
ML<-10
#for different data/model just reset above
out<-arima(z, order=c(p,0,q))
sda<-as.vector(predict(out, n.ahead=ML)$se)
#
phi<-theta<-numeric(0)
if (p>0) phi<-coef(out)[1:p]
if (q>0) theta<-coef(out)[(p+1):(p+q)]
zm<-coef(out)[p+q+1]
sigma2<-out$sigma2
r<-sigma2*tacvfARMA(phi, theta, maxLag=n+ML-1)
sdb<-sqrt(PredictionVariance(r, maxLead=ML))
cbind(sda,sdb)
#Example 8: tacfARMA
#There are two methods: tacvfARMA and ARMAacf.
#tacvfARMA is more general since it computes the autocovariances function
# given the ARMA parameters and the innovation variance whereas ARMAacf
# only computes the autocorrelations. Sometimes tacvfARMA is more suitable
# for what is needed and provides a better result than ARMAacf as in the
# the following example.
#
#general script, just change z, p, q, ML
z<-sqrt(sunspot.year)
n<-length(z)
p<-9
q<-0
ML<-5
#for different data/model just reset above
out<-arima(z, order=c(p,0,q))
phi<-theta<-numeric(0)
if (p>0) phi<-coef(out)[1:p]
if (q>0) theta<-coef(out)[(p+1):(p+q)]
zm<-coef(out)[p+q+1]
sigma2<-out$sigma2
rA<-tacvfARMA(phi, theta, maxLag=n+ML-1, sigma2=sigma2)
rB<-var(z)*ARMAacf(ar=phi, ma=theta, lag.max=n+ML-1)
#rA and rB are slighly different
cbind(rA[1:5],rB[1:5])
#Example 9: ToeplitzInverseUpdate
#In this example we compute the update inverse directly and using ToeplitzInverseUpdate and
#compare the result.
phi<-0.8
sde<-30
n<-30
r<-arima.sim(n=30,list(ar=phi),sd=sde)
r<-phi^(0:(n-1))/(1-phi^2)*sde^2
n1<-25
G<-toeplitz(r[1:n1])
GI<-solve(G) #could also use TrenchInverse
GIupdate<-ToeplitzInverseUpdate(GI,r[1:n1],r[n1+1])
GIdirect<-solve(toeplitz(r[1:(n1+1)]))
ERR<-sum(abs(GIupdate-GIdirect))
ERR
#Example 10: TrenchForecast
#Compare TrenchForecast and predict.Arima
#general script, just change z, p, q, ML
z<-sqrt(sunspot.year)
n<-length(z)
p<-9
q<-0
ML<-10
#for different data/model just reset above
out<-arima(z, order=c(p,0,q))
Fp<-predict(out, n.ahead=ML)
phi<-theta<-numeric(0)
if (p>0) phi<-coef(out)[1:p]
if (q>0) theta<-coef(out)[(p+1):(p+q)]
zm<-coef(out)[p+q+1]
sigma2<-out$sigma2
#r<-var(z)*ARMAacf(ar=phi, ma=theta, lag.max=n+ML-1)
#When r is computed as above, it is not identical to below
r<-sigma2*tacvfARMA(phi, theta, maxLag=n+ML-1)
F<-TrenchForecast(z, r, zm, n, maxLead=ML)
#the forecasts are identical using tacvfARMA
#
#Example 11: TrenchInverse
#invert a matrix of order n and compute the maximum absolute error
# in the product of this inverse with the original matrix
n<-5
r<-0.8^(0:(n-1))
G<-toeplitz(r)
Gi<-TrenchInverse(G)
GGi<-crossprod(t(G),Gi)
id<-matrix(0, nrow=n, ncol=n)
diag(id)<-1
err<-max(abs(id-GGi))
err
#Example 12: TrenchLoglikelihood
#simulate a time series and compute the concentrated loglikelihood using DLLoglikelihood and
#compare this with the value given by TrenchLoglikelihood.
phi<-0.8
n<-200
r<-phi^(0:(n-1))
z<-arima.sim(model=list(ar=phi), n=n)
LD<-DLLoglikelihood(r,z)
LT<-TrenchLoglikelihood(r,z)
ans<-c(LD,LT)
names(ans)<-c("DLLoglikelihood","TrenchLoglikelihood")
#Example 13: TrenchMean
phi<- -0.9
a<-rnorm(100)
z<-numeric(length(a))
phi<- -0.9
n<-100
a<-rnorm(n)
z<-numeric(n)
mu<-100
sig<-10
z[1]<-a[1]*sig/sqrt(1-phi^2)
for (i in 2:n)
z[i]<-phi*z[i-1]+a[i]*sig
z<-z+mu
r<-phi^(0:(n-1))
meanMLE<-TrenchMean(r,z)
meanBLUE<-mean(z)
ans<-c(meanMLE, meanBLUE)
names(ans)<-c("BLUE", "MLE")
ans
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