ltsa-package: Linear Time Series Analysis

Description Details Author(s) References See Also Examples

Description

Linear time series modelling. Methods are given for loglikelihood computation, forecasting and simulation.

Details

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

Author(s)

A. I. McLeod, Hao Yu and Zinovi Krougly.

Maintainer: [email protected]

References

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.

See Also

DHSimulate, DLAcfToAR, DLLoglikelihood, DLResiduals, DLSimulate, exactLoglikelihood, is.toeplitz, PredictionVariance, tacvfARMA, ToeplitzInverseUpdate, TrenchForecast, TrenchInverse, TrenchLoglikelihood, TrenchMean,

Examples

  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

ltsa documentation built on May 2, 2019, 4:01 a.m.