logspec2cov {regspec}

Compute autocovariance values from the basis coefficients for the log spectrum.

Description

This function performs a series of integrals of a spectral density multiplied by sinusoids of increasing frequency in order. The result is a vector of autocovariances for the process values at increasing separation.

The example below shows how this function is useful for informing estimates of missing values in thinned time series data.

Usage

logspec2cov(ebeta, vbeta, SARIMA=list(sigma2=1), lags, subdivisions=100)

Arguments

ebeta

Vector. Expectations for basis coefficients of the log spectrum.

vbeta

Vector. The variance for the basis coefficients of the log spectrum.

SARIMA

List. A list encoding the SARIMA model that acts as the intercept, or base line, for the non-parametric estimate of the log-spectrum. The default is white noise with variance one. The log-spectrum basis coefficients parameterize a deviation away from the SARIMA model's log-spectrum. The contents of the SARIMA list are formatted in line with the format used by the package TSA (see the Examples section for examples).

lags

Integer. The number of lags to which to calculate the autocovariance.

subdivisions

Integer. This number is passed to the numerical integrator that computes the autocovariances from the exponentiated log-spectrum. It is set to 100 by default to reduce CPU demand. For rough spectra is may be advisable to specify a larger value.

Value

autocovariances

A vector of approximate expectations for the process's autocovariances for increasing lags, starting with zero lag (the process variance).

Author(s)

Ben Powell

Examples

#
# Simulate a complete time series.

set.seed(42)
D <- arima.sim(n=150, model=list(ar=c(-0.5,0.4,0.8), ma=0.2))

# Now define indices for a subsampled historical period,
# a fully sampled historical period, and the missing values.

inda <- seq(1, 100, by=3)
indb <- seq(101, 150, by=1)
indc <- (1:150)[-c(inda, indb)]

#
# Adjust our estimate for the spectrum using the two historical periods.
#

adjustment1 <- regspec(D=D[indb], deltat=1, smthpar=0.85,
	plot.spec=FALSE)

adjustment2 <- regspec(D=D[inda], deltat=3, ebeta=adjustment1$ebeta,
	vbeta=adjustment1$vbeta, ylim=c(0,60))

lines(spec.true, col=1, lwd=3, lty=2)

#
# Calculate covariances corresponding to the estimate of the spectrum at
# the data locations.
#

k <- logspec2cov(adjustment2$ebeta, adjustment2$vbeta, lag=150)

#
# Compute linear predictors and variances for the missing data conditional
# on the autocovariances.
#

K <- matrix(0, 150, 150)
K <- matrix(k[1+abs(row(K)-col(K))], 150, 150)
d <- solve(K[c(inda, indb),c(inda, indb)],K[c(inda, indb), indc])
hindcast.exp <- crossprod(d, c(D[inda], D[indb]))
hindcast.var <- K[indc, indc]-crossprod(K[c(inda, indb), indc], d)

#
# Plot the observed historical data and the predictions for the missing data.
#

par(cex=0.7)
plot(NA, NA, xlim=c(0,150), ylim=range(D), xlab="time", ylab="x")

#
#(Observed process values are plotted with black circles.)
#

points(indb, D[indb], type="p", lty=2)
points(c(inda, indb), c(D[inda], D[indb]))

#
# (Hindcasts are plotted with blue circles.)
#
points(indc, hindcast.exp, col=rgb(0.2,0.5,0.7), lwd=2)
for(i in 1:length(indc))	{
	lines(rep(indc[i], 2),
	hindcast.exp[i]+1*c(-1, 1)*hindcast.var[i, i]^0.5,
	col=rgb(0.2,0.5,0.7))
	}

#
# Interpolate the plotted data and predictions.
#

x <- c(inda, indb, indc)
y <- c(D[inda], D[indb], hindcast.exp)
lines(sort(x), y[order(x)], lty=2, col=rgb(0.2,0.5,0.7,0.7))

#
# Reveal the true values.
#

# (The union of observed and unobserved data is plotted with red crosses.)
points(D,col=2,pch=4)