tvvar {TSSS}R Documentation

Time Varying Variance

Description

Estimate time-varying variance.

Usage

tvvar(y, trend.order, tau2.ini = NULL, delta, plot = TRUE, ...)

Arguments

y

a univariate time series.

trend.order

trend order.

tau2.ini

initial estimate of variance of the system noise τ^2. If tau2.ini = NULL, the most suitable value is chosen in τ^2 = 2^{-k}.

delta

search width.

plot

logical. If TRUE (default), 'sm', 'trend' and 'noise' are plotted.

...

further arguments to be passed to plot.tvvar.

Details

Assuming that σ_(2m-1)^2 = σ_(2m)^2, we define a transformed time series s_1,…,s_(N/2) by

s_m = y_(2m-1)^2 + y_(2m)^2,

where y_n is a Gaussian white noise with mean 0 and variance σ_n^2. s_m is distributed as a χ^2 distribution with 2 degrees of freedom, so the probability density function of s_m is given by

f(s) = e^{-s/(2σ^2)} / (2σ^2).

By further transformation

z_m = log{(s_m)/2},

the probability density function of z_m is given by

g(z) = exp{z - e^z/σ^2} / σ^2 = exp{(z-log(σ^2)) - e^(z-log(σ^2))}.

Therefore, the transformed time series is given by

z_m = log(σ^2) + w_m,

where w_m is a double exponential distribution with probability density function

h(w) = exp{w-e^w}.

In the space state model

z_m = t_m + w_m

by identifying trend components of z_m, the log variance of original time series y_n is obtained.

Value

An object of class "tvvar", which is a list with the following elements:

tvv

time varying variance.

nordata

normalized data.

sm

transformed data.

trend

trend.

noise

residuals.

tau2

variance of the system noise.

sigma2

variance of the observational noise.

llkhood

log-likelihood of the model.

aic

AIC.

tsname

the name of the univariate time series y.

References

Kitagawa, G. (2010) Introduction to Time Series Modeling. Chapman & Hall/CRC.

Kitagawa, G. and Gersch, W. (1996) Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics, No.116, Springer-Verlag.

Kitagawa, G. and Gersch, W. (1985) A smoothness priors time varying AR coefficient modeling of nonstationary time series. IEEE trans. on Automatic Control, AC-30, 48-56.

Examples

# seismic data
data(MYE1F)
tvvar(MYE1F, trend.order = 2, tau2.ini = 6.6e-06, delta = 1.0e-06)

[Package TSSS version 1.2.4 Index]