TSSS (version 1.2.3)

tvvar: 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 \(\tau^2\). If tau2.ini = NULL, the most suitable value is chosen in \(\tau^2 = 2^{-k}\).

delta

search width.

plot

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

…

further arguments to be passed to plot.tvvar.

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.

Details

Assuming that \(\sigma_{2m-1}^2 = \sigma_{2m}^2\), we define a transformed time series \(s_1,\dots,s_{N/2}\) by

$$s_m = y_{2m-1}^2 + y_{2m}^2,$$

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

$$f(s) = \frac{1}{2\sigma^2} e^{-s/2\sigma^2}.$$

By further transformation

$$z_m = \log \left( \frac{s_m}{2} \right),$$

the probability density function of \(z_m\) is given by

$$g(z) = \frac{1}{\sigma^2} \exp{ \left\{ z-\frac{e^z}{\sigma^2} \right\} } = \exp{ \left\{ (z-\log\sigma^2) - e^{(z-\log\sigma^2)} \right\} }.$$

Therefore, the transformed time series is given by

$$z_m = \log \sigma^2 + w_m,$$

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.

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

# NOT RUN {
# seismic data
data(MYE1F)
tvvar(MYE1F, trend.order = 2, tau2.ini = 6.6e-06, delta = 1.0e-06)
# }