timsac (version 1.2.7)

tvvar: Time Varying Variance

Description

Estimate time-varying variance.

Usage

tvvar(y, trend.order, tau20=NULL, delta=NULL, plot=TRUE)

Arguments

y

univariate time series.

trend.order

trend order.

tau20

initial estimate of tau2.

delta

search width.

plot

logical. If TRUE (default) normdat, ts, trend and noise are plotted.

Value

• tvvar
  time varying variance.
• normdat
  normalized data.
• ts
  transformed time series $s_m$.
• trend
  trend.
• noise
  residuals.
• tau2
  variance of the system noise tau2.
• sigma2
  variance of the observational noise.
• lkhood
  log-likelihood of the mode.
• aic
  AIC.

Details

Assuming that $\sigma_{2m-1}^2 = \sigma_{2m}^2$, 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 $\sigma_n^2$. $s_m$ 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 $$z_m = log(\frac{s_m}{2}),$$ the probability density function of $z_m$ is given by $$g(z) = \frac{1}{\sigma^2} exp{{z-\frac{e^z}{\sigma^2}}} = exp{{(z-log\sigma^2)} - e^{(z-log\sigma^2)}}.$$ 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. (1993) Time series analysis programing (in Japanese). The Iwanami Computer Science Series. 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

data(MYE1F) # an earthquake wave data
  z <- tvvar(MYE1F, 2, 6.6e-06, 1.0e-06)
  z$lkhood
  z$aic