tvvar {TSSS} | R Documentation |
Estimate time-varying variance.
tvvar(y, trend.order, tau2.ini = NULL, delta, plot = TRUE, ...)
y |
a univariate time series. |
trend.order |
trend order. |
tau2.ini |
initial estimate of variance of the system noise τ^2.
If |
delta |
search width. |
plot |
logical. If |
... |
further arguments to be passed to |
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.
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 |
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.
# seismic data data(MYE1F) tvvar(MYE1F, trend.order = 2, tau2.ini = 6.6e-06, delta = 1.0e-06)