# tvvar

##### Time Varying Variance

Estimate time-varying variance.

- Keywords
- ts

##### 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`

.

##### 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.

##### Value

An object of class `"tvvar"`

, which is a list with the following
elements:

time varying variance.

normalized data.

transformed data.

trend.

residuals.

variance of the system noise.

variance of the observational noise.

log-likelihood of the model.

AIC.

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

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

*Documentation reproduced from package TSSS, version 1.2.3, License: GPL (>= 2)*