# trend

##### Trend Estimation

Estimate the trend by state space model.

- Keywords
- ts

##### Usage

`trend(y, trend.order = 1, 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 (for

`tau2.ini`

is specified (not`NULL`

)) .- plot
logical. If

`TRUE`

(default), '`trend`

' and '`residual`

' are plotted.- …
further arguments to be passed to plot.trend.

##### Details

The trend model can be represented by a state space model

$$x_n = Fx_{n-1} + Gv_n,$$ $$y_n = Hx_n + w_n,$$

where \(F\), \(G\) and \(H\) are matrices with appropriate dimensions. We assume that \(v_n\) and \(w_n\) are white noises that have the normal distributions \(N(0,\tau^2)\) and \(N(0, \sigma^2)\), respectively.

##### Value

An object of class `"trend"`

, which is a list with the following
elements:

trend component.

residuals.

variance of the system noise \(\tau^2\).

variance of the observational noise \(\sigma^2\).

log-likelihood of the model.

AIC.

##### References

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

##### Examples

```
# NOT RUN {
# The daily maximum temperatures for Tokyo
data(Temperature)
trend(Temperature, trend.order = 1, tau2.ini = 0.223, delta = 0.001)
trend(Temperature, trend.order = 2)
# }
```

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