Set up autoregressive or autoregressive moving average trend models
in `mvgam`

. These functions do not evaluate their arguments –
they exist purely to help set up a model with particular autoregressive
trend models.

## Usage

```
RW(ma = FALSE, cor = FALSE)
AR(p = 1, ma = FALSE, cor = FALSE)
CAR(p = 1)
VAR(ma = FALSE, cor = FALSE)
```

## Arguments

- ma
`Logical`

Include moving average terms of order`1`

? Default is`FALSE`

.- cor
`Logical`

Include correlated process errors as part of a multivariate normal process model? If`TRUE`

and if`n_series > 1`

in the supplied data, a fully structured covariance matrix will be estimated for the process errors. Default is`FALSE`

.- p
A non-negative integer specifying the autoregressive (AR) order. Default is

`1`

. Cannot currently be larger than`3`

for`AR`

terms, and cannot be anything other than`1`

for continuous time AR (`CAR`

) terms

## Value

An object of class `mvgam_trend`

, which contains a list of
arguments to be interpreted by the parsing functions in `mvgam`

## Examples

```
if (FALSE) {
# A short example to illustrate CAR(1) models
# Function to simulate CAR1 data with seasonality
sim_corcar1 = function(n = 120,
phi = 0.5,
sigma = 1,
sigma_obs = 0.75){
# Sample irregularly spaced time intervals
time_dis <- c(0, runif(n - 1, -0.1, 1))
time_dis[time_dis < 0] <- 0; time_dis <- time_dis * 5
# Set up the latent dynamic process
x <- vector(length = n); x[1] <- -0.3
for(i in 2:n){
# zero-distances will cause problems in sampling, so mvgam uses a
# minimum threshold; this simulation function emulates that process
if(time_dis[i] == 0){
x[i] <- rnorm(1, mean = (phi ^ 1e-12) * x[i - 1], sd = sigma)
} else {
x[i] <- rnorm(1, mean = (phi ^ time_dis[i]) * x[i - 1], sd = sigma)
}
}
# Add 12-month seasonality
cov1 <- sin(2 * pi * (1 : n) / 12); cov2 <- cos(2 * pi * (1 : n) / 12)
beta1 <- runif(1, 0.3, 0.7); beta2 <- runif(1, 0.2, 0.5)
seasonality <- beta1 * cov1 + beta2 * cov2
# Take Gaussian observations with error and return
data.frame(y = rnorm(n, mean = x + seasonality, sd = sigma_obs),
season = rep(1:12, 20)[1:n],
time = cumsum(time_dis))
}
# Sample two time series
dat <- rbind(dplyr::bind_cols(sim_corcar1(phi = 0.65,
sigma_obs = 0.55),
data.frame(series = 'series1')),
dplyr::bind_cols(sim_corcar1(phi = 0.8,
sigma_obs = 0.35),
data.frame(series = 'series2'))) %>%
dplyr::mutate(series = as.factor(series))
# mvgam with CAR(1) trends and series-level seasonal smooths; the
# State-Space representation (using trend_formula) will be more efficient
mod <- mvgam(formula = y ~ 1,
trend_formula = ~ s(season, bs = 'cc',
k = 5, by = trend),
trend_model = CAR(),
data = dat,
family = gaussian(),
samples = 300,
chains = 2)
# View usual summaries and plots
summary(mod)
conditional_effects(mod, type = 'expected')
plot(mod, type = 'trend', series = 1)
plot(mod, type = 'trend', series = 2)
plot(mod, type = 'residuals', series = 1)
plot(mod, type = 'residuals', series = 2)
}
```