Autoregressive Process with Order-1 Family Function

Maximum likelihood estimation of the three-parameter AR-1 model

Usage

AR1(ldrift = "identitylink", lsd  = "loglink", lvar = "loglink", lrho = "rhobitlink",
    idrift  = NULL, isd  = NULL, ivar = NULL, irho = NULL, imethod = 1,
    ishrinkage = 0.95, type.likelihood = c("exact", "conditional"),
    type.EIM  = c("exact", "approximate"), var.arg = FALSE, nodrift = FALSE,
    print.EIM = FALSE, zero = c(if (var.arg) "var" else "sd", "rho"))

Arguments

ldrift, lsd, lvar, lrho

Link functions applied to the scaled mean, standard deviation or variance, and correlation parameters. The parameter drift is known as the drift, and it is a scaled mean. See Links for more choices.

idrift, isd, ivar, irho

Optional initial values for the parameters. If failure to converge occurs then try different values and monitor convergence by using trace = TRUE. For a $S$-column response, these arguments can be of length $S$, and they are recycled by the columns first. A value NULL means an initial value for each response is computed internally.

ishrinkage, imethod, zero

See CommonVGAMffArguments for more information. The default for zero assumes there is a drift parameter to be estimated (the default for that argument), so if a drift parameter is suppressed and there are covariates, then zero will need to be assigned the value 1 or 2 or NULL.

var.arg

Same meaning as uninormal.

nodrift

Logical, for determining whether to estimate the drift parameter. The default is to estimate it. If TRUE, the drift parameter is set to 0 and not estimated.

type.EIM

What type of expected information matrix (EIM) is used in Fisher scoring. By default, this family function calls AR1EIM, which recursively computes the exact EIM for the AR process with Gaussian white noise. See Porat and Friedlander (1986) for further details on the exact EIM.

If type.EIM = "approximate" then approximate expression for the EIM of Autoregressive processes is used; this approach holds when the number of observations is large enough. Succinct details about the approximate EIM are delineated at Porat and Friedlander (1987).

print.EIM

Logical. If TRUE, then the first few EIMs are printed. Here, the result shown is the sum of each EIM.

type.likelihood

What type of likelihood function is maximized. The first choice (default) is the sum of the marginal likelihood and the conditional likelihood. Choosing the conditional likelihood means that the first observation is effectively ignored (this is handled internally by setting the value of the first prior weight to be some small positive number, e.g., 1.0e-6). See the note below.

Details

The AR-1 model implemented here has $$Y_1 \sim N(\mu, \sigma^2 / (1-\rho^2)), $$ and $$Y_i = \mu^* + \rho Y_{i-1} + e_i, $$ where the $e_i$ are i.i.d. Normal(0, sd = $\sigma$) random variates.

Here are a few notes: (1). A test for weak stationarity might be to verify whether $1/\rho$ lies outside the unit circle. (2). The mean of all the $Y_i$ is $\mu^* /(1-\rho)$ and these are returned as the fitted values. (3). The correlation of all the $Y_i$ with $Y_{i-1}$ is $\rho$. (4). The default link function ensures that $-1 < \rho < 1$.

Warning

Monitoring convergence is urged, i.e., set trace = TRUE.

Moreover, if the exact EIMs are used, set print.EIM = TRUE to compare the computed exact to the approximate EIM.

Under the VGLM/VGAM approach, parameters can be modelled in terms of covariates. Particularly, if the standard deviation of the white noise is modelled in this way, then type.EIM = "exact" may certainly lead to unstable results. The reason is that white noise is a stationary process, and consequently, its variance must remain as a constant. Consequently, the use of variates to model this parameter contradicts the assumption of stationary random components to compute the exact EIMs proposed by Porat and Friedlander (1987).

To prevent convergence issues in such cases, this family function internally verifies whether the variance of the white noise remains as a constant at each Fisher scoring iteration. If this assumption is violated and type.EIM = "exact" is set, then AR1 automatically shifts to type.EIM = "approximate". Also, a warning is accordingly displayed.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

References

Porat, B. and Friedlander, B. (1987). The Exact Cramer-Rao Bond for Gaussian Autoregressive Processes. IEEE Transactions on Aerospace and Electronic Systems, AES-23(4), 537–542.

Author

Victor Miranda (exact method) and Thomas W. Yee (approximate method).

Note

Multiple responses are handled. The mean is returned as the fitted values.

Examples

if (FALSE) { # \dontrun{
### Example 1: using  arima.sim() to generate a 0-mean stationary time series.
nn <- 500
tsdata <- data.frame(x2 =  runif(nn))
ar.coef.1 <- rhobitlink(-1.55, inverse = TRUE)  # Approx -0.65
ar.coef.2 <- rhobitlink( 1.0, inverse = TRUE)   # Approx  0.50
set.seed(1)
tsdata  <- transform(tsdata,
              index = 1:nn,
              TS1 = arima.sim(nn, model = list(ar = ar.coef.1),
                              sd = exp(1.5)),
              TS2 = arima.sim(nn, model = list(ar = ar.coef.2),
                              sd = exp(1.0 + 1.5 * x2)))

### An autoregressive intercept--only model.   ###
### Using the exact EIM, and "nodrift = TRUE"  ###
fit1a <- vglm(TS1 ~ 1, data = tsdata, trace = TRUE,
              AR1(var.arg = FALSE, nodrift = TRUE,
                  type.EIM = "exact",
                  print.EIM = FALSE),
              crit = "coefficients")
Coef(fit1a)
summary(fit1a)

### Two responses. Here, the white noise standard deviation of TS2   ###
### is modelled in terms of 'x2'. Also, 'type.EIM = exact'.  ###
fit1b <- vglm(cbind(TS1, TS2) ~ x2,
              AR1(zero = NULL, nodrift = TRUE,
                  var.arg = FALSE,
                  type.EIM = "exact"),
              constraints = list("(Intercept)" = diag(4),
                                 "x2" = rbind(0, 0, 1, 0)),
              data = tsdata, trace = TRUE, crit = "coefficients")
coef(fit1b, matrix = TRUE)
summary(fit1b)

### Example 2: another stationary time series
nn     <- 500
my.rho <- rhobitlink(1.0, inverse = TRUE)
my.mu  <- 1.0
my.sd  <- exp(1)
tsdata  <- data.frame(index = 1:nn, TS3 = runif(nn))

set.seed(2)
for (ii in 2:nn)
  tsdata$TS3[ii] <- my.mu/(1 - my.rho) +
                    my.rho * tsdata$TS3[ii-1] + rnorm(1, sd = my.sd)
tsdata <- tsdata[-(1:ceiling(nn/5)), ]  # Remove the burn-in data:

### Fitting an AR(1). The exact EIMs are used.
fit2a <- vglm(TS3 ~ 1, AR1(type.likelihood = "exact",  # "conditional",
                                type.EIM = "exact"),
              data = tsdata, trace = TRUE, crit = "coefficients")

Coef(fit2a)
summary(fit2a)      # SEs are useful to know

Coef(fit2a)["rho"]    # Estimate of rho, for intercept-only models
my.rho                # The 'truth' (rho)
Coef(fit2a)["drift"]  # Estimate of drift, for intercept-only models
my.mu /(1 - my.rho)   # The 'truth' (drift)
} # }