# Autoregressive Process with Order-1 Family Function Maximum likelihood estimation of the three-parameter AR-1 model ```r 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 $$ Y_1 \sim N(\mu, \sigma^2 / (1-\rho^2)),%Y(1) ~ N(mu, sigma^2 / (1-rho^2), $$ and $$ Y_i = \mu^* + \rho Y_{i-1} + e_i,%Y(i) = mu^* + rho * Y(i-1) + e(i) $$ 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. ## Returns 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(s) Victor Miranda (exact method) and Thomas W. Yee (approximate method). ## Note Multiple responses are handled. The mean is returned as the fitted values. ## See Also `AR1EIM`, `vglm.control`, `dAR1`, `arima.sim`. ## Examples ```r ## Not run: ### 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) ## End(Not run) ```

AR1 function