# VGLTSMs Family functions: The Order--$(p, d, q)$ Autoregressive Integrated Moving Average Model (ARIMA(p, d, q)) with covariates Maximum likelihood estimation of the drift, standard deviation or variance of the random noise, and coefficients of an autoregressive integrated moving average process of order-$(p, d, q)$ with covariates by MLE using Fisher scoring. No seasonal terms handled yet. No seasonal components handled yet. ```r ARIMAXff(order = c(1, 1, 0), zero = c("ARcoeff", "MAcoeff"), diffCovs = TRUE, xLag = 0, include.current = FALSE, type.EIM = c("exact", "approximate")[1], var.arg = TRUE, nodrift = FALSE, noChecks = FALSE, ldrift = "identitylink", lsd = "loglink", lvar = "loglink", lARcoeff = "identitylink", lMAcoeff = "identitylink", idrift = NULL, isd = NULL, ivar = NULL, iARcoeff = NULL, iMAcoeff = NULL) ``` ## Arguments - `order`: Integer vector with three components, (`p, d, q`): The AR order (`p`), the degree of differencing (`d`), and the MA order (`q`). - `zero`: Integer or character--strings vector. Name(s) or position(s) of the parameters/linear predictors to be modeled as intercept-only. Details at `zero`. - `diffCovs`: Logical. The default is `diffCovs = TRUE`, which means that the order--`d` differences of the covariates (if entered) are internally computed and then incorporated in the conditional--mean model. Otherwise, only the current `actual` values of the covariates are included. - `xLag`: Integer, non--negative. If `xLag` > 0, the covariates at **current** time, $x[t]$, plus the lagged covariates up to order `'xLag'` are embedded into the design matrix (as covariates too). Leave `xLag = 0` and only the **current** value, $x[t]$, will be considered. See more details below. - `include.current`: Logical. Same as `ARIMAX.errors.ff`. - `type.EIM`: The type of expected information matrix (EIM) of the ARMA process to be utilized in Fisher scoring. `type.EIM = "exact"` (default) enables the **exact** IM (Porat, et.al. 1986), otherwise the `approximate` version is utilized. - `var.arg`: Logical. If `FALSE` (default), then the standard deviation of the random noise is estimated. Else, the variance estimate is returned. - `nodrift`: Logical. `nodrift = TRUE` supresses estimation of the intercept (the **drift** in the ARMA case), which is set to zero internally. - `noChecks`: Logical. If `FALSE` (default), this family function internally checks **stationarity** (AR case) and **invertibility** (MA case) of the the estimated model. A `warning` is correspondingly displayed. - `ldrift, lsd, lvar, lARcoeff, lMAcoeff`: Link functions applied to the intercept, the random noise standard deviation (or optionally, the variance), and the coefficients in the ARMA--type conditional--mean model. - `idrift, isd, ivar, iARcoeff, iMAcoeff`: Optional initial values for the intercept (drift), noise SD (or variance), and ARMA coeffcients (a vector of length $p + q$). If failure to converge occurs then try different values and monitor convergence by using `trace = TRUE` in the `vglm()` call. ## Details Let $x[t]$ be a (probably time--varying) vector of suitable covariates. The ARIMAX model handled by `ARIMAXff` is $$ \nabla^d Y_t = \mu^{\star} + \boldsymbol{\beta}^T\nabla^d \boldsymbol{x}_t +\theta_1 \nabla^d Y_{t - 1} + \cdots +\theta_p \nabla^d Y_{t - p} +\phi_1 \varepsilon_{t - 1} + \cdots +\phi_q \varepsilon_{t - q} + \varepsilon,%D(Y[t], d) = \mu[*] + \beta^T * D(x[t], d) +\theta[1] D(Y[t], d)[t - 1] + \ldots +\theta[p] D(Y[t], d)[t - p] +\phi[1] e[t - 1] + \ldots +\phi[q] e[t - q] + e[t], $$ with $D( * , d)$ the operator differencing a series `d` times. If `diffCovs = TRUE`, this function differencing the covariates `d` times too. Similarly, `ARMAXff` manages $$ \nabla^d Y_t = \mu^{\star} + \boldsymbol{\beta}^T \boldsymbol{x}_t +\theta_1 Y_{t - 1} + \cdots + \theta_p Y_{t - p} +\phi_1 \varepsilon_{t - 1} + \cdots +\phi_q \varepsilon_{t - q} + \varepsilon,%Y[t] = \mu[*] + \beta^T * x_t +\theta[1] Y[t - 1] + \ldots + \theta[p] Y[t - p] +\phi[1] e[t - 1] + \ldots +\phi[q] e[t - q] + e[t], $$ where $$ \varepsilon_{t | \Phi_{t - 1}}\sim N(0, \sigma_{\varepsilon_t | \Phi_{t - 1}}^2).%e[t] |\Phi[t - 1] ~ N(0, \sigma[e]^2 | \Phi[t - 1]). $$ Note, c("$%$", "$\n$", "$\\sigma[e]^2 | \\Phi[t - 1]$") is **conditional** on $\Phi[t - 1]$, the information of the joint process c("$\n$", "$(Y[t - 1], x[t])\n$"), at time $t$, and hence may be modelled in terms of $x[t]$, if required. `ARIMAXff()` and `ARMAXff()` handle **multiple responses**, thus a matrix can be used as the response. Note, no seasonal terms handled. This feature is to be incorporated shortly. The default linear predictor is $$ \boldsymbol{\eta} = \left(\mu, \log \sigma^{2}_{\varepsilon_{t | \Phi_{t - 1}}},\theta_1, \ldots, \theta_p,\phi_1, \ldots, \phi_q\right)^T.\eta = (\mu, log \sigma[e]^2 | \Phi[t - 1],\theta[1], \ldots, \theta[p],\phi[1], \ldots, \phi[q] )^T. $$ Other links are also handled. See `Links`. Further choices for the random noise, besides Gaussian, will be implemented over time. As with `ARXff` and `MAXff`, choices for the EIMs are `"exact"` and `"approximate"`. Covariates may be incorporated in the fit for any linear predictor above. Hence, `ARIMAXff` supports non--stationary processes (c("$\n$", "$\\sigma[e]^2 | \\Phi[t - 1]$")) may depend on $X[t]$. Also, constraint matrices on the linear predictors may be entered through `cm.ARMA` or using the argument `constraints`, from `vglm`. Checks on stationarity and invertibility on the esitmated process are performed by default. Set `noChecks = TRUE` to dismiss this step. ## Returns An object of class `"vglmff"` (see `vglmff-class`) to be used by VGLM/VGAM modelling functions, e.g., `vglm` or `vgam`. ## References Miranda, V. and Yee, T.W. (2018) Vector Generalized Linear Time Series Models. **In preparation**. Porat, B., and Friedlander, B. (1986) Computation of the Exact Information Matrix of Gaussian Time Series with Stationary Random Components. **IEEE Transactions on Acoustics, Speech and Signal Processing**. ASSp-34(1) , 118--130. ## Author(s) Victor Miranda and T. W. Yee ## Note No seasonal components handled yet. If no covariates, $x[t]$, are incorporated in the analysis, then `ARIMAXff` fits an ordinary ARIMA model. Ditto with `ARMAXff`. If `nodrift = TRUE`, then the 'drift' is removed from the vector of parameters and is not estimated. By default, an ARMA model of order--$c(1, 0)$ with order--1 differences is fitted. When initial values are entered (`isd`, `iARcoeff`, etc.), they are recycled according to the number of responses. Also, the ARMA coefficients are intercept--only (note, `zero = c("ARcoeff",` `"MAcoeff")`) This may altered via `zero`, or by constraint matrices (See `constraints`) using `cm.ARMA`. Checks on stationarity and/or invertibility can be manually via `checkTS.VGAMextra`. ## Warning `zero` can be a numeric or a character-strings vector specifying the position(s) or the name(s) of the parameter(s) modeled as intercept--only. Numeric values can be set as usual (See `CommonVGAMffArguments`). If names are entered, the parameter names in this family function are: `c("drift.mean", "noiseVar" || "noiseSD", "ARcoeff", "MAcoeff")`. Manually modify this if required. For simplicity, the second choice is recommended. ## See Also `ARXff`, `MAXff`, `checkTS.VGAMextra`, `cm.ARMA`, `CommonVGAMffArguments`, `constraints`, `vglm`. ## Examples ```r set.seed(3) nn <- 90 theta <- c(0.12, 0.17) # AR coefficients phi <- c(-0.15, 0.20) # MA coefficients. sdWNN <- exp(1.0) # SDs mu <- c(1.25, 0.85) # Mean (not drift) of the process. covX <- runif(nn + 1) # A single covariate. mux3 <- mu[1] + covX ## ## Simulate ARMA processes. Here, the drift for 'tsd3' depends on covX. ## tsdata <- data.frame(TS1 = mu[1] + arima.sim(model = list(ar = theta, ma = phi, order = c(2, 1, 2)), n = nn, sd = sdWNN ), TS2 = mu[2] + arima.sim(model = list(ar = theta, ma = phi, order = c(2, 1, 2)), n = nn, sd = exp(2 + covX)), TS3 = mux3 + arima.sim(model = list(ar = theta, ma = phi, order = c(2, 1, 2)), n = nn, sd = exp(2 + covX) ), x2 = covX) ### EXAMPLE 1. Fitting a simple ARIMA(2, 1, 2) using vglm(). # Note that no covariates involved. ## Not run: fit.ARIMA1 <- vglm(TS1 ~ 1, ARIMAXff(order = c(2, 1, 2), var.arg = FALSE, # OPTIONAL INITIAL VALUES # idrift = c(1.5)*(1 - sum(theta)), # ivar = exp(4), isd = exp(2), # iARcoeff = c(0.20, -0.3, 0.1), # iMAcoeff = c(0.25, 0.35, 0.1), type.EIM = "exact"), data = tsdata, trace = TRUE, crit = "log") Coef(fit.ARIMA1) summary(fit.ARIMA1) vcov(fit.ARIMA1, untransform = TRUE) #------------------------------------------------------------------------# # Fitting same model using arima(). #------------------------------------------------------------------------# # COMPARE to EXAMPLE1 ( fitArima <- arima(tsdata$TS1, order = c(2, 1, 2)) ) ### EXAMPLE 2. Here only the ARMA coefficients and drift are intercept-only. # The random noise variance is not constant. fit.ARIMA2 <- vglm(TS2 ~ x2, ARIMAXff(order = c(2, 1, 2), var.arg = TRUE, lARcoeff = "rhobitlink", lMAcoeff = "identitylink", type.EIM = c("exact", "approximate")[1], # NOTE THE ZERO ARGUMENT. zero = c("drift.mean", "ARcoeff", "MAcoeff")), data = tsdata, trace = TRUE) coef(fit.ARIMA2, matrix = TRUE) summary(fit.ARIMA2) constraints(fit.ARIMA2) ### EXAMPLE 3. Here only ARMA coefficients are intercept-only. # The random noise variance is not constant. # Note that the "drift" and the "variance" are "generated" in # terms of 'x2' above for TS3. fit.ARIMA3 <- vglm(TS3 ~ x2, ARIMAXff(order = c(1, 1, 2), var.arg = TRUE, lARcoeff = "identitylink", lMAcoeff = "identitylink", type.EIM = c("exact", "approximate")[1], nodrift = FALSE, zero = c( "ARcoeff", "MAcoeff")), # NOTE THE ZERO ARGUMENT. data = tsdata, trace = TRUE) coef(fit.ARIMA3, matrix = TRUE) summary(fit.ARIMA3) constraints(fit.ARIMA3) ## End(Not run) ```

ARIMAXff function