# Compose and Decompose AR(1) Process `compose.ar1` composes AR(1) process realization by given vector(s) of innovations. `decompose.ar1` extracts AR(1) process residuals from time series. ```r compose.ar1(arcoef, innov, init = 0, xregcoef = 0, xreg = NULL, init.xreg = rep(0, length(xregcoef))) decompose.ar1(arcoef, data, init = NA, xregcoef = 0, xreg = NULL, init.xreg = rep(NA, length(xregcoef))) ``` ## Arguments - `arcoef`: A number specifying autoregression coefficient. - `innov`: A univariate or multivariate time series containing the innovations. - `data`: A univariate or multivariate time series containing the process realization(s). - `init`: A number specifying the value of the process just prior to the start value in `innov`/`data`. - `xregcoef`: A vector specifying coefficients for the external regressors. - `xreg`: A matrix-like object of the same row count as `innov`/`data`, specifying the values of external regressors. The default `NULL` means zeroes. - `init.xreg`: A vector specifying the values of external regressors just prior to the start values in `xreg`. The default `NULL` means zeroes. ## Details Here AR(1) process with external regressors is a linear regresson with AR(1) model for the error term: $$ y_t = b_1 x_{t, 1} + \dots + b_k x_{t, k} + z_t%\code{y[t] = b[1]*x[t, 1] + \dots + b[k]*x[t, k] + z[t]} $$ $$ z_t = a z_{t-1} + e_t%\code{z[t] = a*z[t-1] + e[t]} $$ Use `xreg = NULL` for the regular AR(1) process. ## Returns An object of the same type and dimensions as `innov`/`data` (typically time series). ## See Also `arima` for more general $ARMA(p, q)$ processes. ## Examples ```r ## Simple e <- ts(c(0, 1, 0, 1, 0), freq = 12) compose.ar1(0.1, e) compose.ar1(0.1, e, 1) x <- ts(c(0, 1, 0, 1, 0), freq = 12) decompose.ar1(0.1, x) decompose.ar1(0.1, x, 1) ## Multiseries compose.ar1(0.1, ts(cbind(0, 1))) compose.ar1(0.1, ts(cbind(c(0, 1, 0), c(1, 0, 1)))) decompose.ar1(0.1, ts(cbind(0, 1))) decompose.ar1(0.1, ts(cbind(c(0, 1, 0), c(1, 0, 1)))) ## External regressors xreg1 <- rep(2, 5) xreg2 <- matrix(rep(c(2, 1), each = 5), 5, 2) e <- ts(c(0, 1, 0, 1, 0), freq = 12) compose.ar1(0.1, e, xregcoef = 0.5, xreg = xreg1) compose.ar1(0.1, e, xregcoef = 0.5, init = 0, xreg = xreg1, init.xreg = -2) compose.ar1(0.1, e, xregcoef = c(1, -1), xreg = xreg2) x <- ts(c(0, 1, 0, 1, 0), freq = 12) decompose.ar1(0.1, x, xregcoef = 0.5, xreg = xreg1) decompose.ar1(0.1, x, xregcoef = 0.5, init = 0, xreg = xreg1, init.xreg = -2) decompose.ar1(0.1, x, xregcoef = c(1, -1), xreg = xreg2) ## Back-test a <- 0.5 innov <- ts(rnorm(10), frequency = 12) init <- 1 xrcoef <- seq(-0.1, 0.1, length.out = 3) xreg <- matrix(1:30, 10, 3) init.xreg <- 1:3 x <- compose.ar1(a, innov, init, xrcoef, xreg, init.xreg) r <- decompose.ar1(a, x, init, xrcoef, xreg, init.xreg) stopifnot(all.equal(innov, r)) ```

compose.ar1 function