arima: ARIMA Modeling of Time Series

Description

Fit an ARIMA model to a univariate time series. This function builds on the ARIMA model fitting approach used in stats::arima() by fitting model parameters via a random restart algorithm.

Usage

arima(
  x,
  order = c(0L, 0L, 0L),
  seasonal = list(order = c(0L, 0L, 0L), period = NA),
  xreg = NULL,
  include.mean = TRUE,
  transform.pars = TRUE,
  fixed = NULL,
  init = NULL,
  method = c("CSS-ML", "ML", "CSS"),
  n.cond,
  SSinit = c("Rossignol2011", "Gardner1980"),
  optim.method = "BFGS",
  optim.control = list(),
  kappa = 1e+06,
  diffuseControl = TRUE,
  max_iters = 100,
  max_repeats = 10,
  max_inv_root = 1,
  min_inv_root_dist = 0,
  eps_tol = 1e-04
)

Value

A list of class c("Arima2", "Arima"). This list contains all of the same elements as the output of stats::arima, along with some additional elements. All elements of the output list are:

coef: A vector of AR, MA, and regression coefficients. These can be extracted by the stats::coef method.
sigma2: The MLE of the variance of the innovations.
var.coef: The estimated variance matrix of the coefficients coef, which can be extracted by the stats::vcov method.
mask: A vector containing boolean values, indicating which parameters of the model were estimated.
loglik: The maximized log-likelihood (of the differenced data).
aic: The AIC value corresponding to the log-likelihood.
arma: A compact form of the model specification, as a vector giving the number of AR, MA, seasonal AR and seasonal MA coefficients, plus the period and the number of non-seasonal and seasonal differences.
residuals: The fitted innovations.
call: The matched call.
series: The name of the series x.
code: The convergence value returned by stats::optim.
n.cond: The number of initial observations not used in the fitting.
nobs: The number of observations used for the fitting.
model: A list representing the Kalman Filter used in the fitting.
x: The input time series.
num_starts: Number of restarts before convergence criteria was satisfied.
all_values: Numeric vector of length num_starts containing the loglikelihood of every parameter initialization.

Arguments

x

a univariate time series

order

A specification of the non-seasonal part of the ARIMA model: the three integer components $(p, d, q)$ are the AR order, the degree of differencing, and the MA order.

seasonal

A specification of the seasonal part of the ARIMA model, plus the period (which defaults to frequency(x)). This may be a list with components order and period, or just a numeric vector of length 3 which specifies the seasonal order. In the latter case the default period is used.

xreg

Optionally, a vector or matrix of external regressors, which must have the same number of rows as x.

include.mean

Should the ARMA model include a mean/intercept term? The default is TRUE for undifferenced series, and it is ignored for ARIMA models with differencing.

transform.pars

logical; if true, the AR parameters are transformed to ensure that they remain in the region of stationarity. Not used for method = "CSS". For method = "ML", it has been advantageous to set transform.pars = FALSE in some cases, see also fixed.

fixed

optional numeric vector of the same length as the total number of coefficients to be estimated. It should be of the form $$(\phi_1, \ldots, \phi_p, \theta_1, \ldots, \theta_q, \Phi_1, \ldots, \Phi_P, \Theta_1, \ldots, \Theta_Q, \mu), $$ where $\phi_i$ are the AR coefficients, $\theta_i$ are the MA coefficients, $\Phi_i$ are the seasonal AR coefficients, $\Theta_i$ are the seasonal MA coefficients and $\mu$ is the intercept term. Note that the $\mu$ entry is required if and only if include.mean is TRUE. In particular it should not be present if the model is an ARIMA model with differencing.

The entries of the fixed vector should consist of the values at which the user wishes to “fix” the corresponding coefficient, or NA if that coefficient should not be fixed, but estimated.

The argument transform.pars will be set to FALSE if any AR parameters are fixed. A warning will be given if transform.pars is set to (or left at its default) TRUE. It may be wise to set transform.pars = FALSE even when fixing MA parameters, especially at values that cause the model to be nearly non-invertible.

init

optional numeric vector of initial parameter values. Missing values will be filled in, by zeroes except for regression coefficients. Values already specified in fixed will be ignored.

method

fitting method: maximum likelihood or minimize conditional sum-of-squares. The default (unless there are missing values) is to use conditional-sum-of-squares to find starting values, then maximum likelihood. Can be abbreviated.

n.cond

only used if fitting by conditional-sum-of-squares: the number of initial observations to ignore. It will be ignored if less than the maximum lag of an AR term.

SSinit

a string specifying the algorithm to compute the state-space initialization of the likelihood; see KalmanLike for details. Can be abbreviated.

optim.method

The value passed as the method argument to optim.

optim.control

List of control parameters for optim.

kappa

the prior variance (as a multiple of the innovations variance) for the past observations in a differenced model. Do not reduce this.

diffuseControl

Boolean indicator of whether or initial observations will have likelihood values ignored if controlled by the diffuse prior, i.e., have a Kalman gain of at least 1e4.

max_iters

Maximum number of random restarts for methods "CSS-ML" and "ML". If set to 1, the results of this algorithm is the same as stats::arima() if argument diffuseControl is also set as TRUE. max_iters is often not reached because the condition max_repeats is typically achieved first.

max_repeats

Integer. If the last max_repeats random starts did not result in improved likelihoods, then stop the search. Each result of the optim function is only considered to improve the likelihood if it does so by more than eps_tol.

max_inv_root

positive numeric value less than or equal to 1. This number represents the maximum size of the inverted MA or AR polynomial roots for a new parameter estimate to be considered an improvement to previous estimates. Concerns of numeric stability arise when the size of polynomial roots are near unity circle. The default value 1 means that the the parameter values corresponding with the best log-likelihood will be returned, even if they are near unity. Suitable values of this parameter are near the value 1.

min_inv_root_dist

positive numeric value less than 1. This number represents the minimum distance between AR and MA polynomial roots for a new parameter estimate to be considered an improvement on previous estimates. This is intended to avoid the possibility of returning parameter estimates with nearly canceling roots. Appropriate choices are values near 0.

eps_tol

Tolerance for accepting a new solution to be better than a previous solution in terms of log-likelihood. The default corresponds to a one ten-thousandth unit increase in log-likelihood.

Examples

Run this code

# example code
set.seed(12345)
arima(miHuron_level$Average, order = c(2, 0, 1), max_iters = 100)

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