2.2.5.1.2 Algorithm for ARIMATSA-ARIMA-Algorithm
ARIMA model means an autoregressive integrated moving average model. And it may include autoregressive(AR), moving average (MA) or differencing. In this app, nag function nag_tsa_multi_inp_model_estim (g13bec) is used to fit an ARIMA model [1], and nag function nag_tsa_multi_inp_model_forecast (g13bjc) is used to forecast future values by a known ARIMA model [2].
ARIMA Model
For a general ARIMA model,
,
where is the input time series (t = 1 ... n), P, Q, D, p, q, d are orders of seasonal autoregressive, seasonal moving average, seasonal differencing, autoregressive, moving average and differencing respectively. And s is the seasonal period. c is the mean of the differenced values, , , , are coefficients for seasonal autoregressive, seasonal moving average, autoregressive and moving average. is the residual.
Estimation
Residual series can be obtained by in equation 1. Sum squares of residuals:
Estimation Criterion
Three criteria are available:
Iterate by minimizing D.
are considered as unobserved random variables with known distribution.
where the multiplier M is a function calculated from the ARIMA model arguments.
Minimizing D is equivalent to maximizing the exact likelihood of the data.
but with a different value of M. It is distinct from exact likelihood method only if the mean term is included in the model.
In this app, Marquardt method [4] is used to minimize the objective function.
Quantities
- Residual
Residuals are available at .
- Residual Degrees of Freedom
Differenced series length is: , and .
- Covariance Matrix of Parameters
where H is the linearised least squares matrix in the final iteration.
Forecast
To predict time series at t = n + 1, ... n + L, set for t = n + 1, ... n + L, and calculate the predicted value by reversing Eq 1.
The forecast error variance of can be calculated as:
where Vn is the residual variance of the ARIMA model, and is the "psi-weights" of the model as defined in [3].
Reference
- nag_tsa_multi_inp_model_estim (g13bec)
- nag_tsa_multi_inp_model_forecast (g13bjc)
- George E. P. Box and Gwilym M. Jenkins (1976). Time Series Analysis: Forecasting and Control. (Revised Edition) Holden–Day
- D. W. Marquardt (1963). "An algorithm for least squares estimation of nonlinear parameters". J. Soc. Indust. Appl. Math. 11 431.
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