g13af is an easy-to-use version of (G13AEF not in this release). It fits a seasonal autoregressive integrated moving average (ARIMA) model to an observed time series, using a nonlinear least squares procedure incorporating backforecasting. Parameter estimates are obtained, together with appropriate standard errors. The residual series is returned, and information for use in forecasting the time series is produced for use in g13ag and g13ah.
The estimation procedure is iterative, starting with initial parameter values such as may be obtained using g13ad. It continues until a specified convergence criterion is satisfied or until a specified number of iterations have been carried out. The progress of the iteration can be monitored by means of an optional printing facility.
Public Shared Sub g13af ( _ mr As Integer(), _ par As Double(), _ ByRef c As Double, _ kfc As Integer, _ x As Double(), _ nx As Integer, _ <OutAttribute> ByRef s As Double, _ <OutAttribute> ByRef ndf As Integer, _ sd As Double(), _ nppc As Integer, _ cm As Double(,), _ st As Double(), _ <OutAttribute> ByRef nst As Integer, _ kpiv As Integer, _ nit As Integer, _ <OutAttribute> ByRef itc As Integer, _ isf As Integer(), _ res As Double(), _ <OutAttribute> ByRef nres As Integer, _ <OutAttribute> ByRef ifail As Integer _ )
public: static void g13af( array<int>^ mr, array<double>^ par, double% c, int kfc, array<double>^ x, int nx, [OutAttribute] double% s, [OutAttribute] int% ndf, array<double>^ sd, int nppc, array<double,2>^ cm, array<double>^ st, [OutAttribute] int% nst, int kpiv, int nit, [OutAttribute] int% itc, array<int>^ isf, array<double>^ res, [OutAttribute] int% nres, [OutAttribute] int% ifail )
static member g13af : mr : int * par : float * c : float byref * kfc : int * x : float * nx : int * s : float byref * ndf : int byref * sd : float * nppc : int * cm : float[,] * st : float * nst : int byref * kpiv : int * nit : int * itc : int byref * isf : int * res : float * nres : int byref * ifail : int byref -> unit
- Type: array<System..::..Int32>()An array of size On entry: the orders vector of the ARIMA model whose parameters are to be estimated. , , and refer respectively to the number of autoregressive , moving average , seasonal autoregressive and seasonal moving average parameters. , and refer respectively to the order of non-seasonal differencing, the order of seasonal differencing and the seasonal period.Constraints:
- , , , , , , ;
- if , ;
- if , ;
- Type: array<System..::..Double>()An array of size [dim1]Note: dim1 must satisfy the constraint:On entry: the initial estimates of the values of the parameters, the values of the parameters, the values of the parameters and the values of the parameters, in that order.On exit: contains the latest values of the estimates of these parameters.
- Type: System..::..Double%On entry: if , c must contain the expected value, , of the differenced series.If , c must contain an initial estimate of .
- Type: System..::..Int32On entry: must be set to if the constant, , is to be estimated and if it is to be held fixed at its initial value.Constraint: or .
- Type: array<System..::..Double>()An array of size [nx]On entry: the values of the original undifferenced time series.
- Type: System..::..Int32On entry: , the length of the original undifferenced time series.
- Type: System..::..Double%On exit: the residual sum of squares after the latest series of parameter estimates has been incorporated into the model. If the method exits with a faulty input parameter, s contains zero.
- Type: array<System..::..Double>()An array of size [nppc]On exit: the standard deviations corresponding to the parameters in the model ( autoregressive, moving average, seasonal autoregressive, seasonal moving average and , if estimated, in that order). If the method exits with ifail containing a value other than or , or if the required number of iterations is zero, the contents of sd will be indeterminate.
- Type: System..::..Int32On entry: the number of , , , and parameters to be estimated. if the constant is being estimated and if not.Constraint: .
- Type: array<System..::..Double,2>[,](,)[,][,]An array of size [dim1, nppc]Note: dim1 must satisfy the constraint:On exit: the correlation coefficients associated with each pair of the nppc parameters. These are held in the first nppc rows and the first nppc columns of cm. These correlation coefficients are indeterminate if ifail contains on exit a value other than or , or if the required number of iterations is zero.
- Type: System..::..Int32%
- Type: System..::..Int32On entry: must be nonzero if the progress of the optimization is to be monitored using the built-in printing facility. Otherwise kpiv must contain zero. If selected, monitoring output will be sent to the current advisory message unit defined by (X04ABF not in this release). For each iteration, the heading
G13AFZ MONITORING OUTPUT - ITERATION nfollowed by the parameter values, and residual sum of squares, are printed.
- Type: System..::..Int32On entry: the maximum number of iterations to be performed.Constraint: .
- Type: System..::..Int32%On exit: the number of iterations performed.
- Type: array<System..::..Int32>()An array of size On exit: the first four elements of isf contain success/failure indicators, one for each of the four types of parameter in the model (autoregressive, moving average, seasonal autoregressive, seasonal moving average), in that order.Each indicator has the interpretation:
On entry parameters of this type have initial estimates which do not satisfy the stationarity or invertibility test conditions. The search procedure has failed to converge because the latest set of parameter estimates of this type is invalid. No parameter of this type is in the model. Valid final estimates for parameters of this type have been obtained.
The time series supplied to the method is assumed to follow a seasonal autoregressive integrated moving average (ARIMA) model defined as follows:
where is the result of applying non-seasonal differencing of order and seasonal differencing of seasonality and order to the series , as outlined in the description of g13aa. The differenced series is then of length , where is the generalized order of differencing. The scalar is the expected value of the differenced series, and the series follows a zero-mean stationary autoregressive moving average (ARMA) model defined by a pair of recurrence equations. These express in terms of an uncorrelated series , via an intermediate series . The first equation describes the seasonal structure:
The second equation describes the non-seasonal structure. If the model is purely non-seasonal the first equation is redundant and above is equated with :
Estimates of the model parameters defined by
and (optionally) are obtained by minimizing a quadratic form in the vector .
The minimization process is iterative, iterations being performed until convergence is achieved (see [Description] in g13ae for full details), or until the user-specified maximum number of iterations are completed.
The final values of the residual sum of squares and the parameter estimates are used to obtain asymptotic approximations to the standard deviations of the parameters, and the correlation matrix for the parameters. The ‘state set’ array of information required by forecasting is also returned.
Note: if the maximum number of iterations are performed without convergence, these quantities may not be reliable. In this case, the sequence of iterates should be checked, using the optional monitoring method, to verify that convergence is adequate for practical purposes.
Box G E P and Jenkins G M (1976) Time Series Analysis: Forecasting and Control (Revised Edition) Holden–Day
Marquardt D W (1963) An algorithm for least squares estimation of nonlinear parameters J. Soc. Indust. Appl. Math. 11 431
Note: g13af may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the method:
Some error messages may refer to parameters that are dropped from this interface (NPAR, LDCM) In these cases, an error in another parameter has usually caused an incorrect value to be inferred.
- On entry, , i.e., the number of terms in the differenced series is not greater than the number of parameters in the model. The model is over-parameterised.
On entry, .
- On entry, the required size of the state set array st is greater than nx. This occurs only for very unusual models with long seasonal periods or large numbers of parameters. First check that the orders vector mr has been set up as intended. If it has, change to (G13AEF not in this release) with st dimensioned at least (nst), where nst is the value returned by g13af, or computed using the formula in [Parameters] of this document.
- The search procedure in the algorithm has failed. This may be due to a badly conditioned sum of squares function, or the default convergence criterion may be too strict. Use (G13AEF not in this release) with a less strict convergence criterion.
- The inversion of the Hessian matrix in the calculation of the covariance matrix of the parameter estimates has failed.
- This indicates a failure when solving the equations giving the latest estimates of the backforecasts.
- Satisfactory parameter estimates could not be obtained for all parameter types in the model. Inspect array isf for further information on the parameter type(s) in error.
- An internal error has arisen in partitioning res for use by (G13AEF not in this release). This error should not occur; report it to NAG via your site representative.
The computations are believed to be stable.
The time taken by g13af is approximately proportional to .
This example reads observations from a time series relating to the rate of the earth's rotation about its polar axis. Differencing of order is applied, and the number of non-seasonal parameters is , one autoregressive () and two moving average (). No seasonal effects are taken into account.
The constant is estimated. Up to iterations are allowed.
The initial estimates of , , and are zero.
Some intermediate monitoring output from (G13AFZ not in this release) has been omitted.