Computes the Wiener forecast operator for a stationary stochastic process.
W Vector of length NOBS containing the stationary time series. (Input)
WNADJ White
noise adjustment factor. (Input)
WNADJ must be greater
than or equal to zero.
EPS Bound on
the normalized mean square error. (Input)
EPS must be in the
range (0, 1) inclusive.
MLFOP Maximum
length of the forecast operator. (Input)
MLFOP must be greater
than or equal to one and less than NOBS.
LFOP Length of the estimated forecast operator. (Output)
FOP Vector of length LFOP containing the estimated forecast operator coefficients. (Output)
NOBS Number of
observations in the stationary time series W. (Input)
NOBS
must be greater than or equal to two.
Default: NOBS = size (W,1).
IWMEAN Option
for estimation of the mean of W. (Input)
Default:
IWMEAN = 1.
IWMEAN |
Action |
0 |
WMEAN is user specified. |
1 |
WMEAN is set equal to the arithmetic mean of W. |
WMEAN Estimate
of the mean of the time series W. (Input,
if IWMEAN = 0;
output, if IWMEAN = 1)
WMEAN is used to
center the time series W prior to estimation
of the forecast operator.
Default: WMEAN = 0.0.
Generic: CALL SPWF (W, WNADJ, EPS, MLFOP, LFOP, FOP [, ])
Specific: The specific interface names are S_SPWF and D_SPWF.
Single: CALL SPWF (NOBS, W, IWMEAN, WMEAN, WNADJ, EPS, MLFOP, LFOP, FOP)
Double: The double precision name is DSPWF.
Routine SPWF
performs least-squares estimation of parameters for successive autoregressive
models of a stationary stochastic process given a sample of
n = NOBS
observations {Wt} for
t = 1,
,
n.
Let
be the estimate of the mean μ of the stochastic process {Wt} where
Consider the autoregressive model of order k defined by
where
and
Successive AR(k) models are fit to the centered data using Durbins algorithm (1960) based on the sample autocovariances
Note that the variance
used in the fitting algorithm is adjusted by the amount δ = WNADJ according to
See Robinson (1967, page 96).
Iteration to the next higher order model terminates when
either the expected mean square error of the model is less than EPS
or when k = MLFOP.
The forecast operator ɸ = (ɸ1, ɸ2,
, ɸk*)T for
k* = LFOP
is contained in FOP.
See also Craddock (1969).
1. Workspace may be explicitly provided, if desired, by use of S2WF/DS2WF. The reference is:
CALL S2WF (NOBS, W, IWMEAN, WMEAN, WNADJ, EPS, MLFOP, LFOP, FOP, CW, WK)
The additional arguments are as follows:
CW Vector of length NOBS containing the centered time series W. (Output)
WK Vector of length 2 * MLFOP + 1. (Output)
2. Informational error
Type Code
3 5 No operator could be found of length less than or equal to MLFOP that produced a normalized mean square error less than EPS.
3 The length of the forecast operator is determined by the arguments EPS and MLFOP. Iteration to a longer forecast operator stops when either the normalized mean square error is less than EPS, or the operator reaches the maximum allowable length, MLFOP.
4. The white noise adjustment factor, WNADJ, is used to modify the the estimate of the variance of the time series W used in the computation of the autocorrelation function of W. In the absence of white noise, WNADJ should be set to zero.
Consider the Wφlfer Sunspot Data (Anderson 1971, page 660) consisting of the number of sunspots observed each year from 1749 through 1924. The data set for this example consists of the number of sunspots observed from 1770 through 1869. Application of routine SPWF to these data produces the following results:
USE GDATA_INT
USE SPWF_INT
USE UMACH_INT
IMPLICIT NONE
INTEGER MLFOP, NOBS, NOUT, IMEAN, I
PARAMETER (MLFOP=1, NOBS=100)
! INTEGER I, IMEAN, LFOP, NCOL, NOUT, NROW
REAL EPS, FOP(MLFOP), RDATA(176,2), W(NOBS), WMEAN, WNADJ
REAL NROW, NCOL, LFOP
!
EQUIVALENCE (W(1), RDATA(22,2))
! Wolfer Sunspot Data for
! years 1770 through 1869
CALL GDATA (2, RDATA, NROW, NCOL)
! Center on arithmetic mean
IMEAN = 0
WMEAN = 46.976
! White noise adjustment
WNADJ = 0.0
! Bound on normalized MSE
EPS = 0.1
! Determine autoregressive model
CALL SPWF (W, WNADJ, EPS, MLFOP, LFOP, FOP, IWMEAN=IMEAN, &
WMEAN=WMEAN)
! Print results
CALL UMACH (2, NOUT)
WRITE (NOUT,99997) LFOP
99997 FORMAT (/, 1X, 'Forecast operator length, LFOP = ', I2)
WRITE (NOUT,99998)
99998 FORMAT (/, 1X, ' I FOP(I)')
DO 10 I=1, LFOP
WRITE (NOUT,99999) I, FOP(I)
99999 FORMAT (1X, I2, 2X, F12.4)
10 CONTINUE
!
END
*** WARNING ERROR 5 from SPWF. No operator could
be found of length
less
*** than or equal
to 1 which produced a normalized mean
square
*** error less
than 1.000000E-01.
Forecast operator length, LFOP = 1
I
FOP(I)
1
0.8063
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