Form the correlation matrix
of order order
whose
correlations follow the sar2 pattern, a pattern used in crop competition
models. The resulting matrix
is banded and is a constrained AR3 matrix.
mat.sar2(gamma, order, print = NULL)
A banded correlation matrix
whose elements follow an sar2 pattern.
A numeric
containing the two values of gamma
, being
parameters linked with spatial dependence and competition.
The order of the matrix
to be formed.
A character
giving the object to be printed. Currently,
only the claculated values of the ar3parameters
can be printed.
If NULL, nothing is printed.
Chris Brien
The values of the AR3 parameters, phi, are calculated from the gammas as follows:
phi[1] = gamma[1] + 2 * gamma[2]
; phi[2] = -gamma[2] * (2*gamma[2] + gamma[1])
;
phi[3] = gamma[1] * gamma[2] * gamma[2]
.
Then the correlations in the correlation matrix, corr
say, are calculated
from the correlation parameters, phi.
Let omega = 1 - phi[2] - phi[3] * (phi[1] + phi[3])
.
Then the values in
the diagonal of corr
(k = 1
) are one;
the first subdiagonal band (k = 2
) of corr
are equal to
(phi[1] + phi[2]*phi[3]) / omega
;
the second subdiagonal band (k = 3
) of corr
are equal to
(phi[1] * (phi[1] + phi[3]) + phi[2] * (1 - phi[2])) / omega
;
the subsequent subdiagonal bands, (k = 4:order
), of corr
are equal to
phi[1]*corr[k-1] + phi[2]*corr[k-2] + phi[3]*corr[k-3]
.
mat.I
, mat.J
, mat.cor
, mat.corg
,
mat.banded
, mat.exp
,
mat.gau
, mat.ar1
, mat.ar2
, mat.ar3
, mat.sar
,
mat.ma1
, mat.ma2
, mat.arma
corr <- mat.sar2(gamma = c(-0.4, 0.2), order = 4)
corr <- mat.sar2(gamma = c(-0.4, 0.2), order = 4, print = "ar3")
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