Computation of the Exact EIM of an Order-1 Autoregressive Process
Computation of the exact Expected Information Matrix of the Autoregressive process of order-1 (AR(1)) with Gaussian white noise and stationary random components.
AR1EIM(x = NULL, var.arg = NULL, p.drift = NULL,
WNsd = NULL, ARcoeff1 = NULL, eps.porat = 1e-2)x |
A vector of quantiles. The gaussian time series for which the EIMs are computed. If multiple time series are being analyzed, then |
var.arg |
Logical. Same as with |
p.drift |
A numeric vector with the scaled mean(s) (commonly referred as drift) of the AR process(es) in turn. Its length matches the number of responses. |
WNsd, ARcoeff1 |
Matrices. The standard deviation of the white noise, and the correlation (coefficient) of the AR(1) model, for each observation. That is, the dimension for each matrix is N x NOS, where N is the number of observations and NOS is the number of responses. Else, these arguments are recycled. |
eps.porat |
A very small positive number to test whether the standar deviation
( See below for further details. |
This function implements the algorithm of Porat and Friedlander (1986) to recursively compute the exact expected information matrix (EIM) of Gaussian time series with stationary random components.
Given N consecutive data points, {y[0], y[1], …, y[N - 1]} with probability density f(y), the Porat and Friedlander algorithm calculates the EIMs J(n-1)[θ], for all 1 ≤ n ≤ N. This is done based on the Levinson-Durbin algorithm for computing the orthogonal polynomials of a Toeplitz matrix. In particular, for the AR(1) model, the vector of parameters to be estimated under the VGAM/VGLM approach is
η = ( mu^*, log(sigma^2), rhobit(rho)),
where sigma^2 is the variance of the white noise
and mu^* is the drift parameter
(See AR1 for further details on this).
Consequently, for each observation n = 1, …, N, the EIM, Jn[θ], has dimension 3 x 3, where the diagonal elements are:
J[n, 1, 1] = E[ -δ^2 log f(y) / δ (mu^*)^2 ],
J[n, 2, 2] = E[ - δ^2 log f(y) / δ (σ^2)^2 ],
and
J[n, 3, 3] = E[ -δ^2 log f(y) / δ (rho)^2].
As for the off-diagonal elements, one has the usual entries, i.e.,
J[n, 1, 2] = J[n, 2, 1] = E[ -δ^2 log f(y) / δ σ^2 δ rho ],
etc.
If var.arg = FALSE, then σ instead of σ^2
is estimated. Therefore, J[n, 2, 2],
J[n, 1, 2], etc., are correspondingly replaced.
Once these expected values are internally computed, they are returned in an array of dimension N x 1 x 6, of the form
J[, 1, ] = [ J[ , 1, 1], J[ , 2, 2], J[ , 3, 3], J[ , 1, 2], J[ , 2, 3], J[ , 1, 3] ].
AR1EIM handles multiple time series, say NOS.
If this happens, then it accordingly returns an array of
dimension N x NOS x 6. Here,
J[, k, ], for k = 1, …, NOS, is a matrix
of dimension N x 6, which
stores the EIMs for the kth response, as
above, i.e.,
J[, k, ] = [ J[ , 1, 1], J[ , 2, 2], J[ , 3, 3], … ],
the bandwith form, as per required by
AR1.
An array of dimension N x NOS x 6, as above.
This array stores the EIMs calculated from the joint density as a function of
θ = (mu^*, sigma^2, rho).
Nevertheless, note that, under the VGAM/VGLM approach, the EIMs must be correspondingly calculated in terms of the linear predictors, η.
For large enough n, the EIMs, Jn(θ), become approximately linear in n. That is, for some n0,
Jn(θ) -> Jn0(θ) + (n - n0) * Jbar(θ), (*)
where Jbar(θ) is a constant matrix.
This relationsihip is internally considered if a proper value
of n0 is determined. Different ways can be adopted to
find n0. In AR1EIM, this is done by checking
the difference between the internally estimated variances and the
entered ones at WNsd.
If this difference is less than
eps.porat at some iteration, say at iteration n0,
then AR1EIM takes
Jbar(θ)
as the last computed increment of
Jn(θ), and extraplotates
Jk(θ), for all
k ≥ n0 using (*).
Else, the algorithm will complete the iterations for
1 ≤ n ≤ N.
Finally, note that the rate of convergence reasonably decreases if the asymptotic relationship (*) is used to compute Jk(θ), k ≥ n0. Normally, the number of operations involved on this algorithm is proportional to N^2.
See Porat and Friedlander (1986) for full details on the asymptotic behaviour of the algorithm.
Arguments WNsd, and ARcoeff1 are matrices of dimension
N x NOS. Else, these arguments are accordingly
recycled.
For simplicity, one can assume that the time series analyzed has
a 0-mean. Consequently, where the family function
AR1 calls AR1EIM to compute
the EIMs, the argument p.drift is internally set
to zero-vector, whereas x is centered by
subtracting its mean value.
V. Miranda and T. W. Yee.
Porat, B. and Friedlander, B. (1986). Computation of the Exact Information Matrix of Gaussian Time Series with Stationary Random Components. IEEE Transactions on Acoustics, Speech, and Signal Processing, 54(1), 118–130.
AR1.
set.seed(1)
nn <- 500
ARcoeff1 <- c(0.3, 0.25) # Will be recycled.
WNsd <- c(exp(1), exp(1.5)) # Will be recycled.
p.drift <- c(0, 0) # Zero-mean gaussian time series.
### Generate two (zero-mean) AR(1) processes ###
ts1 <- p.drift[1]/(1 - ARcoeff1[1]) +
arima.sim(model = list(ar = ARcoeff1[1]), n = nn,
sd = WNsd[1])
ts2 <- p.drift[2]/(1 - ARcoeff1[2]) +
arima.sim(model = list(ar = ARcoeff1[2]), n = nn,
sd = WNsd[2])
ARdata <- matrix(cbind(ts1, ts2), ncol = 2)
### Compute the exact EIMs: TWO responses. ###
ExactEIM <- AR1EIM(x = ARdata, var.arg = FALSE, p.drift = p.drift,
WNsd = WNsd, ARcoeff1 = ARcoeff1)
### For response 1:
head(ExactEIM[, 1 ,]) # NOTICE THAT THIS IS A (nn x 6) MATRIX!
### For response 2:
head(ExactEIM[, 2 ,]) # NOTICE THAT THIS IS A (nn x 6) MATRIX!Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.