sar_lndet {spatialprobit}

Approximation of the log determinant \ln{|I_n - \rho W|} of a spatial weight matrix

Description

Compute the log determinant \ln{|I_n - \rho W|} of a spatial weight matrix W using either the exact approach, or using some approximations like the Chebyshev log determinant approximation or Pace and Barry approximation.

Usage

sar_lndet(ldetflag, W, rmin, rmax)
lndetfull(W, rmin, rmax)
lndetChebyshev(W, rmin, rmax)

Arguments

ldetflag

flag to compute the exact or approximate log-determinant (Chebychev approximation, Pace and Barry approximation). See details.

W

spatial weight matrix

rmin

minimum eigen value

rmax

maximum eigen value

Details

This method will no longer provide its own implementation and will use the already existing methods in the package spatialreg (do_ldet).

ldetflag=0 will compute the exact log-determinant at some gridpoints, whereas ldetflag=1 will compute the Chebyshev log-determinant approximation. ldetflag=2 will compute the Barry and Pace (1999) Monte Carlo approximation of the log-determinant.

Exact log-determinant:
The exact log determinant \ln|I_n - \rho W| is evaluated on a grid from \rho=-1,...,+1. The gridpoints are then approximated by a spline function.

Chebychev approximation:
This option provides the Chebyshev log-determinant approximation as proposed by Pace and LeSage (2004). The implementation is faster than the full log-determinant method.

Value

detval

a 2-column Matrix with gridpoints for rho from rmin,...,rmax and corresponding log-determinant

time

execution time

Author(s)

James P. LeSage, Adapted to R by Miguel Godinho de Matos <[e-mail]>

References

Pace, R. K. and Barry, R. (1997), Quick Computation of Spatial Autoregressive Estimators, Geographical Analysis, 29, 232–247

R. Barry and R. K. Pace (1999) A Monte Carlo Estimator of the Log Determinant of Large Sparse Matrices, Linear Algebra and its Applications, 289, 41–54.

Pace, R. K. and LeSage, J. (2004), Chebyshev Approximation of log-determinants of spatial weight matrices, Computational Statistics and Data Analysis, 45, 179–196.

LeSage, J. and Pace, R. K. (2009), Introduction to Spatial Econometrics, CRC Press, chapter 4

See Also

do_ldet for computation of log-determinants

Examples

require(Matrix)

# sparse matrix representation for spatial weight matrix W (d x d) 
# and m nearest neighbors
d <- 10
m <- 3
W <- sparseMatrix(i=rep(1:d, each=m), 
  j=replicate(d, sample(x=1:d, size=m, replace=FALSE)), x=1/m, dims=c(d, d))

# exact log determinant
ldet1 <- sar_lndet(ldetflag=0, W, rmin=-1, rmax=1)

# Chebychev approximation of log determinant
ldet2 <- sar_lndet(ldetflag=1, W, rmin=-1, rmax=1)

plot(ldet1$detval[,1], ldet1$detval[,2], type="l", col="black", 
  xlab="rho", ylab="ln|I_n - rho W|",
  main="Log-determinant ln|I_n - rho W| Interpolations")
lines(ldet2$detval[,1], ldet2$detval[,2], type="l", col="red")
legend("bottomleft", legend=c("Exact log-determinant", "Chebychev approximation"), 
  lty=1, lwd=1, col=c("black","red"), bty="n")