sar_lndet function

Approximation of the log determinant $ln|I_n - rho*W|$ of a spatial weight matrix

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.

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 $\rho=-1,\ldots,+1$. 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.

Returns

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

• time: execution time

See Also

do_ldet for computation of log-determinants

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

Author(s)

James P. LeSage, Adapted to R by Miguel Godinho de Matos miguelgodinhomatos@cmu.edu

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")