The expected number of points and neighbours, E,, n(X) and E,, s(X;R), were calculated for R=0.05 fixed and for various values of (,) in a grid.

Initial experiments showed following:

• The resulting surfaces (,) -> E,, n(X) and (,) -> E,, s(X;R), are relatively smooth such that a rather coarse grid of (,) can be used.

  The resulting grid values used were:
  
  

  {5,25,45,65,...,485}

  {0.01,0.04,0.07,...,1.00}

• Using a smoothing algorithm each estimated mean value can be based on the average of a rather small number of samples with only a low number of spacing between the samples in the MCMC algorithm.
  We used the loess algorithm (local regression), see Cleveland (1993) for very nice graphical as well as mathematical explanations. At this time we use the implementation in the statistical software package Splus.

  Values in MCMC algorithm for estimating each mean value:
  
  

  nburn = 50000 (number of steps to run in chain before reaching equilibrium (burn-in) and begin the sampling)
  

  nsamp = 200 (number of samples to base the means upon)
  

  nspac = 1000 (number of steps in chain between each sample)
  
  

  Note that for each (,) grid point we need to run 50,000+200*1,000=250,000 steps in a chain in order to estimate the mean values, E,, n(X) and E,, s(X;R).

Program for estimating the means in the (,) grid: estimate_means.cpp (uses the C++ library ppunfix5.h)

The predicted values of the means were computed in the same (,) grid points using loess with parameters =0.1 and =2.

We now have 'observed' means and predicted means in each (,) grid point. Furthermore we have observed the CPU times computing the means in each grid points (1.2 GHz AMD Athlon Thunderbird processor and 256 Mb RAM running under Red Hat Linux 7.1). Not surprisingly is the CPU time linear increasing in E,, n(X).

The data can be found in the file data where the columns contain (,, E,, n(X), E,, s(X;R), CPU time, loess-pred(E,, n(X)), loess-pred(E,, s(X;R)))

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This page was last modified on September 28th 2001