Let the beam pointing directions be . Let the beam intensities be modeled by
A sample size of 4096 was created using and selected from uniform distributions in the ranges [10,10], [15,50], respectively. Parameters were , , , . A GM model of 12 modes was trained on the data. To illustrate the ability to create conditional distributions, was computed for a sample of computed for with no additive noise. The result appears in Figure 13.6. The visual effect of this figure is to say to the operator that there are no other values of interest except the peak.

It is also possible to condition on or . The conditional distribution was computed for and . these plots are shown in Figures 13.7,13.8.
Note that the beam output values have distributions symmetric about the value of , as expected. Note also the wider spread of values on outer beams due to the variations in .Estimates of were obtained using formulas (13.8),(13.6). To determine bias, uncorrupted (no noise) values of were created for a range of for fixed at 20, and for a range of for fixed at 2. These two graphs appear in Figures 13.9,13.10.
In each case, the bias error is plotted as a function of the variable parameter. Bias is clearly a function of the operating point. It is also a function of the number of modes and the convergence point of the GM approximation algorithm. Random error was determined by choosing a specific value of and running 300 trials with independent noise added to . The result of 300 trials is shown below.
True Value  Mean  Variance  CR Bound  
2  1.9435  .0550  .0493  
18  18.003  .09756  .0945 
Baggenstoss 20170519