Fitting Linear Regression using Gibbs Sampler

This example taken from Congdon (2001) illustrates a Bayesian Linear Regression of December rainfall on November rainfall based on data for ten years. The data is from Lee (1997), where Y is December rainfall and X is November rainfall.

The full conditional densities take the form

By taking prior distribution parameters as μ₁=0, μ₂=0, σ₁²=10000 , σ₂²=1000, γ =0.001 and δ=0.001, for getting random samples from the full conditionals and computing basic statistics, the input is:

MCMC

USE RAINFALL

GIBBS /SIZE=10000 NSAMP=1 BURNIN=1000 GAP=1 RSEED=53478

FULLCOND / VAR='ALPHA' DIST=Z, PAR1='((0/(10000))+((SUM(Y))*(TAU)))/((10*(TAU))+(1/10000))', PAR2='SQR(1/((10*(TAU))+(1/10000)))' INIT=29.0

FULLCOND / VAR='BETA' DIST=Z, PAR1='((0/(1000))+((SUM(Y*(X-MEAN(X))))*TAU))/(((SUM((X- MEAN(X))^2))*TAU)+(1/1000))', PAR2='SQR(1/(((SUM((X-MEAN(X))^2))*TAU)+(1/1000)))' INIT=0.5

FULLCOND / VAR='TAU' DIST=G, PAR1='(10/2)+0.001', PAR2='1/(((1/2)*(SUM((Y-ALPHA-(BETA*(X- MEAN(X))))^2)))+0.001)' INIT=0.5

SAVE GIBBSYORKRAIN.SYD

GENERATE

USE GIBBSYORKRAIN.SYD

LET SIGSQ=1/TAU1

STATS

CBSTAT ALPHA1 BETA1 SIGSQ/MAXIMUM MEAN,MEDIAN MINIMUM SD VARIANCE N PTILE=2.5 50 97.5

The output is:

 
   ALPHA1  BETA1  SIGSQ  N of cases  10000  10000  10000  Minimum  7.494  -1.014  47.178  Maximum  67.112  0.700  3651.797  Median  40.694  -0.163  207.885  Mean  40.636  -0.161  257.108  Standard Dev  5.078  0.139  181.699  Variance  25.785  0.019  33014.685Method = CLEVELAND  2.5 %  30.639  -0.440  88.040  50 %  40.694  -0.163  207.885  97.5 %  50.732  0.121  748.459

SERIES

TPLOT ALPHA1

TPLOT BETA1

TPLOT SIGSQ

By posterior predictive simulation, the December rainfall can be predicted based on the new November rainfall 46.1.

The input is:

LET THETANEW= ALPHA1+BETA1*(46.1-57.8)

LET YNEW= ZRN(THETANEW, SQR(SIGSQ))

STATS

CBSTAT YNEW/MAXIMUM MEAN,MEDIAN MINIMUM SD VARIANCE N PTILE=2.5 50 97.5

The prediction of December rainfall is 42.553 with standard deviation 16.861.

[此贴子已经被作者于2005-2-17 5:27:42编辑过]

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[此贴子已经被作者于2005-2-17 5:55:42编辑过]

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