- NAME
MLINMIX_ERR
- PURPOSE
Bayesian approach to multiple linear regression with errors in X and Y
- EXPLANATION
- RFORM LINEAR REGRESSION OF Y ON X WHEN THERE ARE MEASUREMENT
- RORS IN BOTH VARIABLES. THE REGRESSION ASSUMES
ETA = ALPHA + BETA ## XI + EPSILON
X = XI + XERR
Y = ETA + YERR
- RE, (ALPHA, BETA) ARE THE REGRESSION COEFFICIENTS, EPSILON IS THE
- TRINSIC RANDOM SCATTER ABOUT THE REGRESSION, XERR IS THE
- ASUREMENT ERROR IN X, AND YERR IS THE MEASUREMENT ERROR IN
EPSILON IS ASSUMED TO BE NORMALLY-DISTRIBUTED WITH MEAN ZERO AND
- RIANCE SIGSQR. XERR AND YERR ARE ASSUMED TO BE
- RMALLY-DISTRIBUTED WITH MEANS EQUAL TO ZERO, COVARIANCE MATRICES
- AR^2 FOR X, VARIANCES YSIG^2 FOR Y, AND COVARIANCE VECTORS
- COV. THE DISTRIBUTION OF XI IS MODELLED AS A MIXTURE OF NORMALS,
- TH GROUP PROPORTIONS PI, MEANS MU, AND COVARIANCES T. BAYESIAN
- FERENCE IS EMPLOYED, AND A STRUCTURE CONTAINING RANDOM DRAWS FROM
- E POSTERIOR IS RETURNED. CONVERGENCE OF THE MCMC TO THE POSTERIOR
MONITORED USING THE POTENTIAL SCALE REDUCTION FACTOR (RHAT,
- LMAN ET AL.2004). IN GENERAL, WHEN RHAT < 1.1 THEN APPROXIMATE
- NVERGENCE IS REACHED.
- MPLE NON-DETECTIONS ON Y MAY ALSO BE INCLUDED
- THOR BRANDON C. KELLY, STEWARD OBS., JULY 2006
- PUTS
- X - THE OBSERVED INDEPENDENT VARIABLES. THIS SHOULD BE AN
[NX, NP]-ELEMENT ARRAY.
- Y - THE OBSERVED DEPENDENT VARIABLE. THIS SHOULD BE AN NX-ELEMENT
VECTOR.
- TIONAL INPUTS
- XVAR - THE COVARIANCE MATRIX OF THE X ERRORS, AND
[NX,NP,NP]-ELEMENT ARRAY. XVAR[I,*,*] IS THE COVARIANCE
MATRIX FOR THE ERRORS ON X[I,*]. THE DIAGONAL OF
XVAR[I,*,*] MUST BE GREATER THAN ZERO FOR EACH DATA POINT.
- YVAR - THE VARIANCE OF THE Y ERRORS, AND NX-ELEMENT VECTOR. YVAR
MUST BE GREATER THAN ZERO.
- XYCOV - THE VECTOR OF COVARIANCES FOR THE MEASUREMENT ERRORS
BETWEEN X AND Y.
- DELTA - AN NX-ELEMENT VECTOR INDICATING WHETHER A DATA POINT IS
CENSORED OR NOT. IF DELTA[i] = 1, THEN THE SOURCE IS
DETECTED, ELSE IF DELTA[i] = 0 THE SOURCE IS NOT DETECTED
AND Y[i] SHOULD BE AN UPPER LIMIT ON Y[i]. NOTE THAT IF
THERE ARE CENSORED DATA POINTS, THEN THE
MAXIMUM-LIKELIHOOD ESTIMATE (THETA) IS NOT VALID. THE
DEFAULT IS TO ASSUME ALL DATA POINTS ARE DETECTED, IE,
DELTA = REPLICATE(1, NX).
- SILENT - SUPPRESS TEXT OUTPUT.
- MINITER - MINIMUM NUMBER OF ITERATIONS PERFORMED BY THE GIBBS
SAMPLER. IN GENERAL, MINITER = 5000 SHOULD BE SUFFICIENT
FOR CONVERGENCE. THE DEFAULT IS MINITER = 5000. THE
GIBBS SAMPLER IS STOPPED AFTER RHAT < 1.1 FOR ALPHA,
BETA, AND SIGMA^2, AND THE NUMBER OF ITERATIONS
PERFORMED IS GREATER THAN MINITER.
- MAXITER - THE MAXIMUM NUMBER OF ITERATIONS PERFORMED BY THE
MCMC. THE DEFAULT IS 1D5. THE GIBBS SAMPLER IS STOPPED
AUTOMATICALLY AFTER MAXITER ITERATIONS.
- NGAUSS - THE NUMBER OF GAUSSIANS TO USE IN THE MIXTURE
MODELLING. THE DEFAULT IS 3.
- TPUT
POST - A STRUCTURE CONTAINING THE RESULTS FROM THE GIBBS
SAMPLER. EACH ELEMENT OF POST IS A DRAW FROM THE POSTERIOR
DISTRIBUTION FOR EACH OF THE PARAMETERS.
ALPHA - THE CONSTANT IN THE REGRESSION.
BETA - THE SLOPES OF THE REGRESSION.
SIGSQR - THE VARIANCE OF THE INTRINSIC SCATTER.
PI - THE GAUSSIAN WEIGHTS FOR THE MIXTURE MODEL.
MU - THE GAUSSIAN MEANS FOR THE MIXTURE MODEL.
T - THE GAUSSIAN COVARIANCE MATRICES FOR THE MIXTURE
MODEL.
MU0 - THE HYPERPARAMETER GIVING THE MEAN VALUE OF THE
GAUSSIAN PRIOR ON MU.
U - THE HYPERPARAMETER DESCRIBING FOR THE PRIOR
COVARIANCE MATRIX OF THE INDIVIDUAL GAUSSIAN
CENTROIDS ABOUT MU0.
W - THE HYPERPARAMETER DESCRIBING THE `TYPICAL' SCALE
MATRIX FOR THE PRIOR ON (T,U).
XIMEAN - THE MEAN OF THE DISTRIBUTION FOR THE
INDEPENDENT VARIABLE, XI.
XIVAR - THE STANDARD COVARIANCE MATRIX FOR THE
DISTRIBUTION OF THE INDEPENDENT VARIABLE, XI.
XICORR - SAME AS XIVAR, BUT FOR THE CORRELATION MATRIX.
CORR - THE LINEAR CORRELATION COEFFICIENT BETWEEN THE
DEPENDENT AND INDIVIDUAL INDEPENDENT VARIABLES,
XI AND ETA.
PCORR - SAME AS CORR, BUT FOR THE PARTIAL CORRELATIONS.
- LLED ROUTINES
RANDOMCHI, MRANDOMN, RANDOMWISH, RANDOMDIR, MULTINOM
- FERENCES
- Carroll, R.J., Roeder, K., & Wasserman, L., 1999, Flexible
Parametric Measurement Error Models, Biometrics, 55, 44
- Kelly, B.C., 2007, Some Aspects of Measurement Error in
Linear Regression of Astronomical Data, ApJ, In press
(astro-ph/0705.2774)
- Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B., 2004,
Bayesian Data Analysis, Chapman & Hall/CRC