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MLINMIX_ERR

MLINMIX_ERR

Name


    MLINMIX_ERR

Purpose


      Bayesian approach to multiple linear regression with errors in X and Y

Explanation


  PERFORM LINEAR REGRESSION OF Y ON X WHEN THERE ARE MEASUREMENT
  ERRORS IN BOTH VARIABLES. THE REGRESSION ASSUMES :
                ETA = ALPHA + BETA ## XI + EPSILON
                X = XI + XERR
                Y = ETA + YERR
  HERE, (ALPHA, BETA) ARE THE REGRESSION COEFFICIENTS, EPSILON IS THE
  INTRINSIC RANDOM SCATTER ABOUT THE REGRESSION, XERR IS THE
  MEASUREMENT ERROR IN X, AND YERR IS THE MEASUREMENT ERROR IN
  Y. EPSILON IS ASSUMED TO BE NORMALLY-DISTRIBUTED WITH MEAN ZERO AND
  VARIANCE SIGSQR. XERR AND YERR ARE ASSUMED TO BE
  NORMALLY-DISTRIBUTED WITH MEANS EQUAL TO ZERO, COVARIANCE MATRICES
  XVAR^2 FOR X, VARIANCES YSIG^2 FOR Y, AND COVARIANCE VECTORS
  XYCOV. THE DISTRIBUTION OF XI IS MODELLED AS A MIXTURE OF NORMALS,
  WITH GROUP PROPORTIONS PI, MEANS MU, AND COVARIANCES T. BAYESIAN
  INFERENCE IS EMPLOYED, AND A STRUCTURE CONTAINING RANDOM DRAWS FROM
  THE POSTERIOR IS RETURNED. CONVERGENCE OF THE MCMC TO THE POSTERIOR
  IS MONITORED USING THE POTENTIAL SCALE REDUCTION FACTOR (RHAT,
  GELMAN ET AL.2004). IN GENERAL, WHEN RHAT < 1.1 THEN APPROXIMATE
  CONVERGENCE IS REACHED.
  SIMPLE NON-DETECTIONS ON Y MAY ALSO BE INCLUDED
  AUTHOR : BRANDON C. KELLY, STEWARD OBS., JULY 2006

Inputs



  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.

Optional 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.

Output



    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.

Called Routines



    RANDOMCHI, MRANDOMN, RANDOMWISH, RANDOMDIR, MULTINOM

References



  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



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