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LINMIX_ERR

LINMIX_ERR

Name


    LINMIX_ERR

Purpose


      Bayesian approach to linear regression with errors in both 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, variances XSIG^2 and
  YSIG^2, respectively, and covariance XYCOV. The distribution of XI
  is modelled as a mixture of normals, with group proportions PI,
  mean MU, and variance TAUSQR. 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.

Calling Sequence



    LINMIX_ERR, X, Y, POST, XSIG=, YSIG=, XYCOV=, DELTA=, NGAUSS=, /SILENT,
                /METRO, MINITER= , MAXITER=

Inputs



  X - THE OBSERVED INDEPENDENT VARIABLE. THIS SHOULD BE AN
      NX-ELEMENT VECTOR.
  Y - THE OBSERVED DEPENDENT VARIABLE. THIS SHOULD BE AN NX-ELEMENT
      VECTOR.

Optional Inputs



  XSIG - THE 1-SIGMA MEASUREMENT ERRORS IN X, AN NX-ELEMENT VECTOR.
  YSIG - THE 1-SIGMA MEASUREMENT ERRORS IN Y, AN NX-ELEMENT VECTOR.
  XYCOV - THE COVARIANCE BETWEEN THE MEASUREMENT ERRORS IN X AND Y,
          AND NX-ELEMENT VECTOR.
  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).
  METRO - IF METRO = 1, THEN THE MARKOV CHAINS WILL BE CREATED USING
          THE METROPOLIS-HASTINGS ALGORITHM INSTEAD OF THE GIBBS
          SAMPLER. THIS CAN HELP THE CHAINS CONVERGE WHEN THE SAMPLE
          SIZE IS SMALL OR IF THE MEASUREMENT ERRORS DOMINATE THE
          SCATTER IN X AND Y.
  SILENT - SUPPRESS TEXT OUTPUT.
  MINITER - MINIMUM NUMBER OF ITERATIONS PERFORMED BY THE GIBBS
            SAMPLER OR METROPOLIS-HASTINGS ALGORITHM. IN GENERAL,
            MINITER = 5000 SHOULD BE SUFFICIENT FOR CONVERGENCE. THE
            DEFAULT IS MINITER = 5000. THE MCMC IS STOPPED AFTER
            RHAT < 1.1 FOR ALL PARAMETERS OF INTEREST, 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 MCMC IS STOPPED
            AUTOMATICALLY AFTER MAXITER ITERATIONS.
  NGAUSS - THE NUMBER OF GAUSSIANS TO USE IN THE MIXTURE
            MODELLING. THE DEFAULT IS 3. IF NGAUSS = 1, THEN THE
            PRIOR ON (MU, TAUSQR) IS ASSUMED TO BE UNIFORM.

Output



    POST - A STRUCTURE CONTAINING THE RESULTS FROM THE MCMC. EACH
          ELEMENT OF POST IS A DRAW FROM THE POSTERIOR DISTRIBUTION
          FOR EACH OF THE PARAMETERS.
            ALPHA - THE CONSTANT IN THE REGRESSION.
            BETA - THE SLOPE 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.
            TAUSQR - THE GAUSSIAN VARIANCES FOR THE MIXTURE MODEL.
            MU0 - THE HYPERPARAMETER GIVING THE MEAN VALUE OF THE
                  GAUSSIAN PRIOR ON MU. ONLY INCLUDED IF NGAUSS >
                  1.
            USQR - THE HYPERPARAMETER DESCRIBING FOR THE PRIOR
                    VARIANCE OF THE INDIVIDUAL GAUSSIAN CENTROIDS
                    ABOUT MU0. ONLY INCLUDED IF NGAUSS > 1.
            WSQR - THE HYPERPARAMETER DESCRIBING THE `TYPICAL' SCALE
                    FOR THE PRIOR ON (TAUSQR,USQR). ONLY INCLUDED IF
                    NGAUSS > 1.
            XIMEAN - THE MEAN OF THE DISTRIBUTION FOR THE
                      INDEPENDENT VARIABLE, XI.
            XISIG - THE STANDARD DEVIATION OF THE DISTRIBUTION FOR
                    THE INDEPENDENT VARIABLE, XI.
            CORR - THE LINEAR CORRELATION COEFFICIENT BETWEEN THE
                    DEPENDENT AND INDEPENDENT VARIABLES, XI AND ETA.

Called Routines



    RANDOMCHI, MRANDOMN, RANDOMGAM, 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, The Astrophysical
    Journal, 665, 1489 (arXiv:0705.2774)
  Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B., 2004,
    Bayesian Data Analysis, Chapman & Hall/CRC

Revision History



    AUTHOR : BRANDON C. KELLY, STEWARD OBS., JULY 2006
  - MODIFIED PRIOR ON MU0 TO BE UNIFORM OVER [MIN(X),MAX(X)] AND
    PRIOR ON USQR TO BE UNIFORM OVER [0, 1.5 * VARIANCE(X)]. THIS
    TENDS TO GIVE BETTER RESULTS WITH FEWER GAUSSIANS. (B.KELLY, MAY
    2007)
  - FIXED BUG SO THE ITERATION COUNT RESET AFTER THE BURNIN STAGE
    WHEN SILENT = 1 (B. KELLY, JUNE 2009)
  - FIXED BUG WHEN UPDATING MU VIA THE METROPOLIS-HASTING
    UPDATE. PREVIOUS VERSIONS DID NO INDEX MUHAT, SO ONLY MUHAT[0]
    WAS USED IN THE PROPOSAL DISTRIBUTION. THANKS TO AMY BENDER FOR
    POINTING THIS OUT. (B. KELLY, DEC 2011)



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