A Covariance Matrix for Total Least Squares with Heteroscedastic Data

Details A Covariance Matrix for Total Least Squares with Heteroscedastic Data A Covariance Matrix for Total Least Squares with Heteroscedastic Data

Apr 1999

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999aj....117.1942b&link_type=abstract

The Astronomical Journal, Volume 117, Issue 4, pp. 1942-1948.

Astronomy and Astrophysics

Astronomy

6

Astrometry, Celestial Mechanics, Stellar Dynamics, Methods: Data Analysis

Scientific paper

Total least squares calculates a least-squares solution when error arises not only in the observations themselves, but also in the equations of condition. The covariance matrix constitutes a vital adjunct to any least-squares solution and should always be calculated. Although covariance matrices exist for total least squares, they assume that the observations, although incorporating normal, statistical scatter, nevertheless embody a single, underlying mean error, an assumption known in statistics as ``homoscedasticity.''Yet, many observations in astronomy violate this assumption, particularly when the observations are made over a long time span and with instruments of different precision. This paper gives a covariance matrix for heteroscedastic observations of this nature. When the observations become homoscedastic, this covariance matrix reduces to the standard covariance matrix for total least squares. When the equations of condition are error-free, the covariance matrix coincides with the standard covariance matrix for ordinary least squares. This covariance matrix is applied to 1140 photographic observations of Pluto that exhibit heteroscedastic scatter in order to estimate the mean errors, the covariances, and the correlations of the solution.

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