Up: Evolution of the "efficiency''

# 1 Introduction

The complete solution of every parameter estimation by the adjustment of data (observations) through minimizing an objective function (e.g., the sum of the squares of the adjustment residuals) yields not only estimates for the target parameters, but also their covariance matrix. This matrix may be interpreted as the coefficient matrix of the quadratic form representing the error ellipsoid of the target-parameter estimates.

There are two quantities which measure the appropriateness of the available equations used for adjusting the data and estimating the target parameters. The first is the condition number of the system of equations from which the parameter estimates were calculated, it is essentially the ratio of the longest to the shortest principal axes of the error ellipsoid. The second is the efficiency, introduced by Eichhorn (1989), which is a measure for how well the main axes of the parameter estimates' error ellipsoid are aligned with those of a rectangular coordinate system whose axes are along the original target parameters. Efficiency and condition number are thus independent of each other: The introduction of new parameters by subjecting the original target parameters to an orthogonal transformation is equivalent to rotating the error ellipsoid with respect to the coordinate system and will always create a set of uncorrelated parameters with efficiency 1, while leaving the condition number unchanged. Independently thereof, the condition number can be rendered equal to 1 by scaling, although it is hard to appreciate any advantage one might thereby derive.

Likewise, the set of orthogonally transformed original set of target parameters, which renders the problem efficiency equal to 1, is not necessarily of interest; if, e.g., one wants to have estimates of the masses of the components of a binary with the highest possible precision, it is little comfort to know instead the sum of these masses, which can be estimated much more precisely than any of the individual masses.

Seen from a different perspective, the efficiency is a measure for the total amount of correlation between the estimates of the original target parameters as they result from the fit of a certain data set. The efficiency was first used by Eichhorn & Xu (1990) in discussing estimations of the parameters of visual binaries and is defined by

where are the eigenvalues of the covariance matrix of the estimated parameters, and qkk are its diagonal elements. In this expression, m denotes the number of parameters in the model.

Other than by introducing more data (observations) of the same type as the original ones, the precision of the original target data (and concomitantly the efficiency) can be improved by introducing data of a different kind, whose adjustment parameters contain a nonempty subset of the original target parameters, and then fitting both sets of data simultaneously in a combined estimation. In this solution, the set of parameters will then be the union of the sets of the parameters of the original problem and those of the new one.

Take, for example, a binary that has been observed both visually as well as for both components spectroscopically. One would then intuitively expect, that those parameters which are necessary for the fit of both data sets will be estimated with greater precision from a unified solution than from the weighted mean of the discrete estimates obtained by having fitted visual and spectroscopic observations separately. In addition, there are system parameters (e.g., the orbital inclination, the radial velocity of the system's center of gravity) that can be estimated only from analyzing one type of observations but not the other. Other parameters, such as the parallax of the system, can be estimated only from analyzing both data sets simultaneously in a unified solution. The efficiency of combined VB-SB2 solutions would therefore increase in comparison to the one achieved in any of the separate solutions.

Our aim is thus to analyze the behavior of the efficiency on a reasonably large set of double-lined spectroscopic-visual systems to check on the conjecture we have just stated, i.e. to ask: Does the efficiency increase as expected when a VB system is upgraded to a VB-SB2 one?

Up: Evolution of the "efficiency''

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