3.1. Overview.

The parameter adjustment module is relatively

straightforward. The processed observations and the

calculated ephemeris data are requested and then

compared, forming the O-C residuals. First, coordi-

nate frame compatibility between the observations

and the synthetic ephemeris is reconciled. The calcu-

lated ephemeris must be transformed to apparent po-

sitions in order to match the observations. The

residuals are characterized, with statistical and de-

scriptive output going to disk as well as to an output

window on-screen. At this point, outlying data

points can be automatically detected and removed.

The core of the module follows with the determina-

tion of parameters via a maximum likelihood estima-

tor. The normal equations are formed and solved,

and the parameters and associated formal error esti-

mates are saved. Finally, the residuals are evaluated,

and the module exits with a solution "acceptability"

code. Figure 4 illustrates the process.

Matrix inversion is accomplished via singular

value decomposition (SVD), which is very robust

and offers useful diagnostics for ill-conditioned ma-

trices. Singularities are automatically detected and

corrected, and the problem parameters are identified.

In essence, if the algorithm encounters an ill-

conditioned matrix, it safely steps around the prob-

lem point(s) and proceeds in such a way as to mine

the matrix for the maximum amount of information.

When a singularity (rare in practice) or degenerate

column (not rare!) is encountered, the combination of

parameters that led to the fault is easily extracted.

Thus, not only are singularities safely handled, but

-- more importantly -- parameter combinations to

which the data are insensitive are automatically iden-

tified. It is unusual to encounter a computational

method that is this reliable and blowup-proof. I have

already developed and tested matrix inversion using

SVD and incorporated it into the Matrix utility class

(Chapter

**??**

). With regard to Newcomb, SVD is a

"plug'n'play" capability.

3.2. Linear Parameter Estimation.

If the observational errors are uncorrelated, then

maximum likelihood estimation becomes a simple

least squares estimation.

*We will assume that the ob-*

*servational errors are uncorrelated and normally*

distributed. More precisely, the data errors are

Chapter 3: The Parameter Adjustment Module

*Chapter 3: Parameter Adjustment*

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Parameter

Adjustment

retrieve

massaged

observations

retrieve

calculated

ephemeris

calculate O-C

quantities

statistical

characterization

of residuals

statistics

form the

normal

equations

solve the

normal

equations

output

parameters

correlations

determine

parameters

evaluate

residuals

exit TRUE

exit FALSE

Figure 4.

Parameter

Adjustment

Module.

removal of

outliers

calculate

apparent

positions