Comparing alternative variance specifications

 

Syntax

BD>compare : B

where B is the name of any base or element.

The COMPARE:  command is used to compare two alternative variance-covariance specifications for a single collection of random quantities given by the base B. The compare  control may be issued before the command is used (see §20.13) to set the sources for the alternative variance specifications.

As far as any expectation specifications are concerned, only the current default expectation store is ever referenced; the remainder are ignored. The default expectation store can be changed by using the e  argument to the CONTROL:  command.

Suppose that these sources are belief stores 1 and 2 respectively, and that we wish to compare to . We require . Optionally, the output consists of a series of variance summaries and the corresponding directions. The variance summaries are output whenever the bcd  option is switched on, and they are also output together with the corresponding canonical directions when the bcd+  option is switched on. The pairs representing the variance summaries are retained for future use as the paired operators bcd1  and bcd2 . The variance summaries are always reported and stored in ascending order, i.e. from smallest to largest eigenvalue.

• If the variance specifications share a common null space, there are eigenvectors such that

  

  These eigenvectors are output with a (0,0) pair representing the zero eigenvalues of each specification. The coefficients are normalised so that their squares sum to unity. The pair (0,0) is retained as and .

• If the variance specifications are such that there exist eigenvectors v such that

  

  then these eigenvectors are output normalised so that together with a (0,1) pair representing their respective variances. The pair (0,1) is retained as and .

• If the variance specifications are such that there exist eigenvectors v such that

  

  then these eigenvectors are output normalised so that , together with a (1,0) pair representing their respective variances. The pair (1,0) is retained as and .

• There remains a grid of directions for which the variances are positive under each specification. These directions are output, together with their variances and their ratio using the larger variance as a numerator. The eigenvectors are normalised so that each direction v has variance . The pair of variance summaries is retained as and .

The directions and the variances reported may be interpreted as described in [27] and [34]. Note that the COMPARE:  command may be used quite independently of any adjustment. Thus, the results pertaining to a previous adjustment remain available, and the previous adjustment remains defined.

The directions generated by the COMPARE:  command may be retained as assignments by using the bcd  argument to the KEEP:  command.