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If 

is the experimental variance of the measured variable v, the variance



) of the estimated variable is:



ðv

Þ¼r

ðM

MÞ

1

r

ð3:36Þ

A variance function can be deﬁned:

VF ¼



ðv

Þ¼Nr

ðM

MÞ

1

r ð3:37Þ

where N is the number of measurements. VF depends on the experimental

design (M and N) and on the location r and can hence be calculated before

doing any measurement.

If VF depends only on the distance to the origin (or the modulus of r),

the experimental design is said to be isovar iant by rotation. If VF is a

constant within the experimental domain, the design gives a uniform accu-

racy. A good experimental design should have a small variance function, as

constant as possible.

If ðM

MÞ

1

is diagonal, the design is orthogonal. In this case, the

variance function is minimum.

Condition of the model matrix

The condition number of the matrix M plays an important role on the

upper bound of the rela tive errors on the result (see Chapter 7, ‘Upper

bound of the errors’). This condition multiplies the experimental errors and

transfers these errors into the result A. It should therefore be as small as

possible. This number depends on the experimental design and on the model

chosen but does not depend on the results of the measur ements. Hence it can

be calculated before doing any measurement and constitutes one more

criterion, which is relatively easy to compute, for the choice of the best

experimental design. This is a much better criterion than the determinant of

Expendability of the experimental design

It may be interesting that the measurements performed to obtain the co-

eﬃcients of a ﬁrst-degree polynomial are not lost and could be used with

other measurements to expand the polynomial to a higher degree. Some

designs are expandable that way, some others are not.

Examples of experimental designs

Several experimental designs for mapping parallelepiped volumes or a

rectangular area were examined (Roulet et al., 1991). Linear, interaction and

quadratic models were tested. Several of these designs were found to be

unusable (singular matrix or too large a condition number for the quadratic

model). Only good examples are given below.

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