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Sharp JX-9400 Informacje Techniczne Strona 73

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The criteria described below are used to establish the most efficient
design.
Criteria for location of the measurement points
The model matrix M
First, let us look at the method used to obtain these coefficients. For each
point, the model is applied, replacing the x
i
by their values given by the
experimental design. A system of equations (one equation for each location)
is obtained this way, which can be written in a matrix notation:
V ¼ MA ð3:32Þ
where:
V (v
1
; v
2
; ...; v
n
) is a vector containing the measured quantities at the n
locations.
M is a matrix, each line of which corresponding to one location. Its first
column is filled with ones and corresponds to a constan t in the model.
The next three columns may contain the coordinates of the locations if
the model conta ins linear terms. The next three columns may contain
the products of these coordinates two by two (for example, x
1
x
2
; x
1
x
3
;
x
2
x
3
) in case of interaction terms and, for a quadratic model, the next
three columns contain the squares of the coordinates. Other models will
produce other matrices.
A is a vector containing the coefficients (e.g. a, b
i
, b
ij
(i 6¼ j) and b
ii
) of the
model.
In the general case, M is rectangular and the least square fit procedure is
used:
A ¼ðM
T
MÞ
1
M
0
V ð3:33Þ
where M
T
is the transposed matrix of M. This equation is also valid if M is
a square matrix, but reduces to the simpler equation:
A ¼ M
1
V ð3:34Þ
In any case, a matrix should be inverted and the determinant of this matrix
should not be zero! Since this determinant can be calculated before making
the measurements, it is a first criterion for the choice of the experimental
design: it should be significantly different from zero.
Variance of the calculated response
If the coefficients are known, an estimate v
e
of the value of the variable v can
be obtained at each location in the enclosure:
v
e
¼ A
T
r ð3:35Þ
where r is the vector (1, x
1
; x
2
; x
3
).
52 Ventilation and Airflow in Buildings
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