Fault Detection and Diagnostics for Rooftop Air Conditioners

where the radial basis function φ can be almost any scalar function of one variable we care
to choose, and the λ
i
’s are computed so that all the known values fit exactly. That is,
y
i
=
m
∑
j=
1
λ φ(| x − x |)
(9)
j
i
j
Writing the matrix Φ with elements
Φ
ij
= φ (| xi
− x
j
|)
(10)
we can rewrite equation (9) as m linear equations in m unknowns:
Φλ = y
(11)
where
y
and
λ are the vectors with elements y
i
and λ
i
,
(11), therefore, determines f completely.
i = 1,
L,
m . Solving equation
Computationally, the significant part of the problem is that of solving the linear equations
(11) for λ . The size of the matrix Φ is the number m of data points and so the
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computational effort, which is of order m , may be large. Fortunately, this calculation is
only performed once for a particular set of data points and φ . The work involved to
interpolate for any given point is then considerably less, of order m .
As well as the difficulty of long computation time, there is a risk that as m grows, Φ will
become ill-conditioned. Fortunately, Dyn and Levin (1983) found some well-conditioned
Φ results form choices of φ .
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