Fault Detection and Diagnostics for Rooftop Air Conditioners

2
Define the scalar function D i
,
D
i T
= ( X − X ) ( X − X ) . (5)
2 i
i
Substituting equtions (4) and (5) into equation (3), yields the following:
yˆ(
X )
n
∑
i T
i
i ( X − X ) ( X − X )
y exp[ −
]
2
2σ
=
i T
i
( X − X ) ( X − X )
exp[ −
]
2
2σ
i = 1
i=
1
=
n
n
∑
i = 1
n
∑
∑
i=
1
2
i Di
y exp( − )
2
2σ
2
Di
exp( − )
2
2σ
(6)
Because the particular estimator, equation (3), is readily decomposed into X and y
factors, the integrations were accomplished analytically. The resulting regression, equation
(6), which involves summations over the observations, is directly applicable to problems
involving numerical data. Parzen and Cacoullos (1966) have shown that density estimators
of the form of equation (2) used in estimating equation (1) by equation (6) are consistent
estimators (asymptotically converging to the underlying probability density function
f ( X , y)
at all points ( X , y) at which the density function is continuous. Provided that
σ = σ (n)
is chosen as a decreasing function of n such that
and
lim σ ( n)
= 0
n→∞
lim nσ p ( n)
= ∞
(7)
n→∞
The estimate
yˆ ( X )
can be visualized as a weighted average of all of the observed values,
i
y
, where each observed value is weighted exponentially according to its Euclidean
distance from
X . When the smoothing parameter σ is made large, the estimated density
is forced to be smooth and in the limit becomes a multivariate Gaussian with
covariance σ
2
I
. On the other hand, a smaller value of σ allows the estimated density to
assume non-Gaussian shapes, but with the hazard that wild points may have too great an
effect on the estimate. As σ becomes very large,
yˆ ( X )
assumes the value of the sample
mean of the
i
i
y and as σ goes to 0, yˆ ( X ) assumes the value of the y associate with the
i
observation closest to X . For intermediate values of σ , all values of y are taken into
account, but those corresponding to points closer to X are given heavier weight.
13