Fault Detection and Diagnostics for Rooftop Air Conditioners

function,
f ( X , y)
, if it can be assumed that the underlying density is continuous and that
the first partial derivatives of the function evaluated at any X are small. The probability
estimator
X and
f ˆ ( X , y)
is based upon sample values
i
X and
y , where n is the number of sample observations,
vector variable X and σ is sample probability width:
i
y
of the random variables
p is the dimension of the
1
fˆ(
X , y)
=
( p+
1) / 2
(2π
) σ
1
( X − X
) ( X − X ) + ( y − y )
n
i T
i
i 2
+ ∑exp[
−
]
( p 1)
2
n i=
1
2σ
(2)
A physical interpretation of the probability estimate
f ˆ(
X , y)
is that it assigns sample
probability of width σ for each sample
y
i
X and
i
y
, and the probability estimate is the
sum of those sample probabilities. Substituting the joint probability estimate
f ˆ ( X , y)
equation (2) into the conditional mean, equation (1), gives the desired conditional mean of
given X . In particular, combining equations (1) and (2) and interchanging the order of
integration and summation yields the desired conditional mean, designated yˆ ( X ) :
in
ŷ(
X ) =
n
∑
i = 1
n
∑
i = 1
i T
i
( X − X ) ( X − X )
exp[ −
]
2
2σ
∫
i T
i
( X − X ) ( X − X )
exp[ −
]
2
2σ
∞
−∞
∫
∞
−∞
i 2
( y − y )
y exp[ − ] dy
2
2σ
i 2
( y − y )
exp[ − ] dy
2
2σ
(3)
Perform the following integration:
∫
=
∞
−∞
∫
( y − y )
y exp[ −
2
2σ
∞
−∞
( y − y
i )
i
2
] dy
( y − y )
exp[ −
2
2σ
i
2
] d(
y −
y i
) +
∫ ∞ −∞
y
i
( y − y )
exp[ −
2
2σ
i
2
] dy
= −σ
= −σ
y
2
2
∫
∫
= 0 + y
=
i
∫
∞
∞
−∞
∞
−∞
i
−∞
∫
( y − y ) ( y − y )
−
y i ∫ ∞ ( y − y )
exp[ ] d[
− ] + exp[ −
2
2
−
2
2σ
2σ
∞
2σ
i 2
i 2
( y − y ) i ( y − y )
d exp[ − ] + y exp[ ] dy
2
2
2
∫ ∞ −
σ
−∞
2σ
i 2
( y − y )
exp[ − ] dy
∞
2
2σ
∞
−
( y − y )
exp[−
2
2σ
i
i
2
2
] dy
i
2
i
2
] dy
(4)
12