Fault Detection and Diagnostics for Rooftop Air Conditioners

Table 2.1 Best model orders for 3D polynomial fits for experimental data
Variable Best Model to Use RMS Error (F) Maximum Error (F)
T evap 1 st order 0.49 0.99
T sh 3 rd order with cross terms 1.39 3.03
T hg 3 rd order with cross terms 1.00 3.24
T cond 1 st order 0.31 0.61
T sc 2 nd order with cross terms 0.46 1.39
∆T ca
1 st order 0.18 0.48
∆T ea
2 nd order with cross terms 0.23 0.56
2.2.2 General Regression Neural Network
Donald [1991] described a memory-based network, general regression neural network
(GRNN), which is a one-pass learning algorithm with a highly parallel structure. This
approach eliminates the necessity of assuming a specific functional form of the model.
Rather, it allows the appropriate form to be expressed as a probability density function
(pdf) which is empirically determined from the observed data using nonparametric
estimators. Thus, this approach is not limited to any particular form and requires no prior
knowledge of the appropriate form. Secondly, the resulting regression equation can be
implemented in a parallel, neural–network-like structure. Since the parameters of the
structure are determined directly from examples rather than iteratively, the structure
“learns” and can begin to generalize immediately. Considering that the idea and algorithm
of GRNN is adopted in our FDD modeling, the derivation of GRNN is repeated and in
order to better understand the original derivation some omitted intermediate derivation is
added.
Assume that f ( X , y)
represents the known joint continuous probability density function
of a vector random variable, X , and a scalar random variable, y . The conditional mean of
y given X (also called the regression of y on X ) is given by
∞
∫
∫
yf ( X , y)
dy
−∞
E[ y | X ] =
(1)
∞
f ( X , y)
dy
−∞
When the density f ( X , y)
is not known, it should usually be estimated from a sample of
observations of X and y . Here the consistent estimators proposed by Parzen (1962) are
adopted. These estimators are a good choice for estimating the probability density
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