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The Normalized Gaussian Category Model
[Shepard 1987] provides an elegant argument for his claim that the
probability of generalization of an existing (known) category to a new
(unknown) stimulus is a monotonic function of the normalized distance in
psychological space of the unknown stimulus to known stimuli belonging to
the category. He further specifies that this function can be approximated
by a simple exponential decay or, under certain circumstances, a Gaussian
function. The distance metric is either Euclidean, resulting in circular
(or spheroid) contours of equal generalization, or a slight variant that
results in elliptic (or ellipsoid) contours of equal generalization.
Following Shepard's suggestion, I have used a variant of the Gaussian
normal distribution as a category model, which I will refer to as the
normalized Gaussian model. The usual normal curve in one variable (as used
in probability theory) is given by
where 
is the standard deviation (determining the ``width'' of the
curve), and 
is the mean or expected value (determining the location
of the maximum) (Fig. 
).
Since the normal curve is used as a probability density function, it has
the special property that 
. The term 
in
equation is the Euclidean distance of the
one-dimensional point 
to the mean . To derive the normalized
Gaussian function, we drop the factor 
,
since we don't need the interpretation as a probability density function,
and we substitute the general N-dimensional Euclidean distance function for
the distance term, which gives us
with symbols as in equation . An example of a
two-dimensional version of this function is shown in
Figure .
This function has a number of interesting properties for use as a basic
color category model:
- The maximum value occurs at and is unity, regardless of the
value of .
- The ``width'' of the curve is a function of the parameter
only.
- The value decreases monotonically (but not linearly) as a function of
the distance to .
- It is strictly positive (non-zero) everywhere except at
infinite distance from .
- It has a simple mathematical definition, using only two
parameters and .
These properties allow us to interpret the normalized Gaussian function as
a category model, with interpreted as the location of the center or
focus of the category, and the function value interpreted as the goodness
value (or alternatively, the fuzzy membership value) of a stimulus
represented by its color space coordinates. By modulating the value of
we can affect the size of the volume of the color space that is
included in the category, relative to some threshold function value. The
maximum goodness (or membership) value is unity, as is common in fuzzy set
models, but we can get a goodness value for any point in the color space,
no matter how far removed from the focus. This is important for our
purpose, as I will show below. The non-linear decrease of goodness with
distance from the focus reflects the general shape of psychological
categories as discussed by [Shepard 1987], and it can result in category
extensions different from those obtained with a simple nearest-neighbor
criterion, since neighboring categories may have different values of
. The representation is also economical, since a category can be
represented by only two parameters.