Partitioning of Data - Spatial Mathematics: Theory and Practice through Mapping - page 135

Table 6.2 Normalization of Data

Merits Limitations

Divide by the sum total of the attribute's
values so that the resulting ratios
represent percentages of the total.
Enables comparison from one region to
the next, using percentages of the total
(region 1 contains 50% of the sales
while region 2 contains a mere 17% of
the sales) rather than absolute totals.
Divide by values in another attribute:
This procedure may take into account
spatial variation influencing the
original attribute. Population density,
dividing total population per unit by
area of the unit, is a common example.

If total count is important, then normalization of data is not appropriate. For
example, it may be more important to know how many members of a
minority group are present in a particular region, to trigger some funding
mandate, than it is to know what the density of population is within that
particular group. In a group of 100, 35 members of a minority group may
appear fairly “dense”; however, if 50 members are required as a floor for
certain programs to be realized, then the density is irrelevant.
Do not normalize data that has already been normalized, such as rates,
attributes per unit area, and so forth.
A GIS makes it so easy to normalize data that caution must be used when
determining what the denominator is to normalize data by. For example, in
working with Census data, dividing the number of Asian Americans by the
number of housing units does not yield the number of Asian American
households. It is critical to understand each data element so that the results of
the normalization are accurate. One can certainly divide “apples by oranges”
but the results will not necessarily mean anything.

A simple closed curve is one that is topologically equivalent to a circle: Like a
rubber band, it can be snapped back into a circular form, no matter how many
bends it might have along its journey from start back to the same point as
endpoint. The following theorem, from topology, about simple closed curves
is attributed to Camille Jordan (1909).

Jordan Curve Theorem: A simple closed curve J , in the plane, separates
the plane into two distinct domains, each with boundary J .

Simple closed curves are often referred to as Jordan curves. The two separate
domains are often called “inside” and “outside” to fit with intuitive ideas. To
determine which the “inside” is, imagine walking along the curve in a coun-
terclockwise direction. Your left hand points to the inside of the curve; you
are functioning as a continuously turning line tangent to the curve. This view
offers precision and replication of results to the otherwise intuitive notion of
“inside.” An example of a curve that is not a Jordan curve is a figure eight.
Walk along it, counterclockwise, with your left hand pointing to the “inside.”
When you walk past the point where the curve crosses itself, what should be
the “inside” has now become the “outside” ( Figure 6.10 ) .

To illustrate this idea in a real-world context, consider a portion of the Chicago
street map in the Hyde Park (University of Chicago) neighborhood. We assume
(contrary to fact but for the sake of demonstration) that all streets in this area
are two-way streets. Suppose that you take a path from the dot placed at 57th
Street and Woodlawn Avenue and travel north on Woodlawn to 56th Street,
then west on 56th to University Avenue, then south on University Avenue to
57th Street, then east on 57th to Kimbark, then south on Kimbark to 58th
Street, then west on 58th to Woodlawn, and then north on Woodlawn to the