Geography Reference
In-Depth Information
17.3.1
Calculus-Based Definitions
The concept of “change” was originally formalized in calculus. When traversing
along a one dimensional function
y
D
f
(
x
)where
y
is a dependent output on an
independent input
x
, the change of
y
near a given locations
x
0
is computed as
the difference between the function values at
x
0
and at its very near location
x
0
C
x
:
Y
D
f
(
x
0
C
x
)
f
(
x
0
), where
x
is the distance between the two
locations.
When computing the change rate of
y
with respect to
x
, differentiation (Wikipedia
2012-D) is often used. It is the change in
y
against the change in
x
:
m
D
y
/
x
.This
rate of change is defined as the derivative of
y
with respect to
x
. For a non-vertical
line, the derivative is also the slope. Slope is a measure of the steepness of the line
and is constant between any pair of locations. For an arbitrary curve, the derivative
may be different from place to place. The derivative at any location is then computed
as the limit when choosing an extremely small x
W
lim
x
!
0
y
x
. Figure
17.6
shows
examples of calculus-based change definitions of change. Figure
17.5
ashowsthree
functions
f
1
,
f
2
and
f
3
,where
f
1
and
f
2
are linear function and
f
3
is a polynomial
function. Their derivative functions are shown in Fig.
17.5
bwhere
f
1
0
,and
f
2
0
are
constant and
f
3
0
is a decreasing line.
Based on the above definition, the sign of the derivative indicates the change
trend (i.e., positive for increase and vice versa). The value of derivative indicates
the rate of change. Figure
17.5
c shows a function with a change of value occurred
between 4 and 6. The derivative function (shown in Fig.
17.5
d) is negative
in this part, indicating the change. For two or more dimensional data, change
between adjacent locations can occur along any direction. In calculus, a vector
representation is used to model a change of value in space. Gradient and divergence
are the measurements of such changes. Change pattern that are defined based on
calculus include spatial boundary analysis (wombling), where a change is defined
as locations with high gradient value and a collection of such locations forms the
boundary area (Barbujani et al.
1989
; Fitzpatrick et al.
2010
). The problem of
change interval discovery (Zhou et al.
2011
) employs a similar idea.
17.3.2
Statistical-Based Definitions
Statistical models assume that the data are samples drawn from an underlying
generative process using certain known or unknown value distributions. Instead of
looking at value changes in the dataset, statistical approaches often model changes
in terms of statistical parameters such as shifts in mean, standard deviation, quantile,
percentile, etc.
For example, Fig.
17.6
a shows the normalized Sahel precipitation index (July-
October precipitation in Sahel region, Africa) from 1900 to 2011 (JISAO
2011
).
Search WWH ::
Custom Search