A Set of Algorithm to Simulate the Dynamics of Land System - Advanced Research on Computer Education, Simulation and Modeling - page 314

4 Explanatory Nonlinear Model

The driving force analysis model for land use pattern in nonlinear form is built based on
land use area percentage grid data. Percentage data was first proposed by Ferrers (1866)
and is becoming increasingly important in statistical analysis. It is usually expressed as
the following vector set:

⎧ ⎫

m

( )

T

∑

m

Ss = , s , " , s ∈ Rs = , 0 < s < 1,

⎨ ⎬

12

m i i

(3)

⎩ ⎭

i = 1

m

∑ (4)
where s i is the i th element of the percentage data, and S i is the original observed value of
s i , or the area of cultivated land and the area of developed land.
Area percentage data is derived from grid data at a certain grid pixel scale. Area
percentage data is constrained by two restriction conditions as follows:

sS = S .

i i j

j = 1

m

∑

s =<< 1, 0 s 1,

(5)

i i

i = 1

m

∑

S =Ω .

i

(6)

i = 1

where Ω is constant and represents the area of the grid pixel.
The grid area percentage data is then obtained for land use categories:

⎧ ⎫

p

⎪ ⎪

( )

T

∑

Yy = , y , " , y ∈ Ry

p

= , 0 < y < 1

⎨ ⎬ 12 p j j

(7)

⎪ ⎪

⎩ ⎭

j = 1

.

Where y j is the area proportion of the j th land use category in the total area of grid
pixels.
Symmetric log-ratio transformation is conducted after the grid is percentage data are
treated to stretch the values of area percentage data from (0, 1) to (-∞, +∞):

⎛ ⎞

p

( )

T

∏

z = ln ⎜ y y ⎟

(8)

" ,

Z = zz , , , z

p

,

j = 1, 2, " , p

⎜ ⎟

j j i

12 p

.

⎝ ⎠

i = 1

∈ (-∞, +∞). Let s j = z j - z p , j = 1, 2… p -1, and through the inverse transformation
we can get Eq (9):

where z j

s

e

j

1

y =

y =

j p − 1 p

p − 1

(9)

∑ ,

s

∑ ,

j = 1, 2, " , p − 1 1 + e

s

1 + e

i

i

.

i = 1 i = 1