(C118) Mass Fractal Dimension

r =                   number of cells on side of box.

n_r =     number of cells of corresponding class in box of size r.

Description

MFRAC equals the slope of the regression line obtained by regressing the logarithm of mean box mass (mean number of cells of focal class in box) against the logarithm of box size (number pixels on a side) for boxes centered on all cells of the focal class. That is, MFRAC is the coefficient b₁ derived from a least squares regression fit to the following equation: ln(box mass) = b₀ + b₁⋅ln(box size).

Units 

None

Range

0 < MFRAC ≤ 2

MFRAC approaches 0 when the percentage of the landscape comprised of the focal class approaches zero. MFRAC increases as an increasing percentage of the landscape is comprised of the focal class and it is maximally aggregated. MFRAC approaches 2 when the focal class comprises 100% of the landscape.

Comments

Mass fractal dimension is based on the scaling relationship between box mass (i.e., the number of pixels of a focal class within a window) and the size of the box defining the window. Specifically, a range of box sizes is used to delineate windows, from 3 pixels on a side to a maximum of approximately ⅓ of the landscape extent. For each box size, the mean number of pixels of the focal class is determined by centering the box on every pixel of that class and counting the number of pixels of that class in the box sample. Mass fractal dimension is equal to the slope derived from regressing the log of the mean number of pixels for each box size on the log of the box lengths. An alternative form of calculating this measure, known as the box dimension, is derived by placing a grid over the entire landscape and counting the number of pixels of the selected class in each grid cell, and repeating this procedure using grids of increasingly finer resolutions. According to Voss (1988), both approaches yield comparable results.