Information statistic (Entropy), I (Shannon’s)

A measure of the amount of pattern, disorder or information, in a set {x_i} where p_i is the proportion of events or values occurring in the i^th class or range. Note that if p_i=0 then p_ilog₂(p_i) is 0. I takes values in the range [0,log₂(k)]. The lower value means all data falls into 1 category, whilst the upper means all data are evenly spread

Information statistic (Diversity), Div

Shannon’s entropy statistic (see above) standardised by the number of classes, k, to give a range of values from 0 to 1

Dimension (topological), D_T

Broadly, the number of (intrinsic) coordinates needed to refer to a single point anywhere on the object. The dimension of a point=0, a rectifiable line=1, a surface=2 and a solid=3. See text for fuller explanation. The value 2.5 (often denoted 2.5D) is used in GIS to denote a planar region over which a single-valued attribute has been defined at each point (e.g. height). In mathematics topological dimension is now equated to a definition similar to cover dimension (see below)

D_T=0,1,2,3,…

Dimension (capacity, cover or fractal), D_C

Let N(h) represent the number of small elements of edge length h required to cover an object. For a line, length 1, each element has length 1/h. For a plane surface each element (small square of side length 1/h) has area 1/h², and for a volume, each element is a cube with volume 1/h³.

More generally N(h)=1/h^D, where D is the topological dimension, so N(h)= h^‑D and thus log(N(h))=‑Dlog(h) and so D_c=‑log(N(h))/log(h). D_c may be fractional, in which case the term fractal is used

D_c>=0