3.3 Landscape representation

### 3.3.1 Unified matrix method

Consider the self-organizing map to be a two-dimensional array with a rectangular lattice topology. Let the matrix of neurons, of size , be denoted and the matrix of weights be denoted . We can now define the three following distances:

;

;

.

The so-called unified matrix method [Ultsch and Siemon, 1989][Ultsch, 1993] has been proposed to combine the three distances into one matrix of size . For the column positions , and , and for row positions , and , the components of the matrix take their value as follow:

.

Mathematically, this gives:

;

;

;

.

with being the median[12] of the surrounding elements, thus

where denotes the surrounding elements arranged in increasing order of magnitude.

[12] This is obviously not the only possibility. The use of the mean value could also make sense.

Cyberspace geography visualization - 15 October 1995

Luc Girardin, The Graduate Institute of International Studies