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The Neighbourhood Function

• The neighbourhood function is greatest for units that are close together (especially the same unit) and decreases as the distance between the two units increases.

• It is important to realize that neighbourhood refers to the physical proximity of the units not how close their weight vectors are.

• In BrainWave, the Neighbourhood function is controlled by the MapWeights between units in the Map layer.

• These weights can be automatically set using the SetNeighbourSize Button.

• The SetNeighbourSize button determines physical proximity by the number of pixels between the units on the screen - so moving the units will affect how much one unit is affected when a nearby unit wins the competition for an input.

• The neighbourhood function used:

  N(j, k) = 1/(1 + (d(j, k) / (20² * n²))

  d(j, k) = (x_j - x_k)² + (y_j - y_k)²

  where x_j is the x screen coordinate of unit j, y_j is the y screen coordinate of unit j, 20 is the width in pixels of a unit (rescaling into unit widths) and n is a neighbourhood size parameter that can be altered to change the rate at which the function decreases.