The urban sprawl dynamics: does a neural network understand the spatial logic better than a cellular automata?

Annexe 2 - Statistical functions for validation of the SANNs

x(p) is the generic output component for the p-th input pattern, and the correspondent
target is t(p). M is the number of patterns considered for the statistical measure.

Root Mean Squared Error (RMSE), evaluating the squared root of the semi-mean of the squared prediction errors:

1^M

2M∑ ∑^{( x}(^p )^{-t ( p} ))²

Ï      p=1

Normalized Root Mean Squared Error (NMSE), evaluating the squared root of the mean of the squared prediction
errors, where target and output values was before normalized between 0 and 1:

Real Error (ERR) evaluating the mean of the prediction error:

1^M

ERR = -M ∑(χ ( p ) -1 ( p )) ■

^M p =1

Relative Error (ABSERR) evaluating the mean of the absolute prediction errors:

1^M

ABSERR = -- ∑ ∣x( p ) -1 ( p )∣ .

^M p =1

Squared R (R2), evaluating the squared of the linear correlation coefficient between target and prediction values:

R²

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