Modeling, Identification and Control of Irrigation Station with Sprinkling: Takagi-Sugeno Approach - Complex System Modelling and Control Through Intelligent Soft Computations - page 478

this paragraph, statistical tests to validate a fuzzy model based on Root Mean
Square Error test, Variance accounting for, the residues autocorrelation function
and on the cross-correlation between residues and other inputs in the system.

Root Mean Square Error test (RMSE) (Troudi et al. 2011 ):
This is an overall measure of the deviation of total points number from the
expected value.

￿

t

1

N X

N

2

ð

y k ^

Þ ð Þ

RMSE ¼ y k 29

k¼1

Variance accounting for test (VAF) (Troudi et al. 2011 ):
This criterion evaluates the quality percentage of a model by measuring the
normalized variance of the difference between two signals.

￿

^

var y ð y Þ

VAF ¼ 100 % 1 ð 30 Þ

var y ðÞ

Autocorrelation function of the residues:

￿

P N s
k¼1

e ð k ; h Þ e e ð k s ; h Þ e

^ r ee ð s Þ ¼ ð 31 Þ

2

P k¼1

; h Þ e

e ð k

Cross-correlation between residues and inputs previous:

￿

P N s
k

s ; h Þ e

1 u

ð ð k Þ u Þ e ð k

¼

^ r ue ð s Þ ¼ r

P k¼1

ð 32 Þ

q

P k¼1 u

2

; h Þ e

2

ð ð k Þ u Þ e ð k

with

N X

N
k¼1 e ð k Þ

1

e ¼

ð 33 Þ

N X

N

1

u ¼ u ð k Þ

k¼1

: Is the prediction error and u(k) is the system input. x(k) can take either the
value

ɛ

or u(k). Ideally, if the model is valid, the result of these correlation tests
gave the following results:

ɛ