Signal Based Fault Detection and Diagnosis for Rotating Electrical Machines: Issues and Solutions - Complex System Modelling and Control Through Intelligent Soft Computations - page 277

work the Asymptotic Mean Integrated Squared Error (AMISE) with plug-in
bandwidth selection procedure is used to choose automatically the bandwidth
h [treated in Wand and Jones ( 1994b )]. In the proposed algorithm, KDE is used to
model a speci

c pattern for each motor condition, indeed the features of the current
signals are mapped in the two-dimensional principal component space, representing
speci c signatures of the motor conditions.

2.1.3 Kullback Leibler Divergence

-

Given two continuous PDFs f 1 ð x Þ and fi 2 ð x Þ , a measure of “ divergence ” or “ dis-

tance ” between fi 1 ð x Þ versus f 2 ð x Þ is given in Kullback and Leibler ( 1951 ), as:

Z

f 1 ð x Þ

I 1 : 2 ð X Þ ¼ f 1 ð x Þ log dx ; ð 8 Þ

f 2 ð x Þ

d

R

and between f 2 ð

Þ

versus f 1 ð

Þ

x x is given by:

Z

f 2 ð x Þ

I 2 : 1 ð X Þ ¼ f 2 ð x Þ log dx : ð 9 Þ

f 1 ð x Þ

d

R

Therefore the K - L divergence between fi 1 ð x Þ and fi 2 ð x Þ is:

J ð f 1 ; f 2 Þ ¼I 1 : 2 ð X Þþ I 2 : 1 ð X Þ

Z

ð 10 Þ

f 1 ð x Þ

¼ ð f 1 ð x Þ f 2 ð x Þ Þ log dx :

f 2 ð x Þ

d

R

L divergence, which repre-
sents a non negative measure between two PDFs. In the present work d is 2 and a
discrete form of K

The above equation is known as the symmetric K -

- L divergence is adopted:

n grid

X X

d

log

f 1 ð x ij Þ

J ð f 1 ; f 2 Þ ¼ f 1 ð x ij Þ f 2 ð x ij Þ

x ij Þ :

ð 11 Þ

f 2 ð

i¼1 j¼1

ne a fault index: if if X is the PDF in the PCs
space estimated by KDE of the oncoming current measurements, the motor con-
dition is that which minimizes the K

The K - L divergence allows to de

- L divergence between fi X and fi i that is the ith

PDF related to each motor condition:

ð

f X ; f i Þ;

ð 12 Þ

c ¼ arg min

i

J

where c is the classi cation output.