
2068 IEEE TRANSACTIONS ON POWER DELIVERY, VOL. 29, NO. 5, OCTOBER 2014
C. Hada mard and Haar Transform s
The wavelet has been popular in various transient analysis
applications, especially in signal synthesis and analysis, pattern
recognition, signal and image processing, denoising, and com-
pression. Hadamard and Haar transforms are computatio nally
advantageous over DCT and DST transform s. The Haar func-
tion is defined as a characteristic function of an interval (0, 1).
The shape of the fam ily of
Haar functions, and of a
given index
depends on two parameters and
(6)
For
, and are determined uniquely, so that
and is the remainder, where .When ,
the Haar function is d efi ned as
(7)
When
otherwise.
An example of a 4
4 Haar transformation matrix is as follows:
This indicates that
(8)
This sho w s the orthog onality of
. The main advantage of
the Haar transform is its provision of a multistage analysis rather
than a unique transform, and its capability for data com pression
at some resolution levels in pattern-recognition problems [24].
Most of the e nergy that is focused in some samples con sists of
a considerable amoun t of information.
The Hadamard transform is regarded as exem plifying a gen-
eralized class of Fourier transforms. The Hadamard transform
is considered as being built ou t of size-2 DFTs through decom-
posing an arbitrary input vector into a su perp osition of Walsh
functions. The Hadamard unitary matrix of order “
”
is a matrix and is generated by the iteration rule as shown
(9)
D. Similarity and Dissimilarity Formulas for Clustering
Fig. 2 shows a clu stering method that is proposed in this work
for PD localization in transformers. The algori thm is mainl y
based on assigning N features vector to M clusters, with
.Thefirst feature vector is allocated to the first cluster. The
second feature of the vector’s similarity with the first one will
base its assignment to the first cluster or a new cluster. The
system parameters d e termined by the user are the amount of
Fig. 2. Proposed clustering method for localization.
dissimilarity between the clusters and the maximum number
of clusters M.
is calculated using
(10)
where
and are extracted features fro
m the recorded signals
at 0 and 1.
The most common dissimilari ty between
real-valued vectors
used in practice is the weighted
dissi
milarity
(11)
where
is th e distance between point s, is the weight, and
and are the points that their distances must be defined. By
setting
1, the unweigh ted metric dissimilarity can be
obtained, w hile setting
2, the w ell- known Euclid ean dis-
tance can be determined.
To test another method of dissimilarity, an inner p roduct for-
mula as in (12) is used
(12)
where
shows the maximum similarity and is equal to 1
and
is the similarity between and .Frommanyfor-
mulas for measuring similarities, the Tanimoto d istan ce formula
is chosen. The similarity
is calculated by using
(13)
The second similarity measure is calculated using
(14)
where
is the distance. The t hird similarity is a fuzzy measure.
The degree of similarity between two real-valued variables
and between 0 and 1 can be calculated using
(15)