HIGHER-ORDER STATISTICS INTERPRETATION. APPLICATION

TO POWER-QUALITY CHARACTERIZATION

Juan Jos

´

e Gonz

´

alez de la Rosa,

´

Africa Luque

Univ. C

´

adiz. Electronics Area. Research Group PAI-TIC-168, EPSA. Av. Ram

´

on Puyol S/N. E-11202-Algeciras-C

´

adiz-Spain

Carlos G. Puntonet, J. M. G

´

orriz

Univ. Granada. Dept. ATC, ESII. Periodista Daniel Saucedo. E-18071-Granada-Spain

Antonio Moreno Mu

˜

noz

Univ. C

´

ordoba. Electronics Area. Research Group PAI-TIC-168

Campus Rabanales. A. Einstein C-2. E-14071-C

´

ordoba-Spain

Keywords:

Higher-Order statistics, Signal processing, Transient Characterization, Power-Quality.

Abstract:

In this paper we perform a practical review on higher-order statistics interpretation. Concretely we focuss

on an unbiased estimate of the 4th-order time-domain cumulants. Some synthetics involving classical noise

processes are characterized using this unbiased estimate, with the goal of checking its performance and to

provide the scientiﬁc community with another result, dealing with the interpretation of this signal processing

tool. A real-life practical example is presented in the ﬁeld of electrical power quality event analysis. The work

also aims to present a set of general advice in order to save memory and gain speed in a real signal processing

frame, dealing with non-stationary processes.

1 INTRODUCTION

Gaussian processes are completely characterized

by the autocorrelation sequence and its associated

Fourier transform, the power spectrum. In the power

spectrum estimation, the information regarding the

phase of the frequency components of the signal is

not present. The information in the power spectrum is

essentially the same as in the autocorrelation (Nandi,

1999).

However, there are numerous situations where

we have to look beyond the autocorrelation in or-

der to get extra information regarding deviations from

the Gaussian behavior and nonlinear characteriza-

tion. These additional characteristics help us distin-

guish among apparently similar measurement data se-

quences; therefore getting the complete characteriza-

tion of the process.

Data sequences, and their associated power spec-

tra, which have been obtained by multiplying more

than two time-series are called higher-order statistics

(HOS). Their associated Fourier transforms are called

poly-spectra. They contain additional information re-

garding the phase of the frequency components of the

signals under study (Nikias and Petropulu, 1993).

The power spectrum (second-order spectrum) is

a particular case of higher-order spectra. The third-

order spectrum is called the bi-spectrum and the

fourth-order spectrum is called the try-spectrum.

They are deﬁned to be the Fourier transforms or the

third and the fourth-order cumulant sequences, re-

spectively.

Poly-spectra are deﬁned as the higher-order mo-

ment spectra and cumulant spectra can be deﬁned for

both deterministic signals and random processes. Mo-

ment spectra can be very useful in the analysis of de-

terministic signals (transient and periodic), whereas

cumulant spectra are of great importance in the anal-

ysis of stochastic signals.

The motivation of the poly-spectral analysis yields

in three applications: (a) To suppress Gaussian noise

processes of unknown spectral characteristics; the bi-

spectrum also suppress noise with symmetrical prob-

ability distribution, (b) to reconstruct the magnitude

and phase response of systems, and (c) to detect and

characterize nonlinearities in time-series.

In this paper we show the application results deal-

ing with the characterization of random processes,

72

José González de la Rosa J., Luque Á., G. Puntonet C., M. Górriz J. and Moreno Muñoz A. (2007).

HIGHER-ORDER STATISTICS INTERPRETATION. APPLICATION TO POWER-QUALITY CHARACTERIZATION.

In Proceedings of the Second International Conference on Signal Processing and Multimedia Applications, pages 72-78

DOI: 10.5220/0002136100720078

Copyright

c

SciTePress

following the indications in (Nandi, 1999) and in

(Nikias and Petropulu, 1993). A real example in-

volving power quality event analysis is then stud-

ied to show the difference in dealing with real time-

series instead of synthetics. Computational intelli-

gence based in neural classiﬁers are pointed as the

classiﬁcation strategy, once the transients have been

characterized.

The paper is structured as follows. Section 2 in-

cludes the motivation of HOS and an amenable math-

ematical foundation. In Section 3 a time-domain

fourth-order cumulant analysis is applied to time se-

ries regarding realizations of noise processes. Section

4 presents the real-life analysis of electrical transients.

Finally, conclusions are shown in Section 5.

2 HOS IN TIME AND

FREQUENCY DOMAINS

2.1 A Preliminary Example

To motivate the use of HOS a ﬁrst application is out-

lined. We consider two cosine waves of the same fre-

quency with a phase shift of π/2 radians. The signals

and their autocorrelation plots are depicted in Fig. 1.

100 200 300 400

-1

-0.5

0

0.5

1

200 400 600 800

-150

-100

-50

0

50

100

150

200

100 200 300 400

-1

-0.5

0

0.5

1

200 400 600 800

-150

-100

-50

0

50

100

150

200

Figure 1: Two sinusoids with the same frequency, shifted

π/2 radians (Two upper graphs). They have the same

second-order statistics (the autocorrelation sequence).

Figs. 2 and 3 show the contour plots associated

to the third-order cumulant and the bi-spectrum of the

former cosine sequences. Qualitatively (but also in

almost every practical case) we only have to pay at-

tention at the coarser differences between graphs. The

main bump in the center of Fig. 2, which corresponds

to the situation of zero phase shift between the sinu-

soids, disappears in Fig. 3, giving rise to others, sec-

ondary bumps. Thanks to the bi-spectrum, these extra

bumps indicate by the way the presence of a phase

shift.

0 2 4 6 8

x 10

-3

-200

-100

0

100

200

Averaged Signal

Time (s )

Signal (V )

3

rd

Order Cumulant (V

3

)

Lag τ

0

(s )

Lag τ

1

(s )

-5 0 5

x 10

-4

-5

0

5

x 10

-4

-1

0

1

2

3

4

x 10

4

Bispectrum Magnitude (V

3

/Hz

2

)

Frequency f

0

(Hz )

Frequency f

1

(Hz )

-5 0 5

x 10

4

-5

0

5

x 10

4

1

2

3

4

x 10

7

Bispectrum Phase (deg )

Frequency f

0

(Hz )

Frequency f

1

(Hz )

-5 0 5

x 10

4

-5

0

5

x 10

4

-150

-100

-50

0

50

100

150

Figure 2: Third-order statistics of the autocorrelation func-

tion (top-left) corresponding to a sinusoid with no phase

shift. A main bump. in the center is observed in the cu-

mulant plot (top-right).

0 2 4 6 8

x 10

-3

-200

-100

0

100

200

300

Averaged Signal

Time (s )

Signal (V )

3

rd

Order Cumulant (V

3

)

Lag τ

0

(s )

Lag τ

1

(s )

-5 0 5

x 10

-4

-5

0

5

x 10

-4

-5000

0

5000

10000

15000

Bispectrum Magnitude (V

3

/Hz

2

)

Frequency f

0

(Hz )

Frequency f

1

(Hz )

-5 0 5

x 10

4

-5

0

5

x 10

4

2

4

6

8

10

12

14

x 10

6

Bispectrum Phase (deg )

Frequency f

0

(Hz )

Frequency f

1

(Hz )

-5 0 5

x 10

4

-5

0

5

x 10

4

-150

-100

-50

0

50

100

150

Figure 3: Third-order statistics of the autocorrelation func-

tion (top-left) corresponding to a sinusoid with a constant

90 phase shift. Several bumps. appear in the cumulant plot

(top-right).

Let’s revise some mathematical foundations.

2.2 Cumulants and Moments

High-order statistics, known as cumulants, are used to

infer new properties about the data of non-Gaussian

processes (Hinich, 1990; Mendel, 1991; Nikias and

Petropulu, 1993). Before cumulants, due to the lack

of analytical tools, such processes had to be treated as

if they were Gaussian. Cumulants and their associated

Fourier transforms, known as poly-spectra, reveal in-

formation about amplitude and phase, whereas sec-

HIGHER-ORDER STATISTICS INTERPRETATION. APPLICATION TO POWER-QUALITY CHARACTERIZATION

73

ond order statistics (power, variance, covariance and

spectra) are phase-blind (Mendel, 1991; Swami et al.,

2001; De la Rosa et al., 2005; De la Rosa and Ruz-

zante, 2007, ).

Before the deﬁnitions, it is convenient to remark

that cumulants of order higher than 2 are all zero in

signals with Gaussian probability density functions.

What is the same, cumulants are blind to any kind of

a Gaussian process. This is the reason why it is not

possible to separate these signals using the statistical

approach (Nikias and Petropulu, 1993).

The relationship among the cumulant of r stochas-

tic signals, {x

i

}

i∈[1,r]

, and their moments of order

p, p ≤ r, can be calculated by using the Leonov-

Shiryaev formula (Mendel, 1991; Nikias and Petrop-

ulu, 1993)

Cum(x

1

, ..., x

r

) =

(−1)

p−1

· (p − 1)! · E{

i∈s

1

x

i

}

· E{

i∈s

2

x

j

} · · · E{

i∈s

p

x

k

}

(1)

where the addition operator is extended over all

the partitions, like one of the form (s

1

, s

2

, · · · , s

p

),

p = 1, 2, · · · , r; and (1 ≤ i ≤ p ≤ r); s

i

is a set be-

longing to a partition of order p, of the set of integers

1,...,r.

For a zero mean variable, using (1), the second-

, third-, and fourth-order cumulants, are particular

cases and are given by:

Cum(x

1

, x

2

) = E{x

1

· x

2

} (2a)

Cum(x

1

, x

2

, x

3

) = E{x

1

· x

2

· x

3

} (2b)

Cum(x

1

, x

2

, x

3

,x

4

) = E{x

1

· x

2

· x

3

· x

4

}

− E{x

1

· x

2

}E{x

3

· x

4

}

− E{x

1

· x

3

}E{x

2

· x

4

}

− E{x

1

· x

4

}E{x

2

· x

3

}

(2c)

Let {x(t)} be an rth-order stationary random real-

valued process. The rth-order cumulant is deﬁned as

the joint rth-order cumulant of the random variables

x(t), x(t+τ

1

),..., x(t+τ

r−1

),

C

r,x

(τ

1

, τ

2

, . . . , τ

r−1

)

= Cum[x(t), x(t + τ

1

), . . . , x(t + τ

r−1

)]

(3)

For stationary random processes the rth-order cu-

mulant is only a function of r-1 lags. If {x(t)} is non-

stationary then the rth-order cumulant includes time

dependency.

The second-, third- and fourth-order cumulants of

zero-mean x(t) can be expressed using (2) and (3), via

C

2,x

(τ) = E{x(t) · x(t + τ)} (4a)

C

3,x

(τ

1

, τ

2

) = E{x(t) · x(t + τ

1

) · x(t + τ

2

)} (4b)

C

4,x

(τ

1

, τ

2

, τ

3

)

= E{x(t) · x(t + τ

1

) · x(t + τ

2

) · x(t + τ

3

)}

− C

2,x

(τ

1

)C

2,x

(τ

2

− τ

3

)

− C

2,x

(τ

2

)C

2,x

(τ

3

− τ

1

)

− C

2,x

(τ

3

)C

2,x

(τ

1

− τ

2

)

(4c)

By putting τ

1

= τ

2

= τ

3

= 0 in Eq. (4), we obtain

γ

2,x

= E{x

2

(t)} = C

2,x

(0) (5a)

γ

3,x

= E{x

3

(t)} = C

3,x

(0, 0) (5b)

γ

4,x

= E{x

4

(t)} − 3(γ

2,x

)

2

= C

4,x

(0, 0, 0) (5c)

The expressions in Eq. (5) are measurements of

the variance, skewness and kurtosis of the distribu-

tion in terms of cumulants at zero lags (the central

cmulants).

Normalized kurtosis and skewness are deﬁned as

γ

4,x

/(γ

2,x

)

2

and γ

3,x

/(γ

2,x

)

3/2

, respectively. We

will use and refer to normalized quantities because

they are shift and scale invariant. If x(t) is symmetri-

cally distributed, its skewness is necessarily zero (but

not vice versa); if x(t) is Gaussian distributed, its kur-

tosis is necessarily zero (but not vice versa).

2.3 Poly-spectra

We will assume in the following that the cumulant se-

quences satisﬁes the bounding condition:

τ

1

=+∞

X

τ

1

=−∞

· · ·

τ

r−1

=+∞

X

τ

r−1

=−∞

|C

r,x

(τ

1

, τ

2

, . . . , τ

r−1

)| < ∞

(6)

Under this assumption, the higher-order spectra

are usually deﬁned in terms of the rth-order cumu-

lants as their (r-1)-dimensional Fourier transforms

S

r,x

(f

1

, f

2

, . . . , f

r−1

)

=

τ

1

=+∞

X

τ

1

=−∞

· · ·

τ

r−1

=+∞

X

τ

r−1

=−∞

C

r,x

(τ

1

, τ

2

, . . . , τ

r−1

)

· exp[−j2π(f

1

τ

1

+ f

2

τ

2

+ · · · + f

r−1

τ

r−1

)]

(7)

The special poly-spectra derived from (7) are

power spectrum (r=2), bi-spectrum (r=3) and try-

spectrum (r=4). Only power spectrum is real, the oth-

ers are complex magnitudes.

SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

74

Poly-spectra are multidimensional functions

which comprise a lot of information. As a con-

sequence, their computation may be impractical

in some cases. To extract useful information

one-dimensional slices of cumulant sequences and

spectra, and bi-frequency planes are employed in

non-Gaussian stationary processes (Jakubowski et al.,

2002), (De la Rosa et al., 2004, ).

3 ESTIMATES AND

CHARACTERIZATION OF

STATISTICAL PROCESSES

In this section we show the computation results ob-

tained from the application of an estimator. In prac-

tice, the computation of the cumulants and the poly-

spectra is based in estimates. In order to asses the

performance of the selected estimator we test it con-

sidering a 2048-point sample register for each noise

process, as it was catalogued in (Nikias and Petrop-

ulu, 1993). These four noise processes are indistin-

guishable from the second-order perspective, as it is

shown in 4.

−20 −15 −10 −5 0 5 10 15 20

−0.5

0

0.5

1

Gaussian

Second−order cumulants: Autocorrelation sequences

−20 −15 −10 −5 0 5 10 15 20

−0.5

0

0.5

1

Uniform

−20 −15 −10 −5 0 5 10 15 20

−0.5

0

0.5

1

Exponential

−20 −15 −10 −5 0 5 10 15 20

−0.5

0

0.5

1

lag

Laplacian

Figure 4: The four second-order cumulants corresponding

to 4 sample registers, each of which is a realization of a dif-

ferent noise process. They exhibit the same autocorrelation

sequence.

If we look into the fourth-order sequences, sub-

stantial differences are observed, specially those cor-

responding to zero time lags. This can be seen in Figs.

form 5 to 8.

200 400 600 800 1000 1200 1400 1600 1800 2000

0

1

2

3

4

5

6

7

Sample number

Exponentially distributed random process

−20 −15 −10 −5 0 5 10 15 20

−1

0

1

2

3

4

5

6

lag, τ

1

c

4,x

(τ

1

,0,0)

c

4,x

(τ

1

,0,0); τ

2

=τ

3

=0

Figure 5: A 2048-point realization of an exponentially dis-

tributed noise process. Maximum value (at zero lag) =

5.9326585626518 (theoretical=6).

200 400 600 800 1000 1200 1400 1600 1800 2000

−5

0

5

10

Sample number

Laplacian distributed random process

−20 −15 −10 −5 0 5 10 15 20

−5

0

5

10

15

τ

1

c

4,x

(τ

1

,0,0)

c

4,x

(τ

1

,0,0); τ

2

=τ

3

=0

Figure 6: A 2048-point realization of an Laplacian dis-

tributed noise process. Maximum value (at zero lag) =

10.7788 (theoretical=12).

4 FEATURE EXTRACTION IN

ELECTRICAL POWER EVENT

ANALYSIS

The aim of the experiment is to differentiate between

two classes of transients (PQ events), named long-

duration and short-duration. The experiment com-

prises two stages. The feature extraction (classiﬁca-

tion) stage is based on the computation of the cumu-

lants (De la Rosa et al., 2007, ). Each vector’s coordi-

nate corresponds to the local maximum and minimum

of the 4th-order central cumulant. This is the feature-

extraction stage. And the classiﬁcation stage is based

on the application of the competitive layer to the fea-

ture vectors, in order to obtain two clusters in the fea-

ture plane. We use a two-neuron competitive layer,

HIGHER-ORDER STATISTICS INTERPRETATION. APPLICATION TO POWER-QUALITY CHARACTERIZATION

75

200 400 600 800 1000 1200 1400 1600 1800 2000

−2

−1

0

1

2

3

Sample number

Gaussian distributed random process

−20 −15 −10 −5 0 5 10 15 20

−0.1

−0.05

0

0.05

0.1

0.15

τ

1

c

4,x

(τ

1

,0,0)

c

4,x

(τ

1

,0,0); τ

2

=τ

3

=0

Figure 7: A 2048-point realization of an Gaussian dis-

tributed noise process. Extreme value not deﬁned; al the

values surrounds zero.

200 400 600 800 1000 1200 1400 1600 1800 2000

−1.5

−1

−0.5

0

0.5

1

1.5

Sample number

Uniformly distributed random process

−20 −15 −10 −5 0 5 10 15 20

−1.5

−1

−0.5

0

lag, τ

1

c

4,x

(τ

1

,0,0)

c

4,x

(τ

1

,0,0); τ

2

=τ

3

=0

Figure 8: A 2048-point realization of a uniformly dis-

tributed noise process. Extreme value (for zero lag) = -

1.15845526517794 (theoretical=-1).

which receives two-dimensional input feature vectors

in this training stage.

We analyze a number of 16 1000-point (roughly)

real-life registers during the feature extraction stage.

Before the computation of the cumulants, two pre-

processing actions have been performed over the sam-

ple signals. First, they have been normalized because

they exhibit very different-in-magnitude voltage lev-

els. Secondly, a high-pass digital ﬁlter (5th-order But-

terworth model with a characteristic frequency of 150

Hz) eliminates the low frequency components which

are not the targets of the experiment. This by the way

increases the non-Gaussian characteristics of the sig-

nals, which in fact are reﬂected in the higher-order

cumulants.

After ﬁltering, a 50-point sliding battery of cen-

tral cumulants (2nd, 3rd and 4th order) are calculated.

The window’s width (50 points) has been selected nei-

ther to be so long to cover the whole signal nor to be

very short. The algorithm calculates the 3 central cu-

mulants over 500 points, and then it jumps to the fol-

lowing starting point; as a consequence we have 98

per cent overlapping sliding windows (49/50=0.98).

Thus, each computation over a window (called a seg-

ment) outputs 3 cumulants.

Fig. 9 shows an example of signal processing

analysis of two sample registers corresponding to

a long-duration and a short-duration events, respec-

tively.

0.02 0.04 0.06 0.08

−0.5

0

0.5

1

Analysis of a long−duration transient

0.02 0.04 0.06 0.08

−0.5

0

0.5

Time(s)

100 200 300 400 500

0.02

0.04

0.06

0.08

0.1

0.12

2

nd

−order sliding cumulant

100 200 300 400 500

−5

0

5

10

x 10

−3

3

rd

−order sliding cumulant

100 200 300 400 500

−5

0

5

10

x 10

−3

Number of segment

4

th

−order sliding cumulant

0.02 0.04 0.06 0.08 0.1

−0.5

0

0.5

1

Analysis of a short−duration transient

0.02 0.04 0.06 0.08 0.1

−0.5

0

0.5

Time(s)

100 200 300 400 500 600

0.02

0.04

0.06

0.08

2

nd

−order cumulant

100 200 300 400 500 600

−0.01

0

0.01

3

rd

−order cumulant

100 200 300 400 500 600

0

5

10

x 10

−3

Number of segment

4

th

−order cumulant

Figure 9: Long-duration vs. short-duration transient anal-

ysis. From top to bottom: the original data record, the ﬁl-

tered sequence, 2nd-3rd-4th-order cumulants sliding win-

dows, respectively.

The 2nd-order cumulant sequence corresponds to

the variance, which clearly indicates the presence of

an event, due to the excess of power. Both types

of transients exhibit an increasing variance in the

neighborhood of the PQ event, that presents the same

shape, with only one maximum. The magnitude of

this maximum is by the way the only available feature

which can be used to distinguish different events from

the second-order point of view. This may suggest the

use of additional features in order to distinguish dif-

ferent types of events.

For this reason the higher-order central cumu-

lants are chosen. An unbiased estimator of the cu-

mulants has been selected. Third-order diagrams

don’t show quite different clusters if we consider a

bi-dimensional space (2 coordinates for each feature

vector) because maxima and minima are similar. It is

possible to differentiate PQ events from the 3rd-order

perspective if we consider more features in the input

vector, like the number of extremes (maxima and min-

ima), and the order in which the maxima and the min-

SIGMAP 2007 - International Conference on Signal Processing and Multimedia Applications

76

ima appear as time increases. In this paper we have

focussed the experience on a bi-dimensional represen-

tation (2-dimensional feature vectors) because we ob-

tain very intelligible 2-D graphs.

Fourth-order sliding cumulants exhibit clear dif-

ferences, not only for the shape of the computation

graph (the bottom graphs in Fig. 9, but also for the

different location of minima, which suggest a cluster-

ing zone for the points.

Fig. 10 presents the results of the training stage,

using the Kohonen rule. The horizontal (vertical) axis

corresponds to the maxima (minima) value. Each

cross in the diagram corresponds to an input vector

and the circles indicate the ﬁnal location of the weight

vector (after learning) for the two neurons of the com-

petitive layer. Both weight vectors point to the aster-

isk, which is the initializing point (the midpoint of the

input intervals).

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

x 10

−3

Cluster classification. 4th−order cumulant

cumulant maxima

cumulant minima

:measured vector

:initial neuron weight vector

:final neuron weight vector

Figure 10: Competitive layer training results over 20

epochs. Upper cluster: Short-duration PQ-events. Down

cluster: Long-duration event.

The separation between classes (inter-class dis-

tance) is well deﬁned. Both types of PQ events are

horizontally clustered. The correct conﬁguration of

the clusters is corroborated during the simulation of

the neural network, in which we have obtained an ap-

proximate classiﬁcation accuracy of 97 percent. Dur-

ing the simulation new signals (randomly selected

from our data base) were processed using the method

described.

The accuracy of the classiﬁcation method in-

creases with the number of data. To evaluate the con-

ﬁdence of the statistics a signiﬁcance test have been

conducted. This informs if the number of experiments

is statistically signiﬁcant according to the ﬁtness test

(

¨

Omer Nezih Gerek and Ece, 2006). As a result of

the test, the number of measurements is signiﬁcatively

correct.

5 CONCLUSION

In this work we have reviewed computation of higher-

order statistics. Concretely we have focussed on a

4-order estimates. We have also proposed a method

to detect and classify two electrical power transients,

named short and long-duration. The method com-

prises two stages. The ﬁrst includes pre-processing

(normalizing and ﬁltering) and outputs the 2-D fea-

ture vectors, each of which coordinate corresponds to

the maximum and minimum of the central cumulants.

The second stage uses a neural network to classify the

signals into two clusters. This stage is different-in-

nature from the one used in (

¨

Omer Nezih Gerek and

Ece, 2006) consisting of quadratic classiﬁers. The

conﬁguration of the clusters is assessed during the

simulation of the neural network, in which we have

obtained an acceptable classiﬁcation accuracy.

ACKNOWLEDGEMENTS

The authors would like to thank the Spanish Min-

istry of Education and Science for funding the project

DPI2003-00878 which involves noise processes mod-

eling, and the PETRI project PTR95-0824-OP in-

volving higher-order statistics. Also thanks to the

Andalusian Government for the trust put in the re-

search group PAI-TIC-168, and for supporting the ex-

cellence project PAI2005-TIC00155, which involves

higher-order statistics.

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