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Analysis of distorted waveforms using
parametric spectrum estimation
methods and robust averaging
Zbigniew LEONOWICZ
13th Workshop on High Voltage Engineering
Söllerhaus Austria 11-15.09.2006
Robust averaging
• Averaging is probably the most widely used basic
statistical procedure in experimental science.
• Estimation of the location of data („central tendency”)
in the presence of random variations among the
observations
• Data variations can be a result of variations in the
phenomenon of interest or of some unavoidable
measuring errors.
• In signal processing terms, this can be considered as
contamination of useful „signal” by useless „noise”
linearly added to it.
• Since the noise usually has zero mean, averaging
minimizes its contribution, while the signal is preserved,
and the signal to noise ratio is improved
Synchronization
• Averaging consists of applying of any statistical
procedure to extract the useful information from
the background noise.
• When useful data are time-locked to some event
and the noise is not time-locked, it allows the
cancellation of the noise by simple point-bypoint data summation.
• This procedure is equivalent to the use of the
arithmetic mean
Review of robust avearging methods
• Sensitivity of an estimator to the presence of
outliers (i.e. data points that deviate from the
pattern set by the majority of the data set)
• Robustness of an estimator is measured by the
breakdown value
• How many data points need to be replaced by
arbitrary values in order to make the estimator
explode (tend to infinity) or implode (tend to
zero) ?
• Arithmetic mean has 0% breakdown
• Median is very robust with breakdown value 50%
Robust location estimators
• Many location estimators can be presented in
unified way by ordering the values of the sample
as
and then applying the weight function
• where
is a function designed to reduce the
influence of certain observations (data points)
in form of weighting and
represents ordered
data.
Examples
• Median
When the data have the size of (2M+1), the median is the
value of the (M +1)th ordered observation.
• Trimmed mean
For the a-trimmed mean (where p = aN) the weights can
be defined as:
p highest and p lowest samples are removed.
Winsorized mean
• Winsorized mean replaces each observation in
each a fraction (p = aN) of the tail of the
distribution by the value of the nearest
unaffected observation.
• 0 p  0,25N usually, depending on the
heaviness of the tails of the distribution.
Weight functions
Weight functions - other
• TL-mean applies
higher weights for the
middle observations
• tanh estimator applies
smoothly changing
weights to the values
close to extreme, it
can be set to ignore
extreme values
Comparison
Investigations
• IEC harmonic and interharmonic subgroups
calculation IEC Std 61000-4-7, 61000-4-30
• DFT with 5 Hz resolution in frequency
characterize the waveform distortions
Parametric methods
• MUSIC
Eigenvalues of the correlation matrix which
correspond to the noise subspace used for
parameter estimation
• ESPRIT
based on naturally existing shift invariance
between the discrete time series, which leads to
rotational invariance between the corresponding
signal subspaces. Uses signal subspace.
Progr. average of harmonic groups
• dc arc furnace supply
• 11th harmonic group
• 2nd interharmonic group
Results MSE
Method
MSE groups
DFT
0.059
MSE
subgroups
0.791
ESPRIT
0.021
0.169
MUSIC
0.027
0.201
Advantage of Winsorized mean
• When comparing values of power quality indices
obtained from different parts of the same recorded
waveform, a high variability of results appears. To
alleviate this problem, winsorized mean was appplied to
compute averages from spectral data. When using the
value of a=0.2 which means that 20% of ordered data
points were discarded and replaced by nearest
unaffected data.
• In such way the outliers were removed and replaced by
data, which are assumed to belong to “true” spectral
content of investigated waveform.
• The use of winsorized mean instead of usual arithmetic
mean allowed reducing the variance of results by nearly
35%.
Conclusions
• Results show that the highest improvement of
accuracy can be obtained by using the ESPRIT
method (especially for interharmonics
estimation), closely followed by MUSIC method,
which outperform classical DFT approach by over
50%.
• Partially stochastic nature of investigated arc
furnace waveforms caused high variability of
calculated power quality indices. The use of
robust averaging (winsorized mean) helped to
reduce this unwanted variability.
Conclusions
Trimmed estimators are a class of robust estimators of data
locations which can help to improve averaging of
experimental data when:
number of experiments is small
data are highly nonstationary
data include outliers.
Their advantages can be understood as a reasonable
compromise between median which is very robust but
discard too much information and arithmetic mean
conventionally used for averaging which use all data but,
due of this, is sensitive to outliers.
Additional improvement of averaging can be gained by
introducing advanced weighting of ordered data