Wavelet Transform as a Preprocessor

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Pre-Processing Using Wavelets
Ashish Babbar Estefan Ortiz
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Wavelet Transform as a Preprocessor

• To use of the discrete wavelet transform to
  reduce the input size and draw out good
  features to train a neural network for fault
  detection.
• Discrete wavelet analysis is the decomposition
  of a signal into the so-called approximations and
  details.
• This is accomplished by using shifted and scaled
  versions of a a mother wavelet. At each
  decomposition level, approximation and detail
  coefficients are calculated.
• Approximation and detail coefficients correspond
  to low-frequency and
• high-frequency components of a signal,
  respectively.
• These coefficients give a measure of how closely
  correlated the modified mother wavelet is with
  the original signal.

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Wavelet Transform as a Preprocessor

• The approximation coefficients give a broad
  picture of the signal highlighting the major
  features.
• In applying wavelet analysis to sampled signals,
  a down sampling operation is performed after each
  level of decomposition. This simply means the
  number of data points in the components at level
  j approximation or detail will be reduced by a
  factor of two compared to the corresponding
  number of data points at level (j-1).
• Thus the advantage of using a wavelet transform
  is it reduces the size of the inputs to a neural
  network while at the same time providing good
  features by using the approximation
  coefficients.

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Brief Overview of Wavelets
A wavelet is a function ?(t) ? L2(R) that has the
following properties Discrete wavelet
analysis refers to the decomposition of a signal
into approximations and details coefficients.
This is accomplished using shifted and scaled
versions of the original (mother) wavelet as
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Brief Overview of Wavelets
Given the basis functions ------ we can represent
f(t) as

Where as are the approximation coefficients And
ds are the detail coefficients and defined to
be
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Brief Overview of Wavelets
In the discrete signal case we compute the
Discrete Wavelet Transform by successive low pass
and high pass filtering of the discrete
time-domain signal. This is called the Mallat
algorithm or Mallat-tree decomposition.
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Wavelet Transform as a Preprocessor
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Example Wavelet Decomposition
Original Signal
Approximation Coefficients of level 3
decomposition
DWT
Data size 66
Data size 500
Five levels of decomposition was used for wavelet
transform
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Example Wavelet Decomposition
Reconstruction using the approximation
coefficients of level 3
Approximation Coefficients
Inverse DWT
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Examples of reconstruction using different levels

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Reduction of the Data set

• The few selected wavelet coefficients serve as
  the reduced size data set obtained from a large
  data set which can be then used for fault
  detection and identification.
• The procedure selects the few important wavelet
  coefficients that represent key features of the
  data and discards the fine scale wavelet
  coefficients that represent the noise or
  secondary characteristics in the data.

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Block Diagram

• The SOM is trained and Clustered using the M
  samples of the wavelet coefficients and not the N
  data samples, where MltltN to reduce the
  computational complexity

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Application to Fault Detection
Summary of investigation, data was taken from two
sensors with the following faults.
Data From 2 SensorsFault 1 100-150, Fault 2
150-200, Fault 3 300-500
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Results Training ( Initial SOM Grid )
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Results (SOM Training)
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Results Testing ( Initial SOM Grid)
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Results (SOM Testing)
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Detection of faulty clusters
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Location of Faulty Clusters (Without Wavelets)
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Location of Faulty Clusters (Using Wavelets)
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Conclusions

• Both methods i.e. with and without using wavelets
  as a pre-processor give us the same faulty
  clusters.
• The Computational load is reduced drastically by
  using wavelets as a pre-processor as we train
  the SOM using only the wavelet coefficients
  instead of the entire data set.
• The centroids of the faulty data clusters
  correspond to the actual faults in the data set
  which were known.

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Further Investigations

• Using Principal Component Analysis to reduce the
  number of features in the data.
• Using a competitive wavelet neural network as
  classifier for fault detection.