End Effects of EMD

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End Effects of EMD

• An unsolved, and perhaps,
• unsolvable problem.

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End Effects

• The end effect problem is a self-inflicted wound
  on EMD.
• Traditional method in dealing with the end is to
  use window, which would mask the ends and force
  the ends to be tapered to zero.

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End Effects

• In EMD, we could do the same, but we decided to
  salvage some thing out of the data near the ends.
• We need at least one data point beyond each end
  to stabilize the spline. This data is an
  extremum, so we have to predict or extrapolate
  the sequence of extrema for an extra point or two
  beyond the end.
• When the end points are not extrema, the spline
  could swing wildly. The effects are not limited
  to the neighborhoods of the ends they could
  propagate into the interior of the data
  especially in the low frequency components.

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End Effects

• As EMD method is designed for nonlinear and
  nonstationary data, forecasting is impossible.
  However, we do have the following alleviating
  conditions
• The forecasting of the extra points are all for
  each IMF, which are relatively narrow band and
  more stationary than the data.
• As the extrema points determine the envelope,
  our task is to extend the envelopes, which have
  much slower variation than the IMF data, and
  therefore more forgiving as far as error is
  concerned.

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Solutions for End Effects

• Mirror images simple mirror image, mirror image
  and rotations.
• Mirror images and tapering adding a taper
  function to force the mirrored data decay to
  zero.
• Adding characteristics waves the extra points
  are determined by the average of n-waves (usually
  n3) in the immediate neighborhood of the ends.
• Extension with linear spline fittings near the
  boundaries.
• Pattern comparison with the interior data points.
• Linear predictions that preserve the power
  spectral shape.
• Extensive search for the points with minimum
  interior perturbations.

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Linear Spline near the Boundary

• Wu and Huang, 2009 AADA 1, 1-41

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Improved end-effects-corrected method
maxima
minima
Red point is the determined extrema. The end
points are always both maxima and minima with
different values.
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Zoom in of end point The green point is
determined by the straight line linking the last
two maxima, if it is less than the data at the
end point, the maxima at the end point is chosen
as the data itself. Envelope is determined using
natural spline.
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Zoom in of end point Please notice when using
Hermite Spline method to determine envelope, all
data will be less than the envelope, this fails
when using natural spline. Envelope is determined
using Hermite spline.
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Minima The red point is determined by the
straight green line linking the last two minima,
if it is less than the data at the end point, the
minima at the end point is chosen as the data
itself. Envelope is determined using Natural
spline
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Notice even using Hermite Spline, the data is
still less than the lower envelope. The problem
may be because the last minima is chosen too
small using the current method. Envelope is
determined using Hermite spline.
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A Variation Here, the minima at the end point
is determined as the mean of the data and the
green point, which is determined the straight
line linking the last two minima. Then, the data
will all larger than the lower envelope.
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End Effects
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End Effects Details Beginning
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End Effects Details End
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Linear Prediction
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Linear Predictive Code I
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Linear Predictive Code II
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Present Status of Solutions

• None of the above methods is totally justifiable.
• All of the above methods have been used in one or
  the other occasions with decent results.
• No solution is in sight the forecasting problem
  in nonlinear and nonstationary processes is ill
  posed and might not have solution at all.
• A wide open question for foreseeable future.