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Re: Moving Least Squares, Anyone?


From: Joe Koski
Subject: Re: Moving Least Squares, Anyone?
Date: Mon, 21 Nov 2005 13:53:36 -0700
User-agent: Microsoft-Entourage/11.2.1.051004

OK, I just tried the references and there are problems. Sorry. The Huang paper has disappeared from the internet. I have a .pdf if anyone needs it. The next best reference on EMD is a .pdf of the paper by Rilling. You can get it by googling ON EMPIRICAL MODE DECOMPOSITION AND ITS ALGORITHMS.

Joe


on 11/21/05 1:22 PM, Joe Koski at address@hidden wrote:

I tried Thorsten’s code on another problem, and it works as advertised. Yes the edges are an issue, but that’s the case with most filtering approaches. Also the code does cause noticeable pauses during execution. I do like the trick of using the 0th derivative to smooth the function itself.

OK, I got many suggestions (Savitzky-Golay, Kalman filters, etc.), but I should clarify what I’m after. There is a recent paper by Blakely (http://
www.cscamm.umd.edu/publications/, Christopher D. Blakely "A Fast Empirical Mode Decomposition Technique for Nonstationary Nonlinear Time Series") that describes a new approach to Empirical Mode Decomposition (EMD). I’m trying to implement the contents of Blakely’s paper in octave.

EMD is a means of decomposing a signal after-the-fact into a series of “intrinsic mode functions” that can be summed together to recover the original signal. The method was described in Norden Huang’s paper (http://www.inrialpes.fr/is2/people/pgoncalv/pub/emd-eurasip03.pdf). The original approach uses cubic splines to connect the signal envelopes. The upper envelope connects all the maxima of the signal, and the lower envelope connects all the minima. A signal that is the mean of the upper and lower envelopes is repetitively calculated and subtracted from the data until there is no change. I have tried the approach on several experimental data sets, and, once the rough edges are knocked off, I think it will be a good tool to add to the signal analysis arsenal. Among other things, it allows a time resolved frequency spectra to be calculated in a manner better than I’ve seen with wavelet transforms. Huang’s paper lists many other advantages.

Blakely has an approach that uses “moving least-squares,” as developed by Fasshauer, to replace the cubic splines. Blakely says that moving least-squares converts the least-squares curve fit into a Lagrangian multiplier problem, which should be near and dear to heart of physicists everywhere. This may be a case of applied mathematicians plowing the same field that the signal analysis people plowed twenty or more years ago, but with different results, and the approach sounds interesting. I also like the looks of the envelopes calculated with moving least-squares. They seem to have fewer over- and undershoots of the extrema than cubic splines.

My question is specific. Has anyone seen an implementation of Fasshauer’s moving least-squares in a language (Fortran, C, etc.) suitable for use with octave? I could then modify the octave EMD code to replace the cubic spline routines and give the approach a try. I contacted Blakely, but he says that he did all his proof-of-concept coding in Java. He’s also now working on his PhD thesis, and this was only a summer diversion.

Joe


on 11/19/05 6:15 AM, Thorsten Meyer at address@hidden wrote:

I have been using the attached function for data smoothing and calculation of smoothed derivatives for non-equally spaced data. It uses moving least-squares polynomial fits of the given data. If I remember it right, the savitzky-golay algorithm does the same, only that the procedure is nicely vectorized for equally spaced data.

My code is terrible:
But the function has been working for me for quite some time now. And you can certainly use it to see if the algorithm does what you need.

regards

Thorsten

function dydx = deriv(x1, y1, nl, nr, m, ld, edge)
% deriv - calculates numerically smoothed derivatives
%
% calling syntax:
%
%     dydx = deriv(x, y, nl, nr, m, ld, edge);
%
% where x      is a vector of x-values
%       y      is a vector of y-values (y(i) = f(x(i)))
%       nl     is the number of leftward data points
%                 used for smoothing
%       nr     is the number of rightward data points
%       m      is the order of the polynomials fitted
%                 to the data
%       ld     is the order of derivative to be calculated
%                 (default is 1). For ld=0, the function
%                 itself is smoothed
%       edge   is a flag that determines the way the
%                 routine deals with the edges of the
%                 vector y.
%                 edge = 1 : the last full polynomial
%                            based on the values y(1:nl+nr+1)
%                            and y(length(y)-nl-nr:length(y))
%                            respectively is used to calculate
%                            all the values dydx(1:nl) and
%                            dydx(length(y)-nl-nr-1:length(y))
%                            respectively.
%                 edge = 2 : the window used for fitting is
%                            linearly shrunk and shifted as
%                            the edges are approached
%                            (default is 2)
%       dydx   is assigned the n-th derivative of y(x)
%

if nargin < 7
  edge = 2;
  if nargin < 6
    ld = 1;
  end;
end;

if m < ld
  m = ld;
end;

faktor = gamma(ld + 1);

xmax = max(abs(x1));
ymax = max(abs(y1));
x    = x1 / xmax;
y    = y1 / ymax;

ly = length(x);

dydx = zeros(size(x));
for i = 1+nl : ly-nr
  xx   = x(i-nl:i+nr) - x(i);
  yy   = y(i-nl:i+nr);
  pp   = polyfit(xx, yy, m);
  dydx(i) = faktor * pp(m+1-ld);
end;

if edge == 1
  xx  = x(1:(nl + nr + 1)) - x(nl+1);
  yy  = y(1:(nl + nr + 1));
  pp  = polyfit(xx, yy, m);
  dpp = polydiff(pp, ld);
  dydx(1:nl) = polyval(dpp, xx(1:nl));
  xx  = x((ly - (nl + nr)) : ly) - x(ly - nr);
  yy  = y((ly - (nl + nr)) : ly);
  pp  = polyfit(xx, yy, m);
  dpp = polydiff(pp, ld);
  dydx((ly - nr + 1) : ly) = polyval(dpp, xx(nl+2:nl+1+nr));
elseif edge == 2
  nn = nl + nr + 1;

  if nl < 2
    nnn = m * 2;
  else
    nnn = round(linspace(m * 2, nl + nr, nl));
  end

  for i = 1:nl
    xx  = x(1:nnn(i)) - x(i);
    yy  = y(1:nnn(i));
    pp  = polyfit(xx, yy, m);
    dydx(i) = faktor * pp(m+1-ld);
  end;

  if nr < 2
    nnn = m * 2;
  else
    nnn = round(linspace(nl + nr, m * 2, nr));
  end
  il  = ly - nnn + 1;
    
  for i = 1:nr
    xx = x(il(i):ly) - x(ly - nr + i);
    yy = y(il(i):ly);
    pp = polyfit(xx, yy, m);
    dydx(ly - nr + i) = faktor * pp(m+1-ld);
  end;
end;

if ld == 0
  dydx = dydx * ymax;
else
  dydx = dydx * ymax / xmax ^ ld;
end

Paul Kienzle wrote:

On Nov 18, 2005, at 5:27 PM, Dmitri A. Sergatskov wrote:
 
 
On 11/18/05, Joe Koski <address@hidden> <mailto:address@hidden>  wrote:
 
Has anyone run across "moving least squares" code in .m, .cc, .f, etc.
formats that would be adaptable for use with octave? Apparently, moving
least squares can be used to create approximating curves for
interpolation/approximation much in the same way that spline curves can be
used. In some situations, the moving least squares approach can reduce the
need to solve large matrices, which is normally associated with the curve
fit process.
 
A Google of "moving least squares" shows several papers, but no code that I
can find. Any ideas?
 
 

Would Kalman filter do the same / better?
 

Savitsky-Golay is a moving least squares filter.  You can use it for
curve smoothing if the data are equally spaced.
 
- Paul
 
 
 
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