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RE: Moving Least Squares, Anyone?
From: |
Hall, Benjamin |
Subject: |
RE: Moving Least Squares, Anyone? |
Date: |
Mon, 21 Nov 2005 16:14:01 -0500 |
Is
this the original (96 page!) paper you were referring to?
There
seems to be some other related at
I
might also be interested in any work someone may have already started
here...I'll have to do some reading
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:
- the function arguments x and y are ordered the
wrong way round
- the calculation is slow because it is not
vectorized (does anybody know a way to vectorize it?)
-
- the treatment of the vector edges can probably
be improved a lot
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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-------------------------------------------------------------
- Moving Least Squares, Anyone?, Joe Koski, 2005/11/18
- Re: Moving Least Squares, Anyone?, Dmitri A. Sergatskov, 2005/11/18
- Re: Moving Least Squares, Anyone?, Paul Kienzle, 2005/11/18
- Re: Moving Least Squares, Anyone?, Thorsten Meyer, 2005/11/19
- Re: Moving Least Squares, Anyone?, Miroslaw Kwasniak, 2005/11/19
- Re: Moving Least Squares, Anyone?, Joe Koski, 2005/11/21
- Re: Moving Least Squares, Anyone?, Joe Koski, 2005/11/21
- Re: Moving Least Squares, Anyone?, Stefan van der Walt, 2005/11/21
- RE: Moving Least Squares, Anyone?,
Hall, Benjamin <=