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Re: Smooth line approximating minima of a data series

From: Carlo de Falco
Subject: Re: Smooth line approximating minima of a data series
Date: Wed, 24 Feb 2010 20:52:10 +0100

On 24 Feb 2010, at 20:04, Matthias Brennwald wrote:

On Feb 24, 2010, at 5:29 PM, Carlo de Falco wrote:

On 24 Feb 2010, at 13:12, Matthias Brennwald wrote:

On Feb 24, 2010, at 10:38 AM, Carlo de Falco wrote:

2010/2/24 Matthias Brennwald <address@hidden>:
Dear all

Consider a series of data values that reflect a smooth function (e.g. a low-degree polynomial), but there might be additional features in the data (e.g. narrow peaks or noise). I'd like to fit a polynomial to this data, whereby this polynomial reflects a smooth approximation of
the minima of the raw data (I call this the "base line"). The
following might help to illustrate what I'm trying to accomplish:

 x = [-1:0.01:1]; % x-axis values
 p = [-3 2 1 0]; yp = polyval (p,x); % make up a polynomial
reflecting the "base line" for illustration
 y = yp + rand(size(x)); % this would be the raw data
 plot (x,y,x,yp); legend ('raw data','base line') % plot the raw
data and the polynomial for illustration

Has anyone an idea of how to accomplish this? Are there standard
methods? I'd appreciate any hints.


does this do what you want?

x = [-1:0.01:1];
p = [-3 2 1 0]; yp = polyval (p,x)
y = yp + randn(size(x));
plot (x,y,x,yp);
yy = polyfit (x, y, 4)
plot (x, polyval (yy, x), x, y, x, yp)

No, this is not what I'm after. I am looking for a polynomial (or some other smooth function) that tracks the (local) minima of the data series (as in the plot of my original example). The polyfit function returns a polynomial wich minimizes the sum of the squares of the residuals relative to each of the the data points. In contrast to this, I would like to do the minimization such that each residuals is positive, i.e. polynomial values never exceed the the values in the data series.


what about the following:
x = [-1:0.01:1];
p = [-3 2 1 0]; yp = polyval (p,x);
y = yp + rand(size(x));
plot (x, y)

vref = 1e-4;
v1   = @(t) interp1 (x, y, t, "nearest");

fun = @(v2, v2dot, t) v2dot - max ((v1(t)-v2)/vref, -10);
yf  = daspk (fun, y(1), 0, x);

plot (x, y, x, yf, x, polyval (polyfit (x(:), yf(:), 4), x))

this actually traces (approximately) the maxima rather than minima but the modifications are trivial

Neat! I do not (yet) understand completetly how it works, but the results sure looks good.

This is just a very rough approximation of the dynamical system corresponding to a rectifying electrical circuit. if this is what you were looking for I am sure someone on this list will know how to design the corresponding digital


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