Re: Smooth line approximating minima of a data series

[Carlo de Falco]

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.
> > >
> > > Thanks
> > > Matthias
> >
> > 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.
>
> Matthias

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
c.