Math Forum Discussions

samar

Posts: 7
Registered: 7/25/11

pchip with 2 variables
Posted: Jan 16, 2013 1:37 PM

Plain Text

Dear Cédric,

I want tu use Pchip function to interpolate my 2-dimensional data set.
I'm actually using spline interpolation (with csapi function) but it doesn't work well.
I think that Pchip will be better for my data set but I have no idea how to use it with 2 variables ( x1 and x2 are my variables they take their values over a given list and V=f(X1,X2) is the matrix that gives the value taken by f on each couple (x1,x2))

The second question can we have the result of pchip in PPFORM?

Thank you very much for you help!

"Cédric Louyot" <cedric.louyot@NOSPAMonera.fr> wrote in message <eee8cbc.3@webx.raydaftYaTP>...
> In fact, I'm not having any problem with the pchip function.
> Actually, it works real fine on the set of data I'm dealing with in
> 2D. However I also have to work on a set of 3D data (that looks
> really similar to the 2D set). Therefore, I try to understand the 2D
> algorithm in details in order to build a pchip3d (or even a pchipnd)
> function that could work on sets of data of the form:
> z(x,y) = f(x,y) (or even z(x1, x2, ..., xn))
>
> I've already tried the interp2 options ('linear','spline' and
> 'cubic') but they don't yield satisfactory results with my data.
> That's why I try to build a pchip3d function.
>
> Is there any chance it has already been done before by someone else ?
> Do you know where I could find the Fritsch and Carlson paper ? I've
> already looked for it in vain.
>
> Thanks,
>
> Cédric
>
> John D'Errico wrote:
> >
> >
> > In article <eee8cbc.-1@webx.raydaftYaTP>,
> > "Cédric Louyot" <cedric.louyot@NOSPAMonera.fr> wrote:
> >
> >> I'm looking for a precision regarding the algorithm that the
> > function
> >> pchip uses. In particular, according to the MATLAB function
> > reference
> >> :
> >> "The slopes at the xj are chosen in such a way that P(x)
> > preserves
> >> the shape of the data and respects monoticity."
> >>
> >> Could anybody explain to me what is hidden behing "in such a
> way"
> > ?
> >> How does pchip computes the slopes at the xj ?
> >>
> >> Thanks for your help,
> >>
> >> Cédric
> >
> > From the comments, we see that pchip is derived from
> > a nice paper by Fritsch and Carlson.
> >
> > % F. N. Fritsch and R. E. Carlson, "Monotone Piecewise Cubic
> > % Interpolation", SIAM J. Numerical Analysis 17, 1980, 238-246.
> >
> > The idea is you choose a decent set of estimates of the
> > slopes at the knots. Then you test to see if they satisfy
> > some approximation to the monotonicity constraints put
> > forth in the F&C paper.
> >
> > These constraints form a boundary, one edge of which is
> > elliptic, around the set of cubic segments. If the choice
> > of slopes falls outside the set of monotone cubic segments,
> > then you adjust the derivatives so this does not happen.
> > Its a nice algorithm that runs quite quickly and does not
> > require the solution of a set of simultaneous linear
> > equations as a simple interpolating spline does.
> >
> > The disadvantages to pchip are
> >
> > 1. It results in a C1 interpolant. (Discontinuous second
> > derivatives, whereas an interpolating cubic spline is C2.)
> > This is rarely a problem, although I have seen cases where
> > the second derivative discontinuities were a significant
> > flaw.
> >
> > 2. Since pchip is designed to produce locally monotone
> > interpolants, it sometimes produces less than desireable
> > results on spiky data. (By locally monotone, I mean it
> > does not introduce any extrema in the curve that is not
> > already in the data.)
> >
> > What happens when you have a spike in your data? Pchip
> > forces the derivative to zero at the maximum (or minimum)
> > of the spike. Again, this is by design.
> >
> > Are you having problems with pchip? Very often this
> > suggests that you may have data which is inappropriate
> > for pchip. For example, I would not use pchip to
> > interpolate an illuminant spectral power curve.
> > Especially not for one derived from a fluorescent light.
> >
> > HTH,
> > John D'Errico
> >
> >
> > --
> > There are no questions "?" about my real address.
> >
> > The best material model of a cat is another, or
> > preferably the same, cat.
> > A. Rosenblueth, Philosophy of Science, 1945
> >