Re: restricting interpolating functions to be positive

• To: mathgroup at smc.vnet.net
• Subject: [mg106607] Re: restricting interpolating functions to be positive
• From: Noqsi <jpd at noqsi.com>
• Date: Mon, 18 Jan 2010 02:35:17 -0500 (EST)
• References: <higdjs\$kfi\$1@smc.vnet.net> <201001131058.FAA06854@smc.vnet.net>

```On Jan 17, 5:11 am, Ray Koopman <koop... at sfu.ca> wrote:
> On Jan 15, 12:18 am, schochet123 <schochet... at gmail.com> wrote:
>
> > On Jan 14, 12:46 pm, DrMajorBob <btre... at austin.rr.com> wrote:
>
> >> For some data, that works pretty well; for other samples
> >> it has HUGE  peaks, reaching far above any of the data:
>
> > This problem can be avoided by using a transformation that
> > is close to the identity for positive values:
>
> > trans[x_] = (Sqrt[1 + x^2] + x)/2
>
> > inv[y_] = (4 y^2 - 1)/(4 y)
>
> There's a whole family of transformations here:
>
> trans[c_,x_] = (Sqrt[4c + x^2) + x)/2
>
> inv[c_,y_] = y - c/y
>
> Extremely small values of c give results that approach those obtained
> by chopping the interpolation at 0.

These transformations have the property that the interpolation behaves
differently for small and large numbers, with the transition at a
scale given by Sqrt[c]. The Log/Exp pair I suggested is scaleless.
{Sqrt[x],y^2} is another scaleless transformation.

All such transformations will exhibit artifacts from some point of
view given some data set: there is no flawless universal interpolation
procedure.

The  {Sqrt[x],y^2} pair is useful for data that's proportional to
counts of independent events (like electric current over a potential
barrier), as it approximately gives each point its proper "weight" in
the analysis, given the correlation of the shot noise variance with
the expectation. However, the interpolation may "bounce" off the x-
axis...

```

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