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Functions

FitPeriodic

This is a notebook on fitting functions to data, or rather which functions to choose for certain types of data. The intelligent part of fitting is the choice of the base functions, not the computation of coefficients. That this choice is not straightforward is shown by the enclosed text, which I had presented on several occasions some time ago.

It both shows the shortcomings of linear regression, and intelligent use of it. It is not a first model for Fit; rather it should come as an application after the treatment of the logistic equation.

Attached I am sending two copies of the Fit notebook, one with and one without outputs. It handles fitting of functions to real data.

I want to stress again that the intelligent part of the experiments is not in the use of the Fit instruction but rather the choice of the family of base functions for the approximation.

The use of graphical data input is gaining interest. Fitting to these is a good way to introduce statistics, since it does not limit the user to standard notions of mean, normal distribution and standard deviation. Mathematica offers an extensive statistics package, but we can implement our own functions. For each type of data, we can look for the (qualitative) distribution format (normal, 1/cosh, 1/(t2 + 1), Poisson, student-t,...) and continue after replacing the data with a best fit to such base functions.

A remark is in order here: Fit uses L2 norms and this may be perceived as restrictive. My experience - and of many people in mathematical analysis - is that in most cases other norms (except maybe L-infinity) relate in some way to L2 norms and therefore taking L2 norms is sufficient for most statistical problems.

Fitting to periodic graphical inputs goes beyond reals: I have already presented a paper at the ATCM Melacca meeting in which (starlike) data in the plane could be modeled by complex trigonometric polynomials. Since few of you were there, I will send copies of this material soon, but it requires polynomial images of the unit circle as a prerequisite (see the Melbourne ATCM paper: "Computer screens and toys").

StirlingFormula

 As an application of both the Growth and the Fit notebooks I am sending you a notebook showing how good the Stirling approximation for the factorial is.

It shows again how reflecting about growth (or decay) helps make the right choice of the family of base functions for the approximation in the Fit instruction.

ColorFormulaDepth

Uses the full power of Mathematica to analyse the structure of expressions, and is a useful tool for educators.

Examples

Color Formula Depth

Fit Periodic Full

Formula Structure

Log Scales

Stirling Formula