Case Study: DataFitting

Submitted: 19 Feb 2008

Operations Research Topics: NonlinearProgramming

Application Areas: Data Analysis

Problem Description

Also known as The Least Squares Problem

When modelling data, scientists often want to fit an analytical model to their experimental data. In this case an experiment has provided some -coordinates that have a growth curve shape, i.e., initially increases quickly with , thens tails off to some maximum value. The scientists in this study have suggested three possible functions to model this behaviour:

$y = f(x) \equiv \displaystyle\frac{1}{a x^2 + b x + c}$
$y = f(x) \equiv a\log(x - b) + c$
$y = f(x) \equiv a\sqrt{x - b} + c$

Given one of the possible functions and the data points $(x_i, y_i), i=1, \ldots, n$ they want to find , and to minimize the squared distance between the data points and the estimates, i.e.,

$\begin{equation*} \min_{a, b, c} \sum_{i=1}^n \left(y_i - f(x_i) \right)^2 \end{equation*}$

The scientists have specified the number of data points and their coordinates in an AMPL data file regression.dat.

The scientists want to know which of the suggested functions provides the best fit to the data.

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Student Tasks

Write AMPL files regression.mod and regression.run that use regression.dat to solve The Data Fitting Problem. Write a management summary of your solution. Be sure to indicate which of the functions fits the data best.
What to hand in Your new AMPL files regression.mod and regression.run. Your management summary.