The Skinny on Least Squares and Polynomial Curve Fitting

THE SKINNY This book gives you the skinny...the inside information, the real facts...about the method of least squares and polynomial curve fitting in a skinny...very thin, compact...format. This is a 'How to...' book presented in a format resembling a cookbook. As a consequence, mathematical derivations, proofs, theory, etc. are relegated to the Appendix. GENERAL BACKGROUND AND REVIEW Curve fitting is generally of interest for one of two reasons: (1) to allow the interpolation of an array or table of data; or (2) to allow the extrapolation of those data. In this book we focus on fitting data with an analytical polynomial in two dimensions, so that the data can either be interpolated or extrapolated. More specifically, we focus on the method of least squares where it is desired to represent the entire data array with a single smooth analytical function. LEAST SQUARES METHOD The least squares method is not new. The method finds the best line to represent a set of data points such that the sum of the squares of the vertical distances of each data point from the proposed line is a minimum. Discussion of the method is frequently limited to a discussion of the use of least squares for establishing a linear trend or straight-line model of the data. In this book, we show how the least squares criterion can be used to find the best quadratic and cubic polynomial data models in addition to the linear model. Higher-order polynomial models also follow from the development presented here. The method is frequently described or classified as a regression analysis technique. Regression implies the application of statistical methods in its analysis. However, there are no statistical methods applied to the development in this book. Indeed, the mathematical relationships derive from a straight-forward application of differential calculus methods and techniques with no application of any statistical methods. We limit our discussion in this book to systems of two-dimensional data; that is, to systems of data that can be represented on a graph with rectangular coordinates with one independent and one dependent variable. IMPLEMENTATION WITH MICROSOFT(R) EXCEL We demonstrate the implementation of the calculations with Microsoft(R) Excel since Excel is readily available on most PC's as one of the components of the Microsoft(R) Office Suite of applications. We have included screen shots of Excel spreadsheets to show enough detail so that the reader can easily reproduce the calculations. An extensive familiarity with Excel is not required to use this book and to understand the screen shots, although familiarity with some fundamental concepts of Excel is assumed and helpful. TABLE OF CONTENTS Chapter 1: Introduction The Skinny General Background and Review Least Squares Method Implementation with Microsoft(R) Excel Where To Go From Here Chapter 2: How To Fit Three or More Points with a Linear Polynomial Chapter 3: How To Fit Four or More Points with a Quadratic Polynomial Chapter 4: How To Fit Five or More Points with a Cubic Polynomial Chapter 5: How To Fit n Points with an (n-1)th Order Polynomial Chapter 6: Quality of the Least Squares Fit Chapter 7: Applications Check Processing Stock Investing Petroleum Fraction Boiling Point Curves Church Growth Steakhouse Restaurant Appendix Appendix 1: Symbols, Nomenclature, Mathematical Notation, Microsoft(R) Excel Functions Appendix 2: Development of and Theoretical Basis for the Method of Least Squares Appendix 3: Summary of Least Squares Relationships Linear Least Squares Quadratic Least Squares Cubic Least Squares Least Squares Relationships Characteristics Appendix 4: Linear (Straight-Line) Fit of Two Points Appendix 5: Quadratic Fit of Three Points Appendix 6: Bibliography, Software, URL's Glossary Index