Practical Handbook of Curve Fitting

Abstract

Worked real-world examples illustrate theoretical aspects of fitting curves to data. TABLE OF CONTENTS: | Introduction--Theoretical Background | Chapter 1, Population Data Analysis: Straight line curve-fitting--least squares; Exponential curve-fitting; Exponential curve-fitting with a lower bound; Logistic curve-fitting; Gompertz curve-fitting. | Chapter 2, Epidemiology Data Analysis: Consistent database construction; Mapping of data; Straight line curve-fitting-least squares; Root mean square error. | Chapter 3, Agriculture Data Analysis: Linear fitting--bounded; Bar charts; Default computer curves; Linear fitting--unbounded--simple use of least squares; Exponenital fitting--unbounded; Extrapolation of fitted curves--prediction. | Chapter 4, Biodiversity Data Analysis: Dot maps; Equal-area map projection; Geographic information systems; Map overlays; Feigenbaum's graphical analysis. | Chapter 5, Soils and Forestry Data Analysis: Simple cubic spline curve fitting; Interpolation using a cubic spline; Feigenbaum's graphical analysis. | Chapter 6, Education Data Analysis: Straight line curve-fitting--least squares; Residual plots; Root mean square error. | Chapter 7, Transportation and Communication Data Analysis: Historical maps; Space-filling measured by density; Rank-ordering; Fitting maps to empirical curves. | Chapter 8, Environmental Toxicity Data Analysis: Straight line curve-fitting--least squares; Fitting curves to maps. | Chapter 9, Urbanization Data Analysis: Straight line curve-fitting--least squares; Partitioning of data--scale transformation; Residual plots; Examination and removal of outliers. | Chapter 10, World Trade Data Analysis: Set-theoretic analysis of numerical structure of data; Partitioning of data; Organization techniques; Geometric self-similarity. | Index of Figure Captions and Table Titles. | Quotation from the Overview section of the Introduction. | "Computers offer even the casual user opportunities to handle data that were seldom dreamed of a decade ago. Most spreadsheets and other software packages that have some sort of analytic capability offer an option to graph input data; indeed, even many hand-held calculators do so. Default graphs generated by the computer often link points in a kind of follow the dots fashion; while this is a useful feature, default curves of this sort do not generally assign an equation to the curve. Thus, these curves cannot be projected nor can they be used for systematic interpolation between values. Many of the same software packages that offer default curves also permit the user to insert equations and to generate other graphs based on these equations. The problem is that the user new to this world often does not have the needed mathematical background. We offer a practical approach to showing how to fit curves; it is practical because it uses global real-world data and exposes the reader, as an important side benefit, to the problems encountered when using first-rate electronic (or other) data bases. The reader who is content to fit curves without some sort of mathematical discussion behind these efforts, but who understands that different choices of curves can forecast highly diverse alternative futures, should be able to use all the techniques in this handbook. This section is included because it is important that it be here should one wish it. | With any data set (presented in electronic or paper format), it is important first to examine the set for interesting or unusual patterns in the display. These patterns often influence decisions in choosing subsets of data and tools to analyze subsets. Thus, we encourage readers to approach any data set with a set of basic guidelines in mind and to browse it in a thoughtful manner. We offer the following as one set of guidelines; it will be repeated throughout, with commentary relating it to the databases selected in particular chapters. | Patterns in Data--What to Look For--| 1. What is the general orgainzation scheme of the entire set? Is it arranged alphabetically, numerically, or in some other fashion? | 2. Are the real-world entries in the Table (nations, states, counties) expressed as comparable units: For example, county data and national data are generally not comparable. | 3. Are the numerical entries in the Table expressed in comparable units? For example, data in one column might measure percentaes while data in another column might measure thousands of dollars--these columns would not be comparable. | 4. Are there gaps in the data? If so, what is their significance to the questions you wish to have the data answer?" |