Data Analysis Using the Method of Least Squares

Extracting the Most Information from Experiments

By: J. Wolberg

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Paperback | 16 December 2005

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Measurements through quantitative experiments are one of the most f- damental tasks in all areas of science and technology. Astronomers a- lyze data from asteroid sightings to predict orbits. Computer scientists - velop models for recognizing spam mail. Physicists measure properties of materials at low temperatures to understand superconductivity. Materials engineers study the reaction of materials to varying load levels to develop methods for prediction of failure. Chemical engineers consider reactions as functions of temperature and pressure. The list is endless. From the very small-scale work on DNA to the huge-scale study of black holes, quantitative experiments are performed and the data must be analyzed. Probably the most popular method of analysis of the data associated with quantitative experiments is least squares. It has been said that the method of least squares was to statistics what calculus was to mathematics. - though the method is hardly mentioned in most engineering and science undergraduate curricula, many graduate students end up using the method to analyze the data gathered as part of f their research. There is not a lot of available literature on the subject. Very few books deal with least squares at the level of detail that the subject deserves. Many books on statistics - clude a chapter on least squares but the treatment is usually limited to the simplest cases of linear least squares.

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Aus den Rezensionen: "... Die ... Methode der kleinsten Quadrate ... ist ... eines der am haufigsten angewendeten statistischen Verfahren zur Aus- und Bewertung der Daten. John Wolberg ... bedauert, dass dieser Methode noch langst nicht die ihr zustehende Aufmerksamkeit ... gewidmet wird ... Die publizistische Lucke ... mochte der Autor mit diesem Buch schliessen. ... Das Buch vermittelt das Wissen fur eine hochst effiziente Methode zur Datenanalyse. Der Autor erlautert die Thematik in dem ... Buch abstrakt konzentriert und untermauert sie durch einfache, uberwiegend hypothetische Beispiele, die zum ... Verstandnis der Methode beitragen ..." (Dr. Norbert Hofert, in: Gefahrstoffe - Reinhaltung der Luft, 2007, Vol. 67, Issue 4, S. 136)

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Introduction	p. 1
Quantitative Experiments	p. 1
Dealing with Uncertainty	p. 5
Statistical Distributions	p. 6
The normal distribution	p. 8
The binomial distribution	p. 10
The Poisson distribution	p. 11
The x[superscript 2] distribution	p. 13
The t distribution	p. 15
The F distribution	p. 16
Parametric Models	p. 17
Basic Assumptions	p. 19
Systematic Errors	p. 22
Nonparametric Models	p. 24
Statistical Learning	p. 27
The Method of Least Squares	p. 31
Introduction	p. 31
The Objective Function	p. 34
Data Weighting	p. 38
Obtaining the Least Squares Solution	p. 44
Uncertainty in the Model Parameters	p. 50
Uncertainty in the Model Predictions	p. 54
Treatment of Prior Estimates	p. 60
Applying Least Squares to Classification Problems	p. 64
Model Evaluation	p. 73
Introduction	p. 73
Goodness-of-Fit	p. 74
Selecting the Best Model	p. 79
Variance Reduction	p. 85
Linear Correlation	p. 88
Outliers	p. 93
Using the Model for Extrapolation	p. 96
Out-of-Sample Testing	p. 99
Analyzing the Residuals	p. 105
Candidate Predictors	p. 115
Introduction	p. 115
Using the F Distribution	p. 116
Nonlinear Correlation	p. 122
Rank Correlation	p. 131
Designing Quantitative Experiments	p. 137
Introduction	p. 137
The Expected Value of the Sum-of-Squares	p. 139
The Method of Prediction Analysis	p. 140
A Simple Example: A Straight Line Experiment	p. 143
Designing for Interpolation	p. 147
Design Using Computer Simulations	p. 150
Designs for Some Classical Experiments	p. 155
Choosing the Values of the Independent Variables	p. 162
Some Comments about Accuracy	p. 167
Software	p. 169
Introduction	p. 169
General Purpose Nonlinear Regression Programs	p. 170
The NIST Statistical Reference Datasets	p. 173
Nonlinear Regression Convergence Problems	p. 178
Linear Regression: a Lurking Pitfall	p. 184
Multi-Dimensional Models	p. 191
Software Performance	p. 196
The Regress Program	p. 198
Kernel Regression	p. 203
Introduction	p. 203
Kernel Regression Order Zero	p. 205
Kernel Regression Order One	p. 208
Kernel Regression Order Two	p. 212
Nearest Neighbor Searching	p. 215
Kernel Regression Performance Studies	p. 223
A Scientific Application	p. 225
Applying Kernel Regression to Classification	p. 232
Group Separation: An Alternative to Classification	p. 236
Generating Random Noise	p. 239
Approximating the Standard Normal Distribution	p. 243
References	p. 245
Index	p. 249
Table of Contents provided by Ingram. All Rights Reserved.

Data Analysis Using the Method of Least Squares

Extracting the Most Information from Experiments

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Paperback

Industry Reviews

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