Designing Quantitative Experiments: Prediction Analysis

by John Wolberg

Overview

Early in my career I was given the task of designing a sub-critical nuclear reactor facility that was to be used to perform basic research in the area of reactor physics. We planned to run a series of experiments to determine fundamental parameters related to the distribution of neutrons in such systems. I felt that it was extremely important to understand how the design would impact upon the accuracy of our results and as a result of this - quirement I developed a design methodology that I subsequently called prediction analysis. After working with this method for several years and applying it to a variety of different experiments, I wrote a book on the subject. Not surprisingly, it was entitled Prediction Analysis and was p- lished by Van Nostrand in 1967. Since the book was published over 40 years ago science and technology have undergone massive changes due to the computer revolution. Not - ly has available computing power increased by many orders of magnitude, easily available and easy to use software has become almost ubiquitous. In the 1960's my emphasis was on the development of equations, tables and graphs to help researchers design experiments based upon some we- known mathematical models. When I reconsider this work in the light of today's world, the emphasis should shift towards applying current techn- ogy to facilitate the design process.

Product Details

ISBN-13:	9783642115882
Publisher:	Springer Berlin Heidelberg
Publication date:	05/06/2010
Edition description:	2010
Pages:	208
Product dimensions:	6.00(w) x 9.10(h) x 0.40(d)

Table of Contents

Chapter 1 Introduction 1

1.1 The Experimental Method 1

1.2 Quantitative Experiments 2

1.3 Dealing with Uncertainty 3

1.4 Parametric Models 5

1.5 Basic Assumptions 11

1.6 Treatment of Systematic Errors 13

1.7 Nonparametric Models 16

1.8 Statistical Learning 18

Chapter 2 Statistical Background 19

2.1 Experimental Variables 19

2.2 Measures of Location 21

2.3 Measures of Variation 24

2.4 Statistical Distributions 27

The normal distribution 28

The binomial distribution 30

The Poisson distribution 32

The x² distribution 34

The t distribution 37

The F distribution 38

The Gamma distributions 39

2.5 Functions of Several Variables 40

Chapter 3 The Method of Least Squares 47

3.1 Introduction 47

3.2 The Objective Function 50

3.3 Data Weighting 55

3.4 Obtaining the Least Squares Solution 60

3.5 Uncertainty in the Model Parameters 66

3.6 Uncertainty in the Model Predictions 69

3.7 Treatment of Prior Estimates 75

3.8 Applying Least Squares to Classification Problems 80

3.9 Goodness-of-Fit 81

3.10 The REGRESS Program 86

Chapter 4 Prediction Analysis 90

4.1 Introduction 90

4.2 Linking Prediction Analysis and Least Squares 91

4.3 Prediction Analysis of a Straight Line Experiment 92

4.4 Prediction Analysis of an Exponential Experiment 98

4.5 Dimensionless Groups 103

4.6 Simulating Experiments 106

4.7 Predicting Calculational Complexity 111

4.8 Predicting the Effects of Systematic Errors 116

4.9 P.A. with Uncertainty in the Independent Variables 118

4.10 Multiple Linear Regression 120

Chapter 5 Separation Experiments 128

5.1 Introduction 128

5.2 Exponential Separation Experiments 129

5.3 Gaussian Peak Separation Experiments 136

5.4 Sine Wave Separation Experiments 144

5.5 Bivariate Separation 150

Chapter 6 Initial Value Experiments 157

6.1 Introduction 157

6.2 A Nonlinear First Order Differential Equation 157

6.3 First Order ODE with an Analytical Solution 162

6.4 Simultaneous First Order Differential Equations 168

6.5 The Chemostat 172

6.6 Astronomical Observations using Kepler's Laws 177

Chapter 7 Random Distributions 186

7.1 Introduction 186

7.2 Revisiting Multiple Linear Regression 187

7.3 Bivariate Normal Distribution 191

7.4 Orthogonality 196

References 205

Index 209