Statistical Methods for Data Analysis

With Applications in Particle Physics
3rd Auflage 2023
ISBN: 978-3-031-19933-2
Verlag: Springer

Buch, Englisch, 334 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 557 g

Reihe: Lecture Notes in Physics

This third edition expands on the original material. Large portions of the text have been reviewed and clarified. More emphasis is devoted to machine learning including more modern concepts and examples. This book provides the reader with the main concepts and tools needed to perform statistical analyses of experimental data, in particular in the field of high-energy physics (HEP).

It starts with an introduction to probability theory and basic statistics, mainly intended as a refresher from readers’ advanced undergraduate studies, but also to help them clearly distinguish between the Frequentist and Bayesian approaches and interpretations in subsequent applications. Following, the author discusses Monte Carlo methods with emphasis on techniques like Markov Chain Monte Carlo, and the combination of measurements, introducing the best linear unbiased estimator. More advanced concepts and applications are gradually presented, including unfolding and regularization procedures, culminating in the chapter devoted to discoveries and upper limits.

The reader learns through many applications in HEP where the hypothesis testing plays a major role and calculations of look-elsewhere effect are also presented. Many worked-out examples help newcomers to the field and graduate students alike understand the pitfalls involved in applying theoretical concepts to actual data.

Zielgruppe

Graduate

Autoren/Hrsg.

Lista, Luca

Fachgebiete

• Mathematik | Informatik Mathematik Stochastik Mathematische Statistik
• Naturwissenschaften Physik Quantenphysik Teilchenphysik
• Naturwissenschaften Physik Physik Allgemein Theoretische Physik, Mathematische Physik, Computerphysik

Inhaltsverzeichnis

Preface to the third edition

Preface to previous edition/s

1 Probability Theory

1.1 Why Probability Matters to a Physicist

1.2 The Concept of Probability

1.3 Repeatable and Non-Repeatable Cases

1.4 Different Approaches to Probability

1.5 Classical Probability

1.6 Generalization to the Continuum

1.7 Axiomatic Probability Definition

1.8 Probability Distributions

1.9 Conditional Probability

1.10 Independent Events

1.11 Law of Total Probability

1.12 Statistical Indicators: Average, Variance and Covariance

1.13 Statistical Indicators for a Finite Sample

1.14 Transformations of Variables

1.15 The Law of Large Numbers

1.16 Frequentist Definition of Probability

References 2 Discrete Probability Distributions

2.1 The Bernoulli Distribution

2.2 The Binomial Distribution

2.3 The Multinomial Distribution

2.4 The Poisson Distribution

References

3 Probability Distribution Functions

3.1 Introduction

3.2 Definition of Probability Distribution Function

3.3 Average and Variance in the Continuous Case

3.4 Mode, Median, Quantiles

3.5 Cumulative Distribution

3.6 Continuous Transformations of Variables

3.7 Uniform Distribution

3.8 Gaussian Distribution

3.9 X^2 Distribution

3.10 Log Normal Distribution

3.11 Exponential Distribution

3.12 Other Distributions Useful in Physics

3.13 Central Limit Theorem

3.14 Probability Distribution Functions in More than One Dimension

3.15 Gaussian Distributions in Two or More Dimensions

References

4 Bayesian Approach to Probability

4.1 Introduction

4.2 Bayes’ Theorem

4.3 Bayesian Probability Definition

4.4 Bayesian Probability and Likelihood Functions

4.5 Bayesian Inference

4.6 Bayes Factors

4.7 Subjectiveness and Prior Choice

4.8 Jeffreys’ Prior

4.9 Reference priors

4.10 Improper Priors

4.11 Transformations of Variables and Error Propagation

References

5 Random Numbers and Monte Carlo Methods

5.1 Pseudorandom Numbers

5.2 Pseudorandom Generators Properties

5.3 Uniform Random Number Generators

5.4 Discrete Random Number Generators

5.5 Nonuniform Random Number Generators

5.6 Monte Carlo Sampling

5.7 Numerical Integration with Monte Carlo Methods

5.8 Markov Chain Monte Carlo

References 6 Parameter Estimate

6.1 Introduction

6.2 Inference

6.3 Parameters of Interest

6.4 Nuisance Parameters

6.5 Measurements and Their Uncertainties

6.6 Frequentist vs Bayesian Inference

6.7 Estimators

6.8 Properties of Estimators

6.9 Binomial Distribution for Efficiency Estimate

6.10 Maximum Likelihood Method

6.11 Errors with the Maximum Likelihood Method

6.12 Minimum X^2 and Least-Squares Methods

6.13 Binned Data Samples

6.14 Error Propagation

6.15 Treatment of Asymmetric Errors

References7 Combining Measurements

7.1 Introduction

7.2 Simultaneous Fits and Control Regions

7.3 Weighted Average

7.4 X^2 in n Dimensions

7.5 The Best Linear Unbiased Estimator

References

8 Confidence Intervals

8.1 Introduction

8.2 Neyman Confidence Intervals

8.3 Binomial Intervals

8.4 The Flip-Flopping Problem

8.5 The Unified Feldman–Cousins Approach

References9 Convolution and Unfolding

9.1 Introduction

9.2 Convolution

9.3 Unfolding by Inversion of the Response Matrix

9.4 Bin-by-Bin Correction Factors

9.5 Regularized Unfolding

9.6 Iterative Unfolding

9.7 Other Unfolding Methods

9.8 Software Implementations

9.9 Unfolding in More Dimensions

References

10 Hypothesis Tests

10.1 Introduction

10.2 Test Statistic

10.3 Type I and Type II Errors

10.4 Fisher’s Linear Discriminant

10.5 The Neyman–Pearson Lemma

10.6 Projective Likelihood Ratio Discriminant

10.7 Kolmogorov–Smirnov Test

10.8 Wilks’ Theorem

10.9 Likelihood Ratio in the Search for a New Signal

References

 

11 Machine Learning

11.1 Supervised and Unsupervised Learning

11.2 Terminology

11.3 Machine Learning Classification from a Statistical Point of View

11.4 Bias-Variance tradeo

11.5 Overtraining

11.6 Artificial Neural Networks

11.7 Deep Learning

11.8 Convolutional Neural Networks

11.9 Boosted Decision Trees

11.10 Multivariate Analysis Implementations

References

 

12 Discoveries and Upper Limits

12.1 Searches for New Phenomena: Discovery and Upper Limits

12.2 Claiming a Discovery

12.3 Excluding a Signal Hypothesis

12.4 Combined Measurements and Likelihood Ratio

12.5 Definitions of Upper Limit

12.6 Bayesian Approach

12.7 Frequentist Upper Limits

12.8 Modified Frequentist Approach: the CLs Method

12.9 Presenting Upper Limits: the Brazil Plot

12.10 Nuisance Parameters and Systematic Uncertainties

12.11 Upper Limits Using the Profile Likelihood

12.12 Variations of the Profile-Likelihood Test Statistic

12.13 The Look Elsewhere Effect

References

 

Index

Über Autor(innen)

Luca Lista is full professor at University of Naples Federico II and Director of INFN Naples Unit. He is an experimental particle physicist and member of the CMS collaboration at CERN. He participated in the BABAR experiment at SLAC and L3 experiment at CERN. His main scientific interests are data analysis, statistical methods applied to physics and software development for scientific applications.