1998 StatisticalDataAnalysis
- (Cowan, 1998) ⇒ Glen Cowan. (1998). “Statistical Data Analysis." Oxford University Press, ISBN:0198501560
Subject Headings: Statistical Data Analysis, ML Estimator, Tikhonov Regularization.
Notes
- Book Homepage: http://www.pp.rhul.ac.uk/~cowan/sda/
- PDF File available at: http://www.hep.phys.soton.ac.uk/~belyaev/papers/books/statistics/Glen_Cowan-Statistical_Data_Analysis.pdf
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Book Overview
This book is a guide to the practical application of statistics to data analysis in the physical sciences. It is primarily addressed at students and professionals who need to draw quantitative conclusions from experimental data. Although most of the examples are taken from particle physics, the material is presented in a sufficiently general way as to be useful to people from most branches of the physical sciences. The first part of the book describes the basic tools of data analysis: concepts of probability and random variables, Monte Carlo techniques, statistical tests, and methods of parameter estimation. The last three chapters then develop more advanced statistical ideas, focusing on interval estimation, characteristic functions, and correcting distributions for the effects of measurement errors (unfolding).
Chapter 1 Fundamental concepts 1
1.1 Probability and random variables 1
1.2 Interpretation of probability 4
1.2.1 Probability as a relative frequency 4
1.2.2 Subjective probability 5
1.3 Probability density functions 7
1.4 Functions of random variables 13
1.5 Expectation values 16
1.6 Error propagation 20
1.7 Orthogonal transformation of random variables 22
Chapter 2 Examples of probability functions 26
2.1 Binomial and multinomial distributions 26
2.2 Poisson distribution 29
2.3 Uniform distribution 30
2.4 Exponential distribution 31
2.5 Gaussian distribution 32
2.6 Log-normal distribution 34
2.7 Chi-square distribution 35
2.8 Cauchy (Breit--Wigner) distribution 36
2.9 Landau distribution 37
Chapter 3 The Monte Carlo method 40
3.1 Uniformly distributed random numbers 40
3.2 The transformation method 41
3.3 The acceptance--rejection method 42
3.4 Applications of the Monte Carlo method 44
Chapter 4 Statistical tests 46
4.1 Hypotheses, test statistics, significance level, power 46
4.2 An example with particle selection 48
4.3 Choice of the critical region using the Neyman--Pearson lemma 50
4.4 Constructing a test statistic 51
=====4.4.1 [[Linear test statistic]]s, the Fisher discriminant function 51 =====
4.4.2 Nonlinear test statistics, neural networks 54
4.4.3 Selection of input variables 56
4.5 Goodness-of-fit tests 57
4.6 The significance of an observed signal 59
4.7 Pearson's chi^2 test 61
Chapter 5 General concepts of parameter estimation 64
5.1 Samples, estimators, bias 64
5.2 Estimators for mean, variance, covariance 66
Chapter 6 The method of maximum likelihood 70
6.1 ML estimators 70
6.2 Example of an ML estimator: an exponential distribution 72
6.3 Example of ML estimators: mu and sigma^2 of a Gaussian 74
6.4 Variance of ML estimators: analytic method 75
6.5 Variance of ML estimators: Monte Carlo method 76
6.6 Variance of ML estimators: the RCF bound 76
6.7 Variance of ML estimators: graphical method 78
6.8 Example of ML with two parameters 80
6.9 Extended maximum likelihood 83
6.10 Maximum likelihood with binned data 87
6.11 Testing goodness-of-fit with maximum likelihood 89
6.12 Combining measurements with maximum likelihood 92
6.13 Relationship between ML and Bayesian estimators 93
Chapter 7 The method of least squares 95
7.1 Connection with maximum likelihood 95
7.2 Linear least-squares fit 97
7.3 Least squares fit of a polynomial 98
7.4 Least squares with binned data 100
7.5 Testing goodness-of-fit with chi^2 103
7.6 Combining measurements with least squares 106
7.6.2 Determining the covariance matrix 112
Chapter 8 The method of moments 114
Chapter 9 Statistical errors, confidence intervals and limits 118
9.1 The standard deviation as statistical error 118
9.2 Classical confidence intervals (exact method) 119
An alternative (and often equivalent) method of reporting the statistical error of a measurement is with a confidence interval, which was first developed by Neyman (Ney37 ]. Suppose as above that one has [math]\displaystyle{ n }[/math] observations of a random variable [math]\displaystyle{ x }[/math] which can be used to evaluate_an estimator [math]\displaystyle{ \hat{\theta}(x_l,...,x_n) }[/math] for a parameter [math]\displaystyle{ \theta }[/math], and that the value obtained is [math]\displaystyle{ \hat{\theta}_{obs} }[/math]. …
9.3 Confidence interval for a Gaussian distributed estimator 123
9.4 Confidence interval for the mean of the Poisson distribution 126
9.5 Confidence interval for correlation coefficient, transformation of parameters 128
9.6 Confidence intervals using the likelihood function or chi^2 130
9.7 Multidimensional confidence regions 132
9.8 Limits near a physical boundary 136
9.9 Upper limit on the mean of Poisson variable with background 139
10.1 Definition and properties of the characteristic function 143
10.2 Applications of the characteristic function 144
10.3 The central limit theorem 147
10.4 Use of the characteristic function to find the p.d.f. of an estimator 149
10.4.1 Expectation value for mean lifetime and decay constant 150
10.4.2 Confidence interval for the mean of an exponential random variable 151
Chapter 11 Unfolding 153
11.1 Formulation of the unfolding problem 154
11.2 Inverting the response matrix 159
11.3 The method of correction factors 164
11.4 General strategy of regularized unfolding 165
11.5 Regularization functions 167
11.5.1 Tikhonov regularization 167
A commonly used measure of smoothness is the mean value of the square of some derivative of the true distribution. This technique was suggested independently by Phillips [Phi62] and Tikhonov [Tik63, Tik77], and is usually called Tikhonov regularization. If we consider the p.d.f. [math]\displaystyle{ f_{true}(y) }[/math] before being discretized as a histogram, then the regularization function is :[math]\displaystyle{ S[f_{true}(y)] = - … (11.41) }[/math] where the integration is over all allowed values of [math]\displaystyle{ y }[/math]. The minus sign comes from the convention taken here that we maximize [math]\displaystyle{ \phi }[/math] as defined by (11.40). That is, greater [math]\displaystyle{ S }[/math] corresponds to more smoothness. (Equivalently one can of course minimize a combination of regularization and log-likelihood functions with the opposite sign; this convention as well is often encountered in the literature.)
11.5.2 Regularization functions based on entropy 169
11.5.3 Bayesian motivation for the use of entropy 170
11.5.4 Regularization function based on cross-entropy 173
11.6 Variance and bias of the estimators 173
11.7 Choice of the regularization parameter 177
11.8 Examples of unfolding 179
11.9 Numerical implementation 184
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