Siegmund Brandt - Data Analysis
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Siegmund Brandt
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Springer International Publishing Switzerland 2014
Siegmund Brandt Data Analysis1. Introduction
Siegmund Brandt 1
(1)
Department of Physics, University of Siegen, Siegen, Germany
1.1 Typical Problems of Data Analysis
Every branch of experimental science, after passing through an early stage of qualitative description, concerns itself with quantitative studies of the phenomena of interest, i.e., measurements . In addition to designing and carrying out the experiment, an important task is the accurate evaluation and complete exploitation of the data obtained. Let us list a few typical problems.
- A study is made of the weight of laboratory animals under the influence of various drugs. After the application of drug A to 25 animals, an average increase of 5 % is observed. Drug B , used on 10 animals, yields a 3 % increase. Is drug A more effective? The averages 5 and 3 % give practically no answer to this question, since the lower value may have been caused by a single animal that lost weight for some unrelated reason. One must therefore study the distribution of individual weights and their spread around the average value. Moreover, one has to decide whether the number of test animals used will enable one to differentiate with a certain accuracy between the effects of the two drugs.
- In experiments on crystal growth it is essential to maintain exactly the ratios of the different components. From a total of 500 crystals, a sample of 20 is selected and analyzed. What conclusions can be drawn about the composition of the remaining 480? This problem of sampling comes up, for example, in quality control, reliability tests of automatic measuring devices, and opinion polls.
- A certain experimental result has been obtained. It must be decided whether it is in contradiction with some predicted theoretical value or with previous experiments. The experiment is used for hypothesis testing .
- A general law is known to describe the dependence of measured variables, but parameters of this law must be obtained from experiment. In radioactive decay, for example, the number N of atoms that decay per second decreases exponentially with time:
. One wishes to determine the decay constant and its measurement error by making maximal use of a series of measured values N 1( t 1), N 2( t 2), . One is concerned here with the problem of fitting a function containing unknown parameters to the data and the determination of the numerical values of the parameters and their errors.
From these examples some of the aspects of data analysis become apparent. We see in particular that the outcome of an experiment is not uniquely determined by the experimental procedure but is also subject to chance: it is a random variable . This stochastic tendency is either rooted in the nature of the experiment (test animals are necessarily different, radioactivity is a stochastic phenomenon), or it is a consequence of the inevitable uncertainties of the experimental equipment, i.e., measurement errors. It is often useful to simulate with a computer the variable or stochastic characteristics of the experiment in order to get an idea of the expected uncertainties of the results before carrying out the experiment itself. This simulation of random quantities on a computer is called the Monte Carlo method , so named in reference to games of chance.
1.2 On the Structure of this Book
The basis for using random quantities is the calculus of probabilities . The most important concepts and rules for this are collected in Chap. Here one considers distributions of random variables, and parameters are defined to characterize the distributions, such as the expectation value and variance. Special attention is given to the interdependence of several random variables. In addition, transformations between different sets of variables are considered; this forms the basis of error propagation .
Generating random numbers on a computer and the Monte Carlo method are the topics of Chap. In addition to methods for generating random numbers, a well-tested program and also examples for generating arbitrarily distributed random numbers are given. Use of the Monte Carlo method for problems of integration and simulation is introduced by means of examples. The method is also used to generate simulated data with measurement errors, with which the data analysis routines of later chapters can be demonstrated.
In Chap. we introduce a number of distributions which are of particular interest in applications. This applies especially to the Gaussian or normal distribution, whose properties are studied in detail.
In practice a distribution must be determined from a finite number of observations, i.e., from a sample . Various cases of sampling are considered in Chap. Computer programs are presented for a first rough numerical treatment and graphical display of empirical data. Functions of the sample, i.e., of the individual observations, can be used to estimate the parameters characterizing the distribution. The requirements that a good estimate should satisfy are derived. At this stage the quantity 2 is introduced. This is the sum of the squares of the deviations between observed and expected values and is therefore a suitable indicator of the goodness-of-fit.
The maximum-likelihood method , discussed in Chap. is devoted to hypothesis testing . It contains the most commonly used F , t , and 2 tests and in addition outlines the general points of test theory.
The method of least squares , which is perhaps the most widely used statistical procedure, is the subject of Chap. various methods are discussed in detail, by which such a minimization can be carried out. The relative efficiency of the procedures is shown by means of programs and examples.
The analysis of variance (Chap.) can be considered as an extension of the F -test. It is widely used in biological and medical research to study the dependence, or rather to test the independence, of a measured quantity from various experimental conditions expressed by other variables. For several variables rather complex situations can arise. Some simple numerical examples are calculated using a computer program.
Linear and polynomial regression , the subject of Chap. For example the determination of confidence intervals for a solution and the relation between regression and analysis of variance are studied. A general program for polynomial regression is given and its use is shown in examples.
In the last chapter the elements of time series analysis are introduced. This method is used if data are given as a function of a controlled variable (usually time) and no theoretical prediction for the behavior of the data as a function of the controlled variable is known. It is used to try to reduce the statistical fluctuation of the data without destroying the genuine dependence on the controlled variable. Since the computational work in time series analysis is rather involved, a computer program is also given.
The field of data analysis, which forms the main part of this book, can be called applied mathematical statistics . In addition, wide use is made of other branches of mathematics and of specialized computer techniques. This material is contained in the appendices.
In Appendix A, titled Matrix Calculations, the most important concepts and methods from linear algebra are summarized. Of central importance are procedures for solving systems of linear equations, in particular the singular value decomposition, which provides the best numerical properties.
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