Data Analysis

Generation by Transformation of the Uniform Distribution.- 4.9.2 Generation with the von Neumann Acceptance-Rejection Technique.- 4.10 Generation of Normally Distributed Random Numbers.- 4.11 Generation of Random Numbers According to a Multivariate Normal Distribution.- 4.12 The Monte Carlo Method for Integration.- 4.13 The Monte Carlo Method for Simulation.- 4.14 Example Programs.- 4.14.1 Main Program E1RN to Demonstrate Subprograms RNMLCG, RNECUY, and RNSTNR.- 4.14.2 Main Program E2RN to Demonstrate Subprogram RNLINE.- 4.14.3 Main Program E3RN to Demonstrate Subprogram RNRADI.- 4.14.4 Main Program E4RN to Simulate Molecular Movement of a Gas.- 4.14.5 Main Program E5RN to Demonstrate Subprograms RNMNPR and RNMNGN.- 4.15 Programming Problems.- 4.15.1 Program to Generate Breit-Wigner-Distributed Random Numbers.- 4.15.2 Program to Generate Random Numbers from a Triangular Distribution.- 4.15.3 Program to Generate Data Points with Errors of Different Size.- 4.15.4 Programs to Simulate Molecular Movement.- 5 Some Important Distributions and Theorems.- 5.1 The Binomial and Multinomial Distributions.- 5.2 Frequency. The Law of Large Numbers.- 5.3 The Hypergeometric Distribution.- 5.4 The Poisson Distribution.- 5.5 The Characteristic Function of a Distribution.- 5.6 The Standard Normal Distribution.- 5.7 The Normal or Gaussian Distribution.- 5.8 Quantitative Properties of the Normal Distribution.- 5.9 The Central Limit Theorem.- 5.10 The Multivariate Normal Distribution.- 5.11 Convolutions of Distributions.- 5.11.1 Folding Integrals.- 5.11.2 Convolutions with the Normal Distribution.- 5.12 Example Programs.- 5.12.1 Main Program E1DS to Simulate Empirical Frequency and Demonstrate Statistical Fluctuations.- 5.12.2 Main Program E2DS to Simulate the Experiment of Rutherford and Geiger.- 5.12.3 Main Program E3DS to Simulate Galton's Board.- 5.13 Problems.- 5.13.1 Binomial Distribution.- 5.13.2 Poisson Distribution.- 5.13.3 Normal Distribution.- 5.13.4 Multivariate Normal Distribution.- 5.13.5 Convolution.- 5.14 Programming Problems.- 5.14.1 Convolution of Uniform Distributions.- 5.14.2 Convolution of Uniform Distribution and Normal Distribution.- 5.15 Hints and Solutions.- 6 Samples.- 6.1 Random Samples. Distribution of a Sample. Estimators.- 6.2 Samples from Continuous Populations. Mean and Variance of a Sample.- 6.3 Graphical Representation of Samples. Histograms and Scatter Plots.- 6.4 Samples from Partitioned Populations.- 6.5 Samples without Replacement from Finite Discrete Populations. Mean Square Deviation. Degrees of Freedom.- 6.6 Samples from Gaussian Distributions. ?2-Distribution.- 6.7 ?2 and Empirical Variance.- 6.8 Sampling by Counting. Small Samples.- 6.9 Small Samples with Background.- 6.10 Determining a Ratio of Small Numbers of Events.- 6.11 Ratio of Small Numbers of Events with Background.- 6.12 Example Programs.- 6.12.1 Main Program E1 SM to Demonstrate Subprogram SMMNVR.- 6.12.2 Main Program E2 SM to Demonstrate Subprograms SMHSIN, SMHSFL, and SMHSGR.- 6.12.3 Main Program E3 SM to Demonstrate Subprogram SMSDGR.- 6.12.4 Main Program E4 SM to Demonstrate Subprogram SMERSS.- 6.12.5 Main Program E5 SM to Demonstrate Subprogram SMERQS.- 6.12.6 Main Program E6 SM to Simulate Experiments with Few Events and Background.- 6.12.7 Main Program E7 SM to Simulate Experiments with Few Signal Events and with Reference Events.- 6.13 Problems.- 6.13.1 Efficiency of Estimators.- 6.13.2 Sample Mean and Sample Variance.- 6.13.3 Samples from a Partitioned Population.- 6.13.4 ?2-distribution.- 6.13.5 Histogram.- 6.14 Hints and Solutions.- 7 The Method of Maximum Likelihood.- 7.1 Likelihood Ratio. Likelihood Function.- 7.2 The Method of Maximum Likelihood.- 7.3 Information Inequality. Minimum Variance Estimators. Sufficient Estimators.- 7.4 Asymptotic Properties of the Likelihood Function and Maximum-Likelihood Estimators.- 7.5 Simultaneous Estimation of Several Parameters. Confidence Intervals.- 7.6 Example Programs.- 7.6.1 Program E1ML to Compute the Mean Lifetime and Asymmetric Errors from a Small Number of Radioactive Decays.- 7.6.2 Program E2ML to Compute the Maximum-Likelihood Estimates of the Parameters of a Bivariate Normal Distribution from a Simulated Sample.- 7.7 Programming Problems.- 7.7.1 Distribution of Lifetimes Determined from a Small Number of Radioactive Decays.- 7.7.2 Distribution of the Sample Correlation Coefficient.- 7.8 Problems.- 7.8.1 Maximum-Likelihood Estimates.- 7.8.2 Information.- 7.8.3 Variance of an Estimator.- 7.9 Hints and Solutions.- 8 Testing Statistical Hypotheses.- 8.1 Introduction.- 8.2 F-Test on Equality of Variances.- 8.3 Student's Test. Comparison of Means.- 8.4 Concepts of the General Theory of Tests.- 8.5 The Neyman-Pearson Lemma and Applications.- 8.6 The Likelihood-Ratio Method.- 8.7 The ?2;-Test for Goodness-of-Fit.- 8.7.1 ?2;-test with Maximal Number of Degrees of Freedom.- 8.7.2 ?2;-test with Reduced Number of Degrees of Freedom.- 8.7.3 ?2;-Test and Empirical Frequency Distribution.- 8.8 Contingency Tables.- 8.9 2 × 2 Table Test.- 8.10 Example Programs.- 8.10.1 Main Program E1TEST to Generate Samples and Test the Equality of their Variances Using the F-Test.- 8.10.2 Main Program E2TEST to Generate Samples and Test the Equality of their Means with a Given Value Using Student's Test.- 8.10.3 Main Program E3TEST to Generate Samples and Compute the Test Statistic ?2; for the Hypothesis that the Samples are Taken from a Normal Distribution with Known Parameters.- 8.11 Problems.- 8.11.1 F-Test.- 8.11.2 Student's Test.- 8.11.3 ?2;-Test for Variance.- 8.11.4 ?2;-TestofGoodness-of-Fit.- 8.11.5 Contingency Table.- 8.12 Hints and Solutions.- 9 The Method of Least Squares.- 9.1 Direct Measurements of Equal or Unequal Accuracy.- 9.2 Indirect Measurements. Linear Case.- 9.3 Fitting a Straight Line.- 9.4 Programs for Fitting Linear Functions of the Unknowns.- 9.4.1 Fitting a Polynomial.- 9.4.2 Fit of an Arbitrary Linear Function.- 9.5 Indirect Measurements. Nonlinear Case.- 9.6 Programs for Fitting Nonlinear Functions.- 9.6.1 Iteration with Step-Size Reduction.- 9.6.2 Marquardt Iteration.- 9.7 Properties of the Least-Squares Solution. ?2;-Test.- 9.8 Confidence Regions and Asymmetric Errors in the Nonlinear Case.- 9.9 Constrained Measurements.- 9.9.1 The Method of Elements.- 9.9.2 The Method of Lagrange Multipliers.- 9.10 The General Case of Least-Squares Fitting.- 9.11 Program for the General Case of Least Squares.- 9.12 Applying the Program for the General Case to Constrained Measurements.- 9.13 Confidence Region and Asymmetric Errors in the General Case.- 9.14 Example Programs.- 9.14.1 Main Program E1LSQ to Demonstrate Subprogram LSQPOL.- 9.14.2 Main Program E2LSQ to Demonstrate Subprogram LSQLIN.- 9.14.3 Main Program E3LSQ to Demonstrate Subprogram LSQNON.- 9.14.4 Main Program E4LSQ to Demonstrate Subprogram LSQMAR.- 9.14.5 Main Program E5LSQ to Demonstrate Subprogram LSQASN.- 9.14.6 Main Program E6LSQ to Demonstrate Subprogram LSQCON.- 9.14.7 Main Program E7LSQ to Demonstrate Subprogram LSQGEN.- 9.14.8 Main Program E8LSQ to Demonstrate Subprograms LSQASG and LSQCOG.- 9.15 Programming Problems.- 9.15.1 Fit of a First-Degree Polynomial to Data that Correspond to a Second-Degree Polynomial.- 9.15.2 Fit of a Power Law (Linear Case).- 9.15.3 Fit of a Power Law (Nonlinear Case).- 9.15.4 Fit of a Breit-Wigner Function to Data Points with Errors.- 9.15.5 Asymmetric Errors and Confidence Region for the Fit of a Breit-Wigner Function.- 9.15.6 Fit of a Breit-Wigner Function to a Histogram.- 9.15.7 Fit of a Circle to Points with Measurement Errors in Abscissa and Ordinate.- 10 Function Minimization.- 10.1 Overview. Numerical Accuracy.- 10.2 Parabola through Three Points.- 10.3 Function of n Variables on a Line in an n-Dimensional Space.- 10.4 Bracketing the Minimum.- 10.5 Minimum Search with the Golden Section.- 10.6 Minimum Search with Quadratic Interpolation.- 10.7 Minimization along a Direction in n Dimensions.- 10.8 Simplex Minimization in n Dimensions.- 10.9 Minimization along the Coordinate Directions.- 10.10 Conjugate Directions.- 10.11 Minimization along Chosen Directions.- 10.12 Minimization in the Direction of Steepest Descent.- 10.13 Minimization along Conjugate Gradient Directions.- 10.14 Minimization with the Quadratic Form.- 10.15 Marquardt Minimization.- 10.16 On Choosing a Minimization Method.- 10.17 Consideration of Errors.- 10.18 Examples.- 10.19 Example Programs.- 10.19.1 Main Program E1MIN to Demonstrate Subprograms MINSIM, MINPOW, MINCJG, MINQDR, and MINMAR.- 10.19.2 Main Program E2MIN to Determine the Parameters of a Distribution from the Elements of a Sample and to Demonstrate Subprogram MINCOV.- 10.19.3 Main Program E3MIN to Demonstrate Subprograms MINASY and MINCNT.- 10.19.4 Main Program E4MIN to Determine the Parameters of a Distribution from a Histogram of a Sample and to Demonstrate Subprograms MINGLP and MINGSQ.- 10.20 Programming Problems.- 10.20.1 Monte Carlo Minimization to Choose a Good First Approximation.- 10.20.2 Determination of the Parameters of a Breit-Wigner Distribution from the Elements of a Sample.- 10.20.3 Determination of the Parameters of a Breit-Wigner Distribution from the Histogram of a Sample.- 10.20.4 Determination of the Parameters the Sum of Two Breit-Wigner Distributions from the Histogram of a Sample.- 11 Analysis of Variance.- 11.1 One-Way Analysis of Variance.- 11.2 Two-Way Analysis of Variance.- 11.3 Example Programs.- 11.3.1 Main Program E1AV to Demonstrate Subprograms AVTBLE and AVOUTP.- 11.3.2 Main Program E2AV to Generate Stimulated Data and Perform an Analysis of Variance.- 11.4 Programming Problems.- 11.4.1 Two-Way Analysis of Variance with Crossed Classification.- 11.4.2 Two-Way Analysis of Variance with Nested Classification.- 12 Linear and Polynomial Regression.- 12.1 Orthogonal Polynomials.- 12.2 Regression Curve. Confidence Interval.- 12.3 Regression with Unknown Errors.- 12.4 Example Programs.- 12.4.1 Main Program E1REG to Demonstrate Subprogram REGPOL.- 12.4.2 Main Program E2REG to Graphically Display the Data and All Polynomials for a Polynomial Regression.- 12.4.3 Main Program E3REG to Demonstrate Subprogram REGCON and Display the Fitted Polynomial and its Confidence Limits.- 12.4.4 Main Program E4REG to Stimulate Data with Measurement Errors and Perform a Polynomial Regression.- 12.5 Programming Problems.- 12.5.1 Stimulation of Data and Display of a Regression Polynomials of Regression Polynomials of Different Degree.- 12.5.2 Stimulation of Data and Display of a Regression Line with confidence Limits.- 13 Time Series Analysis.- 13.1 Time Series Trend.- 13.2 Moving Averages.- 13.3 Edge Effects.- 13.4 Confidence Intervals.- 13.5 Program for Time Series Analysis. Examples.- 13.6 Example Programs.- 13.6.1 Main Program E1TIM to Demonstrate Subprogram TIMSER.- 13.6.2 Main Program E2TIM to Perform a Time Series Analysis and Display the Output.- 13.7 Programming Problems.- 13.7.1 Extrapolation in a Time Series Analysis.- 13.7.2 Discontinuities in Time Series.- A Matrix Calculations.- A.l Definitions. Simple Operations.- A.2 Vector Space, Subspace, Rank of a Matrix.- A.3 Programs for Simple Matrix and Vector Operations.- A.4 Orthogonal Transformations.- A.4.1 Givens Transformation.- A.4.2 Householder Transformation.- A.4.3 Sign Inversion.- A.4.4 Permutation Transformation.- A.5 Determinants.- A.6 Matrix Equations. Least Squares.- A.7 Inverse Matrix.- A.8 Gaussian Elimination.- A.9 LR-Decomposition.- A.10 Cholesky Decomposition.- A.11 Pseudo-inverse Matrix.- A.12 Eigenvalues and Eigenvectors.- A.13 Singular Value Decomposition.- A.14 Singular Value Analysis.- A.15 Algorithm for Singular Value Decomposition.- A.15.1 Strategy.- A.15.2 Bidiagonalization.- A.15.3 Diagonalization.- A.15.4 Ordering of the Singular Values and Permutation.- A.15.5 Singular Value Analysis.- A.16 Least Squares with Weights.- A.17 Least Squares with Change of Scale.- A.18 Modification of Least Squares According to Marquardt.- A.19 Least Squares with Constraints.- A.20 Example Programs for Simple Matrix and Vector Operations.- A.20.1 Main Program E1MTX to Demonstrate Simple Operations of Matrix and Vector Algebra with Subprograms MTXTRA, MTXWRT, MTXADD, MTXSUB, MTXMLT, MTXMBT, MTXMAT, MTXUNT, MTXZER, MTXMSC, MTXTRP, MTXCPV, MTXADV, MTXSBV, MTXDOT, MTXMSV, MTXZRV.- A.20.2 Main Program E2MTX to Demonstrate the Manipulation of Submatrices and Subvectors with Subprograms MTXGSM, MTXPSM, MTXGCL, MTXPCL, MTXGRW, MTXPRW, MTXGSV, MTXPSV.- A.21 Example Programs Related to Orthogonal Transformations.- A.21.1 Main Program E3MTX to Demonstrate the Givens Transformation with Subprograms MTXGVD,MTXGVT,andMTXGVA.- A.21.2 Main Program E4MTX to Demonstrate the Householder Transformation with Subprograms MTXHSD and MTXHST.- A.22 Example Programs Solving Matrix Equations.- A.22.1 Main Program E5MTX to Demonstrate the Gaussian Algorithm with Subprogram MTXEQU.- A.22.2 Main Program E6MTX to Demonstrate Cholesky Decomposition and Cholesky Inversion with Subprograms MTXCHL, MTXCHI, and MTXCHM.- A.22.3 Main Program E7MTX to Demonstrate the Singular Value Decomposition withSubprogramMTXSVD.- A.22.4 Main Program E8MTX to Demonstrate the Solution of Matrix Equations in 9 Different Cases by Singular Value Decomposition withSubprogramMTXDEC.- A.22.5 Main Program E9MTX to Demonstrate Subprogram MTXMAR.- A.22.6 Main Program E10MTX to Demonstrate the Solution of a Least-Squares Problem with Constraints by Subprogram MTXLSC.- B Combinatorics.- C Formulas and Programs for Statistical Functions.- C.1 Binomial Distribution.- C.2 Hypergeometric Distribution.- C.3 Poisson Distribution.- C.4 Normal Distribution.- C.8 Example Programs.- C.8.1 Main Program E1SD to Demonstrate Subprograms SDBINM, SCBINM, SDHYPG, SCHYPG, SDPOIS, SCPOIS, SQPOIS.- C.8.2 Main Program E2SD to Demonstrate Subprograms SDSTNR, SCSTNR, SQSTNR, SDNORM, SCNORM, SQNORM, SDCHI2, SCCHI2, SQCHI2, SDFTST, SCFTST, SQFTST, SCSTUD, SDSTUD, SQSTUD.- C.9 Programming Problems.- C.9.1 Graphical Representation of Statistical Functions of a Discrete Variable.- C.9.2 Graphical Representation of Statistical Functions of a Continuous Variable.- D The Gamma Function and Related Functions. Methods and Programs for their Computation.- D.1 The Euler Gamma Function.- D.2 Factorial and Binomial Coefficients.- D.3 Beta Function.- D.4 Computing Continued Fractions.- D.5 Incomplete Gamma Function.- D.6 Incomplete Beta Function.- D.7 Example Programs.- D.7.1 Main Program E1GAM to Demonstrate Subprograms GGAMMA and GLNGAM.- D.7.2 Main Program E2GAM to Demonstrate Subprogram GBINCO.- D.7.3 Main Program E3GAM to Demonstrate Subprograms GBETAF, GINCGM, and GINCBT.- D.8 Programming Problems.- D.8.2 Graphical Representations of the Beta Function, the Incomplete Gamma Function, and the Incomplete Beta Function.- E Utility Programs.- E.l Numerical Differentiation.- E.2 Numerical Determination of Zero.- F The Graphics Programming Package GRPACK.- F. 1 Introductory Remarks.- F.2 Control Routines.- F.3 Coordinate Systems and Transformations.- F.3.1 Coordinate Systems.- F.3.2 Linear Transformations. Window - Viewport.- F.4 Transformation Routines.- F.5 Drawing Programs.- F.6 GraphicsUtilityRoutines.- F.7 Text within the Plot.- F.8 Programs Yielding a Complete Plot.- F.9 Graphics Workstations.- F.9.1 Types of Workstations. Initialization File. Color Index.- F.9.2 Computer Screen or Window.- F.9.3 Files for Off-Line Printing or Plotting.- F.9.4 Determination of the Workstation Numbers.- F.9.5 Error File. File Handling.- F.10 Example Programs.- F.10.1 Main Program E1GR to Demonstrate Subprograms GRNBWS, GROPEN, GRCLSE, GROPWS, GRCLWS, GRWNCC, GRVWWC, GRWNWC, GRSTFR, GRFRAM, GRBOUN, GRSTCL, GRPLIN, GRBRPL, GRSCLX, GRSCLY, GRTXTF.- F.10.2 Main Program E2GR to Demonstrate Subprogram GRMARK.- F.10.3 Main Program E3GR to Demonstrate Subprogram GRDATP.- F.10.4 Main Program E4GR to Demonstrate Subprogram GRPLCT.- F.10.5 Main Program E5GR to Demonstrate Subprograms GRSCDF and GRCCRS.- F.10.6 Main Program E6GR to Demonstrate Subprogram GRHSCV.- F.10.7 Main Program E7GR to Demonstrate Subprogram GRDTCV.- F.10.8 Main Program E8GR to Demonstrate Subprogram GRDTMC.- G Software Installation and Technical Hints.- G.1 The CD-ROM.- G.2 Different Compilers and Operating Systems.- G.3 Using C Programs.- G.4 Programming for WINDOWS 95/WINDOWS NT.- G.4.1 Introductory Remarks.- G.4.2 Main Program.- G.4.3 Application Program.- G.5 Program Installation.- G.6 The Directory Structure for DOS and WINDOWS.- G.6.1 The Subdirectory \DATAN\DATSRC.- .6.2 The Subdirectory \DATAN\EXASRC.- .6.3 The Subdirectory \DATAN\SOLSRC.- .6.4 The Subdirectory \DATAN\DLIB. The Data Analysis Library.- .6.5 The Subdirectory \DATAN?IB. The Graphics Library.- G.7 The Directory Structure for Linux.- G.7.1 The Directory /usr/ local /datan.- G.7.1.1 The Subdirectory /usr/local/datan/datsrc.- G.7.1.2 The Subdirectory /usr/local/datan/exasrc.- G.7.1.3 The Subdirectory /usr/local/datan/solsrc.- G.7.1.4 The Subdirectory /usr/local/datan/include.- G.7.1.5 The Subdirectory /usr/local/datan/utils.- G.7.2 Libraries.- G.7.3 Shell files.- G.8 Modifying the Data Analysis Library.- G.9 Compiling, Linking, and Executing Programs.- G.10 Array Size Limitations in Some Programs.- H Collection of Formulas.- I Statistical Tables.- Literature.- List of Computer Programs.