Informatics and Particle Physics Experiments I 573 Geoffrey

Informatics and Particle Physics Experiments I 573 Geoffrey Fox Indiana University 27 March 2007

References • http: //www. npac. syr. edu/users/gcf/CPS 713 ST AT/p_Stats. html (20 years old) • http: //monalisa. caltech. edu: 8080/Slides/SC 200 6/SC 2006_HEPNetworks. Grids 111406. ppt by Harvey Newman at Caltech (network focus) • http: //monalisa. caltech. edu: 8080/Slides/CMSP hysics 07/Physicsat. HEFrontier_Ph 10103006 Fi nal. ppt by Harvey Newman at Caltech (physics focus) • ~10 Petabytes data per year • 3 * 109 events per year

Particle Physics Data Typical results from my past “Regression” and Occurrence Probabilities E 350 0 X -t E 260 200 Ge. V hp 0 X 0

Higgs diphoton Analysis using Rootlets

• • • • • 1: Basic Definitions and Results 1. 1 Basic Definitions and Results Is Probability Intuitive or Mathematical? Basic Definitions---Borel Sets Axioms of Probability Immediate Deductions from Axioms of Probability From Mathematical to Intuitive Definition of Probability 1. 2 Bayes Theorem: Conditional Probabilities---1) Frequency Approach Bayes Theorem: Conditional Probabilities---2) Frequency and Probability Bayes Theorem: Conditional Probabilities---3) Bayes Law Example: Estimation of Image Scanning Efficiency Example: Analysis of Unreliable Test The Bayes Controversy Naive Application for Estimation 1. 3 Continuous Distributions General (Borel) Formalism Gaussian Distribution Properties of Gaussian Distribution 1. 4 Functions of a Random Variable Joint Probability Distributions Example of Joint Probability Distributions: Bayes Law for Densities • • • • • • 1. 5 Properties of Joint Distribution Functions Means---Moments, etc. for One-dimensional Distributions 1. 6 Means--Moments--Correlations for Multidimensional Distributions Errors and Moment Matrices Density Matrix Element Examples of Correlations 2 D Track Finding Examples of Correlations Exponential Fits Errors in Linear Combinations of Uncorrelated Variables Error in Lack of Correlation Assumption Parameters of a Gaussian Distribution Addition of Independent Random Variables 1. 7 Leading Up to the Central Limit Theorem Moment Generating Function of a Sum of Tw. Random Variables Moment Generating Function Contd. Central Limit Theorem Statement Laplace Central Limit Theorem Statement---Liapounoff Proof of Central Limit Theorem Laplace Monte-Carlo Integration and the Central Limit Theorem Extension of Central Limit Theorem t. Correlated Random Variables Central Limit Theorem for Quotients (i) Example from Histogramming (Physics) Events (ii) Analysis of Scattering Experiments (iii) Example of Weighted Histograms

2: Visiting Las Vegas: Combinatoric and Discrete Probability Overview • • • • Bernoulli Trials Example of Bernoulli Trials Poisson Limit of Binomial Distribution Example of Poisson Distribution Compound Poisson Distribution Particle Cluster Production Example of Cherenkov Light Additivity Property of Binomial or Poisson Distributions Gaussian Limit of Binomial Distribution Gaussian Limit of Binomial and Poisson Distributions 2. 2 Birth and Death Processes Overview Telephone Calls as a Birth Process Basic Birth Process Formalism General Birth and Death Process Example of Birth/Death Process Markov Chains

3: Estimation of Parameters (Regression) • • • • • Estimation of Parameters---References Overview of Estimation of Parameters 3. 1 Maximum Likelihood Principle Illustrated by Measurement Errors Enter Mr. Bayes for Conditional Probablities Interpretation of Bayes Formulation of Likelihood The Likelihood Maximizes the Quivering Rod General Maximum Likelihood Methods Warning Example of Misuse of Likelihood Method Practical Advice on Use of Maximum Likelihood Determination of Lifetime Example of Maximum Likelihood Central Limit Estimate of Error Maximum Likelihood Estimate of Error in Lifetime Comparison of Tw. Error Estimates in Maximum Likelihood Theorem on Asymptotic Validity of Maximum Likelihood Method---Conditions and Statement of Theorem on Asymptotic Validity of Maximum Likelihood Method Conditions---First Assertion Theorem on Asymptotic Validity of Maximum Likelihood Method Conditions---Second and Third Assertions Proof of Maximum Likelihood Theorem Part (a) Proof of Maximum Likelihood Part (b): Formula for Error Multiplication of Experiments • • • • • • 3. 2 Overview of chisq Method Derivation of chisq from Maximum Likelihood Mean and Standard Deviation of chisq Interpretation of Value of chisq Correlated Observations in chisq Example of Multiple Moments Example of Use of Linear fits to Cross Section Data: Overview Formulation of Linear Fit for Cross Section Data Solution and Errors from Linear Equations Estimate of Errors to be Used in Fits 3. 3 Counter Physics---Introduction and Poisson Process Unnormalized Counter Experiment Normalized Counter Experiment Gaussian Limit of Counter Physics 3. 4 Bubble Chamber or Spectrometer Physics Bubble/Spectrometer---Physics Individual Events with Continuous Distributions Individual Events Become Binned Data Method of Moments Example of Moment Method Comparison of Moment Method with Maximum Likelihood Several Parameters in Method of Moments Remarks on Counting Errors in Moments

4: Minimization • • • • • 4. 1 Basic Problem 4. 2 Derivative Formalism for chisq Derivative Calculation for Maximum Likelihood 4. 3 The Derivative Method for chisq Error Calculation from Expansion 4. 4 The Real World and Minimization Example of Two Variables Pictures of Two Variable Case The Real World and Minimization Example of Two Variables Contd. Eigenfunction Analysis Transformation of M to Eigendirections What I Used to Do! Why is M Poorly Conditioned? Marquardt's Method Estimate of Marquardt Parameter Q Experience with Marquardt's Method

5: Goodness of Fit • • • • • Distribution for One Degree of Freedom General Distribution Characteristic Function 5. 2 Confidence Level Typical Confidence Level Distributions Real World Confidence Levels Error Estimate from Value 5. 3 Bartlett's Function 5. 4 Hypothesis Testing 5. 5 Student's t Distribution for Hypothesis Testing 5. 6 Robust Estimation--Motivation Robust Determination of Mean Trimmed Mean Likelihood Based Robust Estimation of Means Particular Robust Estimates for Likelihood---Huber Particular Robust Estimates for Likelihood---Andrews Comparison of Methods

6: Generation of Random Numbers • • • • Overview of Generation of Random Numbers---Genuinely Random Congruential Pseudorandom Numbers Congruential Random Numbers in Monte Carlo A Practical Congruential Pseudorandom Number Generator Shift Register Pseudorandom Numbers---Overview Initialization of Pseudorandom Numbers Correlations Decorrelation of Pseudorandom Numbers Generation of Gaussian Random Numbers Generation of General Random Number Distributions Generation of Exponentially Distributed Random Numbers Interpolation t. Find General Distributions Accept/Reject Method for General Probability Distributions Accept-Reject Methods for Event Generation