UIUC Physics 598AEM
Analysis of Experimental Measurements
Lecture Notes and Other Hand-Outs:

P598AEM Lecture Notes:

• Lecture 1
  Introduction/Course Goals, Applications, Intro Concepts, Conditional Probability, Bayes Theorem, Law of Total Probability
  
• Lecture 2
  Some examples, def'n of a random variable, PDFs and CDFs, the Uniform dist'n, change of variables, random vs. systematic uncertainties
  
• Lecture 3
  Expectation value/true mean, variance, std. dev, skewness, kurtosis, Uniform/Cauchy/Gaussian Dist'ns, multi-dimensional Gaussian dist'ns, marginal dist'n, change of variables.
  
• Lecture 4
  Expectation value, variance, covariance, correlation coefficient of two (or more) random variables, sample mean, variance and std. dev. of sample mean.
  
• Lecture 5
  Taylor series expansion, matrix formalism for N random variables, covariance matrix - error propagation.
  
• Lecture 6
  Continuation of Lect. 5: Change of Variables, the Inverse Transformation, Examples.
  
• Lecture 7
  Approximation of a Gaussian/Normal Dist'n; Characteristic Functions; the Central Limit Theorem; Calculating Moments of a PDF f(x) via the Characteristic Fcn.
  
• Lecture 8
  Herschel's method of 2-D Gaussian/normal dist'n; Probability interpretation of Gaussian/normal dist'n - Confidence Intervals; The Law of Large Numbers; Experimental sampling of a PDF, Biased and Unbiased Estimators of Low-Order Moments of a PDF; Median and Mode; Probable Error, Reliable Error.
  
• Lecture 9
  Multivariate Gaussian Dist'n; Orthogonal Transformations; Binomial Probability Dist'n; Multinomial Probability Dist'n;
  
• Lecture 10
  The Poisson Probability Distribution, Poisson Double and Single-Sided Confidence Intervals, Relationships between Binomial, Poisson and Gaussian Probability Distributions.
  
• Lecture 11
  Hypothesis Testing, Likelihood Functions, The Maximum Likelihood Method (MLM) and Parameter Estimation, Information and Information Inequality, Biased and Un-Biased Estimators
  
• Lecture 12
  More information on Information, Minimum Variance Estimators (MVE's), MVE's for Gaussian and Poisson Processes
  
• Lecture 13
  The MLM vs. the Method of Moments, the MLM with many Parameters, Correlations of Parameters, the Chi2 PDF and CDF
  
• Lecture 14
  The Extended MLM (EMLM), Use of the MLM with Binned Data, Use of {Univariate} MLM/EMLM (ANOVA) to Determine Signal vs. Background, Generalization to Multivariate case (MANOVA), Combining Experimental Results using Log-Likelihood Functions
  
• Lecture 15
  The Method of Least Squares, the Least Squares Principal, Surveyor's "Failure-to-Close" problem, Least Squares Method with constraints - use of Lagrange Multipliers, properties of the Chi2 PDF, p-value, upper/lower CL's, Pearson's Chi2 Test
  
• Lecture 16
  More on Chi2, p-values, upper/lower CL's, Examples: Combining results of different experiments, comparing data vs. theory prediction
  
• Lecture 17
  More on Chi2 and "errors", Example: LSQ 2-parameter fit to a straight line (y = mx + b), LSQ fit interpolation and extrapolation
  
• Lecture 18
  General Formulation of the Least Squares Method, General Formulation of the Linear Least Squares Method, Example: LSQ Fit to a Parabola, LSQ Polynomial Fits, Linear LSQ 2-parameter fit to a straight line (y = mx + b) when have uncertainties on both x and y.
  
• Lecture 19
  More on the General Formulation of the Linear Least Squares Method, Example: LSQ Fit to a Parabola (II)
  
• Lecture 20
  Combining Results via the Linear Least Squares Method, Least Squares Method vs. Maximum Likelihood Method, The Non-Linear Least Squares Method - Newton-Raphson Method, Linear Least Squares Fits with Linear Constraints, the use of Lagrange Multipliers,
• Lecture 21
  General Least Squares Method with General Constraints
  
• Lecture 22
  Function Minimization, Finding Local Minima, Grid Search Method, Single Parameter Variation Method
  
• Lecture 23
  Function Minimization (Continued), the Simplex Method, Gradient Methods, Method of Steepest Descent,
  
• Lecture 24
  Function Minimization (continued), the Variable Metric Method, Function Minimization with Constraints, Finding Multiple Minima and the Global Minimum, the Metropolis-Hastings Algorithm and Simulated Annealing
  
• Invariance, Covariance and Contravariance
  Auxilliary Lecture on Invariance, Contravariance and Covariance of Vectors and Tensors - Coordinate Transformations
  
• Lecture 25
  Hypothesis Testing, the Neyman-Pearson Test, Critical Regions, "Goodness-of-Fit" Chi2 Tests, Kolmogorov-Smirnov Test(s), the Smirnov-Cramer-Von Mises Test
  
• Lecture 26
  Monte Carlo Methods/Techniques, the Basis for Monte Carlo Integration, Properties of the MC Integral Estimate, Example: Buffon's Needle,
  
• Lecture 27
  Random Numbers, Quasi-Random/Pseudo-Random Numbers, Multiplicative Congruential Random Number Generators, Testing Random Number Generators, DIEHARD, TESTU01 Battery of Random Number Tests, Gaussian Random Number Generators, Triangular Random Number Generators, Square-Root Random Number Generators, General Method for Non-Uniform Random Numbers, Exponential Random Number Generators, Uniform Cos(Theta) Distribution, Random Isotropic Distribution in 3-D, Random Uniform Distribution on a 2-D Disk, the Von Neumann Acceptance-Rejection Method, Random Numbers from Histogram Distributions, Propagation of Large Errors,
  
• Lecture 28
  Obtaining CL Limits Near a Physical Boundary, Bayes Theorem and Bayesian Posterior and Prior Probability Densities, Example: Upper Limit on the Mean of a Poisson Variable with Background
  

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