PHYS 6710: Nuclear and Particle Physics II

Transcription

1 Data Analysis Content (~7 Classes) Uncertainties and errors Random variables, expectation value, (co)variance Distributions and samples Binomial, Poisson, and Normal Distribution Student's t-distribution Data fitting by minimization -Tests Parameter error Covariance and multidimensional fitting Likelihood techniques (?)

2 Examples Fit of Photoproduction Data (from Jlab, VA; MAMI, GE; ELSA, GE)

3 Photoproduction Reaction type Observable: Differential cross section Data from different experiments Models from different analysis groups (GW-based and others) Excluded regions large variations ~ 2 orders of magnitude

4 Challenges A theory/model inspires a fit hypothesis with potentially many free parameters to be adapted ( fitted ) to data. Possibly describing different reactions with the same underlying theory Search for universal properties in data physical model. Model free parameters adapted simultaneously to data of reactions 1,2 predict reaction 3 from model confirm/reject model & inspire new measurements. Problems: Uncertainties in data/ Incompatibility of data Deficient model (may or may not result in bad fit what is good, what is bad? Regions without applicability of the model Difficult data behavior (large variations etc.) to find a good model/fit hypothesis is a challenge of theoretical physics

5 Uncertainty of data Statistical Uncertainties (random errors) This course Photoproduction: Counting experiment (taggers, calorimeters,.) More counted events smaller error bars on data (how much smaller? this course) Uncertainty from thermal noise in measuring devices: inherent source of uncertainties that cannot be corrected for; Noise density per 1 Hz; k: Boltzmann's constant; T: Temperature Examples: electronic circuit, camera sensor Finite sample size vs. Systematic Errors miscalibration of detectors; correctable or uncorrectable (but not random). Difficult to rigorously treat in general.

6 Random Variables A random variable X is a measurable function from the set of possible outcomes to a set Image of X countably infinite: discrete random variable associated with probability mass function Image of X uncountable infinite: continuous random variable associated with probability density function

7 Examples: Coins and dices X: $1 payoff for every successful bet on heads Probability mass function X: possible outcome Probability mass function

8 Example: Two dices

9 Example: Two dices

10 Example: Two dices Normalization: x Source: Wikipedia

11 Continuous probability distributions 6-sided dice 20 sided dice sphere scattering angle in pion photoproduction: Let probability density function be is the probability that X takes a value in Normalization:

12 Expectation value Expectation value of a function g(x) with respect to probability distribution f(x): (Continuous case) (discrete case) Mean of x

13 (continued) Variance: standard deviation (s.d.) or root-mean-square deviation (r.m.s.): s.d. of the uncertainty distribution of an experimental result: standard uncertainty or standard error or r.m.s. error See [Berendsen] for definitions of n-th moment, skewness, kurtosis, characteristic function

14 Properties of Expectation Values Notation: Here: we will use with distributions and samples in experiments (explained later). With c constant, Apply properties of expectation value to show in connection with Variance is the mean of the square minus square of the mean (used in quantum mechanics)

15 Several dimensions Probability density function (pdf) in several variables: Normalization: Marginal pdf f(x) of x: Conditional probability of having x given y: if x,y independent : if dependent: Show: Tower property

16 Experiments A probability distribution is a theoretical construct In an actual experiment with finite number of counts, or with finite number of measurements of the same quantity (a random variable) one cannot reconstruct the underlying probability distribution, but only sample it. Colloquially, this is referred to as finite statistics. This necessary shortcoming induces uncertainties. The sampling of probability distributions is called stochastics, or stochastic estimates thereof (or of other quantities such as mean value and variance). There is a fundamental difference between probability distributions and their estimate from experiment. Estimators are used to estimate mean value and variance.

17 Unbiased and biased estimators Mean value of underlying distribution function: Estimate of the mean value from an experiment in which have been measured: is a unbiased estimator of expectation value): ( unbiased defined through

18 Estimator of the Variance Is Show: meaning that The term an unbiased estimator of? is a biased estimator of the variance. is the variance of the mean. This term can be evaluated. For this we make the assumption that the data are uncorrelated. Thus, for :

19 (continued) With this, the variance of the mean is:

20 Wrapping up We considered the estimator We found: Meaning that is an unbiased estimator of the sample variance. As shown, if X and Y are independent, The measure of dependence is the covariance

21 Remarks The variance of the mean, becomes smaller as n Infinity. In experimental data of counting experiments, the sample size is given by the counts; the error bar in plots of those data represents the standard error But the exact variance might not known in general this will change the result a bit (see upcoming lectures). It can be shown (not here) that is an unbiased estimator of the standard error if the underlying probability distribution is normal (no general proof).