PHYS 6710: Nuclear and Particle Physics II

Transcription

1 The ellipse of constant chi-square is described by The error ellipse To describe the tilt of the error ellipse, we need to consider the following: For two parameters a,b, there are new coordinates a', b' in which the error ellipse has no tilt. There, the covariance vanishes and the covariance matrix is diagonal (see previous considerations about general form of the covariance matrix). This is equivalent to As C is symmetric and positive semidefinite, there is an orthogonal transformation Q such that C -1 is diagonal in the new coordinates, Eigenvectors build columns of Q, is diagonal matrix with eigenvalues as diagonal elements. Eigenvectors point in the directions of the half axes of the eror ellipse Tilt in (a,b) calculated from this Largest eigenvalue: shortest half-axis Smallest eigenvalue: longest half-a. Useful to find linear combinations of parameters that are most sensitive to the data eliminate superflous parameters!

2 Graphical representation Eigenvalues : Length of half axis i.

3 Correlations a general warning Anscombe's quartet: Source: Wikipedia All four data sets share: It is indispensable to make a graphical representation of your data Be aware of outliners in your data which may grossly change the analysis

4 Correlated chi-square fits So far: we have calculated the covariance matrix to get access to parameter uncertainties confidence regions linear combinations of most/least significant parameters The covariance matrix has another application: whenever data points are not independent (we have assumed independent data points so far). In that case, the covariance matrix transports the full information beyond the individual error bars on each data point

5 (continued) The correct chi-square function gets replaced according to where y is the vector of data points, f is the vector of values of f evaluated at x, and C is the covariance matrix. In case of no correlations, and the original chis-square function is recovered. Example 1: Hadronic physics on the lattice. Researcher A has produced two data points with uncertainties: Energy o o Also, researcher A has determined the correlations between the points, using, e.g., bootstrap. Researcher B would like to use these data points for further analysis. He could fit to the data using normal chi-square, but would lose the correlations. If researcher A provides the covariance matrix, researcher B can perform a correlated chi-square fit without losing information.

6 A real world example A. Alexandru, C. Pelissier, Phys.Rev. D87 (2013) 1, Resonance parameters of the rho-meson from asymmetrical lattices xl uncorrelated for different x (different lattices) correlated at same x (same lattice) x Measurements of Eigenvalues of QCD Hamiltonian in an asymmetric box with periodic boundary conditions Figure shows error bars, but behind it is a 6x6 covariance matrix with 3 blocks of size 2x2 Input to fit hadronic models More in Nuclear Seminar, March 24, 2015, D. Guo, B. Hu, R. Molina

7 Another example [M. Doring, U. G. Meissner, JHEP01(2009) 009] Fit lattice eigenvalues and observe how derived quantities (phase shift and pole positions) behave: Towards a determination of resonance properties directly from QCD. Here, the resonance: Some physics is used to stabilize fit. Known properties from Effective Field Theory in V_2 (see Griesshammer, this course), plus unknown piece V_4 expanded as polynomial in energy much in physics is about finding the best fit hypothesis! Include the Known explicitly and fit the Unknown!

8 (continued) Phase shifts and amplitudes can be continued to complex energies and pole positions (resonance properties) can be determined. Such quantities are complicated, only numerically known functions of the internal paramters (more in second part of this lecture). Bootstrap can be used to calculate confidence regions of pole positions: Actual Pole position unknown Adding parameters and observe how estimated pole position changes Convergence, but quantitative answer is given by F-test. Note the non-linear dependence of pole position on parameters confidence regions are not ellipses any more Bootstrap is used to determine confidence regions.

9 Derived quantities variable transformations In the previous example, the pole position is a complicated function of the internal parameters, used to parameterize a fit function that is used to fit the data points (the latter being correlated or uncorrelated): Internal parameters of fit hypothesis complicated but known nonlinear dependence (un)correlated data complicated but known nonlinear dependence How do uncertainties from data propagate? complicated, not explicitly known,nonlinear dependence Derived quantity of physical interest ( resonance pole position ) Another derived quantity (phase shift delta) Bootstrap solves the problem: Almost automatic workflow: Generate synthetic data ensembles around the real data Perform fits on all enselmbles, save fit parameters of converged fits For every such set, evaluate your quantity of physical interest Perform statistical analysis on set of resulting points (mean, variance, covariance matrix, confidence regions) Applicable to not necessarily linear problems (banana shaped confidence regions are not a problem).

10 Variable transformations explicitly In previous example, coordinates are implicitly transformed: Internal parameterization (no physical meaning of parameters) Derived quantities with physical meaning Bootstrap can be replaced by explicit coordinate transformation. How does the covariance matrix transform? We have already met one very special transformation: Diagonalization of covariance matrix == rotation in the system in which parameters are uncorrelated. General case is of interest, but be aware that you may make a non-linear problem out of your linear one! The resulting covariance matrix is then only an approximation close to the minimum.

11 Transformation in multiple variables consider the new variables as function of old ones: y=f(x) Dimension m of y in general unequal dimension n of x f not necessarily linear Show: The covariance matrix in y,, is given by the covariance matrix in x,, through the transformation Proof: blackboard Allows to determine covariance matrix in derived quantities

12 Example Where do two linear regression lines intersect, and what is the uncertainty in intersection point? Thomson, Univ. of Cambridge, UK]: [inspired by Prof. M.A. Uncertainty in intersection point (derived quantity) (unrelated issue: note how fluctuations are identical don't forget to reset you random number generator!)