Multivariate Fitting and the Error Matrix in Global Analysis of Data

TL;DR

Global fits with many parameters often render standard error matrices unreliable. The authors introduce an iterative Hessian procedure to robustly compute the curvature of $\chi^{2}$ and a Lagrange multiplier method to map $\chi^{2}$ as a function of observables without relying on linear or quadratic approximations. They demonstrate improved Hessian stability for a global analysis of parton distribution functions and show how the Lagrange method can yield robust uncertainty bounds, such as a few percent on $\sigma_{W}$ for $W$ production with $\Delta\chi^{2}$ around 100. Implemented as an extension to MINUIT, these methods provide practical, cross-checked tools for rigorous uncertainty quantification in large-scale, multi-experiment analyses.

Abstract

When a large body of data from diverse experiments is analyzed using a theoretical model with many parameters, the standard error matrix method and the general tools for evaluating errors may become inadequate. We present an iterative method that significantly improves the reliability of the error matrix calculation. To obtain even better estimates of the uncertainties on predictions of physical observables, we also present a Lagrange multiplier method that explores the entire parameter space and avoids the linear approximations assumed in conventional error propagation calculations. These methods are illustrated by an example from the global analysis of parton distribution functions.

Figures (5)

• Figure 1: Variation of $\chi^{2}$ with distance along a typical direction in parameter space. The dotted curve is the exact $\chi^{2}$ and the solid curve is the quadratic approximation based on the Hessian. The quadratic form is seen to be a rather good approximation over the range shown.
• Figure 2: Difference between $\chi^{2}$ and its quadratic approximation (\ref{['eq:taylor']}), both of which are shown in Fig. \ref{['fig:one']}. A cubic contribution can be seen, along with a noticeable amount of numerical noise. The fine structure revealed here is small compared to the main variation of $\chi^{2}$ itself, which rises by $20$ over the region shown, as can be seen in Fig. 1.
• Figure 3: Frequency distribution of $\Delta \chi^2$ according to the Hessian approximation (\ref{['eq:taylor']}) for displacements in random directions for which the true value is $\Delta \chi^2 = 5.0\,$. Solid histogram: using Hessian calculated by iterative method of Section 3; Dotted histogram: using Hessian calculated by MINUIT.
• Figure 4: Same as Fig. \ref{['fig:three']}, except that the displacements are restricted to the parameter subspace spanned by the 10 steepest directions.
• Figure 5: Minimum $\chi^{2}$ as a function of the predicted cross section for $W^{\pm}$ production in $p\overline{p}$ collisions. Parabolic curve is the prediction of the iteratively improved Hessian method. Points are from the Lagrange multiplier method.