Reformulation of the Gaussian error propagation for a mixture of dependent and independent variables

Sándor Kristyán

doi:10.3311/ppch.7313

Abstract

The Gaussian error propagation is a state of the art expression in error analysis for estimating standard deviation for an expression f(x1,…,xn,z) via its variables. One of its basic assumptions is the independence of the measurable variables in its argument. However, in practice, measurable quantities are correlated somehow, and sometimes, z depends on some of the xi’s. We provide the generalized version of the Gaussian error propagation formula in this case. We will prove this with the formula for total derivative of a general multivariable function for which some of its variables are not independent from the others; a counterpart to the probability approach of this subject.

Highlights

1 Introduction Frequently, the final result of an experiment cannot be measured directly, rather, it is calculated from several measurable physical quantities, each of which has a mean value and an error, and we are interested in the resulting error in the final result of such an experiment
To the best of our knowledge, this case has been commonly formulated with algorithms using the concept of covariance via probability theory approach, still there is no compact expression formulated via calculus – here we do this
(∂w / ∂x) + (∂w / ∂z)(dz / dx) = 0 or, and dz / dx can be expressed.) If ∂zj / ∂xi = 0 for all i = 1,...,n and all j = 1,...,m in Eqs. 6,7, all zj fall into the independent set of {x1,.., xn}, and Eqs. 6,7 reduce to Eq 3 or Eq 1, i.e. to the general expression of Gaussian error propagation for independent variables, as expected

Summary

Introduction

The final result of an experiment cannot be measured directly, rather, it is calculated from several measurable physical quantities, each of which has a mean value and an error, and we are interested in the resulting error in the final result of such an experiment. The measurement protocol is very complex and the set of measured physical quantities is a mix of variables in which some are independent of others and some are not. Selecting only independent physical quantities to be measured is not always possible. These difficulties occur in data analysis after collecting the outcome of measurements, for example: in weather observation or meteorology, astro- or high-energy physics, physical-, chemical- or biological measurements, as well as economics. Statisticians use a procedure commonly called the delta method [1,2,3] to obtain an estimator of the variance when the estimator is not a simple sum of observations. To the best of our knowledge, this case has been commonly formulated with algorithms using the concept of covariance via probability theory approach, still there is no compact expression formulated via calculus – here we do this

Problem formulation

The way to the reformulation via calculus