kafe2 - a Modern Tool for Model Fitting in Physics Lab Courses Johannes G aler1 G unter Quast1 Daniel Savoiu2 Cedric Verstege1 1Karlsruhe Institute of Technology KIT2Hamburg University

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kafe2 - a Modern Tool for Model Fitting in Physics Lab Courses

Johannes G¨aßler(1), G¨unter Quast(1), Daniel Savoiu(2), Cedric Verstege(1)

(1) Karlsruhe Institute of Technology (KIT), (2) Hamburg University

October 25, 2022

Abstract

Fitting models to measured data is one of the standard tasks in the natural sciences, typically

addressed early on in physics education in the context of laboratory courses, in which statistical methods

play a central role in analysing and interpreting experimental results. The increased emphasis placed on

such methods in modern school curricula, together with the availability of powerful free and open-source

software tools geared towards scientiﬁc data analysis, form an excellent premise for the development

of new teaching concepts for these methods at the university level. In this article, we present kafe2,

a new tool developed at the Faculty of Physics at the Karlsruhe Institute of Technology, which has

been used in physics laboratory courses for several years. Written in the Python programming language

and making extensive use of established numerical and optimization libraries, kafe2 provides simple

but powerful interfaces for numerically ﬁtting model functions to data. The tools provided allow for

ﬁne-grained control over many aspects of the ﬁtting procedure, including the speciﬁcation of the input

data and of arbitrarily complex model functions, the construction of complex uncertainty models, and

the visualization of the resulting conﬁdence intervals of the model parameters.

1 Model ﬁtting in lab courses: pitfalls and limitations

A primary goal of data analysis in physics is to check whether an assumed model adequately describes

the experimental data within the scope of their uncertainties. A regression procedure is used to infer the

causal relationships between independent and dependent variables, the abscissa values xiand the ordinate

values yi. The dependency is typically represented by a functional relation yi=f(xi;{pk}) with a set of

model parameters {pk}. Both the yiand the xiare the results of measurements and hence are aﬀected

by unavoidable measurement uncertainties that must be taken into account when determining conﬁdence

intervals for the model parameters. It has become a common standard in the natural sciences to determine

conﬁdence intervals for the parameter values based on a maximum-likelihood estimation (MLE). In practice,

one typically uses the negative logarithm of the likelihood function (NLL), which is minimised by means of

numerical methods.

In the basic laboratory courses, parameter estimation is traditionally based on the method of least

squares (LSQ), which in many cases is equivalent to the NLL method. For example, ﬁtting a straight line

to data aﬀected by static Gaussian uncertainties on the ordinate values yiis an analytically solvable problem

and the necessary calculations can be carried out by means of a pocket calculator or a simple spreadsheet

application. For simplicity, non-linear problems are often converted to linear ones by transforming the

yvalues. However, the Gaussian nature of the uncertainties is lost in this process, which is why the

transformed problem only gives accurate results if the uncertainties are suﬃciently small. Simple error

propagation is often used to transform the ﬁtted parameters like the slope or yintercept to the model

parameters of interest. As a result, the parameter uncertainties obtained in this way are no longer Gaussian,

and it is non-trivial to determine sensible conﬁdence intervals. When the independent variables are also

aﬀected by measurement uncertainties, or when relative uncertainties with respect to the true values are

present, this traditional approach reaches its limits.

Such simple analytical methods are very limited in their general applicability to real-world problems. For

instance, the presence of uncertainties on the abscissa values in addition to those on the ordinate values,

as they would be encountered when measuring a simple current-voltage characteristic of an electronic

component, already results in a problem that can generally no longer be solved analytically. Unfortunately,

arXiv:2210.12768v1 [physics.ed-ph] 23 Oct 2022

most of the widely used numerical tools for parameter estimation provide only limited support for such

non-linear problems, even though they occur frequently in practice.

Prior to drawing any conclusions on model parameters, the most important task in physics is to check

the validity of a model hypothesis assumed to describe a set of measurement data. The full task therefore

consists of the following steps:

1. carefully quantifying the uncertainties of all input measurements;

2. deﬁning a model hypothesis for the measured data;

3. testing whether the measurements are compatible with the model hypothesis;

4. if so, determining the model parameters, e.g. the slope and intercept of a straight line.

It is precisely the hypothesis testing in step 3 that drives progress in scientiﬁc understanding by providing

the relation between measurements and theoretical models. A simple example for such a hypothesis test is

the χ2test, which results directly from the frequently used method of least squares. A test for the validity

of the model assumption should deﬁnitely be performed before drawing any conclusions from the values and

uncertainties of the ﬁtted model parameters. Unfortunately, many of the common tools come with a default

setting that assumes the correctness of the given parameterization and scales the parameter uncertainties

so that the model perfectly describes the data within the parameter uncertainties so determined. This

behaviour can usually be switched oﬀ by choosing appropriate options, but in practice this is often neglected.

The parameters of a complex model can be highly correlated. In this case it is no longer suﬃcient to

consider the distribution of just a single parameter. Instead the shared distribution of multiple parameters

needs to be considered, for example when conducting hypothesis testing. Correlations between model

parameters can also be problematic for numerical optimisation. For this reason strategies for choosing

appropriate parameterizations to reduce these correlations also need to be addressed. It is then indispensable

that the correlations are also shown when presenting the ﬁt results. Ideally the correlation between two

model parameters can be expressed with a simple correlation coeﬃcient. However, if non-linear models are

ﬁtted, this is often not suﬃcient for the evaluation of the result; in this case the corresponding conﬁdence

regions should be determined and presented as contour graphs for pairs of parameters.

The distinction between ”statistical” and ”systematic” uncertainties leads to many misunderstandings

among students. For example, the systematic error of a measuring instrument becomes a statistical one

if several measuring instruments of the same type are used. A much more suitable approach is to diﬀer-

entiate whether uncertainties aﬀect all measured values or groups of measured values equally or whether

they are independent. This approach, however, requires dealing with the covariance matrix of measure-

ment uncertainties and constructing it from the individual uncertainties on a problem-by-problem basis.

Unfortunately, there are very few simple tools that allow a full covariance matrix to be considered in the

ﬁtting process. None of the tools commonly used in physics lab courses allow students to take into account

correlated uncertainties of both the abscissa and ordinate values.

While there are tools with which the requirements listed here could be partially implemented, these do

not provide the simplicity needed for undergraduate physics education. To address the issues described

above it became necessary to write a tool of our own, kafe2, an open-source Python package designed to

provide a ﬂexible Python interface for the estimation of model parameters from measured data.

2 The Open-Source Python Package kafe2

After designing a prototype package, kafe, development continued with kafe2 [1]. Said tools make use

of contemporary methods for visualization and evaluation of measurement data and are based on freely

available, open-source software that is also used in scientiﬁc contexts and thus relevant for later professional

practice.

2.1 Implementation

Due to its widespread use in scientiﬁc communities as well as in the burgeoning professional ﬁeld of data

science, Python [2] was chosen as the programming language for kafe2. As a high-level language with

particular emphasis on clear and intuitive syntax, and convenient features such as dynamic typing and

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摘要：

kafe2-aModernToolforModelFittinginPhysicsLabCoursesJohannesGaler(1),GunterQuast(1),DanielSavoiu(2),CedricVerstege(1)(1)KarlsruheInstituteofTechnology(KIT),(2)HamburgUniversityOctober25,2022AbstractFittingmodelstomeasureddataisoneofthestandardtasksinthenaturalsciences,typicallyaddressedearlyoninph...

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kafe2 - a Modern Tool for Model Fitting in Physics Lab Courses Johannes G aler1 G unter Quast1 Daniel Savoiu2 Cedric Verstege1 1Karlsruhe Institute of Technology KIT2Hamburg University

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