Measurements.jl

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Image credit: "xkcd: Error Bars" (CC-BY-NC 2.5)

Physical measures are typically reported with an error, a quantification of the uncertainty of the accuracy of the measurement. Whenever you perform mathematical operations involving these quantities you have also to propagate the uncertainty, so that the resulting number will also have an attached error to quantify the confidence about its accuracy. Measurements.jl relieves you from the hassle of propagating uncertainties coming from physical measurements, when performing mathematical operations involving them. The linear error propagation theory is employed to propagate the errors.

This library is written in Julia, a modern high-level, high-performance dynamic programming language designed for technical computing.

When used in the Julia interactive session, it can serve also as an easy-to-use calculator.

• Support for most mathematical operations available in Julia standard library and special functions from SpecialFunctions.jl package, involving real and complex numbers. All existing functions that accept AbstractFloat (and Complex{AbstractFloat} as well) arguments and internally use already supported functions can in turn perform calculations involving numbers with uncertainties without being redefined. This greatly enhances the power of Measurements.jl without effort for the users
• Functional correlation between variables is correctly handled, so x-x ≈ zero(x), x/x ≈ one(x), tan(x) ≈ sin(x)/cos(x), cis(x) ≈ exp(im*x), etc...
• Support for arbitrary precision (also called multiple precision) numbers with uncertainties. This is useful for measurements with very low relative error
• Define arrays of measurements and perform calculations with them. Some linear algebra functions work out-of-the-box
• Propagate uncertainty for any function of real arguments (including functions based on C/Fortran calls), using @uncertain macro
• Function to get the derivative and the gradient of an expression with respect to one or more independent measurements
• Functions to calculate standard score and weighted mean
• Parse strings to create measurement objects
• Easy way to attach the uncertainty to a number using the ± sign as infix operator. This syntactic sugar makes the code more readable and visually appealing
• Extensible in combination with external packages: you can propagate errors of measurements with their physical units, perform numerical integration with QuadGK.jl, numerical and automatic differentiation, and much more.
• Integration with Plots.jl.

The method used to handle functional correlation is described in this paper:

• M. Giordano, 2016, "Uncertainty propagation with functionally correlated quantities", arXiv:1610.08716 (Bibcode: 2016arXiv161008716G)

A current limitation of the package is that it is not yet possible to define quantities related by a correlation matrix.

If you use use this package for your research, please cite it.

The complete manual of Measurements.jl is available at https://juliaphysics.github.io/Measurements.jl/stable/. There, people interested in the details of the package, in order integrate the package in their workflow, can find a technical appendix explaining how the package internally works.

The latest version of Measurements.jl is available for Julia v1.0 and later releases, and can be installed with Julia built-in package manager. In a Julia session, after entering the package manager mode with ], run the command

pkg> update
pkg> add Measurements

Older versions of this package are also available for Julia 0.4-0.7.

After installing the package, you can start using it with

using Measurements

The module defines a new Measurement data type. Measurement objects can be created with the two following constructors:

measurement(value, uncertainty)
value ± uncertainty

where

• value is the nominal value of the measurement
• uncertainty is its uncertainty, assumed to be a standard deviation.

Here is a quick taster of the functionalities of the package:

julia> using Measurements

julia> a = measurement(4.5, 0.1)
4.5 ± 0.1

julia> b = 3.8 ± 0.4
3.8 ± 0.4

julia> 2a + b
12.8 ± 0.4472135954999579

julia> x = 8.4 ± 0.7

julia> x - x
0.0 ± 0.0

julia> x/x
1.0 ± 0.0

julia> x*x*x - x^3
0.0 ± 0.0

julia> sin(x)/cos(x) - tan(x)
-2.220446049250313e-16 ± 0.0 # They are equal within numerical accuracy

For more details about the use of the package read the documentation, in particular the Usage and Examples sections.

The Measurements.jl package is licensed under the MIT "Expat" License. The original author is Mosè Giordano.

Please, cite the paper Giordano 2016 (http://arxiv.org/abs/1610.08716) if you employ this package in your research work. For your convenience, a BibTeX entry is provided in the CITATION.bib file.