Usage¶
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 measurementuncertainty
is its uncertainty, assumed to be a standard deviation.
They are both subtype of AbstractFloat
. Some keyboard layouts provide an
easy way to type the ±
sign, if your does not, remember you can insert it in
Julia REPL with \pm
followed by TAB
key. You can provide value
and
uncertainty
of any subtype of Real
that can be converted to
AbstractFloat
. Thus, measurement(42, 33//12)
and pi ± 0.1
are
valid.
measurement(value)
creates a Measurement
object with zero uncertainty,
like mathematical constants. See below for further examples.
Note
Every time you use one of the constructors above you define a new
independent measurement. Instead, when you perform mathematical operations
involving Measurement
objects you create a quantity that is not
independent, but rather depends on really independent measurements.
Most mathematical operations are instructed, by operator overloading, to accept
Measurement
type, and uncertainty is calculated exactly using analityc
expressions of functions’ derivatives.
It is also possible to create a Complex
measurement with
complex(measurement(real_part_value, real_part_uncertainty), measurement(imaginary_part_value, imaginary_part_uncertainty))
In addition to making the code prettier, the fact that the ±
sign can be
used as infix operator to define new independent Measurement
s makes the
printed representation of these objects valid Julia syntax, so you can quickly
copy the output of an operation in the Julia REPL to perform other calculations.
Note however that the copied number will not be the same object as the
original one, because it will be a new independent measurement, without memory
of the correlations of the original object.
This module extends many methods defined in Julia’s mathematical standard
library, and some methods from widespread thirdparty packages as well. This is
the case for most special functions in SpecialFunctions.jl package, and the
quadgk
integration routine from QuadGK.jl package.
Those interested in the technical details of the package, in order integrate the package in their workflow, can have a look at the technical appendix.

measurement
(string)
measurement
function has also a method that enables you to create a
Measurement
object from a string. See the “Examples” section for details.
Caution
The ±
infix operator is a convenient symbol to define quantities with
uncertainty, but can lead to unexpected results if used in elaborate
expressions involving many ±
s. Use parantheses where appropriate to
avoid confusion. See for example the following cases:
julia> 7.5±1.2 + 3.9±0.9 # This is wrong!
11.4 ± 1.2 ± 0.9 ± 0.0
julia> (7.5±1.2) + (3.9±0.9) # This is correct
11.4 ± 1.5
Correlation Between Variables¶
The fact that two or more measurements are correlated means that there is some sort of relationship beetween them. In the context of measurements and error propagation theory, the term “correlation” is very broad and can indicate different things. Among others, there may be some dependence between uncertainties of different measurements with different values, or a dependence between the values of two measurements while their uncertainties are different.
Here, for correlation we mean the most simple case of functional relationship:
if \(x = \bar{x} \pm \sigma_x\) is an independent measurement, a quantity
\(y = f(x) = \bar{y} \pm \sigma_y\) that is function of \(x\) is not
like an independent measurement but is a quantity that depends on \(x\), so
we say that \(y\) is correlated with \(x\). The package
Measurements.jl
is able to handle this type of correlation when propagating
the uncertainty for operations and functions taking two or more arguments. As a
result, \(x  x = 0 \pm 0\) and \(x/x = 1 \pm 0\). If this correlation
was not accounted for, you would always get nonzero uncertainties even for
these operations that have exact results. Two truly different measurements that
only by chance share the same nominal value and uncertainty are not treated as
correlated.
Propagate Uncertainty for Arbitrary Functions¶

@uncertain f(x, ...)
Existing functions implemented exclusively in Julia that accept
AbstractFloat
arguments will work outofthebox with Measurement
objects as long as they internally use functions already supported by this
package. However, there are functions that take arguments that are specific
subtypes of AbstractFloat
, or are implemented in such a way that does not
play nicely with Measurement
variables.
The package provides the @uncertain
macro that overcomes this limitation and
further extends the power of Measurements.jl
.
This macro allows you to propagate uncertainty in arbitrary functions, including
those based on C/Fortran calls,
that accept any number of real arguments. The macro exploits derivative
and
gradient
functions from Calculus package in order to perform
numerical differentiation.
Derivative and Gradient¶

Measurements.
derivative
(y::Measurement, x::Measurement)¶
In order to propagate the uncertainties, Measurements.jl
keeps track of the
partial derivative of an expression with respect to all independent measurements
from which the expression comes. For this reason, the package provides a
convenient function, Measurements.derivative
, to get the partial derivative
and the gradient of an expression with respect to independent measurements.
Uncertainty Contribution¶

Measurements.
uncertainty_components
(x::Measurement)¶
You may want to inspect which measurement contributes most to the total
uncertainty of a derived quantity, in order to minimize it, if possible. The
function Measurements.uncertainty_components
gives you a dictonary whose
values are the components of the uncertainty of x
.
Standard Score¶

stdscore
(measure::Measurement, expected_value) → standard_score¶

stdscore
(measure_1::Measurement, measure_2::Measurement) → standard_score
The stdscore
function is available to calculate the standard score between a measurement and its
expected value (not a Measurement
). When both arguments are Measurement
objects, the standard score between their difference and zero is computed, in
order to test their compatibility.
Weighted Average¶

weightedmean
(iterable) → weighted_mean¶
weightedmean
function gives the weighted mean of a set of
measurements using inverses of variances as weights. Use mean
for
the simple arithmetic mean.
Access Nominal Value and Uncertainty¶

Measurements.
value
(x)¶

Measurements.
uncertainty
(x)¶
As explained in the technical appendix, the nominal value and the uncertainty of
Measurement
objects are stored in val
and err
fields respectively,
but you do not need to use those field directly to access this information.
Functions Measurements.value
and Measurements.uncertainty
allow you to
get the nominal value and the uncertainty of x
, be it a single measurement
or an array of measurements. They are particularly useful in the case of
complex measurements or arrays of measurements.
Error Propagation of Numbers with Units¶
Measurements.jl
does not know about units of measurements, but can be easily
employed in combination with other Julia packages providing this feature.
Thanks to the type system of Julia programming
language this integration is seamless and comes for free, no specific work has
been done by the developer of the present package nor by the developers of the
above mentioned packages in order to support their interplay. They all work
equally good with Measurements.jl
, you can choose the library you prefer and
use it. Note that only algebraic functions are allowed to operate
with numbers with units of measurement, because transcendental functions operate on
dimensionless quantities. In the Examples
section you will find how this feature works with a couple of packages.
Printing to TeX and LaTeX MIMEs¶
You can print Measurement
objects to TeX and LaTeX MIMES ("text/xtex"
and "text/xlatex"
), the ±
sign will be rendered with \pm
command:
julia> display("text/xtex", 5±1)
5.0 \pm 1.0
julia> display("text/xlatex", pi ± 1e3)
3.141592653589793 \pm 0.001