# Usage

After installing the package, you can start using it with

julia> using Measurements

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

Measurements.measurementFunction
measurement(val::Real, [err::Real]) -> Measurement
measurement(::Missing, [err::Union{Real,Missing}]) -> Missing
val ± err -> Measurement

Return a Measurement object with val as nominal value and err as uncertainty. err defaults to 0 if omitted.

The binary operator ± is equivalent to measurement, so you can construct a Measurement object by explicitely writing 123 ± 4.

If val is missing, missing is returned.

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where

• val is the nominal value of the measurement
• err 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 val and err 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 analytic 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 third-party 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.

Measurements.measurementMethod
measurement(string) -> Measurement{Float64}

Parse the string and return a Measurement{Float64} object.

Examples of valid strings and the resulting Measurement{Float64} are:

julia> using Measurements

julia> measurement("-123.4(56)")
-123.4 ± 5.6

julia> measurement("+1234(56)e-1")
123.4 ± 5.6

julia> measurement("12.34e-1 +- 0.56e1")
1.2 ± 5.6

julia> measurement("(-1.234 ± 0.056)e2")
-123.4 ± 5.6

julia> measurement("1234e-1 +/- 5.6e0")
123.4 ± 5.6

julia> measurement("-1234e-1")
-123.4 ± 0.0
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measurement function has also a method that enables you to create a Measurement object from a string.

Caveat about the ± sign

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 parentheses where appropriate to avoid confusion. See for example the following cases:

julia> using Measurements

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 between 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 non-zero 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

Measurements.@uncertainMacro
@uncertain f(value ± stddev, ...)

A macro to calculate f(value) ± uncertainty, with uncertainty derived from stddev according to rules of linear error propagation theory.

Function f can accept any number of real arguments.

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Existing functions implemented exclusively in Julia that accept AbstractFloat arguments will work out-of-the-box 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.derivativeFunction
derivative(x::Measurement, y::Measurement)

Return the value of the partial derivative of x with respect to the independent measurement y, calculated on the nominal value of y. Return zero if x does not depend on y.

Use Measurements.derivative.(x, array) to calculate the gradient of x with respect to an array of independent measurements.

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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_componentsFunction
Measurements.uncertainty_components(x::Measurement)

Return the components to the uncertainty of the dependent quantity x in the form of a Dict for all the independent Measurements from which x is derived.

The key of each entry of the dictionary is the triplet (value, uncertainty, tag) of an independent Measurement, and the value is the absolute value of the product between its uncertainty and the partial derivative of x with respect to this Measurement.

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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 dictionary whose values are the components of the uncertainty of x.

## Standard Score

Measurements.stdscoreFunction
stdscore(measure::Measurement, expected_value::Real) -> standard_score

Gives the standard score between a measure, with uncertainty, and its expected value:

(measure.val - expected_value)/measure.err
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stdscore(measure_1::Measurement, measure_2::Measurement) -> standard_score

Gives the standard score between two measurements (both with uncertainty) computed as the standard score between their difference and 0:

stdscore(measure_1 - measure_2, 0)
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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

Measurements.weightedmeanFunction
weightedmean(iterable) -> measurement(weighted_mean, standard_deviation)

Return the weighted mean of measurements listed in iterable, using inverse-variance weighting. NOTA BENE: correlation is not taken into account.

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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.valueFunction
Measurements.value(x::Measurement)
Measurements.value(x::Complex{Measurement})
Measurements.value(x::Missing)

Return the nominal value of measurement x.

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Measurements.uncertaintyFunction
Measurements.uncertainty(x::Measurement)
Measurements.uncertainty(x::Complex{Measurement})
Measurements.uncertainty(x::Missing)

Return the uncertainty of measurement x.

source

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 fields 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.

## Representation of Measurements

### Measurements in the REPL

When working in the Julia REPL, Measurement objects are shown truncated in order to present two significant digits for the uncertainty:

julia> using Measurementsjulia> -84.32 ± 5.6-84.3 ± 5.6julia> 7.9 ± 18.67.9 ± 19.0

Note that truncation only affects the numbers shown in the REPL:

julia> using Measurementsjulia> Measurements.value(7.9 ± 18.6)7.9julia> Measurements.uncertainty(7.9 ± 18.6)18.6

### Printing to TeX and LaTeX MIMEs

You can print Measurement objects to TeX and LaTeX MIMES ("text/x-tex" and "text/x-latex"), the ± sign will be rendered with \pm command:

julia> using Measurementsjulia> repr("text/x-tex", 5±1)"5.0 \\pm 1.0"julia> repr("text/x-latex", pi ± 1e-3)"3.1416 \\pm 0.001"