Introduction
The GUM Tree calculator (GTC) is a data processing tool that uses uncertain numbers to represent measured quantities. GTC automates evaluation of uncertainty in derived quantities when they are calculated from measured data.
Measurement error
A measurement obtains information about a quantity, but the quantity itself (the measurand) is never determined exactly. There is always some measurement error involved. This can be expressed as an equation, where the unknown measurand is \(Y\) and the measurement result is \(y\), we have
where \(E_y\) is the measurement error. So, the result, \(y\), is only an approximate value for the quantity of interest \(Y\).
This is how ‘uncertainty’ arises. After any measurement, we are faced with uncertainty about what will happen if we take the measured value \(y\) and use it for the (unknown) value \(Y\).
For example, suppose the speed of a car is measured by a law enforcement officer. The officer needs to decide whether, in fact, a car was travelling faster than the legal limit but this simple fact cannot be determined, because the actual speed \(Y\) remains unknown. The measured value \(y\) might indicate that the car was speeding when in fact it was not, or that it was not speeding when in fact it was. In practice, a decision rule that takes account of the measurement uncertainty must be used. In this example, the rule will probably err on the side of caution (a few speeding drivers will escape rather than unfairly accusing good drivers of speeding).
Like the measurand, the measurement error \(E_y\) will never be known. At best, its behaviour can be described in statistical terms. This leads to technical meanings of the word ‘uncertainty’. For instance, the term ‘standard uncertainty’ refers to the standard deviation of a statistical distribution associated with an unpredictable quantity.
Measurement models
A measurement error comes about because there are unpredictable factors that influence the outcome of a measurement process. In a formal analysis, these factors must be identified and included in a measurement model, which defines the measurand in terms of all other significant influence quantities. In mathematical terms, we write
where the \(X_i\) are influence quantities.
Once again, the actual quantities \(X_1, X_2, \cdots\) are not known; only estimates \(x_1, x_2, \cdots\) are available. These are used to calculate a measured value that is approximately equal to the measurand
Uncertain Numbers
An uncertain number is a data-type designed to represent a measured quantity. It encapsulates information about the measurement, including the measured value and its uncertainty.
Uncertain numbers are used when processing measurement data; that is, to evaluate measurement models. The inputs to a model (like \(X_1, X_2, \cdots\) above) will be defined as uncertain numbers using measurement data. Calculations then produce an uncertain number for the measurand (\(Y\)).
There are two types of uncertain number: one for real-valued quantities and one for complex-valued quantities. At the very least, two pieces of information are needed to define an uncertain number: a value (that is, a measured, or approximate, value of the quantity) and the uncertainty associated with the error in the measured value.
Uncertain real numbers
The function ureal() is usually the preferred way to define uncertain numbers representing real-valued quantities.
Example: an electrical circuit
Suppose the current flowing in an electrical circuit \(I\) and the voltage across a circuit element \(V\) have been measured.
The measured values are \(x_V = 0.1\, \mathrm{V}\) and \(x_I = 15\,\mathrm{mA}\), with standard uncertainties \(u(x_V) = 1\, \mathrm{mV}\) and \(u(x_I) = 0.5\,\mathrm{mA}\), respectively.
Uncertain numbers for \(V\) and \(I\) are defined by
>>> V = ureal(0.1,1E-3)
>>> I = ureal(15E-3,0.5E-3)
and then the resistance can be calculated directly using Ohm’s law
>>> R = V/I
>>> print(R)
6.67(23)
The measured value of resistance \(x_R = 6.67 \,\Omega\) is an estimate (approximation) for \(R\), the standard uncertainty in \(x_R\) as an estimate of \(R\) is \(0.23 \,\Omega\).
Example: height of a flag pole
Suppose a flag is flying from a pole that is 15 metres away from an observer (with an uncertainty of 3 cm). The angle between horizontal and line-of-sight to the top of the pole is 38 degrees (with an uncertainty of 2 degrees). How high is the top of the pole?
A measurement model should express a relationship between the quantities involved: the height of the pole \(H\), the distance to the base of the pole \(B\) and the line-of-sight angle \(\Phi\),
To calculate the height, we create uncertain numbers representing the measured quantities and use the model
>>> B = ureal(15,3E-2)
>>> Phi = ureal(math.radians(38),math.radians(2))
>>> H = B * tan(Phi)
>>> print(H)
11.72(84)
The result \(x_H = 11.7\) metres is our best estimate of the height \(H\). The standard uncertainty of this value, as an estimate of the actual height, is 0.8 metres.
It is important to note that uncertain-number calculations are open ended. In this case, for example, we can keep going and evaluate what the observer angle would be at 20 metres from the pole (the uncertainty in the base distance remains 3 cm)
>>> B_20 = ureal(20,3E-2)
>>> Phi_20 = atan( H/B_20 )
>>> print(Phi_20)
0.530(31)
>>> Phi_20_deg= Phi_20 * 180./math.pi
>>> print(Phi_20_deg)
30.4(1.8)
The angle of 30.4 degrees at 20 metres from the pole has a standard uncertainty of 1.8 degrees.
Uncertain complex numbers
The function ucomplex() is usually preferred for defining uncertain complex numbers.
Example: AC electric circuit
Suppose measurements have been made of: the alternating current \(i\) flowing in an electrical circuit, the voltage \(v\) across a circuit element and the phase \(\phi\) of the voltage with respect to the current. The measured values are: \(x_v \approx 4.999\, \mathrm{V}\), \(x_i \approx 19.661\,\mathrm{mA}\) and \(x_\phi \approx 1.04446\,\mathrm{rad}\), with standard uncertainties \(u(x_v) = 0.0032\, \mathrm{V}\), \(u(x_i) = 0.0095\,\mathrm{mA}\) and \(u(x_\phi) = 0.00075\,\mathrm{rad}\), respectively.
Uncertain numbers for the quantities \(v\), \(i\) and \(\phi\) can be defined
>>> v = ucomplex(complex(4.999,0),(0.0032,0))
>>> i = ucomplex(complex(19.661E-3,0),(0.0095E-3,0))
>>> phi = ucomplex(complex(0,1.04446),(0,0.00075))
Note, the uncertainty argument is a pair of numbers in these definitions. These are the standard uncertainties associated with measured values of the real and imaginary components.
The complex impedance is
>>> z = v * exp(phi) / i
>>> print(z)
(+127.73(19)+219.85(20)j)
We see that our best estimate of the impedance is the complex value \((127.73 +\mathrm{j}219.85) \,\Omega\). The standard uncertainty in the real component is \(0.19 \,\Omega\) and the standard uncertainty in the imaginary component is \(0.20 \,\Omega\). There is also a small correlation between our estimates of the real and imaginary components
>>> get_correlation(z)
0.0582038103158399...
If a polar representation of the impedance is preferred,
>>> print(magnitude(z))
254.26(20)
>>> print(phase(z))
1.04446(75)
Uncertain Number Attributes
Uncertain number objects have attributes that provide access to: the measured value (the estimate), the uncertainty (of the estimate) and the degrees of freedom (associated with the uncertainty) (see UncertainReal).
Continuing with the flagpole example, the attributes x, u, df obtain the value, the uncertainty and the degrees-of-freedom (which is infinity), respectively
>>> H.x
11.71928439760076...
>>> H.u
0.84353295110757...
>>> H.df
inf
Alternatively, there are functions that return the same attributes
>>> value(H)
11.71928439760076...
>>> uncertainty(H)
0.84353295110757...
>>> dof(H)
inf
Uncertain numbers and measurement errors
It is often is helpful to to formulate measurement models that explicitly acknowledge measurement errors. As we said above, these errors are not known exactly; many will be residual quantities with estimates of zero or unity. However, errors have a physical meaning and it is often useful to identify them in the model.
In the example above, errors associated with measured values of \(B\) and \(\Phi\) were not identified but we can do so now by introducing the terms \(E_b\) and \(E_\phi\). The measured values \(b=15\,\mathrm{m}\) and \(\phi=38 \, \mathrm{deg}\) are related to the quantities of interest as
Our best estimates of these errors are trivial, \(E_b \approx 0\) and \(E_\phi \approx 0\), but the actual values are unpredictable and give rise to uncertainty in the height of the pole. It is appropriate to attribute the standard uncertainties \(u(E_b)=3\times 10^{2}\, \mathrm{m}\) and \(u(E_\phi)=2\, \mathrm{deg}\) to measurement errors, rather than associate uncertainty with the fixed quantities \(B\) and \(\Phi\).
The calculation becomes
>>> B = 15 - ureal(0,3E-2,label='E_b')
>>> Phi = math.radians(38) - ureal(0,math.radians(2),label='E_phi')
>>> H = B*tan(Phi)
>>> print(H)
11.72(84)
This reflects our understanding of the problem better: the numbers \(b=15\) and \(\phi=38\) are known, there is nothing ‘uncertain’ about their values. What is uncertain are the unknown measurement errors \(E_b\) and \(E_\phi\).
When defining uncertain numbers, setting labels allows an uncertainty budget to be displayed later (see budget()). For instance,
>>> for cpt in rp.budget(H):
... print("{0.label}: {0.u:.3f}".format(cpt))
...
E_phi: 0.843
E_b: 0.023