User Guide

Basic setup

Basic mathematical operations involving numbers with uncertainties only require a simple import:

>>> from uncertainties import ufloat

The ufloat() function creates numbers with uncertainties. Existing calculation code can usually run with no or little modification and automatically produce results with uncertainties.

The uncertainties module contains other features, which can be made accessible through

>>> import uncertainties

The uncertainties package also contains sub-modules for advanced mathematical functions, and arrays and matrices.

Creating and handling numbers with uncertainties

Numbers with uncertainties can be input either numerically, or through one of many string representations, so that files containing numbers with uncertainties can easily be parsed. Thus, x = 0.20±0.01 can be expressed in many convenient ways:

>>> x = ufloat(0.20, 0.01)  # x = 0.20+/-0.01
>>> x = ufloat_fromstr("0.20+/-0.01")
>>> x = ufloat_fromstr("0.20(1)")
>>> x = ufloat_fromstr("20(1)e-2")  # Exponential notation supported
>>> x = ufloat_fromstr("0.20")  # Automatic uncertainty of +/-1 on last digit

The available representations can be listed with pydoc uncertainties.ufloat_fromstr. Representations that are invalid raise a ValueError exception.

Basic math

Calculations can be performed directly, as with regular real numbers:

>>> square = x**2
>>> print square
0.04+/-0.004

Mathematical operations

Besides being able to apply basic mathematical operations to numbers with uncertainty, this package provides generalizations of most of the functions from the standard math module. These mathematical functions are found in the uncertainties.umath module:

>>> from uncertainties.umath import *  # Imports sin(), etc.
>>> sin(x**2)
0.039989334186634168+/-0.003996800426643912

The list of available mathematical functions can be obtained with the pydoc uncertainties.umath command.

Arrays of numbers with uncertainties

It is possible to put numbers with uncertainties in NumPy arrays and matrices:

>>> arr = numpy.array([ufloat(1, 0.01), ufloat(2, 0.1)])
>>> 2*arr
[2.0+/-0.02 4.0+/-0.2]
>>> print arr.sum()
3.0+/-0.100498756211

Thus, usual operations on NumPy arrays can be performed transparently even when these arrays contain numbers with uncertainties.

More complex operations on NumPy arrays and matrices can be performed through the dedicated uncertainties.unumpy module.

Correlations

Correlations between variables are automatically handled whatever the number of variables involved, and whatever the complexity of the calculation. Thus, each calculation result keeps track of how it is correlated to random variables. For example, when x is the number with uncertainty defined above,

>>> square = x**2
>>> print square
0.04+/-0.004
>>> square - x*x
0.0
>>> y = x*x + 1
>>> y - square
1.0

The last two printed results above have a zero uncertainty despite the fact that x, y and square have a non-zero uncertainty: the calculated functions give the same value for all samples of the random variable x.

Thanks to the tracking of dependencies on random variables, calculations can therefore be performed in as many steps as necessary, exactly as with simple floats. When various quantities are combined through mathematical operations, the result is calculated by taking into account all the correlations between the quantities involved. All of this is done completely transparently.

Access to the uncertainty and to the nominal value

The nominal value and the uncertainty (standard deviation) can also be accessed independently:

>>> print square
0.04+/-0.004
>>> print square.nominal_value
0.04
>>> print square.std_dev
0.004

Access to the individual sources of uncertainty

The various contributions to an uncertainty can be obtained through the error_components() method, which maps the independent variables a quantity depends on to their contribution to the total uncertainty. According to the linear error propagation theory implemented in uncertainties, the sum of the squares of these contributions is the squared uncertainty.

The individual contributions to the uncertainty are more easily usable when the variables are tagged:

>>> u = ufloat(1, 0.1, "u variable")  # Tag
>>> v = ufloat(10, 0.1, "v variable")
>>> sum_value = u+2*v
>>> sum_value
21.0+/-0.22360679774997899
>>> for (var, error) in sum_value.error_components().items():
...     print "%s: %f" % (var.tag, error)
...
u variable: 0.100000
v variable: 0.200000

The variance (i.e. squared uncertainty) of the result (sum_value) is the quadratic sum of these independent uncertainties, as it should be (0.1**2 + 0.2**2).

The tags do not have to be distinct. For instance, multiple random variables can be tagged as "systematic", and their contribution to the total uncertainty of result can simply be obtained as:

>>> syst_error = math.sqrt(sum(  # Error from *all* systematic errors
...     error**2
...     for (var, error) in result.error_components().items()
...     if var.tag == "systematic"))

The remaining contribution to the uncertainty is:

>>> other_error = math.sqrt(result.std_dev**2 - syst_error**2)

The variance of result is in fact simply the quadratic sum of these two errors, since the variables from result.error_components() are independent.

Comparison operators

Comparison operators behave in a natural way:

>>> print x
0.2+/-0.01
>>> y = x + 0.0001
>>> y
0.2001+/-0.01
>>> y > x
True
>>> y > 0
True

One important concept to keep in mind is that ufloat() creates a random variable, so that two numbers with the same nominal value and standard deviation are generally different:

>>> y = ufloat(1, 0.1)
>>> z = ufloat(1, 0.1)
>>> print y
1.0+/-0.1
>>> print z
1.0+/-0.1
>>> y == y
True
>>> y == z
False

In physical terms, two rods of the same nominal length and uncertainty on their length are generally of different sizes: y is different from z.

More detailed information on the semantics of comparison operators for numbers with uncertainties can be found in the Technical Guide.

Covariance and correlation matrices

Covariance matrix

The covariance matrix between various variables or calculated quantities can be simply obtained:

>>> sum_value = u+2*v
>>> cov_matrix = uncertainties.covariance_matrix([u, v, sum_value])

has value

[[0.01, 0.0,  0.01],
 [0.0,  0.01, 0.02],
 [0.01, 0.02, 0.05]]

In this matrix, the zero covariances indicate that u and v are independent from each other; the last column shows that sum_value does depend on these variables. The uncertainties package keeps track at all times of all correlations between quantities (variables and functions):

>>> sum_value - (u+2*v)
>>> 0.0

Correlation matrix

If the NumPy package is available, the correlation matrix can be obtained as well:

>>> corr_matrix = uncertainties.correlation_matrix([u, v, sum_value])
>>> corr_matrix
array([[ 1.        ,  0.        ,  0.4472136 ],
       [ 0.        ,  1.        ,  0.89442719],
       [ 0.4472136 ,  0.89442719,  1.        ]])

Correlated variables

Reciprocally, correlated variables can be created transparently, provided that the NumPy package is available.

Use of a covariance matrix

Correlated variables can be obtained through the covariance matrix:

>>> (u2, v2, sum2) = uncertainties.correlated_values([1, 10, 21], cov_matrix)

creates three new variables with the listed nominal values, and the given covariance matrix:

>>> sum_value
21.0+/-0.22360679774997899
>>> sum2
21.0+/-0.22360679774997899
>>> sum2 - (u2+2*v2)
0.0+/-3.8337185686225597e-09

The theoretical value of the last expression is exactly zero, like for sum - (u+2*v), but numerical errors yield a small uncertainty (3e-9 is indeed very small compared to the uncertainty on sum2: correlations should in fact cancel the uncertainty on sum2).

The covariance matrix is the desired one:

>>> uncertainties.covariance_matrix([u2, v2, sum2])

reproduces the original covariance matrix cov_matrix (up to rounding errors).

Use of a correlation matrix

Alternatively, correlated values can be defined through a correlation matrix (the correlation matrix is the covariance matrix normalized with individual standard deviations; it has ones on its diagonal), along with a list of nominal values and standard deviations:

>>> (u3, v3, sum3) = uncertainties.correlated_values_norm(
...     [(1, 0.1), (10, 0.1), (21, 0.22360679774997899)], corr_matrix)
>>> print u3
1.0+/-0.1

The three returned numbers with uncertainties have the correct uncertainties and correlations (corr_matrix can be recovered through correlation_matrix()).

Making custom functions accept numbers with uncertainties

This package allows code which is not meant to be used with numbers with uncertainties to handle them anyway. This is for instance useful when calling external functions (which are out of the user’s control), including functions written in C or Fortran. Similarly, functions that do not have a simple analytical form can be automatically wrapped so as to also work with arguments that contain uncertainties.

It is thus possible to take a function f() that returns a single float, and to automatically generalize it so that it also works with numbers with uncertainties:

>>> wrapped_f = uncertainties.wrap(f)

The new function wrapped_f() accepts numbers with uncertainties as arguments wherever a Python float is used for f(). wrapped_f() returns the same values as f(), but with uncertainties.

With a simple wrapping call like above, uncertainties in the function result are automatically calculated numerically. Analytical uncertainty calculations can be performed if derivatives are provided to wrap().

More details are available in the documentation string of wrap() (accessible through the pydoc command, or Python’s help() shell function).

Miscellaneous utilities

It is sometimes useful to modify the error on certain parameters so as to study its impact on a final result. With this package, the uncertainty of a variable can be changed on the fly:

>>> sum_value = u+2*v
>>> sum_value
21.0+/-0.22360679774997899
>>> prev_uncert = u.std_dev
>>> u.std_dev = 10
>>> sum_value
21.0+/-10.001999800039989
>>> u.std_dev = prev_uncert

The relevant concept is that sum_value does depend on the variables u and v: the uncertainties package keeps track of this fact, as detailed in the Technical Guide, and uncertainties can thus be updated at any time.

When manipulating ensembles of numbers, some of which contain uncertainties while others are simple floats, it can be useful to access the nominal value and uncertainty of all numbers in a uniform manner. This is what the nominal_value() and std_dev() functions do:

>>> print uncertainties.nominal_value(x)
0.2
>>> print uncertainties.std_dev(x)
0.01
>>> uncertainties.nominal_value(3)
3
>>> uncertainties.std_dev(3)
0.0

Finally, a utility method is provided that directly yields the standard score (number of standard deviations) between a number and a result with uncertainty: with x equal to 0.20±0.01,

>>> x.std_score(0.17)
-3.0

Derivatives

Since the application of linear error propagation theory involves the calculation of derivatives, this package automatically performs such calculations; users can thus easily get the derivative of an expression with respect to any of its variables:

>>> u = ufloat(1, 0.1)
>>> v = ufloat(10, 0.1)
>>> sum_value = u+2*v
>>> sum_value.derivatives[u]
1.0
>>> sum_value.derivatives[v]
2.0

These values are obtained with a fast differentiation algorithm.

Additional information

The capabilities of the uncertainties package in terms of array handling are detailed in Uncertainties in arrays.

Details about the theory behind this package and implementation information are given in the Technical Guide.