Transparent calculations with uncertainties on the quantities involved (aka error propagation); fast calculation of derivatives

Overview

uncertainties allows calculations such as (2 +/- 0.1)*2 = 4 +/- 0.2 to be performed transparently. Much more complex mathematical expressions involving numbers with uncertainties can also be evaluated directly.

The uncertainties package takes the pain and complexity out of uncertainty calculations.

Basic examples

>>> from uncertainties import ufloat

>>> x = ufloat(2, 0.25)
>>> x
2.0+/-0.25

>>> square = x**2  # Transparent calculations
>>> square
4.0+/-1.0
>>> square.nominal_value
4.0
>>> square.std_dev  # Standard deviation
1.0

>>> square - x*x
0.0  # Exactly 0: correlations taken into account

>>> from uncertainties.umath import *  # sin(), etc.
>>> sin(1+x**2)
-0.95892427466313845+/-0.2836621854632263

>>> print (2*x+1000).derivatives[x]  # Automatic calculation of derivatives
2.0

>>> from uncertainties import unumpy  # Array manipulation
>>> random_vars = unumpy.uarray([1, 2], [0.1, 0.2])
>>> print random_vars
[1.0+/-0.1 2.0+/-0.2]
>>> print random_vars.mean()
1.50+/-0.11
>>> print unumpy.cos(random_vars)
[0.540302305868+/-0.0841470984808 -0.416146836547+/-0.181859485365]

Main features

• Transparent calculations with uncertainties: no or little modification of existing code is needed. Similarly, the Python (or IPython) shell can be used as a powerful calculator that handles quantities with uncertainties (print statements are optional, which is convenient).

• Correlations between expressions are correctly taken into account. Thus, x-x is exactly zero, for instance (most implementations found on the web yield a non-zero uncertainty for x-x, which is incorrect).

• Almost all mathematical operations are supported, including most functions from the standard math module (sin,…). Comparison operators (>, ==, etc.) are supported too.

• Many fast operations on arrays and matrices of numbers with uncertainties are supported.

• Extensive support for printing numbers with uncertainties (including LaTeX support and pretty-printing).

• Most uncertainty calculations are performed analytically.

• This module also gives access to the derivatives of any mathematical expression (they are used by error propagation theory, and are thus automatically calculated by this module).

Installation instructions are available on the main web site for this package.

Contact

Please send feature requests, bug reports, or feedback to Eric O. LEBIGOT (EOL).

Version history

Main changes:

• 3.0: Massive speedup for some operations involving large numbers of numbers with uncertainty, like sum(ufloat(1, 1) for _ in xrange(100000)) (this is about 5,000 times faster than before).

• 2.4.8: Friendlier completions in Python shells, etc.: internal functions should not appear anymore (for the user modules: uncertainties, uncertainties.umath and uncertainties.unumpy). Parsing the shorthand notation (e.g. 3.1(2)) now works with infinite values (e.g. -inf(inf)); this mirrors the ability to print such numbers with uncertainty. The Particle Data Group rounding rule is applied in more cases (e.g. printing 724.2±26.2 now gives 724±26). The shorthand+LaTeX formatting of numbers with an infinite nominal value is fixed. uncertainties.unumpy.matrix now uses .std_devs instead of .std_devs(), for consistency with floats with uncertainty (automatic conversion of code added to uncertainties.1to2).

• 2.4.7: String formatting now works for (-)inf+/-... numbers.

• 2.4.5: String formatting now works for NaN+/-... numbers.

• 2.4.4: The documentation license now allows its commercial use.

• 2.4.2: NumPy 1.8 compatibility.

• 2.4.1: In uncertainties.umath, functions ceil(), floor(), isinf(), isnan() and trunc() now return values of the same type as the corresponding math module function (instead of generally returning a value with a zero uncertainty ...+/-0).

• 2.4: Extensive support for the formatting of numbers with uncertainties. A zero uncertainty is now explicitly displayed as the integer 0. The new formats are generally understood by ufloat_fromstr(). Abbreviations for the nominal value (n) and the standard deviation (s) are now available.

• 2.3.6: Full support for limit cases of the power operator umath.pow().

• 2.3.5: Uncertainties and derivatives can now be NaN (not-a-number). Full support for numbers with a zero uncertainty (sqrt(ufloat(0, 0)) now works). Full support for limit cases of the power operator (x**y).

• 2.3: Functions wrapped so that they accept numbers with uncertainties instead of floats now have full keyword arguments support (improved wrap() function). Incompatible change: wrap(..., None) should be replaced by wrap(...) or wrap(..., []).

• 2.2: Creating arrays and matrices of numbers with uncertainties with uarray() and umatrix() now requires two simple arguments (nominal values and standard deviations) instead of a tuple argument. This is consistent with the new, simpler ufloat() interface. The previous usage will be supported for some time. Users are encouraged to update their code, for instance through the newly provided code updater, which in addition now automatically converts .set_std_dev(v) to .std_dev = v.

• 2.1: Numbers with uncertainties are now created more directly like ufloat(3, 0.1), ufloat(3, 0.1, "pi"), ufloat_fromstr("3.0(1)"), or ufloat_fromstr("3.0(1)", "pi"). The previous ufloat((3, 0.1)) and ufloat("3.0(1)") forms will be supported for some time. Users are encouraged to update their code, for instance through the newly provided code updater.

• 2.0: The standard deviation is now obtained more directly without an explicit call (x.std_dev instead of x.std_dev()). x.std_dev() will be supported for some time. Users are encouraged to update their code. The standard deviation of a variable can now be directly updated with x.std_dev = 0.1. As a consequence, x.set_std_dev() is deprecated.

• 1.9.1: Support added for pickling subclasses of UFloat (= Variable).

• 1.9: Added functions for handling correlation matrices: correlation_matrix() and correlated_values_norm(). (These new functions mirror the covariance-matrix based covariance_matrix() and correlated_values().) UFloat.position_in_sigmas() is now named UFloat.std_score(), so as to follow the common naming convention (standard score). Obsolete functions were removed (from the main module: NumberWithUncert, num_with_uncert, array_u, nominal_values, std_devs).

• 1.8: Compatibility with Python 3.2 added.

• 1.7.2: Compatibility with Python 2.3, Python 2.4, Jython 2.5.1 and Jython 2.5.2 added.

• 1.7.1: New semantics: ufloat("12.3(78)") now represents 12.3+/-7.8 instead of 12.3+/-78.

• 1.7: ufloat() now raises ValueError instead of a generic Exception, when given an incorrect string representation, like float() does.

• 1.6: Testing whether an object is a number with uncertainty should now be done with isinstance(..., UFloat). AffineScalarFunc is not imported by from uncertainties import * anymore, but its new alias UFloat is.

• 1.5.5: The first possible license is now the Revised BSD License instead of GPLv2, which makes it easier to include this package in other projects.

• 1.5.4.2: Added umath.modf() and umath.frexp().

• 1.5.4: ufloat does not accept a single number (nominal value) anymore. This removes some potential confusion about ufloat(1.1) (zero uncertainty) being different from ufloat("1.1") (uncertainty of 1 on the last digit).

• 1.5.2: float_u, array_u and matrix_u renamed ufloat, uarray and umatrix, for ease of typing.

• 1.5: Added functions nominal_value and std_dev, and modules unumpy (additional support for NumPy arrays and matrices) and unumpy.ulinalg (generalization of some functions from numpy.linalg). Memory footprint of arrays of numbers with uncertainties divided by 3. Function array_u is 5 times faster. Main function num_with_uncert renamed float_u, for consistency with unumpy.array_u and unumpy.matrix_u, with the added benefit of a shorter name.

• 1.4.5: Added support for the standard pickle module.

• 1.4.2: Added support for the standard copy module.

• 1.4: Added utilities for manipulating NumPy arrays of numbers with uncertainties (array_u, nominal_values and std_devs).

• 1.3: Numbers with uncertainties are now constructed with num_with_uncert(), which replaces NumberWithUncert(). This simplifies the class hierarchy by removing the NumberWithUncert class.

• 1.2.5: Numbers with uncertainties can now be entered as NumberWithUncert("1.23+/-0.45") too.

• 1.2.3: log(x, base) is now supported by umath.log(), in addition to log(x).

• 1.2.2: Values with uncertainties are now output like 3+/-1, in order to avoid confusing 3+-1 with 3+(-1).

• 1.2: A new function, wrap(), is exposed, which allows non-Python functions (e.g. Fortran or C used through a module such as SciPy) to handle numbers with uncertainties.

• 1.1: Mathematical functions (such as cosine, etc.) are in a new uncertainties.umath module; they do not override functions from the math module anymore.

• 1.0.12: Main class (Number_with_uncert) renamed NumberWithUncert so as to follow PEP 8.

• 1.0.11: origin_value renamed more appropriately as nominal_value.

• 1.0.9: correlations() renamed more appropriately as covariance_matrix().

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