Testing whether an object is a number with uncertainty¶
The recommended way of testing whether
value carries an
uncertainty handled by this module is by checking whether
value is an instance of
The quantities with uncertainties created by the
package can be pickled
(they can be stored in a file, for instance).
If multiple variables are pickled together (including when pickling NumPy arrays), their correlations are preserved:
>>> import pickle >>> x = ufloat(2, 0.1) >>> y = 2*x >>> p = pickle.dumps([x, y]) # Pickling to a string >>> (x2, y2) = pickle.loads(p) # Unpickling into new variables >>> y2 - 2*x2 0.0+/-0
The final result is exactly zero because the unpickled variables
y2 are completely correlated.
However, unpickling necessarily creates new variables that bear no relationship with the original variables (in fact, the pickled representation can be stored in a file and read from another program after the program that did the pickling is finished). Thus
>>> x - x2 0.0+/-0.14142135623730953
which shows that the original variable
x and the new variable
are completely uncorrelated.
Linear propagation of uncertainties¶
Constraints on the uncertainties¶
This package calculates the standard deviation of mathematical expressions through the linear approximation of error propagation theory.
The standard deviations and nominal values calculated by this package are thus meaningful approximations as long as uncertainties are “small”. A more precise version of this constraint is that the final calculated functions must have precise linear expansions in the region where the probability distribution of their variables is the largest. Mathematically, this means that the linear terms of the final calculated functions around the nominal values of their variables should be much larger than the remaining higher-order terms over the region of significant probability (because such higher-order contributions are neglected).
For example, calculating
x = 5±3 gives a
perfect result since the calculated function is linear. So does
x = 0±1, since only the
final function counts (not an intermediate function like
Another example is
sin(0+/-0.01), for which
yields a meaningful standard deviation since the sine is quite linear
over 0±0.01. However,
cos(0+/-0.01), yields an approximate
standard deviation of 0 because it is parabolic around 0 instead of
linear; this might not be precise enough for all applications.
More precise uncertainty estimates can be obtained, if necessary,
with the soerp and mcerp packages. The soerp package performs
second-order error propagation: this is still quite fast, but the
standard deviation of higher-order functions like f(x) = x3
for x = 0±0.1 is calculated as being exactly zero (as with
uncertainties). The mcerp package performs Monte-Carlo
calculations, and can in principle yield very precise results, but
calculations are much slower than with approximation schemes.
As a consequence, it is possible for uncertainties to be
>>> umath.sqrt(ufloat(0, 1)) 0.0+/-nan
This indicates that the derivative required by linear error propagation theory is not defined (a Monte-Carlo calculation of the resulting random variable is more adapted to this specific case).
uncertainties package correctly handles
perfectly precise numbers, in this case:
>>> umath.sqrt(ufloat(0, 0)) 0.0+/-0
gives the correct result, despite the fact that the derivative of the square root is not defined in zero.
Mathematical definition of numbers with uncertainties¶
Mathematically, numbers with uncertainties are, in this package, probability distributions. They are not restricted to normal (Gaussian) distributions and can be any distribution. These probability distributions are reduced to two numbers: a nominal value and an uncertainty.
Thus, both independent variables (
Variable objects) and the
result of mathematical operations (
contain these two values (respectively in their
The uncertainty of a number with uncertainty is simply defined in this package as the standard deviation of the underlying probability distribution.
The numbers with uncertainties manipulated by this package are assumed to have a probability distribution mostly contained around their nominal value, in an interval of about the size of their standard deviation. This should cover most practical cases.
A good choice of nominal value for a number with uncertainty is thus the median of its probability distribution, the location of highest probability, or the average value.
Probability distributions (random variables and calculation results) are printed as:
nominal value +/- standard deviation
but this does not imply any property on the nominal value (beyond the fact that the nominal value is normally inside the region of high probability density), or that the probability distribution of the result is symmetrical (this is rarely strictly the case).
Comparison operations (>, ==, etc.) on numbers with uncertainties have a pragmatic semantics, in this package: numbers with uncertainties can be used wherever Python numbers are used, most of the time with a result identical to the one that would be obtained with their nominal value only. This allows code that runs with pure numbers to also work with numbers with uncertainties.
The boolean value (
if x …) of a number with
x is defined as the result of
x != 0, as usual.
However, since the objects defined in this module represent probability distributions and not pure numbers, comparison operators are interpreted in a specific way.
The result of a comparison operation is defined so as to be essentially consistent with the requirement that uncertainties be small: the value of a comparison operation is True only if the operation yields True for all infinitesimal variations of its random variables around their nominal values, except, possibly, for an infinitely small number of cases.
>>> x = ufloat(3.14, 0.01) >>> x == x True
because a sample from the probability distribution of
x is always
equal to itself. However:
>>> y = ufloat(3.14, 0.01) >>> x != y True
y are independent random variables that
almost always give a different value. Note that this is different
from the result of
z = 3.14; t = 3.14; print z != t, because
y are random variables, not pure numbers.
>>> x = ufloat(3.14, 0.01) >>> y = ufloat(3.00, 0.01) >>> x > y True
x is supposed to have a probability distribution largely
contained in the 3.14±~0.01 interval, while
y is supposed to be
well in the 3.00±~0.01 one: random samples of
most of the time be such that the sample from
x is larger than the
y. Therefore, it is natural to consider that for all
x > y.
Since comparison operations are subject to the same constraints as other operations, as required by the linear approximation method, their result should be essentially constant over the regions of highest probability of their variables (this is the equivalent of the linearity of a real function, for boolean values). Thus, it is not meaningful to compare the following two independent variables, whose probability distributions overlap:
>>> x = ufloat(3, 0.01) >>> y = ufloat(3.0001, 0.01)
In fact the function (x, y) → (x > y) is not even continuous over the region where x and y are concentrated, which violates the assumption of approximate linearity made in this package on operations involving numbers with uncertainties. Comparing such numbers therefore returns a boolean result whose meaning is undefined.
However, values with largely overlapping probability distributions can sometimes be compared unambiguously:
>>> x = ufloat(3, 1) >>> x 3.0+/-1.0 >>> y = x + 0.0002 >>> y 3.0002+/-1.0 >>> y > x True
In fact, correlations guarantee that
y is always larger than
y-x correctly satisfies the assumption of linearity,
since it is a constant “random” function (with value 0.0002, even
x are random). Thus, it is indeed true
uncertainties package automatically calculates the
derivatives required by linear error propagation theory.
Almost all the derivatives of the fundamental functions provided by
uncertainties are obtained through a analytical formulas (the
few mathematical functions that are instead differentiated through
numerical approximation are listed in
The derivatives of mathematical expressions are evaluated through a
fast and precise method:
uncertainties transparently implements
automatic differentiation with reverse accumulation. This method
essentially consists in keeping track of the value of derivatives, and
in automatically applying the chain rule. Automatic differentiation
is often faster than symbolic differentiation and more precise than
numerical differentiation (when used with analytical formulas, like in
The derivatives of any expression can be obtained with
uncertainties in a simple way, as demonstrated in the User
Tracking of random variables¶
This package keeps track of all the random variables a quantity depends on, which allows one to perform transparent calculations that yield correct uncertainties. For example:
>>> x = ufloat(2, 0.1) >>> a = 42 >>> poly = x**2 + a >>> poly 46.0+/-0.4 >>> poly - x*x 42+/-0
x*x has a non-zero uncertainty, the result has a zero
uncertainty, because it is equal to
If the variable
a above is modified, the value of
is not modified, as is usual in Python:
>>> a = 123 >>> print poly 46.0+/-0.4 # Still equal to x**2 + 42, not x**2 + 123
Random variables can, on the other hand, have their uncertainty
updated on the fly, because quantities with uncertainties (like
poly) keep track of them:
>>> x.std_dev = 0 >>> print poly 46+/-0 # Zero uncertainty, now
As usual, Python keeps track of objects as long as they are used.
Thus, redefining the value of
x does not change the fact that
poly depends on the quantity with uncertainty previously stored
>>> x = 10000 >>> print poly 46+/-0 # Unchanged
These mechanisms make quantities with uncertainties behave mostly like regular numbers, while providing a fully transparent way of handling correlations between quantities.
Python classes for variables and functions with uncertainty¶
Numbers with uncertainties are represented through two different classes:
- a class for independent random variables (
Variable, which inherits from
- a class for functions that depend on independent variables
AffineScalarFunc, aliased as
Documentation for these classes is available in their Python docstring, which can for instance displayed through pydoc.
The factory function
ufloat() creates variables and thus returns
>>> x = ufloat(1, 0.1) >>> type(x) <class 'uncertainties.Variable'>
Variable objects can be used as if they were regular Python
numbers (the summation, etc. of these objects is defined).
Mathematical expressions involving numbers with uncertainties
AffineScalarFunc objects, because they
represent mathematical functions and not simple variables; these
objects store all the variables they depend on:
>>> type(umath.sin(x)) <class 'uncertainties.AffineScalarFunc'>