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Uncertainty Calculations

The uncertainties package allows you to perform calculations on numbers with uncertainties. It handles the propagation of uncertainties for you. The following line can be used to install (or upgrade) the package in Anaconda or Google Colabaoratory.

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pip install --upgrade uncertainties

Individual Numbers with Uncertainties

The following import statements make the parts of the package for inidividual numbers available.

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from uncertainties import ufloat, ufloat_fromstr
from uncertainties.umath import *

There are multiple ways to create a variable like $2.0 \pm 0.2$ . With the ufloat function, the two arguments are the number and its uncertainty.

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r = ufloat(2.0, 0.2)
print(r)

2.00+/-0.20

Using the ufloat_fromstr function, the the number and its uncertainty are entered as strings (between double quotes). The second example is equivalent to $(4.1 \pm 0.3)\times 10^{-4}$ .

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s = ufloat_fromstr("3.2 +/- 0.4")
t = ufloat_fromstr("(4.1 +/- 0.3)e-4")  # factored exponential
print(s)
print(t)

3.2+/-0.4
0.000410+/-0.000030

Calculations can be performed just as for ordinary numbers, but the uncertainties are propagated automatically.

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square = r**2
print(square)
product = s*t
print(product)

4.0+/-0.8
0.00131+/-0.00019

You can also access the number (nominal value) and the uncertainty (standard deviation) separately.

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print(square.nominal_value)
print(square.std_dev)

4.0
0.8

It is possible to perform calculations involving many mathematical functions (see a list a the end of this tutorial).

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print(sin(1+r**2))

-0.96+/-0.23

Arrays of Numbers with Uncertainties

The following import statement makes it possible to use arrays of numbers with uncertainties.

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from uncertainties import unumpy

The unumpy.uarray function takes lists of the nominal values and their uncertainties.

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y = unumpy.uarray([1, 4, 9, 16], [0.1, 0.2, 0.3, 0.4])
print(y)

[1.0+/-0.1 4.0+/-0.2 9.0+/-0.3 16.0+/-0.4]

The numbers (nominal values) and the uncertainties (standard deviations) can be accessed as follows.

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print(unumpy.nominal_values(y))
print(unumpy.std_devs(y))

[ 1.  4.  9. 16.]
[0.1 0.2 0.3 0.4]

You can perform calculations on the elements of the array. The list of functions that can follow "unumpy" are listed at the end of this tutorial.

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print(unumpy.sqrt(y))

[1.0+/-0.05 2.0+/-0.05 3.0+/-0.049999999999999996 4.0+/-0.05]

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print(unumpy.cos(y))

[0.5403023058681398+/-0.08414709848078966
 -0.6536436208636119+/-0.15136049906158566
 -0.9111302618846769+/-0.12363554557252697
 -0.9576594803233847+/-0.11516132666602613]

It is often useful to find an average (mean) of the elements of an array.

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print(y.mean())

7.50+/-0.14

As an example, suppose that you have data for variables $x$ and $y$ , where there are uncertainties for the values of $y$ . It is easy to make a plot of $\log(y)$ vs. $\log(x)$ with appropriate error bars.

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import pylab as pl

x = pl.array([1,2,3,4]) 
logx = pl.log10(x)
logy = unumpy.log10(y)
pl.figure()
pl.errorbar(logx, unumpy.nominal_values(logy), unumpy.std_devs(logy), 
         ls='None', marker='o', ms=4)
pl.show()

Additional Documentation

More information is available at: http://pythonhosted.org/uncertainties/

A list of the mathematical functions that work on numbers with uncertainties is below. If there is one argument it is called $x$ in the description. If there is a second argument, it is called $y$ .

Individual Number	Array of Numbers	Function Description
acos	unumpy.arccos	Return the arc cosine (in radians)
acosh	unumpy.arccosh	Return the inverse hyperbolic cosine
asin	unumpy.arcsin	Return the arc sine (in radians)
asinh	unumpy.asinh	Return the inverse hyperbolic sine
atan	unumpy.arctan	Return the arc tangent (in radians)
atan2	unumpy.arctan2	Return the arc tangent (in radians) of y/x (signs of both x and y are considered)
atanh	unumpy.arctanh	Return the inverse hyperbolic tangent
ceil	unumpy.ceil	Return the smallest integer >= x
copysign	unumpy.copysign	Return a float with the magnitude (absolute value) of x but the sign of y
cos	unumpy.cos	Return the cosine of x (measured in radians)
cosh	unumpy.cosh	Return the hyperbolic cosine
degrees	unumpy.degrees	Convert angle from radians to degrees
erf	unumpy.erf	Returns the error function
erfc	unumpy.erfc	Returns the complementary error function
exp	unumpy.exp	Return e raised to the power of x
expm1	unumpy.expm1	Return exp(x)-1 avoiding the loss of precision for small x
fabs	unumpy.fabs	Return the absolute value of the float x
factorial		Return x! (error if x is negative or non-integer)
floor	unumpy.floor	Returns the largest integer <= x
fmod	unumpy.fmod	Returns x modulo y
frexp		Return the mantissa (m) and exponent (e) of x such that x = m * 2**e
gamma	unumpy.gamma	Returns the Gamma function
hypot	unumpy.hypot	Return the Euclidean distance, sqrt(xx + yy)
isinf	unumpy.isinf	Return True if x is a positive or negative infinity, and False otherwise
isnan	unumpy.isnan	Return True if x is a NaN (not a number), and False otherwise
ldexp	unumpy.ldexp	Return x * (2 **i)
lgamma	unumpy.lgamma	Return the atural logarithm of absolute value of Gamma function
log	unumpy.log	Return the logarithm of x to the base y; if the base not specified, returns the natural logarithm (base e)
log10	unumpy.log10	Return the base 10 logarithm
log1p	unumpy.log1p	Return the natural logarithm of 1+x (base e) in a way which is accurate for x near zero
modf	unumpy.modf	Return the fractional and integer parts of x
pow	unumpy.pow	Return x to the power of y
radians	unumpy.radians	Convert angle from degrees to radians
sin	unumpy.sin	Return the sine of x (measured in radians)
sinh	unumpy.sinh	Return the hyperbolic sine
sqrt	unumpy.sqrt	Return the square root
tan	unumpy.tan	Return the tangent of x (measured in radians)
tanh	unumpy.tanh	Return the hyperbolic tangent
trunc	unumpy.trunc	Truncates x to the nearest integer toward 0

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Uncertainty Calculations

Individual Numbers with Uncertainties

Arrays of Numbers with Uncertainties

Additional Documentation

Product

Resources

Company

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more, all in one place. Commercial Alternative to JupyterHub.

Uncertainty Calculations

Individual Numbers with Uncertainties

Arrays of Numbers with Uncertainties

Additional Documentation

Real-time collaboration for Jupyter Notebooks, Linux Terminals, LaTeX, VS Code, R IDE, and more,
all in one place. Commercial Alternative to JupyterHub.