Tutorial 3: More Detail on Units and Uncertainty

In order to understand how to use units and uncertainty, let’s take a careful look at the following example of a simple formula:

from uncertainty_wrapper import *
from pint import UnitRegistry
import numpy as np

UREG = UnitRegistry()  # Pint unit registry

def f(a, b):
    """
    Pythagorean theorem

    :param a: first leg
    :type a: int, float, np.ndarray
    :param n: second leg
    :type a: int, float, np.ndarray
    :returns: hypotenuse
    :type a: np.float64, np.ndarray
    """
    return np.sqrt(a * a + b * b)

# test with scalar
f(5., 12.)
# 13.0  <- returns np.float64

# test with vector (a Python list will raise TypeError)
f(np.array([3., 5.]), np.array([4., 12.]))
# array([  5.,  13.])  <- returns np.ndarray

If this formula were used in Carousel, and its attributes were set to propagate uncertainty for all arguments by setting isconstant=[], then the formulas would be automatically wrapped by uncertainty_wrapper.core.unc_wrapper_args(). We’ll show how this is done explicitly to understand more about UncertaintyWrapper.

# wrap function with uncertainty wrapper
# NOTE: set `covariance_keys=None` or else uncertainaty wrapper assumes
# arguments are grouped into a 2-D NumPy array
g = unc_wrapper_args(None)(f)

# test with scalar
cov = np.array([[0.01, 0], [0, 0.01]])  # covariance fractions w/ no-units
g(3., 4., __covariance__=cov)
# 5.0  <- hypotenuse np.float64
# array([[[ 0.1348]]])  <- variance = (stdev [same units as hypotenuse])^2
# array([[[ 0.6,  0.8]]])  <- sensitivities df/da and df/db

# test with vector
cov = np.array([[[0.01, 0], [0, 0.01]], [[0.01, 0], [0, 0.01]]])
try:
    g(np.array([3., 5.]), np.array([4., 12.]), __covariance__=cov)
except ValueError as err:
    err
# ValueError: could not broadcast input array from shape (2) into shape (2,1)

Uncertainty wrapper thinks there are two return values and only one observation, so it tries to calculate the derivatives with respect to two return values but there’s actually the opposite. The number of observations refers to vectorized calculations that repeat the same calculation for each independent observation, ie: observations do not depend on each other. The uncertainty wrapper always looks at the second dimension of the output to determine the number of observations. Therefore to fix the formulas in our example, all we have to do is reshape the output.

def f_fixed(a, b):
    """
    fixed Pythagorean function for uncertainty wrapper with nobs > 1
    """
    # cast a, b to NumPy arrays so we can do vector multiplication on lists
    a, b = np.asarray(a), np.asarray(b)
    # reshape output so that it is 1 X Nobs, the number of observations
    return f(a, b).reshape(1, -1)

# test Python list
f_fixed([3., 5.], [4., 12.])
# array([[  5.,  13.]])  <- returns 1 X 2 np.ndarray

# wrap fixed function
# NOTE: remember to set `covariance_keys=None` or else!
g = unc_wrapper_args(None)(f_fixed)

# test scalar
cov = np.array([[0.01, 0], [0, 0.01]])
g(5., 12., __covariance__=cov)
# [13.0]  <- returns 1-D np.ndarray
# array([[[ 1.2639645]]])
# array([[[ 0.38461538,  0.92307692]]])

# test vector
# NOTE: specify covariance for each observation or else uncertainty
# wrapper assumes there's only one argument
cov = np.array([[[0.01, 0], [0, 0.01]], [[0.01, 0], [0, 0.01]]])
g([3., 5.], [4., 12.], __covariance__=cov)
# [5.0, 13.0]  <- returns 1-D np.ndarray
# array([[[ 0.1348   ]],
#        [[ 1.2639645]]])
# array([[[ 0.6       ,  0.8       ]],
#        [[ 0.38461538,  0.92307692]]])

Now that we’ve got uncertainty wrapper working the way we want for both scalars and vectors, for multiple arguments, and possibly multiple return values, we can use the Pint unit wrapper:

# wrap the wrapped function with Pint units wrapper
# NOTE: Carousel adds `None` units for covariance and sensitivity for you
# but in this example we have to do it ourselves
h = UREG.wraps(('=A', None, None), ['=A', '=A'])(g)
# make some quantities
a, b = [3., 5.] * UREG.cm, [4., 12.] * UREG.cm
# don't forget to specify covariance for each observation
cov = np.array([[[0.01, 0], [0, 0.01]], [[0.01, 0], [0, 0.01]]])
h(a, b, __covariance__=cov)
# <Quantity([  5.  13.], 'centimeter')>
# array([[[ 0.1348   ]],
#        [[ 1.2639645]]])
# array([[[ 0.6       ,  0.8       ]],
#        [[ 0.38461538,  0.92307692]]])

So the key takeaway is that vectorized calculations should always return a 2-D array with the number of observations in the 2nd dimension.