Visualizing uncertainties in plotted data

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The equation for the temperature-dependence of the diffusion of hydrogen in tungsten may be written in Arrhenius form: $$ k = A\exp\left(-\frac{E}{T}\right) \quad \Rightarrow \; \ln k = \ln A - \frac{E}{T}, $$ where the temperature, $T$, and activation energy, $E$, are expressed in eV and the pre-exponential Arrhenius parameter, $A$, and rate constant, $k$, take units of $\mathrm{m^2\,s^{-1}}$. From the study of Frauenfelder [1] the parameters $A$ and $E$ may be associated with uncertainties as follows: $$ \begin{align*} A & = (4.1 \pm 0.5) \times 10^{-7}\;\mathrm{m^2\,s^{-1}}, \\ E &= 0.39 \pm 0.08 \;\mathrm{eV}. \end{align*} $$ These uncertainties can be propagated to the expression for $\ln k$: $$ \sigma_{\ln k} \approx \sqrt{ \left( \frac{\sigma_A}{A} \right)^2 + \left( \frac{\sigma_E}{T} \right)^2 }. $$ If we assume the uncertainty remains normally-distributed, Matplotlib's imshow function can be used to illustrate the Arrhenius equation for this data.

enter image description here

The following code produces the above plot

import numpy as np
from scipy.constants import k as kB, e
import matplotlib.pyplot as plt
from matplotlib import rc
rc('font', **{'family': 'serif', 'serif': ['Computer Modern'], 'size': 14})
rc('text', usetex=True)

# Temperature limits (K), number of temperature points
Tmin, Tmax = 1000, 3000
nT = 10

# Equally-spaced inverse temperature grid, K^-1
invT = np.linspace(1/Tmax, 1/Tmin, nT)
# Convert to eV^-1
invTeV = e * invT / kB

# Activation energy (eV), Arrhenius parameter (m2.s-1) and their
# 1-sigma uncertainties
E, A = 0.39, 4.1e-7
E_unc, A_unc = 0.08, 0.5e-7

def calc_lnk(A, E, invTeV):
    """Calculate the log of the rate coefficient."""
    return np.log(A) - E * invTeV

lnk = calc_lnk(A, E, invTeV)

fig = plt.figure()
ax = fig.add_subplot()

# Arrhenius plot: ln(k) vs. 1/T
plt.plot(invT, lnk, 'k')

# Propagate the errors in A, E to the error in ln(k). NB doesn't really
# preserve normally-distributed errors...
lnk_unc = np.sqrt((A_unc/A)**2 + (E_unc*invTeV)**2)
# Plot the 1-sigma uncertainty intervals
plt.plot(invT, lnk - lnk_unc, 'k:', lw=1)
plt.plot(invT, lnk + lnk_unc, 'k:', lw=1)

minlnk, maxlnk = ax.get_ylim()
ymin, ymax = minlnk, maxlnk
xmin, xmax = invT[0], invT[-1]

# Produce an array of the unnormalized gaussian error distribution about
# a large number of ln(k) points for each of the temperatures 
nlnk = 100
_lnk = np.linspace(minlnk, maxlnk, nlnk)
arr = np.exp(-((_lnk[:,np.newaxis] - lnk) / lnk_unc)**2 / 2)

# Plot the error distribution as a grey-scale image: get the orientation right,
# interpolate to make it look smooth, fill the plot area and use a greyscale
# colourmap but max-out at midscale, not full black.
plt.imshow(arr, origin='lower', extent=(xmin,xmax,ymin,ymax),
           interpolation='bicubic', aspect='auto', cmap='Greys',
           vmin=np.min(arr), vmax=np.max(arr)*1.5)

# Set the temperature ticks in Kelvin, not inverse eV, and corresponding label.
Tticks = np.linspace(Tmin, Tmax, 3)
ax.set_xticks(1/Tticks)
ax.set_xticklabels([str(int(tick)) for tick in Tticks])
ax.set_xlabel(r'$1/T \; /\mathrm{eV}^{-1}$')

# Set the ln(k) ticks and label, converting logarithm base from e to 10.
yticks = ax.get_yticks() / np.log(10)
ytick_min = int(np.round(np.min(yticks)))
ytick_max = int(np.round(np.max(yticks)))
nyticks = ytick_max - ytick_min + 1
yticks = np.linspace(ytick_min, ytick_max, nyticks)
ax.set_yticks(yticks * np.log(10))
ax.set_yticklabels([r'$10^{{ {} }}$'.format(int(ytick)) for ytick in yticks])
ax.set_ylabel(r'$k \; /\mathrm{{ m^2\,s^{-1} }}$')

plt.savefig('plot-rate-unc.png')
plt.show()
  1. R. Frauenfelder, Solution and diffusion of Hydrogen in Tungsten, Journal of Vacuum Science and Technology 6, 388 (1969)
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