⚠️ 2. Distribution gallery - Uncertainty Quantification in Inverse Problems with CUQIpy

⚠️ Contents of the notebook to be reviewed and expanded to include other examples from the distribution gallery (list names of all examples in the gallery, and plot one or two more besides the donut distribution)

from cuqi.distribution import DistributionGallery, Gaussian, JointDistribution
from cuqi.testproblem import Poisson1D
from cuqi.problem import BayesianProblem
import inspect
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as sps

def plot2d(val, x1_min, x1_max, x2_min, x2_max, N2=201, **kwargs):
    # plot
    pixelwidth_x = (x1_max-x1_min)/(N2-1)
    pixelwidth_y = (x2_max-x2_min)/(N2-1)

    hp_x = 0.5*pixelwidth_x
    hp_y = 0.5*pixelwidth_y

    extent = (x1_min-hp_x, x1_max+hp_x, x2_min-hp_y, x2_max+hp_y)

    plt.imshow(val, origin='lower', extent=extent, **kwargs)
    plt.colorbar()


def plot_2D_density(distb, x1_min, x1_max, x2_min, x2_max, N2=201, **kwargs):
    N2 = 201
    ls1 = np.linspace(x1_min, x1_max, N2)
    ls2 = np.linspace(x2_min, x2_max, N2)
    grid1, grid2 = np.meshgrid(ls1, ls2)
    distb_pdf = np.zeros((N2,N2))
    for ii in range(N2):
        for jj in range(N2):
            distb_pdf[ii,jj] = np.exp(distb.logd(np.array([grid1[ii,jj], grid2[ii,jj]]))) 
    plot2d(distb_pdf, x1_min, x1_max, x2_min, x2_max, N2, **kwargs)

The “donut” distribution ¶

In CUQIpy, we provide a set of bi-variate distributions for illustrative purposes. One of these is the “donut” distribution, which is a bi-variate distribution of a donut-shaped. The distribution is defined as follows:

\begin{aligned} log (p(\mathbf{x})) \propto - \frac{1}{\sigma_\text{donut}^2} \left( \left\| \mathbf{x} \right\| - r_\text{donut} \right)^2 \end{aligned}

Where $\mathbf{x} = (x_1, x_2)$ is a 2D vector, $\left\| \mathbf{x} \right\|$ is the Euclidean norm of $\mathbf{x}$ , $r_\text{donut}$ is the radius of the donut, and $\sigma_\text{donut}$ is a scalar value that controls the width of the “donut”.

To load the “donut” distribution, we use the following:


target_donut = DistributionGallery("donut")

print(target_donut)

CUQI DistributionGallery.

We can plot the distribution probability density function (pdf):

plot_2D_density(target_donut, -4, 4, -4, 4)

/tmp/ipykernel_2789/1264361712.py:23: DeprecationWarning: Conversion of an array with ndim > 0 to a scalar is deprecated, and will error in future. Ensure you extract a single element from your array before performing this operation. (Deprecated NumPy 1.25.)
  distb_pdf[ii,jj] = np.exp(distb.logd(np.array([grid1[ii,jj], grid2[ii,jj]])))