I apologize for linking to images, but I can't put them inline because this is my first post.
To start off with, consider the 1-D case of transforming a probability distribution defined on the line $[-1, 1]$ to the semicircle. You can uniformly sample from a semicircle by randomly sampling $\theta \in [0, \pi]$. Here are some samples. Letting $x = \cos(\theta)$, we can see that the $x$s follow an arcsine distribution. If we want to calculate $p_\theta(\theta)$ given the density provided for $x$, we can do $p_{\theta}(\theta) = p_{x}(\cos(\theta)) |-\sin(\theta)|$. Indeed, if I approximate the distribution for $x$ by binning, I can approximately recover the true uniform density for $\theta$ using this transformation.
So what I want to know is how I can reproduce these steps for a 2-ball and a 2-hemisphere and a 3-ball and a 3-"hemisphere". When I follow this same recipe for two dimensions using spherical coordinates, the results aren't right. Here are some randomly sampled points from a hemisphere and here's the distribution of the $(x, y)$ coordinates on the circle. The determinant of the Jacobian for $x$ and $y$ from $\theta$ and $\phi$ using spherical coordinates is $-\frac{\sin(2\theta)}{2}$. However, when I use this scaling factor, I don't get the expected $\frac{1}{2\pi}$ (i.e., one over the surface area of the hemisphere) value when mapping to the hemisphere and the distribution looks pretty wonky.
I'm wondering if the fact that the area element of the circle is one while the area element of the sphere is $\sin(\theta)$ means I have to take different steps. Anyway, intuitively I understand that I need to "stretch" the density when going from the ball to the hemisphere, but I don't know how to accomplish that. Googling around, it seems like there should be a straightforward answer using some kind of change of variables, but this is math I'm not familiar with, so I'm struggling to come up with the right approach. Any help is appreciated!
Here's the Python code to generate all the plots:
import matplotlib.pyplot as plt
import numpy as np
plt.rcParams.update({"font.size": 20})
def one_dimension():
xs = np.linspace(-1, 1, 500)
ys = np.sqrt(1 - xs**2)
skip = 50
samp_xs = xs[skip::skip]
samp_ys = ys[skip::skip]
plt.plot(xs, ys)
plt.hlines(0, -1, 1)
plt.vlines(samp_xs, 0, samp_ys)
plt.plot(samp_xs, samp_ys, "o", label="$S^{1}$ Manifold Sample")
plt.plot(samp_xs, np.zeros_like(samp_xs), "o", label="$\mathbb{R}^{1}$ Sample")
plt.axis("equal")
plt.axis("off")
plt.legend(loc="upper right")
plt.show()
N = 1000000
thetas = np.random.uniform(0, np.pi, size=N)
(fig, axs) = plt.subplots(2, 2)
axs[0, 0].hist(thetas)
axs[0, 0].set_title(r"Samples of $\theta$")
axs[0, 0].set_xlabel(r"$\theta$")
axs[0, 0].set_ylabel("Count")
xs = np.cos(thetas)
axs[0, 1].hist(xs)
axs[0, 1].set_title(r"Samples of $x = \cos(\theta)$")
axs[0, 1].set_xlabel(r"$x$")
axs[0, 1].set_ylabel("Count")
n_bins = 500
bins = np.linspace(-1, 1, n_bins + 1)[1:]
labels = np.digitize(xs, bins, right=True)
(unique_labels, counts) = np.unique(labels, return_counts=True)
ps = counts / counts.sum()
entropy = -np.sum(ps * np.log(ps))
print(entropy)
# See: https://nathancarter.github.io/how2data/site/how-to-plot-discrete-probability-distributions-in-python-using-scipy/.
axs[1, 1].plot(unique_labels, ps, "o")
axs[1, 1].vlines(unique_labels, 0, ps)
axs[1, 1].set_ylim(bottom=0)
axs[1, 1].set_title("Bin Probabilities for $x$")
axs[1, 1].set_xlabel("$x$")
axs[1, 1].set_ylabel("$p$")
new_thetas = np.linspace(0, np.pi, 100)
new_xs = np.cos(new_thetas)
new_bins = np.digitize(new_xs, bins, right=True)
bin_width = 2 / n_bins
new_ps = ps[new_bins] * 1 / bin_width
theta_ps = new_ps * np.abs(-1 * np.sin(new_thetas))
axs[1, 0].plot(new_thetas, theta_ps, "o")
axs[1, 0].vlines(new_thetas, 0, theta_ps)
axs[1, 0].hlines(1 / np.pi, 0, np.pi, color="red", label=r"$\frac{1}{\pi}$")
axs[1, 0].set_ylim(bottom=0)
axs[1, 0].set_title(
r"$p_{\theta}(\theta) = p_{\mathrm{Bin}(x)}(\cos(\theta)) \frac{1}{w} |-\sin(\theta)|$"
)
axs[1, 0].set_xlabel(r"$\theta$")
axs[1, 0].set_ylabel(r"$p_{\theta}(\theta)$")
axs[1, 0].legend(loc="upper center")
plt.show()
def two_dimensions():
N = 1000000
# See: https://stats.stackexchange.com/questions/7977/how-to-generate-uniformly-distributed-points-on-the-surface-of-the-3-d-unit-sphe.
points = np.random.normal(size=(N, 3))
points /= np.linalg.norm(points, axis=1, keepdims=True)
points = points[points[:, 2] > 0]
samps = 1000
fig = plt.figure()
ax = fig.add_subplot(projection="3d")
ax.scatter(*np.hsplit(points[:samps], 3))
# See: https://stackoverflow.com/a/72928548/1316276.
limits = np.array([getattr(ax, f"get_{axis}lim")() for axis in "xyz"])
ax.set_box_aspect(np.ptp(limits, axis=1))
plt.show()
n_bins = 25
bins = np.linspace(-1, 1, n_bins + 1)[1:]
xy_labels = np.digitize(points[:, :2], bins, right=True)
(unique_labels, counts) = np.unique(xy_labels, return_counts=True, axis=0)
all_counts = np.zeros((n_bins, n_bins))
for ((col, row), count) in zip(unique_labels, counts):
all_counts[row, col] = count
plt.imshow(all_counts)
plt.show()
ps = all_counts / all_counts.sum()
entropy = -np.sum(ps[ps > 0] * np.log(ps[ps > 0]))
print(entropy)
phis = np.linspace(0, 2 * np.pi, 100)
thetas = np.linspace(np.pi / 2, np.pi, 100)
xs = np.cos(thetas) * np.sin(phis)
ys = np.sin(thetas) * np.sin(phis)
new_xys = np.stack([xs, ys]).T
new_bins = np.digitize(new_xys, bins, right=True)
bin_width = 2 / n_bins
new_ps = ps[new_bins[:, 0], new_bins[:, 1]] * 1 / bin_width**2
cov_factor = np.abs(np.sin(2 * thetas) / 2)
phi_theta_ps = new_ps * cov_factor
plt.plot(np.arange(len(phi_theta_ps)), phi_theta_ps)
plt.hlines(
1 / (2 * np.pi), 0, len(phi_theta_ps), color="red", label=r"$\frac{1}{2\pi}$"
)
plt.legend()
plt.show()
def main():
one_dimension()
two_dimensions()
if __name__ == "__main__":
main()
The general recipe is to calculate the norm of the "formal determinant" [1, 2] derived from the Jacobian, which gives us a measure of the "area" of the surface of the n-sphere "above" the n-ball. This works for 1-D, 2-D, and 3-D. Here's the relevant snippet for the 3-D case:
which produces this distribution.