Figure 4.10:

The multivariate normal, marginals, marginal box, and bounding box.

Code for Figure 4.10

Text of the GNU GPL.

main.py

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import numpy as np
from numpy.linalg import inv
from scipy.stats import chi2
from scipy.linalg import sqrtm
import matplotlib.pyplot as plt
from misc import ellipse

# Set parameters
m = np.array([[1], [2]])
n = len(m)
P = np.array([[2, 0.75], [0.75, 0.5]])
nsam = 1000
np.random.seed(0)

# Generate samples
x = np.tile(m, (1, nsam)) + sqrtm(P) @ np.random.randn(n, nsam)

# Compute the 95% confidence interval ellipse
alpha = 0.95
A = inv(P)
b = chi2.ppf(alpha, n)
xe, ye, *_ = ellipse(A, b, 100, m)

# Count how many samples are outside the ellipse
e = x - np.tile(m, (1, nsam))
outside_ellipse = np.sum(np.diag((e.T @ A) @ e) > b) / nsam

# Compute the 95% bounding box
xd = np.sqrt(b * np.diag(P))
bbox = np.array([
    [-xd[0], -xd[1]],
    [xd[0], -xd[1]],
    [xd[0], xd[1]],
    [-xd[0], xd[1]],
    [-xd[0], -xd[1]]
])
bbox = bbox + np.tile(m.T, (5, 1))

# Count samples outside the bounding box
z = x - np.tile(m, (1, nsam))
outside_bbox = sum( sum(np.abs(z) > xd[:,None]) > 0 ) / nsam

# Compute 95% marginal intervals
b2 = chi2.ppf(alpha, 1)
xd2 = np.sqrt(b2 * np.diag(P))
mbox = np.array([
    [-xd2[0], -xd2[1]],
    [xd2[0], -xd2[1]],
    [xd2[0], xd2[1]],
    [-xd2[0], xd2[1]],
    [-xd2[0], -xd2[1]]
])
mbox = mbox + np.tile(m.T, (5, 1))

# Count samples outside the marginal box
outside_mbox = sum( sum(np.abs(z) > xd2[:,None]) > 0 ) / nsam

# Prepare mtab data (xplot, fx for both components)
xrange = np.round([np.min(x, axis=1), np.max(x, axis=1)]).astype(int).T
nplot = 100
xplot = np.linspace(xrange[:,0], xrange[:,1], nplot).T
z = xplot - m
diagP = np.diag(P)[:,None];
fx = np.exp( -(1/2) * (z / diagP * z)) / np.sqrt(2*np.pi*diagP)

# compute closed curves for using filledcurve in gnuplot
ind = abs(z) <= xd2[:,None]
xp1 = xplot[0,:]
xp2 = xplot[1,:]
fx1 = fx[0,:]
fx2 = fx[1,:]
ind1 = ind[0,:]
ind2 = ind[1,:]

fill1 = np.row_stack( (xp1[ind1],  fx1[ind1]) )
fill1 = np.column_stack( (np.row_stack( (fill1[0,0], 0) ), fill1, np.row_stack( (fill1[0,-1], 0) ) ) ).T
fill2 = np.row_stack( (xp2[ind2], fx2[ind2]) )
fill2 = np.column_stack( (np.row_stack( (fill2[0,0], 0) ), fill2, np.row_stack( (fill2[0,-1], 0) ) ) ).T

nbins = 20
# Histogram for x[0,:]
n1, bins1 = np.histogram(x[0,:], bins=nbins, density=True)
bin_centers1 = 0.5 * (bins1[:-1] + bins1[1:])
# make bar plot data for gnuplot
barx1 = np.kron(bins1,[1,1,1])[1:-1]
bary1 = np.append(np.kron(n1, [0, 1, 1]), 0)

# Histogram for x[1,:]
n2, bins2 = np.histogram(x[1,:], bins=nbins, density=True)
bin_centers2 = 0.5 * (bins2[:-1] + bins2[1:])
# make bar plot data for gnuplot
barx2 = np.kron(bins2,[1,1,1])[1:-1]
bary2 = np.append(np.kron(n2, [0, 1, 1]), 0)

plt.figure()
plt.plot(x[0,:], x[1,:], 'o', mfc='none')
plt.plot(xe, ye)
plt.plot(bbox[:,0], bbox[:,1])
plt.plot(mbox[:,0], mbox[:,1])

plt.figure()
plt.plot(barx1, bary1, label='Histogram x[1,:]')

plt.figure()
plt.plot(barx2, bary2, label='Histogram x[2,:]')

plt.figure()
plt.plot (xplot[0,:], fx[0,:])
plt.plot (xplot[1,:], fx[1,:])

plt.show(block=False)

samtab = x.T
eltab = np.column_stack([xe, ye])
mtab = np.column_stack( (xplot.T, fx.T) )

# Create the .dat file with correct formatting for Gnuplot
with open("confmarg.dat", "w") as f:
    # Write sample matrix (samtab)
    np.savetxt(f, samtab, fmt='%f', header="samtab")
    f.write("\n\n")

    # Write ellipse data (eltab)
    np.savetxt(f, eltab, fmt='%f', header="eltab")
    f.write("\n\n")

    # Write ellipse data (eltab)
    np.savetxt(f, mtab, fmt='%f', header="mtab")
    f.write("\n\n")

    # Write bounding box data (bbox)
    np.savetxt(f, bbox, fmt='%f', header="bbox")
    f.write("\n\n")

    # Write marginal box data (mbox)
    np.savetxt(f, mbox, fmt='%f', header="mbox")
    f.write("\n\n")

    # Write fill1 and fill2 data
    np.savetxt(f, fill1, fmt='%f', header="fill1")
    f.write("\n\n")
    np.savetxt(f, fill2, fmt='%f', header="fill2")