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#!/usr/bin/python
# -*- coding: utf-8 -*-
"""
=========================================================
Principal Component Analysis
=========================================================
These figures aid in illustrating how a the point cloud
can be very flad in one direction - which is where PCA
would come in to choose a direction that is not flat.
"""
print __doc__
# Code source: Gael Varoqueux
# Modified for Documentation merge by Jaques Grobler
# License: BSD
import pylab as pl
import numpy as np
from scipy import stats
from mpl_toolkits.mplot3d import Axes3D
e = np.exp(1)
np.random.seed(4)
def pdf(x):
return 0.5 * (stats.norm(scale=0.25 / e).pdf(x)
+ stats.norm(scale=4 / e).pdf(x))
y = np.random.normal(scale=0.5, size=(30000))
x = np.random.normal(scale=0.5, size=(30000))
z = np.random.normal(scale=0.1, size=len(x))
density = pdf(x) * pdf(y)
pdf_z = pdf(5 * z)
density *= pdf_z
a = x + y
b = 2 * y
c = a - b + z
norm = np.sqrt(a.var() + b.var())
a /= norm
b /= norm
###############################################################################
# Plot the figures
def plot_figs(fig_num, elev, azim):
fig = pl.figure(fig_num, figsize=(4, 3))
pl.clf()
ax = Axes3D(fig, rect=[0, 0, .95, 1], elev=elev, azim=azim)
ax.scatter(a[::10], b[::10], c[::10], c=density, marker='+',
alpha=.4)
Y = np.c_[a, b, c]
U, pca_score, V = np.linalg.svd(Y, full_matrices=False)
x_pca_axis, y_pca_axis, z_pca_axis = V.T * pca_score / pca_score.min()
x_pca_axis, y_pca_axis, z_pca_axis = 3 * V.T
x_pca_plane = np.r_[x_pca_axis[:2], - x_pca_axis[1::-1]]
y_pca_plane = np.r_[y_pca_axis[:2], - y_pca_axis[1::-1]]
z_pca_plane = np.r_[z_pca_axis[:2], - z_pca_axis[1::-1]]
x_pca_plane.shape = (2, 2)
y_pca_plane.shape = (2, 2)
z_pca_plane.shape = (2, 2)
ax.plot_surface(x_pca_plane, y_pca_plane, z_pca_plane)
ax.w_xaxis.set_ticklabels([])
ax.w_yaxis.set_ticklabels([])
ax.w_zaxis.set_ticklabels([])
elev = -40
azim = -80
plot_figs(1, elev, azim)
elev = 30
azim = 20
plot_figs(2, elev, azim)
pl.show()
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