pca.ipynb

Kernel: Python 3

# Standard Python libraries
from __future__ import absolute_import, division, print_function, unicode_literals

import os
import time
import numpy as np
import glob
import matplotlib.pyplot as plt
import PIL
import imageio

from IPython import display

import sklearn

import seaborn as sns

sns.set(style="ticks", color_codes=True)

import pandas as pd

pd.set_option("precision", 2)  # 2 decimal places
pd.set_option("display.max_rows", 20)
pd.set_option("display.max_columns", 30)
pd.set_option("display.width", 100)  # wide windows

Dimensionality reduction of iris data using PCA

In this section, we show how to find low dimensional structure in an unlabeled version of the Iris dataset by fitting a PCA model. We will use sklearn.

# Visualize raw 3d data

from sklearn.datasets import load_iris


from mpl_toolkits import mplot3d
from mpl_toolkits.mplot3d import Axes3D

# https://jakevdp.github.io/PythonDataScienceHandbook/04.12-three-dimensional-plotting.html

iris = load_iris()
X = iris.data
y = iris.target

fig = plt.figure().gca(projection="3d")
fig.scatter(X[:, 0], X[:, 1], X[:, 2])
fig.set_xlabel("sepal length")
fig.set_ylabel("sepal width")
fig.set_zlabel("petal length")
plt.show()

# 2d projection of points

from sklearn.decomposition import PCA

X = iris.data[:, 0:3]
pca_xy = PCA(n_components=2).fit_transform(X)
fig, ax = plt.subplots()
ax.scatter(pca_xy[:, 0], pca_xy[:, 1])
# save_fig("iris-pca")
plt.show()

# plot latent 2d projection of points in ambient 3d feature space

pca = PCA(n_components=2)
mu = np.mean(X, axis=0)
Xc = X - mu  # center the data
pca.fit(Xc)
W = pca.components_.T  # D*K
Z = np.dot(Xc, W)  # N * K latent scores
Xrecon = np.dot(Z, W.T) + mu  # N*D

# span the latent space in area covered by data
a = np.min(Z[:, 0])
b = np.max(Z[:, 0])
c = np.min(Z[:, 1])
d = np.max(Z[:, 1])
z0 = np.linspace(a, b, 10)
z1 = np.linspace(c, d, 10)
ZZ0, ZZ1 = np.meshgrid(z0, z1)
Zgrid = np.c_[ZZ0.ravel(), ZZ1.ravel()]  # 100x2
plane = np.dot(Zgrid, W.T) + mu  # N*D

latent_corners = np.array([[a, c], [a, d], [b, c], [b, d]])  # 4x2
recon_corners = np.dot(latent_corners, W.T) + mu  # 4x3

fig = plt.figure().gca(projection="3d")
scatterplot = fig.scatter(X[:, 0], X[:, 1], X[:, 2], color="red")
# recon = fig.scatter(Xrecon[:,0], Xrecon[:,1], Xrecon[:,2], marker='*', color='green')
lineplot = fig.scatter(plane[:, 0], plane[:, 1], plane[:, 2], color="black", alpha=0.5)
fig.set_xlabel("sepal length")
fig.set_ylabel("sepal width")
fig.set_zlabel("petal length")
# save_fig("iris-pca-3d")
plt.show()