Friday, October 17, 2014

Andrews curves

Andrews curves are a method for visualizing multidimensional data by mapping each observation onto a function. This function is defined as

It has been shown the Andrews curves are able to preserve means, distance (up to a constant) and variances. Which means that Andrews curves that are represented by functions close together suggest that the corresponding data points will also be close together. Now, we will demonstrate the effectiveness of the Andrew curves on the iris dataset (which we already used here). Let's create a function to compute the values of the functions give a single sample:
import numpy as np
def andrew_curve4(x,theta):
    # iris has 4 four dimensions
    base_functions = [lambda x : x[0]/np.sqrt(2.), 
                      lambda x : x[1]*np.sin(theta), 
                      lambda x : x[2]*np.cos(theta), 
                      lambda x : x[3]*np.sin(2.*theta)]
    curve = np.zeros(len(theta))
    for f in base_functions:
        curve = curve + f(x)
    return curve
At this point we can load the dataset and plot the curves for a subset of samples:
samples = np.loadtxt('iris.csv', usecols=[0,1,2,3], delimiter=',')
#samples = samples - np.mean(samples)
#samples = samples / np.std(samples)
classes = np.loadtxt('iris.csv', usecols=[4], delimiter=',',dtype=np.str)
theta = np.linspace(-np.pi,np.pi,100)
import pylab as pl
for s in samples[:20]: # setosa
    pl.plot(theta, andrew_curve4(s,theta), 'r')

for s in samples[50:70]: # versicolor
    pl.plot(theta, andrew_curve4(s,theta), 'b')

for s in samples[100:120]: # virginica
    pl.plot(theta, andrew_curve4(s,theta), 'g')


In the plot above, the each color used represents a class and we can easily note that the lines that represent samples from the same class have similar curves.