## 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/np.sqrt(2.),
lambda x : x*np.sin(theta),
lambda x : x*np.cos(theta),
lambda x : x*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)
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')

pl.xlim(-np.pi,np.pi)
pl.show()
```

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.

1. This way is faster, especially with lot of data or data with large dimensionality

def andrew_curve4_vec(x,theta):
# iris has 4 four dimensions
tarr=np.array([1/np.sqrt(2.), np.sin(theta), np.cos(theta), np.sin(2. * theta)])
return x.dot(tarr)

...

for s in samples[:20]: # setosa
pl.plot(theta, np.array([andrew_curve4_vec(s,t) for t in theta]), 'r')

...

2. Hi, I know I'm coming late to the party (almost 2 years later, jeje). I've found some cool stuffs on your blog (this post being one of them).

I needed to use Andrew's Curves and made a generalized implementation in Python:

. https://gist.github.com/ryuzakyl/12c221ff0e54d8b1ac171c69ea552c0a

I thought it could be helpful for you. Any comments will be highly appreciated ;).

1. thank you!