PCA of high dimensional random walks

In [1]:

    

%pylab inline
import sklearn.decomposition


    

    




Populating the interactive namespace from numpy and matplotlib

In [2]:

    

n_dims = 10000
n_steps = 1000


    

In [3]:

    

walk = np.zeros((n_steps, n_dims))
for i in range(1, n_steps):
    walk[i] = walk[i-1] + np.random.normal(scale=1/np.sqrt(n_dims), size=n_dims)


    

In [4]:

    

pca = sklearn.decomposition.PCA()


    

In [5]:

    

pca_walk = pca.fit_transform(walk)


    

In [6]:

    

# The observed variance in the first PCA component
pca.explained_variance_ratio_[0]


    

    
Out[6]:





0.6084576781019368

In [7]:

    

# Predicted value
6 / np.pi**2


    

    
Out[7]:





0.6079271018540267

In [8]:

    

# Plot the projection of the trajectory onto the first few PCA components
for i in range(5):
    plot(pca_walk[:, i], label='PCA %d' % i)

xlabel('Step')
ylabel('Distance along component')
legend(loc='upper right', fontsize=10);