Vector Space Models

Imports

import numpy as np
import scipy as sp
import matplotlib.pyplot as plt
import pandas as pd
from sklearn.decomposition import PCA
import seaborn as sns

sns.set()

PCA implementation

np.random.seed(1)
n_components = 2
A = np.random.random([3,10]) # shape 3x10
Am = (A - A.mean(axis=0)) # S1 - Mean centering
cv = np.cov(Am,  rowvar=False); cv.shape # S2 - Calculating Covariance
ev, evecs = np.linalg.eigh(cv); ev.shape, evecs.shape # S3 - Calculating Eigenvalues
idx_sort = np.argsort(ev)[::-1] # S4- Sorting
ev_sort = ev[idx_sort]
evecs_sort = evecs[:,idx_sort]
evecs_subset = evecs_sort[:,0:n_components]; evecs_subset # S5 - Taking Column subset
# ev[idx_sort]
# evecs_subset=evecs[idx_sort][:,:n_components]; evecs_subset.shape
comps = (evecs_subset.T@Am.T).T # S6 - Reducing components
comps

array([[ 0.43437323,  0.49820384],
       [ 0.42077249, -0.50351448],
       [-0.85514571,  0.00531064]])

pca = PCA(n_components=2)

pca.fit_transform(A)

array([[-0.43437323, -0.49820384],
       [-0.42077249,  0.50351448],
       [ 0.85514571, -0.00531064]])

Cosine similarity

def cos_sim(a,b):
    return a@b/(np.linalg.norm(a)*np.linalg.norm(b))
def euclid_sim(a,b):
    return np.linalg.norm(a-b)
cos_sim(np.array([1,2,3]), -1*np.array([2,4,6]))

-1.0

cos_sim(np.array([1,2,3]), np.array([3,1,4])), cos_sim(np.array([4,7,2]), np.array([3,1,4]))

(0.8910421112136306, 0.6374594222773456)

euclid_sim(np.array([1,2,3]), np.array([3,1,4])), euclid_sim(np.array([4,7,2]), np.array([3,1,4]))

(2.449489742783178, 6.4031242374328485)

cos_sim(np.array([1,0,-1]), np.array([2,8,1]))

0.08512565307587484

usa = np.array([5,6])
wash = np.array([10,5])
turkey = np.array([3,1])
ankara = np.array([9,1])
russian = np.array([5,5])
japan = np.array([4,3])
usa, wash, turkey, ankara, russian, japan

diff = usa - wash


cos_sim(turkey, ankara+diff), cos_sim(russian, ankara+diff), cos_sim(japan, ankara+diff)

(0.9899494936611664, 0.9486832980505138, 0.9838699100999074)

euclid_sim(turkey, ankara+diff), euclid_sim(russian, ankara+diff), euclid_sim(japan, ankara+diff)

(1.4142135623730951, 3.1622776601683795, 1.0)