6. Similarity Functions

• Created by Andrés Segura Tinoco
• Created on May 20, 2019
• Updated on Mar 19, 2021

In statistics and related fields, a similarity measure or similarity function is a real-valued function that quantifies the similarity between two objects. In short, a similarity function quantifies how much alike two data objects are [1].

6.1. Common similarity functions

# Load the Python libraries
from math import *
from decimal import Decimal
from scipy import stats as ss
import sklearn.metrics.pairwise as sm
import math

\begin{align} similarity(X, Y) = d(X, Y) = \sqrt{\sum_{i=1}^n (X_i - Y_i)^2} \tag{1} \end{align}

# (1) Euclidean distance function
def euclidean_distance(x, y):
    return sqrt(sum(pow(a-b,2) for a, b in zip(x, y)))

\begin{align} similarity(X, Y) = d(X, Y) = \sum_{i=1}^n |X_i - Y_i| \tag{2} \end{align}

# (2) manhattan distance function
def manhattan_distance(x, y):
    return sum(abs(a-b) for a,b in zip(x,y))

\begin{align} similarity(X, Y) = d(X, Y) = (\sum_{i=1}^n |X_i - Y_i|^p)^\frac{1}{p} \tag{3} \end{align}

# (3) Minkowski distance function
def _nth_root(value, n_root):
    root_value = 1/float(n_root)
    return round(Decimal(value) ** Decimal(root_value),3)

def minkowski_distance(x, y, p = 3):
    return float(_nth_root(sum(pow(abs(a-b), p) for a,b in zip(x, y)), p))

\begin{align} similarity(X, Y) = cos(\theta) = \frac{\vec{X}.\vec{Y}}{\|\vec{X}\|.\|\vec{Y}\|} = \frac{\sum_{i=1}^n X_i.Y_i}{\sqrt{\sum_{i=1}^n X_i^2}.\sqrt{\sum_{i=1}^n Y_i^2}} \tag{4} \end{align}

# (4) Cosine similarity function
def _square_rooted(x):
    return round(sqrt(sum([a*a for a in x])),3)

def cosine_similarity(x, y):
    numerator = sum(a*b for a,b in zip(x,y))
    denominator = _square_rooted(x) * _square_rooted(y)
    return round(numerator/float(denominator),3)

\begin{align} similarity(X, Y) = \frac{cov(X, Y)}{\sigma_X . \sigma_Y} = \frac{\sum_{i=1}^n (X_i - \bar{X}).(Y_i - \bar{Y})}{\sqrt{\sum_{i=1}^n (X_i - \bar{X})^2 . (Y_i - \bar{Y})^2}} \tag{5} \end{align}

# (5) Pearson similarity function
def _avg(x):
    assert len(x) > 0
    return float(sum(x)) / len(x)

def pearson_similarity(x, y):
    assert len(x) == len(y)
    n = len(x)
    assert n > 0
    avg_x = _avg(x)
    avg_y = _avg(y)
    diffprod = 0
    xdiff2 = 0
    ydiff2 = 0
    for idx in range(n):
        xdiff = x[idx] - avg_x
        ydiff = y[idx] - avg_y
        diffprod += xdiff * ydiff
        xdiff2 += xdiff * xdiff
        ydiff2 += ydiff * ydiff

    return diffprod / math.sqrt(xdiff2 * ydiff2)

\begin{align} similarity(X, Y) = J(X, Y) = \frac{|X \cap Y|}{|X \cup Y|} = \frac{|X \cap Y|}{|X| + |Y| - |X \cap Y|} \tag{6} \end{align}

# (6) Jaccard similarity function
def jaccard_similarity(x, y):
    intersection_cardinality = len(set.intersection(*[set(x), set(y)]))
    union_cardinality = len(set.union(*[set(x), set(y)]))
    return intersection_cardinality / float(union_cardinality)

6.2. Manual examples

# Vectors
x = [-4.593481, -5.478033, 1.127111,  1.252885, -2.286953]  # Messi
y = [-4.080334, -3.406618, 4.334073, -0.485612, -2.817897]  # CR
z = [-4.048185, -5.546171, 0.505673,  0.616553, -1.730906]  # Neymar

Euclidean distance

euclidean_distance(x, y)

4.259455195846412

euclidean_distance(x, z)

1.1841800466723797

Manhattan distance

manhattan_distance(x, y)

8.060965

manhattan_distance(x, z)

2.4272509999999996

Minkowski distance

minkowski_distance(x, y)

3.619

minkowski_distance(x, z)

0.941

Cosine similarity

cosine_similarity(x, y)

0.842

cosine_similarity(x, z)

0.99

Pearson similarity

pearson_similarity(x, y)

0.8214001476231276

pearson_similarity(x, z)

0.9888645775446726

Jaccard similarity

a = [0,  1, 2, 3, 4, 5]
b = [-1, 1, 2, 0, 3, 5]

jaccard_similarity(a, b)

0.7142857142857143

6.3. Sklearn examples

corr = sm.euclidean_distances([x], [y])
float(corr[0])

4.259455195846413

corr = sm.manhattan_distances([x], [y])
float(corr[0])

8.060965

corr = sm.cosine_similarity([x], [y])
float(corr[0])

0.841904969009294

corr, p_value = ss.pearsonr(x, y)
corr

0.8214001476231275

Reference

[1] Wikipedia - Similarity measure.

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