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Measures of
Similarity and
Dissimilarity
Unit - II
Datamining

Measures of Similarity and
Dissimilarity
● Similarity and dissimilarity are important because they are used by
a number of data mining techniques
○ such as
■ clustering,
■ nearest neighbor classification, and
■ anomaly detection.
● Proximity is used to refer to either similarity or dissimilarity.
○ proximity between objects having only one simple attribute, and
○ proximity measures for objects with multiple attributes.

Measures of Similarity and Dissimilarity
● Similarity between two objects is a numerical measure of the
degree to which the two objects are alike.
○ Similarity - high -objects that are more alike.
○ Non-negative
○ between 0 (no similarity) and 1 (complete similarity).
● Dissimilarity between two objects is a numerical measure of the
degree to which the two objects are different.
○ Dissimilarity - low - objects are more similar.
○ Distance - synonym for dissimilarity

Transformations
● Transformations are often applied to
○ convert a similarity to a dissimilarity,
○ convert a dissimilarity to a similarity
○ to transform a proximity measure to fall within a particular range, such as [0,1].
● Example
○ Similarities between objects range from 1 (not at all similar) to 10 (completely similar)
○ we can make them fall within the range [0, 1] by using the transformation
■ s’ = (s 1)/9
−
■ s - Original Similarity
■ s’ - New similarity values

Dissimilarities between Data Objects
Euclidean Distance

If d(x, y) is the distance between two points, x and y, then the following properties hold.
1. Positivity
(a) d(x, x) 0 for all x and y,
≥
(b) d(x, y) = 0 only if x = y.
2. Symmetry
d(x, y) = d(y, x) for all x and y.
3. Triangle Inequality
d(x, z) d(x, y) + d(y, z) for all points x, y, and z.
≤
Note:-Measures that satisfy all three properties are known as metrics.

Non-metric Dissimilarities: Set Differences
A = {1, 2, 3, 4} and B = {2, 3, 4},
then A − B = {1} and
B − A = , the empty set.
∅
If d(A, B) = size(A − B), then it does not satisfy the second part of the
positivity property, the symmetry property, or the triangle inequality.
d(A, B) = size(A − B) + size(B − A) (modified which follows all properties)

Non-metric Dissimilarities: Time
Dissimilarity measure that is not a metric,but still useful.
d(1PM, 2PM) = 1 hour
d(2PM, 1PM) = 23 hours
● Example:- when answering the question: “If an event occurs at 1PM
every day, and it is now 2PM, how long do I have to wait for that event to
occur again?”

Similarities between Data Objects
● Typical properties of similarities are the following:
○ 1. s(x, y) = 1 only if x = y. (0 ≤ s ≤ 1)
○ 2. s(x, y) = s(y, x) for all x and y. (Symmetry)
● A Non-symmetric Similarity Measure
○ Classify a small set of characters which is flashed on a screen.
○ Confusion matrix - records how often each character is classified as itself,
and how often each is classified as another character.
○ “0” appeared 200 times but classified as
■ “0” 160 times,
■ “o” 40 times.
○ ‘o’ appeared 200 times and was classified as
■ “o” 170 times
■ “0” only 30 times.
● similarity measure can be made symmetric by setting
○ S`(x, y) = S`(y, x) = (s(x, y)+s(y, x))/2,

Examples of proximity measures
● Similarity Measures for Binary Data
○ Similarity measures between objects that contain only binary
attributes are called similarity coefficients
○ Let x and y be two objects that consist of n binary attributes.
○ The comparison of two objects (or two binary vectors), leads to
the following four quantities (frequencies):
f00 = the number of attributes where x is 0 and y is 0

Simple Matching Coefficient(SMC)
Jaccard Coefficient

Cosine similarity (Document similarity)
If x and y are two document vectors, then

cosine similarity (Document similarity)

# import required libraries
import numpy as np
from numpy.linalg import norm
# define two lists or array
A = np.array([2,1,2,3,2,9])
B = np.array([3,4,2,4,5,5])
print("A:", A)
print("B:", B)
# compute cosine similarity
cosine = np.dot(A,B)/(norm(A)*norm(B))
print("Cosine Similarity:", cosine)

● Cosine similarity - measure of angle between x and y.
● Cosine similarity = 1 (angle is 0◦
, and x & y are same (except magnitude or length))
● Cosine similarity = 0 (angle is 90
◦
, and x & y do not share any terms (words))

Note:-
Dividing x and y by their lengths normalizes them to have a length of 1 ( means magnitude is not
considered)

Extended Jaccard Coefficient (Tanimoto Coefficient)

Pearson’s correlation

Pearson’s correlation
● The more tightly linear two variables X and Y
are, the closer Pearson's correlation
coefficient(PCC)
○ PCC = -1, if the relationship is negative,
○ PCC=+1, if the relationship is positive.
■ an increase in the value of one variable increases the value of another variable
○ PCC = 0 Perfectly linearly uncorrelated
numbers
■ an increase in the value of one decreases the value of another variable.

Pearson’s correlation ( scipy.stats.pearsonr() - automatic)

Pearson’s correlation (manual in python)

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