Jaccard index and dice coifficient

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本文深入探讨了用于比较样本集相似性和多样性的Jaccard指数及其与Dice系数的关系。通过分析二元属性组合，阐述了如何计算Jaccard相似系数，并介绍了Dice系数作为另一种衡量集合相似度的方法。文章还对比了两者在不同场景下的应用特点。

he Jaccard index, also known as the Jaccard similarity coefficient (originally coined coefficient de communauté by Paul Jaccard), is a statistic used for comparing the similarity and diversity of sample sets.

The Jaccard coefficient measures similarity between sample sets, and is defined as the size of the intersection divided by the size of the union of the sample sets:

$J(A,B) = {{|A \cap B|}\over{|A \cup B|}}.$

The Jaccard distance, which measures dissimilarity between sample sets, is complementary to the Jaccard coefficient and is obtained by subtracting the Jaccard coefficient from 1, or, equivalently, by dividing the difference of the sizes of the union and the intersection of two sets by the size of the union:

$J_{\delta}(A,B) = 1 - J(A,B) = { { |A \cup B| - |A \cap B| } \over |A \cup B| }.$

This distance is a proper metric^[1] .^[2]

Binary Properties

Given two objects, A and B, each with n binary attributes, the Jaccard coefficient is a useful measure of the overlap that A and B share with their attributes. Each attribute of A and B can either be 0 or 1. The total number of each combination of attributes for both A and B are specified as follows:

$M_{11}$ represents the total number of attributes where A and B both have a value of 1.

$M_{01}$ represents the total number of attributes where the attribute of A is 0 and the attribute of B is 1.

$M_{10}$ represents the total number of attributes where the attribute of A is 1 and the attribute of B is 0.

$M_{00}$ represents the total number of attributes where A and B both have a value of 0.

Each attribute must fall into one of these four categories, meaning that

$M_{11} + M_{01} + M_{10} + M_{00} = n.$

The Jaccard similarity coefficient, J, is given as

$J = {M_{11} \over M_{01} + M_{10} + M_{11}}.$

Dice's coefficient, named after Lee Raymond Dice^[1] and also known as the Dice coefficient or Dice similarity coefficient (DSC), is a similarity measure over sets:

$s = \frac{2 | X \cap Y |}{| X | + | Y |}$

It is identical to the Sørensen similarity index, and is occasionally referred to as the Sørensen-Dice coefficient. It is not very different in form from the Jaccard index but has some different properties.

The function ranges between zero and one, like Jaccard. Unlike Jaccard, the corresponding difference function

$d = 1 - \frac{2 | X \cap Y |}{| X | + | Y |}$

is not a proper distance metric as it does not possess the property of triangle inequality. The simplest counterexample of this is given by the three sets {a}, {b}, and {a,b}, the distance between the first two being 1, and the difference between the third and each of the others being one-third.