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In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures are in some sense the inverse of distance metrics : they take on large values for similar ...
Given any function in variables with values in an abelian group, a symmetric function can be constructed by summing values of over all permutations of the arguments. . Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permut
Functional notation: if the first is the name (symbol) of a function, denotes the value of the function applied to the expression between the parentheses; for example, (), (+). In the case of a multivariate function , the parentheses contain several expressions separated by commas, such as f ( x , y ) {\displaystyle f(x,y)} .
Given two jointly distributed random variables and , the conditional probability distribution of given is the probability distribution of when is known to be a particular value; in some cases the conditional probabilities may be expressed as functions containing the unspecified value of as a parameter.
The corresponding expression was valid for two variables as well (it suffices to set X 3 to zero), but since it involves p 3, it could not be used to illustrate the statement for n = 2. The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational ...
Typically, in a polynomial expression, like terms are those that contain the same variables to the same powers, possibly with different coefficients. More generally, when some variable are considered as parameters, like terms are defined similarly, but "numerical factors" must be replaced by "factors depending only on the parameters".
But the terms of P which contain only the variables X 1, ..., X n − 1 are precisely the terms that survive the operation of setting X n to 0, so their sum equals P(X 1, ..., X n − 1, 0), which is a symmetric polynomial in the variables X 1, ..., X n − 1 that we shall denote by P̃(X 1, ..., X n − 1). By the inductive hypothesis, this ...
For example, consider a supermarket with 1000 products and two customers. The basket of the first customer contains salt and pepper and the basket of the second contains salt and sugar. In this scenario, the similarity between the two baskets as measured by the Jaccard index would be 1/3, but the similarity becomes 0.998 using the SMC.