Can anyone provide a deeper insight between variance and variation(from variational calculus) from statistics and calculus perspective? Is there any relationship between these two terms? I want to study with regard to spectral clustering and principle component analysis(PCA).
Variance is a term used in Statistics (branch of mathematics), It measures the indefinition or spread of some variable.
Say you measure the height of persons. Then you get a mean height but also the variance of this height.
A variation is intended to be some small change, as in the calculus of variations, say to prove the principle
of least action in mechanics. Also used in the "principle of virtual work" also in mechanics.
The term "variance" is one of the measures of a random variable over a Borel sigma algebra with a probability measure - that is a space B with a measure P where P(B)=1. The variance of a random variable X is given by the second moment around the mean or E(X- E(X))^2) where E is the expected value. That is E[X]=Integral(X dP). The variance is a measure of the randomness of the random variable X.
The term variation in mathematics appears in several places in mathematical analysis and is in general related to change. One important concept is in the definition of functions of bounded variation. See https://en.wikipedia.org/wiki/Bounded_variation for the definition of the total variation of a function. In general in analysis the concept of "variation" is one that is used to study the change of a function over the real numbers f(x) as it relates to the change in the independent variable x. The definition of the derivative and its measure theoretic extensions is an example. In the calculus of variations the function f is replaced by a functional F[g] defined over the appropriate function space S containing g and relates to how the functional varies as the independent g varies in S.
https://en.wikipedia.org/wiki/Calculus_of_variations
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