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Covariance:
Covariance calculates that how much two random variables change together. If small value of one variable corresponds with smaller value of other variable and the same thing for greater values, it means covariance is positive. On the other hand, when smaller value of one variable corresponds with greater value of other variable and vice versa, then in such case covariance is negative. These mathematical signs show the linear relationship between these variables. The magnitude of covariance is not simple to interpret. Magnitude of strength of the linear correlation determines the covariance coefficient.
A more general identity for covariance matrices: Let A be a random vector with covariance matrix sigma(A).Let X be a matrix that can act on The covariance matrix of vector XA is:
Sigma (XA) = X sigma (A) X^T.
It is a direct result of linearity of expectation and it is useful when applying linear transformation to vector.
Uncorrelatedness and Independence: IfA and B are independent, then their covariance is zero.
E(AB) = E(A)*E(B).
Correlation:
Correlation means relationship between two 2 or more variables.
Correlation analysis deals with association of two or more variables-Simpson&Kafka
Steps:
For ex:
Significance of Correlation:
Types of Correlation
1. Positive and Negative: Correlation is said to be positive when both the variables moves in the same direction. For ex.supply and price, age of husband and wife.
Correlation is said to be negative when both variables moves in the opposite direction. For ex. demand and price, expenses and income.
2. Linear and Non-Linear: When the rate of change in both the variables is same, then it is said to be linear correlation. When the rate of change between both the variables is different, then it is said to be non-linear correlation.
3. Simple, Partial &Multiple: When we studies correlation between only two variables, not more than two is called simple correlation.
When we studies 3 or more variables, then it is called multiple correlation.When we studies 3 or more variable, but consider only two at a time, then it is called partial correlation.
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