Correlation
In probability theory and statistics, the correlation, also called correlation coefficient, between two random variables is found by dividing their covariance by the product of their standard deviations. (It is only defined if these standard deviations are finite.) It is a corollary of the Cauchy-Schwarz inequality that the correlation cannot exceed 1 in absolute value.The correlation is 1 in the case of an increasing linear relationship, -1 in the case of a decreasing linear relationship, and some value in between in all other cases, indicating the degree of linear dependence between the variables.
If the variables are independent then the correlation is 0, but the converse isn't true because the correlation coefficient detects only linear dependencies between two variables. Here is an example: Suppose the random variable X is uniformly distributed on the interval from -1 to 1, and Y = X2. Then Y is completely determined by X, so that X and Y are as far from being independent as two random variables can be, but their correlation is zero; they are uncorrelated.
If several values of X and Y have been measured, then the Pearson product-moment correlation coefficient can be used to estimate the correlation of X and Y. The coefficient is especially important if X and Y are both normally distributed and follow the linear regression model.
Pearson's correlation coefficient is a parametric statistic, and it is less useful if the underlying assumption of normality is violated. Less powerful non-parametric correlation methods, such as Spearman's ρ may be useful when distributions are not normal.
To get a measure for more general dependencies in the data (also nonlinear) it is better to use so called correlation ratio[?] which is able to detect almost any functional dependency, or mutual information[?] which detects even more general dependencies.
She was returning from her
naturally entered and walked up the stairs together. She had
door of her parlor she invited him to enter.
He hesitated a moment. Nothing was farther from his intention than
entanglements had proved fatal before to ambitious men; moreover he
reasonable excuse, he had manifestly been sauntering when they met,
the hotel parlor she had made unlike any other room in San Francisco,
in Paris and given her as a wedding present. A log fire was blazing.
aside her hat and lifted her shining crushed hair with both hands,
handsome, but quite useless as it was not even lined with cedar.
"I keep them in here," she exclaimed as gleefully as a naughty
isolated them on a little oasis of their own in this lawless waste of
dozen. On the whole I think I'd rather you'd read a poem to me. Do
delightfully."
"I don't mind in the least," he said gallantly. (At all events he
shall it be?"
And, alas, she chose "The Statue and the Bust."
He was disconcerted, but his sense of humor come to his rescue, and
hesitant lovers, with the proper emphasis and as much feeling as he
he finished he dropped the book and glanced over at his companion.
her breath coming unevenly. But there was neither challenge nor
was their innocence, not their melting, that stirred him profoundly.
compliment on his reading, then asked him if he believed that.
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