Taken somewhere from Internet.
Variance, covariance and autocovariance:
- Variance is one of the parameter to describe the distribution. It tells how the distribution extends from the mean value.
- Moments are always used to describe the distribution due to its computational simplicity (i.e. the power series). 1st moment is mean value, 2nd moment is variance, 3rd moment is skewness.
- Let
and
denote two random variables. Covariance
is a measure of how much
change together.
- In this case, variance is a special case of the covariance when two variables are identical.
- Given a stochastic process
, the autocovariance is the covariance of
.
Correlation and Autocorrelation:
- Correlation is the measure of the strength of association between two variables. For example, there is a strong correlation between black eyes and and black hair as you may seem them moving together all the time. But, there is a weak correlation between blue wallpaper and puppies. Bear in mind that correlation is not causation. Though the death of babies increase with the cases of vehicle accidents. They have a strong correlation. But we cannot say the death of babies causes the happen of vehicle accidents. It does not make sense. [Taken from Yahoo! Answer]
- Autocorrelation is the cross-correlation of the variable itself, in which separated by certain amount of time delay.
Random Variable:
- Say we run an experiment in which it gives a finite number of possible outcome, named as sample space
.
- For the convenience of mathematical calculation, each outcome (denoted as
) of
where
. We name
if the context is well understood.
Stationary Random Process:
- Let
denote a random process, in which
and
denote its mean value and autocorrelation function.
varies.
- Otherwise,
- For weakly stationary process,
and
.
- For strongly stationary process, all possible moments and join moments are time-invariant.
titanlab.org 