Variables and are said to be uncorrelated if their covariance is zero:Independent statistics are always uncorrelated,but the converse is not necessarily true.
A sequence of uncorrelated numbers developed by Wiener (1926-1927). The numbers are constructed by beginning with(1)then forming the outer product with to obtain(2)This row is repeated twice, and its outer product is then taken to give(3)This is then repeated four times. The procedure is repeated, and the result repeated eight times, and so on. The sequences from each stage are then concatenated to form the sequence 1, , 1, 1, 1, , , 1, , , 1, 1, 1, , , 1, , , ....
where is the entropy and is the joint entropy. Linear redundancy is defined aswhere are eigenvalues of the correlation matrix.
Predictability at a time in the future is defined byand linear predictability bywhere and are the redundancy and linear redundancy, and is the entropy.
A time series , , ... is nonstationary if, for some , the joint probability distribution of , , ..., is dependent on the time index .