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Serial Correlation Model & Notation: ARMA(1,1)
¥ Denote noise value at time index i by xi for i=0..N1
¥ Variance is average (AKA expected) value of noise squared:
H                      where E [á] means Òexpected value of áÓ
¥ Covariance is similar to variance, measured between different time points:
H                          which depends on time difference between time points i and j
¥ Correlation is covariance with variance factored out
H                            (with r0=1)
oN.B.: rk measures predictability of noise value at time j+k given value at time j
¥ For entire time series, express variance/correlation as a matrix
H 
¥ Need to have a simplified model for R (i.e., the rk for k =1, 2, É , N-1)
H Otherwise, have too many parameters to estimate
H My choice: ARMA(1,1) = AutoRegressive order 1 + Moving Average order 1
H parameter a = decay rate of the rk as k increases: for FMRI,     0 £ a < 1
H parameter b = determines correlation at lag 1 (r1):                   -1 < b < 1
o 
H For a > 0 and -a < b < 0, ARMA(1,1) noise can be thought of as a sum of AR(1) noise and white noise, with variance proportions determined by b
oThis feature is one reason I prefer ARMA(1,1) as a noise correlation model over AR(1)