5
Mathematical Model for Serial Correlation
¥My choice: ARMA(1,1) = AutoRegressive order 1 + Moving Average order 1
H Notation: rk = correlation at time lag #k for k =1, 2, É , N-1
¥ parameter a = decay rate of the rk as k increases: for FMRI,     0 £ a < 1
¥ parameter b = affects correlation at lag 1 (r1):                          -1 < b < 1
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
H Why I prefer 2 parameter ARMA(1,1) over easier 1 parameter AR(1) model (b=0)
Red: a = 0.7  b =    0.0  =  pure AR(1) model
Green:   a = 0.7  b = +0.6
Blue: a = 0.7  b = 0.6
k
b = 0 gives r1= a
b < 0 reduces r1 
   (as additive WN does)
b > 0 increases r1
a determines decay rate
   for larger lags
r1