–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
