–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
a = 0.7 b = 0.0 = pure AR(1) modelGreen: a = 0.7 b = +0.6
Blue:
a = 0.7 b = –0.6
a = 0.7 b = –0.6k
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