¥ Denote noise value at time index i by xi for i=0..N–1
¥ 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)