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IPS61 Rituparna S. et al.
2.2 Time Series of Funcitonal Data
In this section we show that an ARMA (p,q) structure on the principal
components scores. Starting with the setup as described in (1), we assume that
the series of functions follows a stationary ARMAH (p,q) model with mean ∈
ℋ:
(∙) − = ( −1 (∙) − ) + ⋯ + ( − (∙) − ) + ∈ (∙) . (6)
1

Where ∈ (∙) = ղ + ղ −1 (∙) + ⋯ + ղ − (∙), ղ (∙) is ℌ white

1

noise. , ⋯, , ⋯ , are linear continuous functions. Combining (6)
1

1

and (3) we have,
∞ ∞ ∞
∑ (∙) = ∑ − 1(∙)) + ⋯ + (∑ − (∙)) + ∈ (∙) . (7)

1

=1 =1 =1

Using linearity and continuity of , ⋯ , , this implies,

1
∞ ∞ ∞
( (∙)) ∈ (∙). (8)
∑ (∙) = ∑ − 1 1 ((∙)) + ⋯ + ∑ −
=1 =1 =1

Using vector notation, we have:
Φ(∙)Ξ = (Φ(∙))Ξ −1 + ⋯ + (Φ(∙))Ξ − +∈ (∙). (9)

1

T
where Φ = ( , , ⋯ ) and Ξ = ( ⋯ ) . Since the eigenfuncitons are
2
1
1, 2,
T
orthonormal, we can premultiply equation (10) by Φ to get:
T T
Ξ = Φ (Φ(∙))Ξ −1 + ⋯ + Φ (Φ(∙))Ξ − + Z . (10)

1

T
It remains to show that Z = Φ ∈ (∙) is an MAH(q) process. This can be proved

by verifying that the autocovariances of Z vanish for lags of order greater than
q. This is immediate as
T
Cov(Z , Z ) = Φ Cov(∈ , ∈ ) = Φ

and ∈ is itself an MAH(q) process.

This implies a VARMA(p,q) structure on the vector of principal component
score Ξ .

T
Moreover, since Ξ = Φ , stationarity of implies stationarity of Ξ.

2.3 Granger Causality
The concept of the Granger causality was originally introduced by
Wiener(1956) and formulated by Granger(1969). Given two stationary time
series, if the variance of the prediction error for the second time series at
the present time is reduced by including past measurements from the first
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