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IPS61 Rituparna S. et al.
their functional principal component scores, which are available for
subsequent statistical analysis. This often leads to substantial dimension
reduction.
Most of the development in FDA has been with independent and identical
replications of function valued data. This permits the use of information from
multiple curves to identify patterns. However, in certain situations, it is
unrealistic to assume that the functions are independent. We do need some
process structure. One idea to follow up here is to work with the replication
principle implicit in stationary time series, where the values of the process are
functions. This problem of dependent functional observations is gaining
popularity only recently. Besse et al. (2000) develop an AR(1) model for
forecasting climatic variations. Kargin and Onatski (2008) use an AR(1) model
for forecasting Eurodollar futures. H¨ormann and Kokoszka (2010) study weakly
dependent functional processes, but they ignore the issue of smoothing. This
is common to a lot of work following Bosq (2012), eg. Bosq (2014) and Aue et
al. (2015), where the theory is developed assuming that the functions are
observed continuously. In practice, however, we only observe the functions at
a dense but discrete subset of the support, often with measurement error. Then
we need to interpolate smoothly to infer about the whole function. This raises
new questions about the behavior of the estimators. We develop the theory,
where the functions follow a stationary ARMA(p; q) model and are observed
discretely with noise. We start with kernel smoothing, followed by dimension
reduction using FPCA. Based on the time series of the first few significant
principal components, we fit a VAR or VARMA model. We provide techniques
for estimation of the model parameters and selection of the optimal model.
The methods developed are applied to the modeling and forecast of yield
curves. Hays et al. (2012) have previously used FDA for yield curve modeling.
The method in Hays et al. (2012) is factor analysis with penalized likelihood for
estimation, assuming an AR(p) model for the factors. In this paper the method
is PCA with state space modeling for time series, assuming VARMA(p,q)
evolution of the components.

2. Methodology
Consider a sample of smooth random trajectories ( ()) ∈

for = 1, …, generated from a process . Throughout we assume that
2
is an element of the Hilbert space ℌ := L (Ƭ) endowed with the inner product
〈, 〉 ℌ = ∫Ƭ()() and the norm ‖‖ = √〈, 〉 < ∞ a.s.. The observe
ℋ
measurement are support point on the domain Ƭ = [ , ] with additive
2

1
white noise error which is independent of the underlying process. The

measurements are for = 1, . . . , and = 1, . . . , :
( ) = ( ) + with ( ) = 0, ( ) = (1)
2
̃

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