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IPS61 Rituparna S. et al.
2.1 Principal Components Analysis of Functional Data
We represent the smooth functional in terms of its decomposition into
functional principal components, a common approach in FDA. For the domain
Ƭ, setting
(, ) = ((), ()), (()) = (), , Ƭ, (2)

the functional principal components are the eigenfunctions of the auto-
covariance operator ∶ ℋ ↦ ℝ, a linear operator on the space ℋ, that is

given by
()() = ∫ Ƭ (, )().

We denote the orthonormal eigenfuncitons by , with associated

eigenvalues for = 1, 2, . . ., such that ≥ ≥ ⋯ and Σ < ∞. The

1
2

Karhunen-Loève theorem (see Rice and Silverman (1991)) provides a
representation of individual random trajectories of the functional , given by
() = () + ∑ ∞ (), ϵ Ƭ, (3)
=1

Where the are uncorrelated random variables that satisfy

= ∫(() − ()) (), = 0, ( ) = λ (4)
.

Under the data generating mechanism in (1), one has with indicator function
I(.)
̃
̃
E( ()) = (), Cov( (), ()) = (, ) + I ( = ). (5)
̃
2

This implies that the smooth mean function and the smooth covariance

surface can be consistenly estimated from available data by pooling the

sample of trajectories and smoothing the resulting scatterplot. The

exception for targeting points on with = in (5) is necessitated by the

presence of . This does not pose a problem, since it follows from the
smoothness of the surface that the areas of (, ), for which = , can

still be consistently estimated. Well-known procedures exist to infer
eigenfunctions and eigenvalues (Rice and Silverman (1991); Müller et al.
(2006)).
Processes are then approximated by substituting estimates and using a
judiciously chosen finite number K of terms in sum (3). This choice can be
made using one-curve-leave out cross-validation (Rice and Silverman (1991)),
pseudo-AIC criteria Yao et al. (2005) or a scree plot, a tool from multivariate
analysis, where one uses estimated eigenvalues to obtain a prespecified
fraction of variance explained as a function of K or looks for a change-point.
The above procedure is also known in numerical analysis under the
acronym proper orthogonal decomposion and as such it is used to price and
hedge financial derivatives on forward curves; see Hepperger (2010) for
examples from the energy market and further references.
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