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A Review on Functional Data Analysis for Cox processes

221

of FPCA is the reduction of the dimension of the problem. Furthermore, this

technique derives a continuous estimation of a stochastic process from observed

sample paths and weak theoretical restrictions.

A stochastic process is usually observed on a discrete set of time points. The

modeling and forecasting by means of FDA from observed discrete data of the

process is usually addressed by a first reconstruction of the sample paths. A

different procedure within FDA is presented by Wu et al. (2013) using kernel

density estimation. In this paper, the research line under review propose not to

model the CP itself, which has also been addressed in Bouzas et al. (2002), but

to estimate the mean or the intensity as this provides more information and is

useful for any statistic of the CP.

This section reviews the methodology of estimating the mean process of a

CP proposed in Bouzas et al. (2006c) and posterior forecasting in Bouzas et al.

(2010b). It consists in adapting the FPCA methodology to the characteristics of

the mean sample paths. An

ad hoc

inference method preserving the monotonic-

ity of these sample paths is built by means of functional principal component

analysis. The steps are the following.

1. Having observed

k

sample paths of the CP, each one can be split up into

r

shorter ones due to the independence of increments. The usual point

estimation for the mean can be applied to every set of

r

sample paths and

so, an estimated sample path of the mean is derived. Figure 1 illustrates

how the splitting is carried out.

N

ω

(

t

)

!

t

0

t

0

+

T

t

0

+ 2

T

t

0

+ (

r

1)

T

!

t

0

+

rT

N

ω

1

(

t

)

!

t

0

• • •

!

t

0

+

T

.

.

.

N

ωr

(

t

)

!

t

0

+ (

r

1)

T

• • •

!

t

0

+

rT

=

N

ω

1

(

t

)

!

t

0

t

1

t

2

t

p

1

!

t

p

=

t

0

+

T

.

.

.

N

ωr

(

t

)

!

t

0

t

1

t

2

t

p

1

!

t

p

=

t

0

+

T

Figure 1: Sketch of how to split a sample path,

N

ω

(

t

), of the CP.

2. Instead of using any of the usual presmoothing techniques, the

k

sample

paths of the mean are reconstructed preserving its monotonicity using the

monotone piecewise cubic interpolation from Fritsch and Carlson (1980).