220

P. R. Bouzas, N. Ruiz-Fuentes

t

is deduced in Bouzas et al.(2010b) as

⎧⎪⎪⎪⎨ ⎪⎪⎪⎩

n

B

max

(

t

) = Λ(

t

)

!

B

P

u

(

dU

)

−

1

,

Λ(

t

)

!

B

P

u

(

dU

)

∈

N

n

B

max

(

t

) =

⎧⎨ ⎩

int

+

Λ(

t

)

!

B

P

u

(

dU

)

−

1

,

or

int

+

Λ(

t

)

!

B

P

u

(

dU

)

−

1

,

+ 1

,

Λ(

t

)

!

B

P

u

(

dU

)

/

∈

N

n

B

max

(

t

) = 0

Λ(

t

)

!

B

P

u

(

dU

)

<

1

Note:

int

= integer part.

As seen above, all the statistics presented can be expressed in terms of the

intensity and of the mean process of the CP formed by all the points. The only

difference for each

N

(

t, B

) is the constant given in the representation theorems.

This observation is essential because it proves the interest of the inference for

the mean and for the intensity so that the CP or the CCP are inferred as well. It

should be noted that a CCP generalizes important particular types of processes

such as:

•

A CCP is a CP if the mark space is denumerable with one mark

U

= 1 or a

CP with random deletions if the mark space is

{

0

,

1

}

marking the deleted

points with 0.

•

By relaxing the property of orderliness, a CCP can represent a CP with

simultaneous occurrences when the mark space is

{

1

,

2

, . . .

}

(positive inte-

gers) indicating the number of occurrences.

•

A multichannel CP can be considered a CCP in which the mark indicates

the region where the point occurs.

•

A time-space Cox process can also be modeled by a CCP in time where

the mark on a point is continuously distributed and indicates the spatial

location of the point.

3. Modelling the mean process

Functional data analysis (FDA), including functional principal component

analysis (FPCA), provides a variety of techniques to model a continuous-time

stochastic process. Principal component prediction models are used to forecast

the mean process of a Cox process in a future interval from its evolution in the

past. They are based on a double FPCA, one for representing the mean process in

the past and other for its representation in the future. Then, a functional linear

regression approach based on linear regression of the principal components in the

future in terms of the functional component in the past is considered. A general

overview about FDA and PCP methods can be found in Ramsay and Silverman

(1997), Aguilera et al. (1997) and Valderrama et al. (2000). The importance