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A Review on Functional Data Analysis for Cox processes

223

p

j

principal components in the past, given by

p

j

7

i

=1

b

j

i

ξ

i

.

This expression can be used to forecast the probability mass function of the

CP or the CCP simply replacing the mean, Λ, by the mean in the future, ˜Λ

q

.

Similarly, this can be done with the mean or the mode. Their corresponding

expressions for the mean and the mode of the

N

(

s, B

) process (chosen as in the

general case) become the following.

The number of points in

B

from

T

1

until

s

(

T

1

, T

2

) is

E

[

N

(

s, B

)] =

E

x

*

Λ

2

(

s, x

(

s

))

$

B

P

u

(

dU

)

-

=

µ

2

Λ

(

s

)

$

B

P

u

(

dU

)

The most probable number of points in

B

from

T

1

until

s

(

T

1

, T

2

), is

⎧⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎩

n

max

= ˜Λ

q

(

s

)

!

B

P

u

(

dU

)

1

,

˜Λ

q

(

s

)

!

B

P

u

(

dU

)

N

n

max

=

⎧⎪⎨ ⎪⎪⎩

int

9

˜Λ

q

(

s

)

!

B

P

u

(

dU

)

1

:

or

int

9

˜Λ

q

(

s

)

!

B

P

u

(

dU

)

1

:

+ 1

,

˜Λ

q

(

s

)

!

B

P

u

(

dU

)

/

N

n

max

= 0

,

˜Λ

q

(

s

)

!

B

P

u

(

dU

)

<

1

Bouzas et al. (2010b) illustrates the method with several simulations of a

CCP with different intensities and marks. Ruiz-Fuentes (2011) applies it to the

forecasting of the number of turning points of the price tendency of shares in

the stock exchanges of Athens and New York. The examples also illustrates the

importance of the representation theorems; the mean process is estimated and

forecasted regardless of the marks and consequently, the statistics of

N

(

t, B

) for

any

B

.

4. Inference for the intensity process

As previously mentioned, the interest of estimating and forecasting the in-

tensity process is that it permits to do the same with the CP as the intensity

characterizes it. Moreover, the intensity itself is relevant to several knowledge

fields including optics, radiopharmacy, medicine, and economics. Bouzas et al.

(2006b) made a first attempt to apply FDA to the intensity of a CP based on

the point estimation of the intensity in a discrete set of point times and the

direct application of FPCA. However, the method is improved using the estima-

tion of the mean process in Bouzas et al. (2012) as it preserves the theoretical

characteristic of monotonicity. The method proposed in Bouzas et al. (2012)