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. RuizFuentes (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)