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222

P. R. Bouzas, N. Ruiz-Fuentes

Then, the basis functions are adapted to the needs of the methodology.

The reconstruction of the

ω

sample path of the mean process estimated

from the points

{

(

t

0

,

Λ

ω

(

t

0

))

, . . . ,

(

t

p

,

Λ

ω

(

t

p

))

}

is the following

I

Λ

ω

(

t

) =

p

7

j

=0

Λ

ω

(

t

j

j

(

t

) +

p

7

j

=0

d

ωj

Ψ

j

(

t

)

, t

[

t

0

, t

0

+

T

)

, ω

= 1

, . . . , k

where the basis is the usual Lagrange basis of cubic splines

Φ

j

(

t

) =

"

φ

(

t

t

j

1

h

j

1

)

, t

[

t

j

1

, t

j

]

φ

(

t

j

+1

t

h

j

)

, t

[

t

j

, t

j

+1

]

,

j

̸

= 0

, p

Ψ

j

(

t

) =

"

h

j

1

ψ

(

t

t

j

1

h

j

1

)

, t

[

t

j

1

, t

j

]

h

j

ψ

(

t

j

+1

t

h

j

)

, t

[

t

j

, t

j

+1

]

,

j

̸

= 0

, p

Φ

0

(

t

) =

φ

(

t

1

t

h

0

)

,

t

[

t

0

, t

1

]

,

Φ

p

(

t

) =

φ

(

t

t

p

1

h

p

1

)

,

t

[

t

p

1

, t

p

]

,

Ψ

0

(

t

) =

h

0

ψ

(

t

1

t

h

0

)

,

t

[

t

0

, t

1

]

,

Ψ

p

(

t

) =

h

p

1

ψ

(

t

t

p

1

h

p

1

)

,

t

[

t

p

1

, t

p

]

where

h

j

=

t

j

+1

t

j

,

ψ

(

x

) = 3

x

2

2

x

3

,

φ

(

x

) =

x

3

x

2

and

d

ωj

=

dp

ωj

(

t

)

dt

8 8 8

t

=

t

j

and

d

ωj

+1

=

dp

ωj

(

t

)

dt

8 8 8

t

=

t

j

+1

are the values that control the

monotonicity as in Fritsch and Carlson (1980).

3. Functional principal components analysis is applied using the basis derived

in the former step.

4. Finally, a PCP model can be used for predicting the Cox process in a future

interval (

T

1

, T

2

]. The expression for the prediction of the mean process is

˜Λ

q

(

s

) =

µ

2

Λ

(

s

) +

q

7

j

=1

ˆ

η

j

g

j

(

s

);

s

(

T

1

, T

2

)

(3.1)

where

µ

2

Λ

(

s

) is the mean in the future,

q

is the number of principal com-

ponents used for estimating the process in the future,

g

j

are the weight

functions (eigenfunctions) in the future and ˆ

η

j

is the estimations of each

principal component in the future provided by linear regression on the first