19th International Conference on Computational Statistics COMPSTAT - - PowerPoint PPT Presentation

19th International Conference

• n Computational Statistics

COMPSTAT 2010

Nuria Ruiz-Fuentes University of Jaén, Spain Paula R. Bouzas, Juan E. Ruiz-Castro University of Granada, Spain

Cox Process ≡ N(t)

Information process

x(t)

t t t

λ(t)

t t t

Intensity process

N(t)

t t t

Cox process (CP) Notation: λ(t, x(t)) ≡ λ(t)

Compound Cox Process ≡ N(t)

CCP

t0 w1 w2 w3 wi wi+1

U

t

u1 u2 u3 ui ui+1

λ(w1) λ(w2) λ(w3) λ(wi) λ(wi+1)

wi = i-th ocurrence time λ(wi) = intensity in time wi U r.v. in Υ = mark space ui = mark of the i-th ocurrence

Compound Cox Process with marks in a given subset ≡ N(t,B)

CCP, ui  B

t0 w1 w2 w3 wi wi+1

U

t

u1 u2 u3 ui ui+1

λ(w1) λ(w2) λ(w3) λ(wi) λ(wi+1)

wi = i-th ocurrence time λ(wi) = intensity in time wi B  U r.v. in Υ = mark space ui = mark of the i-th ocurrence

Representation theorems of a CCP

(Bouzas et al., 2007)

is a with inten sity ( ) P CC

(

• r mean

( ) )

N t

t t λ Λ

is a with intensit ( , y

• r mean

) CP ( ) ( )

( ) ( )

u B u B

N t B P dU P dU

t t λ Λ

∫ ∫

Examples:

• Earthquakes of a certain magnitude interval or in a certain zone, ...
• Number of telephone calls with length in a given range
• Number of maximum prices of a stock beyond a given threshold
• Etc.

• Probability mass function

[ ]

{ }

1 ( , ) exp ( ( ! ) )

( ) ( ) ; 0,1,2,

n u u B B

P N t P dU E dU n P B n

t t n

= =

    Λ −Λ =    

∫ ∫

 Counting statistics

[ ]

( , )

( ) ( )

u B

N t B E

E t P dU

=

  Λ  

∫

• Mean
• Mode

Estimation of the mean process of N(t,B)

Estimation of the mean process

• f a CP by an ad hoc FPCA

(Bouzas et al., 2006)

1

( ) ( ) ( ),

q q j j j

t t f t t I µ ξ

Λ =

Λ = ∈

∑

Representation theorems

• f a CCP

(Bouzas et al., 2007)

is a with mean ( , ) CP ( )

( )

u B

N t B P dU

t Λ ∫

Forecasting the mean process of N(t,B)

Principal Components Prediction

[ ) [ )

1 1 2 2

1 1 1 2 1 2 1

( ) ( ) ( ); , ( ) ( ) ( ); s ,

q q j j j q q j j j

t t f t t T T s s g s T T µ ξ µ η

Λ = Λ =

Λ = ∈ Λ = ∈

∑ ∑

( )

( )

2 2

2 1 2 1 1

( ) ( ) ( ); s ,

j

q p q j i i j j i

s s b g s T T µ ξ

Λ = =

Λ = ∈

∑ ∑

 Λ(t)

t t t

T1 T0 T2 T0 T0 T1 T1 T2 T2 Past Future

Forecasting the counting statistics of N(t,B)

Λ(t)

t

T1 T0 T2 Past Future

?

[ ]

2

( , )

( ( ) )

u B q

N s B E

s E P dU

=

 Λ   

∫



max max

ˆ

(2;3,1) 9.02 9.8 ˆ 9 8

s s

E E

PCP n n = = = =

max max

ˆ

(4;2,1,2,1) 7.88 8.87 ˆ 7 8

s s

E E

PCP n n = = = =

Simulations

100 + 1 sample paths in [0,10] T1 = 7, s = 8 T1 = 5, s = 7 λ(t) ∼ Γ(5,0.4); U ∼Β(10,0.4) and B = {u; 4 ≤ u ≤ 6}

( ) 0.5630

u B P dU

p ⇒ = =

∫

1

Notation:

max max

ˆ

Real mean , Estimated mean , ˆ Real mode , Estimated mode

s s

E E

n n

= = =

=

max max

ˆ

(3;2,2,1) 3.11 3.93 ˆ 3 1,3,4

s s

E E

PCP n n = = = =

max max

(2;3,2) ˆ 5.64 6.33 ˆ 5 4,7

s s

E

PCP E n n = = = =

Simulations

T1 = 4, s = 5 T1 = 5, s = 9

Notation:

max max

ˆ

Real mean , Estimated mean , ˆ Real mode , Estimated mode

s s

E E

n n

= = =

=

Boolean vector of four CP with λ(t) ∼ Υ(0,1); U ∼lgn(1,0.5) and B = {u; 2 ≤ u ≤ 5}

( ) 0.6188

u B P dU

p ⇒ = =

∫

2

100 + 1 sample paths in [0,10]

Conclusions ≈ Basis

CCP with marks in a subset N(t,B)

Representation theorems PCP models

• Prediction of the mean

process

• Prediction of the mean
• Prediction of the mode
• Application to particular cases:

CP with simultaneous ocurrences, multichannel CP, time-space CP,…

• Application to real data:

turning points of a stock price, earthquakes with restrictives characteristics, …

References

Aguilera et al., 1997. An aproximated principal component prediction model for continuous-time stochastic processes. Appl. Stoch. Model. Data. Anal., 13, 61 – 72. Aguilera et al., 1999. Forecasting time series by functional PCA. Discussion of several weighted approaches. Comput. Stat., 14, 443 – 467. Bouzas et al., 2006. Modelling the mean of a doubly stochastic Poisson process by functional data analysis. Comput. Stat. Data Anal., 50, 2655 – 2667. Bouzas et al., 2007. Functional approach to the random mean of a compound Cox

• process. Comput. Stat., 22, 467 – 479.

Ocaña et al., 1999. Functional principal components analysis by choice of norm. J.

• Multiv. Analysis, 71, 262 – 276.