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

19th International Conference on Computational Statistics COMPSTAT 2010 Nuria Ruiz-Fuentes Paula R. Bouzas, Juan E. Ruiz-Castro University of Jaén, Spain University of Granada, Spain

Cox Process ≡ N (t) Information process Intensity process t t x ( t ) λ ( t ) t t t t Cox process (CP) t N (t) t t Notation: λ (t, x(t)) ≡ λ (t)

Compound Cox Process ≡ N (t) CCP U u i+1 u 2 u i u 1 u 3 t t 0 w 1 w 2 w 3 w i w i+1 λ (w i+1 ) λ (w 1 ) λ (w 2 ) λ (w 3 ) λ (w i ) w i = i-th ocurrence time λ (w i ) = intensity in time w i U r.v. in Υ = mark space u i = mark of the i-th ocurrence

Compound Cox Process with marks in a given subset ≡ N (t, B ) CCP, u i  B U u i+1 u 2 u i u 1 u 3 t t 0 w 2 w 3 w i w i+1 w 1 λ (w i+1 ) λ (w 1 ) λ (w 2 ) λ (w 3 ) λ (w i ) λ (w i ) = intensity in time w i w i = i-th ocurrence time B  U r.v. in Υ = mark space u i = mark of the i-th ocurrence

Representation theorems of a CCP (Bouzas et al., 2007) λ Λ ( t ) or mean ( ) t N t ( ) is a CC P with inten sity ∫ λ ( t ) N t B ( , ) is a CP with intensit y P dU ( ) u B ∫ Λ ( ) t or mean P dU ( ) u B 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.

Counting statistics • Probability mass function { } 1 [ ] n  ∫   ∫  Λ −Λ = = =  ( ) ( ) ; 0,1,2, t t n P N t ( , ) B n E P dU ( ) exp P ( dU )     u u n ! B B • Mean [ ]   ∫ Λ = E ( ) t P dU ( ) N t B ( , ) E   u B • Mode

Estimation of the mean process of N (t, B ) Estimation of the mean process of a CP by an ad hoc FPCA (Bouzas et al., 2006) ∑ q Λ = µ ξ ∈ q ( ) t ( ) t f t ( ), t I Λ = j j j 1 Representation theorems of a CCP (Bouzas et al., 2007) N t B ( , ) is a CP with mean Λ ∫ ( ) t P dU ( ) u B

Forecasting the mean process of N (t, B ) t T 0 T 2 T 1 Λ ( t ) t T 0 T 1 T 2 t T 0 T 1 T 2 Future Past Principal Components Prediction [ ∑ ) Λ = µ q ξ ∈ q 1 1 ( ) t ( ) t f t ( ); t T T , 1 Λ = j j 0 1 j 1 ∑ [ ) Λ = µ q η ∈ q 2 2 ( ) s ( ) s g ( ); s s T T , 2 Λ = j j 1 2 j 1 ( ) ∑ ∑ ( )  q p Λ = µ ξ ∈ q 2 j 2 j ( ) s ( ) s b g ( ); s s T T , 2 Λ = = i i j 1 2 j 1 i 1

Forecasting the counting statistics of N (t, B ) [ ]   ∫  Λ = q E ( s ) P dU ( ) ( , ) N s B E 2   u B ? Λ ( t ) t T 0 T 1 T 2 Past Future

Simulations 100 + 1 sample paths in [0,10] λ (t) ∼ Γ (5,0.4); U ∼Β (10,0.4) and B = {u; 4 ≤ u ≤ 6} 1 ∫ ⇒ = = B P dU ( ) p 0.5630 u T 1 = 5, s = 7 T 1 = 7, s = 8 PCP (4;2,1,2,1) PCP (2;3,1) = = = = ˆ ˆ 9.02 9.8 7.88 8.87 E E E E s s s s = = = = ˆ n 9 n 8 ˆ n 7 n 8 max max max max = = ˆ Notation: Real mean , Estimated mean , E E s s = = ˆ Real mode n , Estimated mode n max max

Simulations 100 + 1 sample paths in [0,10] Boolean vector of four CP with λ (t) ∼ Υ (0,1); 2 U ∼ lgn (1,0.5) and B = {u; 2 ≤ u ≤ 5} ∫ ⇒ = = B P dU ( ) p 0.6188 u T 1 = 4, s = 5 T 1 = 5, s = 9 PCP (3;2,2,1) PCP (2;3,2) = = = = ˆ ˆ 3.11 3.93 E E 5.64 E 6.33 E s s s s = = = = ˆ n 3 n 1,3,4 ˆ n 5 n 4,7 max max max max = = ˆ Notation: Real mean , Estimated mean , E E s s = = ˆ Real mode n , Estimated mode n max max

Conclusions ≈ Basis Representation theorems • Prediction of the mean CCP process with marks in a subset • Prediction of the mean N (t,B) • Prediction of the mode PCP models  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.