Change point in trending regression

Transcription

1 Charles University, Prague ROBUST 2010 joint work with A. Aue, L.Horváth, J. Picek (Charles University, Prague)

2 1 (Charles University, Prague)

3 1 2 model-formulation (Charles University, Prague)

4 1 2 model-formulation 3 (Charles University, Prague)

5 1 2 model-formulation 3 4 (Charles University, Prague)

6 1 2 model-formulation (Charles University, Prague)

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9 Procedures on stability of statistical models - structural breaks, disorder, stability, segmented, switching, change point problem, etc. Observations Y 1,..., Y n obtained at the ordered time points t 1 < < t n such that Y 1,..., Y k - model I Y k,..., Y n - model II k change point unknown The problem: to detect ( to test H 0 : no change & H 1 : there is a change), to identify k ( to estimate k ) to estimate the model before after the change. (Charles University, Prague)

10 Many variants multiple changes, abrupt changes, gradual changes, changes in various parameters, changes in distributions, independent observations, dependent observations. Construction of tests estimators various approaches as in most of the statistical problems. Theoretical problems theoretical problems. s (meteorology, climatology, hydrology or environmental studies, econometric time series, statistical quality control, etc.) (Charles University, Prague)

11 1 model-formulation 2 model-formulation (Charles University, Prague)

12 model-formulation model- formulation Y 1,..., Y n are observed at time points t 1 < < t n : Y i = x T i β + e i, i = 1..., k = x T i β + x T i δ + e i, i = k , n, e 1,..., e n rom errors i.i.d., zero mean, nonzero variance σ 2 finite E e i 2+ with some > 0 β, δ 0 parameters k... change point (Charles University, Prague)

13 model-formulation x 1,..., x n p-dim. design points (rom or nonrom): nontrending : 1 n k i=1 x ix T i k n C, k n (Charles University, Prague)

14 model-formulation x 1,..., x n p-dim. design points (rom or nonrom): nontrending : 1 n k i=1 x ix T i k n C, k n trending : x i = h(i/n), i = 1,..., n, h smooth nonconstant vector function Main problems: H 0 : no jump in & H 1 : at most one jump estimators of change points (Charles University, Prague)

15 model-formulation Test statistics T n = max p k<n p β k LSE of β based on Y 1,..., Y k {( βk β 0 k) T Σ 1 k ( βk β 0 k)} β 0 k LSE of β based on Y k+1,..., Y n 1 Σ k is an estimator of the variance matrix of β k β 0 k Equivalently T n = { max S T k p k<n p C 1 k C n(c 0 1 } k ) 1 S k σ n 2, S k = k i=1 x iê i, k = 1,..., n, ê i = Y i x T i β n, i = 1,..., n - residuals C k = k i=1 x ix T i, C 0 k = C n C k (Charles University, Prague)

16 model-formulation Test statistics T n = max p k<n p {( βk β 0 k) T Σ 1 k ( βk β 0 k)} β k LSE of β based on Y 1,..., Y k β 0 k LSE of β based on Y k+1,..., Y n 1 Σ k is an estimator of the variance matrix of β k β 0 k Equivalently T n = { max S T k p k<n p C 1 k C n(c 0 1 } k ) 1 S k σ n 2, S k = k i=1 x iê i, k = 1,..., n, ê i = Y i x T i β n, i = 1,..., n - residuals C k = k i=1 x ix T i, C 0 k = C n C k σ n 2 is an estimator of var e i = σ 2 (Charles University, Prague)

17 model-formulation Critical regions c n (α) critical value α level T n > c n (α) Approximation of the critical values: (i) limit distribution of T n under H 0 ; (ii) resampling methods (bootstrap) Estimator k of the change point k defined as such k it maximizes w.r.t. k { ( β k β 0 k) T Σ 1 k ( β k β 0 } k) (Charles University, Prague)

18 1 2 model-formulation (Charles University, Prague)

19 Test statistics: T n = max p k<n p T n (ε) = {( βk β 0 k) T Σ 1 k ( βk β 0 k)} { } max εn k (1 ε)n..., 0 < ε < 1/2 nontrending polynomial : a(y) = (2 log y) 1/2, lim P(a(log n)(t n) 1/2 t + b p (log n)) n = exp{ 2 exp{ t}}, t R 1, b p (y) = 2 log y + p 2 log log y log(2γ(p/2)), y > 1, x i = (1, i/n, (i/n) 2,..., (i/n) p 1 ) T, i = 1,..., n (Charles University, Prague)

20 nontrending { p T n (ε) d i=1 sup B2 i (t) } ε<t<1 ε t(1 t) {B j (t); t (0, 1)}, j = 1,..., p, independent Brownian bridges trending x i = h(i/n) S(t) = T n (ε) d t 0 sup S T (t)c(t)c 1 (1)C 0 (t)s(t) ε<t<1 ε h(x)db(x) C(t)C 1 (1) 1 with {B(x), x [0, 1]} being a Brownian bridge, C(t) = lim n 1 n C nt. 0 h(x)db(x), t [0, 1] (Charles University, Prague)

21 1 2 model-formulation (Charles University, Prague)

22 Assumptions (A.1) The sequence (e i : i 1) satisfy E[e i ] = 0, E[e 2 i ] = σ 2 > 0 (1) (A.2) There are independent stard Brownian motions (W 1,n (t): t 0) (W 2,n (t): t 0) such that 1 k max 1 k n/2 k 1/ν e i τw 1,n (k) = O P(1) (n ) (2) max n/2<k<n i=1 1 (n k) 1/ν n i=k+1 with some ν > 2 τ > 0. (Charles University, Prague) e i τw 2,n (n k) = O P(1) (n ) (3)

23 (A.3) The components of h(.) are continuous on [0, 1]. The matrices t 0 h(x)ht (x)dx 1 t h(x)ht (x)dx are regular for all t (t 0, 1 t 0 ) for all t 0 (0, 1/2). (A.4) There are p linearly independent p-dimensional vectors a 01,..., a 0p nonnegative 0 γ 01 <... < γ 0p such that lim sup t t γ 0p+1 h(t) p a 0l t γ 0l <. (4) l=1 (A.5) There are p linearly independent p-dimensional vectors a 11,..., a 1p nonnegative 0 γ 11 <... < γ 1p such that lim sup t 1 1 t γ 1p+1 h(t) p a 1l t γ 1l <. (5) l=1 (Charles University, Prague)

24 Particular cases: polynomial h(t) = (t γ 1,..., t γp ) T, t [0, 1], harmonic 0 γ 1 <... < γ p h(t) = (cos(2πtω 1 ), sin(2πtω 1 ),..., cos(2πtω p ), cos(2πtω p )) T, t [0, 1] ω 1,..., ω p known. independent e 1,..., e n : τ = σ. dependent e 1,..., e n : the estimator with flat top kernel used: τ 2 n = 1 n ê j = Y j x T j n (ê j e n ) q n n j=1 β n, e n = 1 n n i=1 êi j=1 n j w j i=1 (ê i e n )(ê i+j e n ) w j = 1I {1 j q n /2} + 2(1 j/q n )I {q n /2 < j q n } (Charles University, Prague)

25 1 2 model-formulation (Charles University, Prague)

26 Recall notation: T n = max p k<n p V k (τ 2 / τ 2 n ) V k = S T k C 1 k C n(c 0 k ) 1 S k /τ 2 S k = k i=1 x iê i, k = 1,..., n, ê i = Y i x T i β n, i = 1,..., n - residuals C k = k i=1 x ix T i, C 0 k = C n C k x i = h(i/n) (Charles University, Prague)

27 Main step of the proof The limit behavior of T n is the same as max(t n0, T n1 ) T n0 = T n1 = max p k<ns n V k max p n k<ns n V k for some s n 0 +, T n0 T n1 are asymptotically independent. T n0 has the same limit distribution as T 0 n0 = max Z T k p k<ns M 1 k Z k/τ 2 n Z k = k i=1 x ie i, M k = k i=1 x ix T i. T 0 n0 will not change if x i is replaced by x 0 i = Ax i with arbitrary nonsingular p p matrix A. max 1 k n/2 Z k τ k j=1 x j(w 1,n (j) W 1,n (j 1)) = o P (n κ ) for some κ > 0. (Charles University, Prague)

28 After some steps we get that Tn0 0 has the same distribution as 1 max k k Q k n k<ns n kτ QT k R 1 with Q k = k i=1 ((i/k)γ 01,..., (i/k) γ 0p ) T e i R k = ( 1 k k j=1 (j/k)γ 0v +γ 0r ) p v,r=1. Finally, we get that Tn0 0 has the same distribution as sup Q(t) T V(t) 1 Q(t) k n t ns n ( 1 t ) Q(t) = t γ x γ 0l dw n,1 (x); l = 1,..., p 0l+1/2 0 ( 1 t ) V(t) = t γ x γ 0l+γ 0l dx; l, l = 1,..., p 0l+γ 0l +1 0 After exponential transformation Q(t) T V(t) 1 Q(t) is a norm of stationary Gaussian process results of Albin, Piterbarg, Jaruskova etc. are applied. (Charles University, Prague)

29 1 2 model-formulation (Charles University, Prague)

30 Figure 1: Monthly air carrier traffic in the United Stated from January 1996 to March (Charles University, Prague)

31 Y j = 4+ 6j n + 3 2πj cos( 2 12n )+ 3 2π sin( 2 12n ) cos(2π 4n sin(2π 4n )+e j, j = 1,..., 200 e j either AR(1) or MA(1), normal distr. k = 100 either in the intercept or in one harmonic regressor 1000 repetitions critical value obtained through circular block bootstrap H (1) A H (1) A change in intercept, AR (1) or MA(1) change in one regressor, AR (1) or MA(1) (Charles University, Prague)

32 Figure 4: Time series plots of the processes under H (1) A with δ = 2 (upper panel) ROBUST H(2) A (lower 2010 joint work with A. Aue, L.Hor (Charles University, Prague)

33 Figure 2: Circular bootstrap distribution versus finite sample distribution for AR(1) innovations (left) MA(1) innovations (right) with scalings ϕ 2 = 2 (- - -), 1.5 ( - - -), 1.0 ( ), 0.5 (- - -). (Charles University, Prague)

34 Figure 3: Size-power curves for H (1) A (upper half) H(2) A (lower half) for the asymptotic modification (first third line) the the circular bootstrap (second fourth line) with AR(1) innovations (left) MA(1) innovations (right). (Charles University, Prague) 17

35 ROBUST joint work with A. Aue, L.Hor (Charles University, Prague)

36 ROBUST joint work with A. Aue, L.Hor (Charles University, Prague)

37 Data Monthly air traffic data model through the root of data, n = 159 Y j = β 0 + β 1 j/n + q ( l=1 β c l cos(2πω ellj/n) + β (s) l ) sin(2πω ell j/n) + e j q = 4, ω 1 = 2/160, ω 3 = 13/160, ω 3 = 40/160, ω 1 = 80/160 ω 2 annual cycle ω 3 quarterly cycle ω 4 two months cycle k = 69 (Charles University, Prague)

38 bwidth = Figure 6: periodogram (Charles University, (lower panel). Prague) Square root transformation of the monthly air carrier traffic data (upper panel) its

39 Figure 7: The fitted model based on the proposed data segmentation procedure (upper panel) based (Charles on auniversity, global fit (lower Prague) panel). The Change original point datainis trending shown both plots as dashed line.