A Bayesian nonparametric dynamic AR model for multiple time series analysis

Transcription

1 A Bayesian nonparametric dynamic AR model for multiple time series analysis Luis E. Nieto Barajas (joint with Fernando Quintana) Department of Statistics ITAM, Mexico COBAL V 8 June 2017 Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

2 Outline 1 Model 2 Dependent Pólya trees 3 Analysis 4 References Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

3 Destationalized Series ITAEE Time Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

4 Model The economy of the de 32 Mexican states depends on the global economy of the whole country The 32 time series interact among each other Need a model that respects the individual evolution of each time series but considers the country dependence Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

5 Model The economy of the de 32 Mexican states depends on the global economy of the whole country The 32 time series interact among each other Need a model that respects the individual evolution of each time series but considers the country dependence Let X i = {X ti, t 1}, i = 1,..., n. Propose with X ti = 0 w.p.1 for t < 0, and X ti = β 1i X t 1,i + + β pi X t p,i + ε ti, ε ti F t iid F t, for i = 1,..., n {F 1, F 2,...} θ dpt q(π θ, a, ρ, C) θ f (θ). Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

6 Pólya tree m=0 B Y0=p(B0 B) Y1 = p(b1 B) = 1 Y0 m=1 B0 B1 Y00=P(B00 B0) Y01=P(B01 B0) Y10 Y11 m=2 B00 B01 B10 B11 Y000 Y001 Y010 Y011 Y100 Y101 Y110 Y111 m=3 B000 B001 B010 B011 B100 B101 B110 B111 Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

7 Pólya tree Density y Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

8 Pólya tree Formally, a random probability measure F in (IR, B) has a Pólya tree distribution with parameters (Π, A), In notation F PT(Π, A), if there exists a sequence of non-negative numbers A = {α mj } and a family of r.v. Y = {Y mj } s.t. a) All r.v. in Y are independent ; b) For each (m, j), j = 1,..., 2 m 1 and m = 1, 2,..., Y m,2j 1 Be(α m,2j 1, α m,2j ) and Y m,2j = 1 Y m,2j 1 ; and, c) For each m = 1, 2,... and each j = 1,..., 2 m, where j (m,j) k 1 F(B mj ) = m k=1 Y (m,j), m k+1,j m k+1 = j(m,j) /2 is a recursive formula with initial value j (m,j) k m = j Typically α m,j = a ρ(m) F PT(Π, a, ρ) Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

9 Pólya tree Randomness in the tree depends on Y m+1,2j 1 = P(B m+1,2j 1 B mj ) Introduce dependence across several trees by defining a sequence of dependent variables Y t = {Y t,m,j } How? Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

10 Pólya tree Randomness in the tree depends on Y m+1,2j 1 = P(B m+1,2j 1 B mj ) Introduce dependence across several trees by defining a sequence of dependent variables Y t = {Y t,m,j } How? Through a beta process (Jara et al., 2013) w u 1 u 2 u 3 u 4 u 5 y 1 y 2 y 3 y 4 y 5 q q ind y t u t, u t 1,..., u t q Be a + u t j, b + (c t j u t j ), j=0 j=0 u t w ind Bin(c t, w), t = 1, 2,... w Be(a, b) Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

11 Order q beta process (BeP q ) BeP BDM Year Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

12 Dependent Pólya trees F = {F 1, F 2,...} are dependent Pólya trees s.t. F dpt q(π, a, ρ, C), with C = {c t,m,j }, a > 0 and ρ(m) = m δ, δ > 1 to ensure contonuity. Usually δ = 2 but we suggest δ = 1.1 (Watson et al., 2017) Properties : Corr{F t (B mj ), F t+s (B mj )} = m k=1 { } ψ (m,j) σ 2 t,s,m k+1,j m k+1 + 1/4 (1/4) m m k+1 }, m k=1 {σm k /4 (1/4) m with ψ t,s,k,j (m,j) k = 2aρ(k) ( ) q s l=0 c t l,k,j (m,j) + ( k 2aρ(k) + q l=0 c t l,k,j (m,j) k ( ) ( ) q q l=0 c t l,k,j (m,j) l=0 c t+s l,k,j (m,j) ) ( k ) k, σ 2 k = 1 4{2aρ(k) + 1}. 2aρ(k) + q l=0 c t+s l,k,j (m,j) k Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

13 Correlation in dpt Corr Lag Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

14 Mixtures in dependent Pólya trees It is well known that PT have discontinuities at the partition boundaries Generally Π = {B mj } is defined via the quantiles of F 0 ( ( ) ( )] B mj = F 1 j m, F 1 j 0 2 m It is possible to diminish the partition effect if we mix with respect to a parameter θ, i.e. Π θ = {B θ mj } defined by F 0( θ) with θ f (θ) which implies F dpt q(π θ, a, ρ, C)f (θ)dθ Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

15 Analysis Recall our model X ti = β 1i X t 1,i + + β pi X t p,i + ε ti, with X ti = 0 w.p.1 for t < 0, and Specifications : F 0 ( θ) = N(0, θ 2 ) ε ti F t iid F t, for i = 1,..., n {F 1, F 2,...} θ dpt q(π θ, a, ρ, C) θ f (θ). Fix median at zero : Take B 11 = (, 0] and B 12 = (0, ) with F t (B 11 ) = F t (B 12 ) = 1/2 Y t,1,1 = Y t,1,2 = 1/2 w.p.1. Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

16 Bayesian inference Priors For C : c t,m,2j 1 λ m,2j 1 ind Po(λ m,2j 1 ), λ m,2j 1 iid Ga(b λ 1, bλ 2 ) for t = 1,..., T, m = 1, 2,... and j = 1,..., 2 m 1 For θ : For the AR coefficients β : θ Ga 1/2 (b θ 1, bθ 2 ) β ki iid N(0, σ 2 β ), for k = 1,..., p and i = 1,..., n Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

17 Data analysis Let Y ti be the ITAEE observation for state i at time t Remove level, tendency and seasonality via second differences, i.e. X t = (Y t Y t 1 ) (Y t 1 Y t 2 ), for t = 3,..., 46 Plots of the partial autocorrelation of X ti suggest an autoregressive dependence of order between 2 and 4 Prior specifications : (b1 λ, bλ 2 ) = (1, 1) ; (bθ 1, bθ 2 ) = (0.1, 0.1) ; σ2 β = 100 Take a finite tree with M = 5 levels, a = 1, and ρ(m) = m δ with δ = 1.1 The values (p, q) are determined via the DIC Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

18 Data analysis Let Y ti be the ITAEE observation for state i at time t Remove level, tendency and seasonality via second differences, i.e. X t = (Y t Y t 1 ) (Y t 1 Y t 2 ), for t = 3,..., 46 Plots of the partial autocorrelation of X ti suggest an autoregressive dependence of order between 2 and 4 Prior specifications : (b1 λ, bλ 2 ) = (1, 1) ; (bθ 1, bθ 2 ) = (0.1, 0.1) ; σ2 β = 100 Take a finite tree with M = 5 levels, a = 1, and ρ(m) = m δ with δ = 1.1 The values (p, q) are determined via the DIC q p Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

19 Second differences Density Time nd. diff Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

20 Estimated c t,m,j parameters c Time FIGURE : m = 2 and j = 3 (solid line) ; m = 3 and j = 5 (dashed line) ; and m = 4 and j = 9 (dotted-dashed line) Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

21 Estimated error distribution Density Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

22 Quantiles of the error distributions Quantiles Time Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

23 P(β ki > 0 data) beta beta i i Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21

24 References 1 Jara, A., Nieto-Barajas, L.E. & Quintana, F. (2013). A time series model for responses on the unit interval. Bayesian Analysis 8, Nieto-Barajas, L.E. & Quintana, F.A. (2016). A Bayesian nonparametric dynamic AR model for multiple time series analysis. Journal of Time Series Analysis 37, Watson, J., Nieto-Barajas, L.E. & Holmes, C. (2017). Characterising variation of nonparametric random probability measures using the Kullback-Leibler divergence. Statistics 51, Luis E. Nieto Barajas BNP dynamic AR COBAL V 8 June / 21