Modelling Non-linear and Non-stationary Time Series

Transcription

1 Modelling Non-linear and Non-stationary Time Series Chapter 7(extra): (Generalized) Hidden Markov Models Henrik Madsen Lecture Notes September 2016 Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

2 Hidden Markov Model First Order Markov Property p(x t X t 1 ) = p(x t X (t 1) ), t N (1) p(y t X t ) = p(y t X (t), Y (t 1) ), t N (2) Y 1 Y 2 Y 3 Y 4 X 1 X 2 X 3 X 4 Figure : Directed graph of basic HMM. The index denotes time. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

3 Markov Chains Discrete state vector at time t, X t, with m states. Transition probability One-step transition probability p(x t = j X t s = i) (3) γ ij,t = p(x t = j X t 1 = i) (4) One-step transition probability matrix from time t 1 to t γ 11,t γ 1m,t Γ t =..... (5) γ m1,t γ mm,t where the rows must sum to 1. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

4 Generalized State Space Models (GSSM) Parameter driven - model evolve independently of the past observation process. p(y t x t ) = p ( y t x t, X (t 1), Y (t 1)) p(x t+1 x t ) = p ( x t+1 x t, X (t 1), Y (t)) (6a) (6b) Observation driven - model depends on the past observation process. p(y t x t ) = p ( y t x t, X (t 1), Y (t 1)) p ( x t+1 Y (t)) = p ( x t+1 x t, X (t 1), Y (t)) (7a) (7b) Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

5 Model Validation Likelihood Setting Forward Selection Maximum of 5 states and model chosen by AIC Forecast Pseudo Residuals AIC = 2L + 2p (8) z t = Φ 1 ( p ( Y t y t Y (t 1))) (9) Marginal distribution Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

6 Data IC-Meter Table : 5-Minute Variables. Dataset Feature (type) Unit Description Indoor-Minutes Datetime Date and time Indoor-Minutes Temperature In C Indoor temperature Indoor-Minutes Humidity In % Indoor humidity Indoor-Minutes CO2 In ppm Indoor CO 2 content Indoor-Minutes Noise Avg DB Average noise level (5 min) Indoor-Minutes Noise Peak DB Largest average of 3 subsequent measurement (15 sec per measurement) Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

7 Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

8 Homogen HMM Original Scale Setting Results y t = CO 2,t p(x t x t 1 ) Γ p(y t x t ) N ( µ i, σi 2 ) for i = 1, 2,, m Table : Comparison of univariate homogen HMMs for 2 to 5 states. L p AIC BIC 2 states states states states Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

9 Homogen HMM Original Scale Table : Fit of the HMM (CO 2 ) with 5 states. δ i µ i σ i γ i1 γ i2 γ i3 γ i4 γ i5 State State State State State Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

10 Figure : Global Decoding of the HMM (CO 2 ) with 5 states. Top is the entire time series. Bottom is zoomed in on one day. Vertical lines indicate 00:00. Horizontal lines indicate the state dependent mean. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

11 Figure : Fit of the HMM (CO 2) with 5 states. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

12 Homogen HMM Transformed Scale h(y) = log(y 350) Setting y t = h(co 2,t ) p(x t x t 1 ) Γ p(y t x t ) N ( µ i, σi 2 ) for i = 1, 2,, m Results Table : Comparison of univariate (log transformed CO 2 ) homogen HMMs for 2 to 5 states. L p AIC BIC 2 states states states states Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

13 Figure : Fit of the HMM (log CO 2) with 5 states. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

14 Generalized Form Hierarchical Model p(y t x t, θ) p ( x t+1 x t, θ ) Hierarchical model by random effects ( ) Y (U = u) f Y u y; u, β ( ) U f U u; Ψ The likelihood is given by L ( β, Ψ; y, u ) = f ( y; θ ) = f ( y, u; β, Ψ ) ( ) ( ) = f Y u y; u, β fu u; Ψ (10a) (10b) (11) The distribution f ( y, u; β, Ψ ) is given by an exponential dispersion model with density given by Equation (??) with canonical link η = h(µ). The random effect is given by f U ( u; Ψ ) and the conditional likelihood by fy u ( y; u, β ). Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

15 Transition Probability Matrix with Covariates Γ t = exp(x T D,tβ11 + ZU11) exp(x T D,t β11 + ZU11) + j 1 exp(τ1j). exp(τ m1) exp(x T D,t βmm + ZUmm) + j m exp(τmj) exp(τ 1m) exp(x T D,t β11 + ZU11) + j 1 exp(τ1j) exp(x T D,tβmm + ZUmm) exp(x T D,t βmm + ZUmm) + j m exp(τmj) (12) Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

16 Figure : Periodic Splines. The knots are indicated on the x-axis. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

17 Inhomogen HMM Transformed Scale Setting y t = h(co 2,t ) p(x t x t 1 ) Γ t p(y t x t ) N ( µ i, σi 2 ) for i = 1, 2,, m Results Table : Comparison of univariate (log transformed CO 2 ) inhomogen HMMs for 2 to 5 states. L p AIC BIC 2 states states states states Same trend in residuals - not appropriate Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

18 Markov Switching AR(1) p(x t+1 x t ) = p ( x t+1 x t, X (t 1), Y (t)) p(y t x t, y t 1 ) = p ( y t x t, X (t 1), Y (t 1)) (13a) (13b) Y t 1 Y t Y t+1 Y t+2 X t 1 X t X t+1 X t+2 Figure : Directed graph of Markov switching AR(1). Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

19 Inhomogen Markov Switching AR(1) Transformed Scale Setting y t = h(co 2,t ) p(x t x t 1 ) Γ t p(y t x t, y t 1 ) N ( c i + φ i y t 1, σi 2 ) for i = 1, 2,, m Results Table : Comparison of univariate (log transformed CO 2 ) inhomogen AR(1) for 2 to 5 states. L p AIC BIC 2 states states 16204* states states Residuals - appropriate for 4 and 5 state model!! Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

20 Final Model - 5 states Residuals Figure : Model diagnostics of the final model. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

21 Final Model Marginal Distribution Figure : Comparison of the distribution of the measured CO 2 and the simulated distribution. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

22 Final Model Interpretation of the states State 1: Absence or sleeping State 2: Long term absence State 3: Outdoor interaction State 4: Presence (high activity) State 5: Presence (long term, low activity) Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

23 Figure : Transition probabilities over the day of the final model. The lower right plot is the stationary distribution. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

24 Figure : Profile of the states over the course of the day. I.e. Stacked stationary probabilities over the course of the day of the final model. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15

25 Figure : Global Decoding of the final model. Top is the entire time series. Bottom is zoomed in on one day. Vertical lines indicate 00:00. Horizontal lines indicate the state dependent mean. Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September / 15