Modelling Non-linear and Non-stationary Time Series · 2016-09-28 · Modelling Non-linear and Non-stationary Time Series Chapter 7(extra): (Generalized) Hidden Markov Models Henrik

Modelling Non-linear and Non-stationary TimeSeries

Chapter 7(extra):(Generalized) Hidden Markov Models

Henrik Madsen

Lecture Notes

September 2016

Henrik Madsen (02427 Adv. TS Analysis) Lecture Notes September 2016 1 / 15

Hidden Markov ModelFirst Order Markov Property

p(Xt|Xt−1) = p(Xt|X (t−1)), t ∈ N (1)

p(Yt|Xt) = p(Yt|X (t),Y(t−1)), t ∈ N (2)

X1 X2 X3 X4

Y1 Y2 Y3 Y4

Figure : Directed graph of basic HMM. The index denotes time.


Markov Chains

Discrete state vector at time t, Xt, with m states.

Transition probabilityp(Xt = j|Xt−s = i) (3)

One-step transition probability

γij,t = p(Xt = j|Xt−1 = i) (4)

One-step transition probability matrix from time t − 1 to t

Γt =

γ11,t · · · γ1m,t...

. . ....

γm1,t · · · γmm,t

(5)

where the rows must sum to 1.


Generalized State Space Models (GSSM)

Parameter driven - model evolve independently of the pastobservation process.

p(yt|xt) = p(yt|xt,X (t−1),Y(t−1)

)(6a)

p(xt+1|xt) = p(xt+1|xt,X (t−1),Y(t)

)(6b)

Observation driven - model depends on the past observationprocess.

p(yt|xt) = p(yt|xt,X (t−1),Y(t−1)

)(7a)

p(xt+1|Y(t)

)= p

(xt+1|xt,X (t−1),Y(t)

)(7b)


Model ValidationLikelihood Setting

Forward SelectionMaximum of 5 states and model chosen by AIC

AIC = −2L+ 2p (8)

Forecast Pseudo Residuals

zt = Φ−1 (p (Yt ≤ yt|Y(t−1)))

(9)

Marginal distribution


DataIC-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 CO2 content

Indoor-Minutes Noise Avg DB Average noise level (5 min)

Indoor-Minutes Noise Peak DB Largest average of 3 subse-quent measurement (15 secper measurement)



Homogen HMMOriginal Scale

Setting

yt = CO2,t

p(xt|xt−1) ∼ Γ

p(yt|xt) ∼ N(µi, σ

2i)

for i = 1, 2, · · · ,m

Results

Table : Comparison of univariate homogen HMMs for 2 to 5 states.

L p AIC BIC2 states -97903 6 195818 1958643 states -91239 12 182502 1825954 states -87492 20 175023 1751785 states -83968 30 167995 168227


Homogen HMMOriginal Scale

Table : Fit of the HMM (CO2) with 5 states.

δi µi σi γi1 γi2 γi3 γi4 γi5

State 1 0.21 410.79 6.88 0.99 0.01 0.01 0.00 0.00State 2 0.18 432.00 7.42 0.02 0.94 0.04 0.00 0.00State 3 0.20 477.92 24.17 0.00 0.05 0.90 0.05 0.00State 4 0.23 618.45 68.26 0.00 0.00 0.05 0.92 0.03State 5 0.18 1013.61 267.72 0.00 0.00 0.00 0.04 0.96


Figure : Global Decoding of the HMM (CO2) with 5 states. Top is the entire time series. Bottom iszoomed in on one day. Vertical lines indicate 00:00. Horizontal lines indicate the state dependentmean.


Figure : Fit of the HMM (CO2) with 5 states.


Homogen HMMTransformed Scale

h(y) = log(y− 350)

Setting

yt = h(CO2,t)

p(xt|xt−1) ∼ Γ


2i)

for i = 1, 2, · · · ,m

Results

Table : Comparison of univariate (log transformed CO2) homogen HMMs for 2 to 5states.

L p AIC BIC2 states -9378 6 18768 188143 states -4292 12 8609 87014 states -800 20 1640 17955 states 2181 30 -4303 -4071


Figure : Fit of the HMM (log CO2) with 5 states.


Generalized FormHierarchical Model

p(yt|xt,θ)

p(xt+1|xt,θ

)Hierarchical model by random effects

Y|(U = u) ∼ fY|u(y; u,β

)(10a)

U ∼ fU(u;Ψ

)(10b)

The likelihood is given by

L(β,Ψ; y, u

)= f

(y;θ

)= f

(y, u;β,Ψ

)= fY|u

(y; u,β

)fU

(u;Ψ

)(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 fU

(u;Ψ

)and the conditional likelihood by fY|u

(y; u,β

).


Transition Probability Matrix with Covariates

Γt =

exp(XT

D,tβ11 + ZU11)

exp(XTD,tβ11 + ZU11) +

∑j6=1 exp(τ1j)

· · ·exp(τ1m)

exp(XTD,tβ11 + ZU11) +

∑j6=1 exp(τ1j)

.... . .

...exp(τm1)

exp(XTD,tβmm + ZUmm) +

∑j6=m exp(τmj)

· · ·exp(XT

D,tβmm + ZUmm)

exp(XTD,tβmm + ZUmm) +

∑j6=m exp(τmj)

(12)


Figure : Periodic Splines. The knots are indicated on the x-axis.


Inhomogen HMMTransformed Scale

Setting

yt = h(CO2,t)

p(xt|xt−1) ∼ Γt


2i)

for i = 1, 2, · · · ,m

Results

Table : Comparison of univariate (log transformed CO2) inhomogen HMMs for 2 to 5states.

L p AIC BIC2 states -9319 14 18667 187753 states -4229 24 8506 86924 states -742 36 1556 18355 states 2258 50 -4417 -4030

Same trend in residuals - not appropriate


Markov Switching AR(1)

p(xt+1|xt) = p(xt+1|xt,X (t−1),Y(t)

)(13a)

p(yt|xt, yt−1) = p(yt|xt,X (t−1),Y(t−1)

)(13b)

Xt−1 Xt Xt+1 Xt+2

Yt−1 Yt Yt+1 Yt+2

Figure : Directed graph of Markov switching AR(1).


Inhomogen Markov Switching AR(1)Transformed Scale

Setting

yt = h(CO2,t)

p(xt|xt−1) ∼ Γt

p(yt|xt, yt−1) ∼ N(ci + φiyt−1 , σ

2i)

for i = 1, 2, · · · ,m

Results

Table : Comparison of univariate (log transformed CO2) inhomogen AR(1) for 2 to 5states.

L p AIC BIC2 states 15845 16 -31659 -315353 states 16204* 27 -32354 -321454 states 16964 40 -33848 -335385 states 17336 55 -34561 -34136

Residuals - appropriate for 4 and 5 state model!!


Final Model - 5 statesResiduals

Figure : Model diagnostics of the final model.


Final ModelMarginal Distribution

Figure : Comparison of the distribution of the measured CO2 and the simulated distribution.


Final ModelInterpretation of the states

State 1: Absence or sleepingState 2: Long term absenceState 3: Outdoor interactionState 4: Presence (high activity)State 5: Presence (long term, low activity)


Figure : Transition probabilities over the day of the final model. The lower right plot isthe stationary distribution.


Figure : Profile of the states over the course of the day. I.e. Stackedstationary probabilities over the course of the day of the final model.


Figure : Global Decoding of the final model. Top is the entire time series. Bottom iszoomed in on one day. Vertical lines indicate 00:00. Horizontal lines indicate the statedependent mean.


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Modelling Non-linear and Non-stationary Time Series · 2016-09-28 · Modelling Non-linear and Non-stationary Time Series Chapter 7(extra): (Generalized) Hidden Markov Models Henrik