Nonparametric inference in hidden Markov and related … · Nonparametric inference in hidden...

Nonparametric inference in hidden Markov and related models

Roland Langrock, Bielefeld University

Roland Langrock | Bielefeld University 1 / 47

Introduction and motivation


Figure: Haggis (the dish).


Figure: Wild Haggis (Dux magnus gentis venteris saginati).


Introducing hidden Markov models using Wild Haggis movement

simulated movement track

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Pr(St = j | St−1 = j) = 0.95 for j = 1, 2

where St : state at time t


Introducing hidden Markov models using Wild Haggis movement

simulated movement track

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0 5 10 15 20 25 30

0.0

0.1

0.2

0.3

0.4

step length distributions

step length

dens

ity

−3 −2 −1 0 1 2 3

0.0

0.2

0.4

0.6

0.8

turning angle distributions

turning angle

dens

ity

Pr(St = j | St−1 = j) = 0.95 for j = 1, 2,

where St : state at time t


Examples of HMM-type models/doubly stochastic processes

St−1 0St 0 St+1

Xt−1 0Xt 0 Xt+1

. . . . . . (hidden)

(observed)

hidden Markov models (HMMs)

(general) state-space models

Markov-switching regression

Cox point processes

In each case two components:

an observable state-dependent process1.) in animal movement: e.g. step lengths & turning angles2.) in financial time series: some economic indicator, e.g. GDP values3.) in disease progression: e.g. blood samples

a latent (nonobservable) state process/system processin 1.): behavioural statein 2.): the nervousness of the marketin 3.): the disease stage


Inference in HMM-type models

Why nonparametric?

1. specifying a suitable model can be hard — lots of ways to get it wrong!

2. more flexibility, perhaps leading to models that are more parsimonious,e.g. in terms of the number of states

3. as an exploratory tool

A strategy applicable in many scenarios combines

the simple yet powerful HMM machinery ...

... and the conceptual simplicity and general advantages of P-splines


1 Some basics on hidden Markov models

2 Nonparametric inference in hidden Markov models

3 Markov-switching generalized additive models

4 Concluding remarks


Some basics on hidden Markov models


HMMs — summary/definition

St−1 0St 0 St+1

Xt−1 0Xt 0 Xt+1

. . . . . . (hidden)

(observed)

two (discrete-time) stochastic processes, one of them hidden

distribution of observations determined by underlying state

hidden state process is an N-state Markov chain


Building blocks of HMMs

{St}t=1,2,...,T is (usually) assumed to be an N-state Markov chain:• state transition probabilities: γij = Pr(St = j |St−1 = i)• transition probability matrix (t.p.m.):

Γ =

γ11 . . . γ1N...

. . ....

γN1 . . . γNN

• initial state distribution: δ =

(Pr(S1 = 1), . . . ,Pr(S1 = N)

)

State-dependent distributions f (xt | st = j):• specify suitable class of parametric distributions• e.g. normal, Poisson, Bernoulli, multivariate normal, gamma, Dirichlet, ...• one set of parameters for each state


HMMs — likelihood calculation using brute force

L(θ) = f (x1, . . . , xT )

=N∑

s1=1

. . .

N∑sT =1

f (x1, . . . , xT , s1, . . . , sT )

=N∑

s1=1

. . .N∑

sT =1

f (x1, . . . , xT |s1, . . . , sT )f (s1, . . . , sT )

=N∑

s1=1

. . .N∑

sT =1

δs1

T∏t=1

f (xt |st )T∏

t=2

γst−1,st

Simple form, but O(TNT ), numerical maximiz. of this expression thus infeasible.


HMMs — likelihood calculation via forward algorithm

Consider instead the so-called forward probabilities,

αt (j) = f (x1, . . . , xt , st = j)

These can be calculated using an efficient recursive scheme:

α1 = δQ(x1)

αt = αt−1ΓQ(xt )

with Q(xt ) = diag(f (xt |st = 1), . . . , f (xt |st = N)

)

⇒ L(θ) =N∑

j=1

αT (j) = δQ(x1)ΓQ(x2) · . . . · ΓQ(xT )1

Computational effort: O(TN2) — linear in T !


Further inference — a brief overview

uncertainty quantification→ (parametric) bootstrap or Hessian-based

model selection→ criteria such as the AIC

model checking→ quantile residuals, simulation-based, ...

state decoding→ Viterbi algorithm


Related model classes

state-space models −→ can be approximated arbitrarily accurately byHMMs by finely discretizing the state space

Markov-switching regression models −→ HMMs with covariates

Markov-modulated Poisson processes −→ can be regarded as HMMs(with slightly modified dependence structure)

The corresponding likelihoods can be written as easy-to-evaluate matrix products!


Nonparametric inference in hidden Markov models


HMMs — motivation for a nonparametric approach

distribution of observations selected by underlying state

state-dependent distributions usually from a class of parametric distributions

finding the “right” distribution, or even a suitable one, can be difficult

an unfortunate choice can lead to ...• ... a poor fit and hence poor predictive power• ... a bad performance of the state decoding• ... invalid inference e.g. on the number of states

0 200 400 600 800 1000

−40

−20

020

40

observed time series

time

obse

rvat

ions

histogram of observations

observations

Fre

quen

cy

−60 −40 −20 0 20 40 60

020

4060

What family of distributions to use for the state-dependent process?


Nonparametric estimation based on P-splines

represent densities of state-dep. distributions using standardized B-splinebasis densities:

f (xt |st = i) =∑K

k=−K ai,kφk (xt )

transform constrained parameters ai,−K , . . . , ai,K :

ai,k =exp(βi,k )∑K

j=−K exp(βi,j )with βi,0 = 0

numerically maximize the penalized log-likelihood:

lp(θ,λ) = log(L(θ)

)−

[N∑

i=1

λi

2

K∑k=−K+2

(∆2ai,k

)2

]


Inference

identifiability holds under fairly weak conditions(essentially there needs to be serial correlation)

generalized cross-validation or AIC-type statistic for(i) choosing λ from N-dimensional grid(ii) model selection on the number of states

parameter estimation by numerical maximization of lp(θ,λ)

local maxima can be an issue→ use many different initial values in the maximization

uncertainty quantification via parametric bootstrap

model checking via pseudo-residuals (standard)

state decoding using Viterbi (standard)


A simple simulation experiment

simulate T = 800 observations from 2-state HMM

Γ =

(0.9 0.10.1 0.9

)

−60 −40 −20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

true densities of the state−dep. distributions

dens

ity




Γ =

(0.9 0.10.1 0.9

)

−60 −40 −20 0 20 40 60 80

0.00

00.

005

0.01

00.

015

0.02

0

marginal distribution of obs.

dens

ity




Γ =

(0.9 0.10.1 0.9

)K = 15, thus 2K + 1 = 31 B-spline basis functions

−60 −40 −20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04

true (black) and estimated densities of the state−dep. distributions

dens

ity

lambdas about right




Γ =

(0.9 0.10.1 0.9


−60 −40 −20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04


dens

ity

lambdas too big




Γ =

(0.9 0.10.1 0.9


−60 −40 −20 0 20 40 60 80

0.00

0.01

0.02

0.03

0.04


dens

ity

lambdas too small


Blainville’s beaked whale dive data−

6−

4−

20

24

observed time series

time in hours

log(

|dep

th d

ispl

acem

ent|

in m

eter

s)

10 20 30 40

histogram of the observations

log(|depth displacement| in meters)

Den

sity

val

ue

−8 −6 −4 −2 0 2 4 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

sample ACF

lag

AC

F


Blainville’s beaked whale — parametric HMMs

Table: Results of fitting HMMs with normal state-dependent distributions.

#states p AIC BIC

3 12 9784.00 9855.594 20 9498.16 9617.475 30 9400.30 9579.276 42 9294.88 9545.437 56 9208.04 9542.118 72 9129.15 9558.679 90 9090.98 9627.8710 110 9064.53 9720.74


Blainville’s beaked whale — parametric HMM, N = 7

fitted state−dependent distributions (3−state parametric HMM)

log(absolute depth displacement)

Den

sity

−6 −4 −2 0 2 4 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

state 1state 2state 3state 4state 5state 6state 7marginal

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−3 −2 −1 0 1 2 3

−3

−2

−1

01

23

qq−plot of residuals against standard normal

quantiles of the standard normal

sam

ple

quan

tiles

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0

sample ACF for series of residuals

lagA

CF


Blainville’s beaked whale — parametric HMM, N = 3

fitted state−dependent distributions (3−state parametric HMM)


Den

sity

−6 −4 −2 0 2 4 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

state 1state 2state 3marginal

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−3 −2 −1 0 1 2 3

−3

−2

−1

01

23



sam

ple

quan

tiles

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0


lagA

CF


Blainville’s beaked whale — nonparametric HMM with N = 3

fitted state−dependent distributions (3−state nonparametric HMM)


Den

sity

−6 −4 −2 0 2 4 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

state 1state 2state 3marginal

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−3 −2 −1 0 1 2 3

−3

−2

−1

01

23



sam

ple

quan

tiles

0 5 10 15 20 25 30

0.0

0.2

0.4

0.6

0.8

1.0


lagA

CF


Blainville’s beaked whale — Viterbi for nonparametric HMM with N = 3

time in hours

−4

−2

0

2

4

2 4 6 8

state 3

state 2

state 1

1200

1000

800

600

400

200

0

● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●

●●●●●●●●●● ●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●● ●● ●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●● ●●●●●●●●●

●●●●●●●●●●●● ●●●● ● ●●●●●●●●●●● ●●●●●●●●●●● ●●●●●●●●●●●●●●●● ●●●●●●●● ●●● ●●●● ●●●●●●●●●●●●●● ●●●●●●●●●● ●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●

d

epth

s in

met

ers

l

og(|

dept

h di

spla

cem

ent|

in m

eter

s) d

ecod

ed s

tate

s


Markov-switching generalized additive models


Markov-switching regression — a basic model

A simple Markov-switching (linear) regression model:

Yt = β(st )0 + β

(st )1 xt + σst εt ,

with

a time series {Yt}t=1,...,T

associated covariates x1, . . . , xT (including the possibility of xt = yt−1)

εtiid∼ N (0, 1)

st : state at time t of an unobservable N-state Markov chain


Markov-switching regression — remarks on the basic model

commonly used in economics to deal with parameter instability over time(key references: Goldfeld and Quandt, 1973; Hamilton, 1989)

linear form of the predictor is usually assumed with little investigation (if any!)into the absolute or relative goodness of fit

we consider nonparametric methods for estimating the form of the predictor(in analogy to the extension of linear models to GAMs)


Markov-switching regression — more general model formulation

More general model formulation:

g(E(Yt | st , x·t )︸︷︷︸

µ(st )t

)= η(st )(x·t ),

where

Yt follows some distribution from the exponential family

x·t = (x1t , . . . , xPt ) is the covariate vector at time t

g is a suitable link function

η(st ) is the predictor function given state st

(the form of which we do not yet specify)

(φ(st ): any additional state-dependent dispersion parameters)


Likelihood evaluation using the forward recursion

Define, analogously as for HMMs, the forward variable

αt (j) = f (y1, . . . , yt ,St = j | x·1 . . . x·t )

Then the following recursive scheme can be applied:

α1 = δQ(y1) , αt = αt−1ΓQ(yt ) (t = 2, . . . ,T )

whereQ(yt ) = diag

(pY (yt ;µ

(1)t , φ(1)), . . . , pY (yt ;µ

(N)t , φ(N))

)

⇒ L(θ) =N∑

j=1

αT (j) = δQ(x1)ΓQ(x2) · . . . · ΓQ(xT )1

This form applies for any form of the conditional density pY (yt ;µ(st )t , φ(st ))


Nonparametric modelling of the predictor

here we consider a GAM-type framework:

η(st )(x·t ) = β(st )0 + f (st )

1 (x1t ) + f (st )2 (x2t ) + . . .+ f (st )

P (xPt ),

we represent each f (i)p as a linear combination of B-spline basis functions:

f (i)p (x) =K∑

k=1

γipk Bk (x)

... and numerically maximize the penalized log-likelihood:

lp(θ,λ) = log(L(θ)

)−

N∑i=1

P∑p=1

λip

2

K∑k=3

(∆2γipk )2

inference analogous as for nonparametric HMMs

notably, parametric models are nested special cases (for λ→∞)



simulate T = 300 observations from 2-state Markov-switching regr. model:

Yt ∼ Poisson(eβ0+f (st )(xt )), Γ =

(0.9 0.10.1 0.9

)

−3 −2 −1 0 1 2 3

−6

−4

−2

02

46

xt

f(st) (x

t)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)





(0.9 0.10.1 0.9


smoothing parameter selection from a grid using AIC-type statistic

−3 −2 −1 0 1 2 3

−6

−4

−2

02

46

xt

f(st) (x

t)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)





(0.6 0.40.4 0.6


smoothing parameter selection from a grid using AIC-type statistic

−3 −2 −1 0 1 2 3

−6

−4

−2

02

46

xt

f(st) (x

t)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)

st=1 (state 1)

st=2 (state 2)


Example Lydia Pinkham sales1.

01.

52.

02.

53.

03.

5

year

sale

s (in

mill

ion

US

D)

1910 1920 1930 1940 1950 1960


Example Lydia Pinkham sales

Model MS-LIN: salest = β(st )0 + β

(st )1 advertisingt + β

(st )2 salest−1 + σst εt

Model MS-GAM: salest = β(st )0 + f (st )(advertisingt ) + β

(st )1 salest−1 + σst εt

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Advertising

Sal

es

Figure: Estimated state-dependent mean sales as functions of advertising expenditure(state 1 in green, state 2 in red). Displayed are the predictor values when fixing theregressor salest−1 at its overall mean, 1.84.



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Figure: Sales figures and decoded states underlying the MS-GAM model.



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Concluding remarks


Concluding remarks

bringing together HMMs & P-splines gives lots of modelling options

while inference is slightly more involved, resulting models often substantiallyincrease the goodness of fit, and may in fact be more parsimonious thanparametric alternatives

various other such models can be formulated (and fitted), e.g. MS-GAMLSSmodels — but does anyone need this kind of thing??

we’re currently working on alternative, less computer-intensive methods forselecting the smoothing parameters


References

Langrock, R., Kneib, T., Sohn, A., DeRuiter, S. (2015), “Nonparametricinference in hidden Markov models using P-splines”, Biometrics

Langrock, R., Glennie, R., Kneib, T., Michelot, T. (2016). “Markov-switchinggeneralized additive models”, Statistics and Computing

Thank you!


Nonparametric inference in hidden Markov and related … · Nonparametric inference in hidden...

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