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Longitudinal multicategorical processes :
Generalized state space models
Sadeq Awad Kadhim
Work joint with Professeur: Joseph Ngatchou-Wandji
Institut ÉLIE CARTAN (IECL)
Universite de Lorraine.
3-7 Avril 2017
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Longitudinal multicategorical processes : Generalized state space models
Introduction
1
We are interested to latent variables Xi (t) produced by an
individual i, (i = 1, · · · , n), at time t, (t = 1, · · · , T ).
The Xi (t)0 s may be the patients health, a latent
trait,ect....
2
We only observe Y (t) instead of Xi (t).
The Y (t)0 s may be the responses of the individuals in the
interviews typically involve filling out a questionnaire in
which they are asked multiple choice questions.
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
what does it mean?
t
1
d/m/y
1
1
i
Q1 · · · Qq
··· ,
··· ,
..
.
Q1 · · · Qq
Q1 · · · Qq
Q1 · · · Qq
··· ,
··· ,
..
.
··· ,
..
.
n
d/m/y
T
Q1 · · · Qq
Q1 · · · Qq
Q1 · · · Qq
..
.
··· ,
..
.
··· ,
d/m/y
Q1 · · · Qq
..
.
··· ,
Q1 · · · Qq
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Example to questionnaire
The STAI (State-Trait Anxiety Questionnaire)
Non
Plutôt non
Plutôt oui
Oui
1- je me sens calme
2-je me sens en sécurité
3- je me sens tendue
4- je me sens surmenée
5- je me sens tranquille,bien dans ma peau
6- je me sens bouleversée
7- je me sens préoccupée par tout les malheurs possibles
8- je me sens comblée
9- je me sens effrayée
10- je me sens à l’aise
11- je me sens que j’ai confiance en moi
12- je me sens nerveuse
13- je suis affolée
14- je me sens indécise
15- je suis détendue
16- je me sens satisfaite
17- je suis préoccupée
18- je ne sais plus où j’en suis, je me sens
perturbée
19- je me sens solide, posée
20- je me sens de bonne humeur,aimable
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The Path
1
c
c
1
1
1 } · · · , {Y
k
{Y11,t · · · Y11,t
1k,t · · · Y1k ,t }
|
{z
}
|
{z
}
↑ ······ ↑
|
{z
}
item 1 · · ·
|
↑ ······ ↑
|
{z
}
{z
↑
X1,t
item k
}
c
1
1 }··· ,
{Yn1,t · · · Yn1,t
|
{z
}
···
···
↑ ······ ↑
|
{z
}
item 1 · · ·
|
c
{Ynk,t · · · Ynkk,t }
|
{z
}
↑ ······ ↑
|
{z
}
{z
item k
}
↑
Xn,t
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Goal and methods
Goal: Estimate the latent variables (the Patients health,
latent trait,ect....).
Methods:
1
2
3
Using state space models, we estimate the latent variables
by posterior mode via the working extended Kalman
filtering recursions.
We find the posterior distribution p[Xi,t | Yik,t ] through the
Auxiliary Iterated Extended Kalman Particle filter (AIEKPF)
algorithm.
The model parameters are estimated by the Maximum
Likelihood method (MLE) via the EM algorithm.
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The model
State space models are characterized by a state equation and
an observation equation.
The observation equation:
for an individual i and item k at time t, one has:
ck
1
p(Yik,t = (yik,t
, · · · yik,t
) | Xi,t = xi,t ) =
ck h
Y
s
πik,t
iy s
ik ,t
,
(1)
s=1
πiks ,t =
exp[ηiks ,t ]
Pck
j
1 + j=1 exp[ηik,t
]
πikck,t =
where
s
ηik,t
=
s )
logit(πik,t
if s < ck ,
1
1+
j
j=1 exp[ηik,t ]
Pck
= log
s
πik,t
c
k
πik,t
,
= u0i,t βks + Xi,t ,
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The model
The state equation:
Xi,t = F [Xi,t−1 , ui,t , γ] + H[Xi,t−1 , ui,t , δ] × εi,t ,
(2)
Where:
1
(ui,t ) is covariate variable , ui,t ∈ R r .
2
γ, δ are the parameters model .
3
(εi,t ) is the noise for state process has an exponential
family distribution,with E[εi,t ] = 0, Var [εi,t ] = R 2 . R is a
positive scalar parameter.
4
F (.) : R × Rl × Rr −→ R, is a non-linear function.
5
H(.) : R × Rl × Rr −→ R, is non-linear function.
0
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The model
We assume that the latent process (Xi,t : 1 ≤ t ≤ T ) satisfies.
p(Xi,t | Xi,t−1 , Xi,t−2 , · · · , Xi,1 ) = p(Xi,t | Xi,t−1 ) ∼ expf (υi,t , φi,t ).
The joint law of (Xi,1 , · · ·n, Xi,T ) h: is given by
io
PT
Xi,t υi (t)−b[υi,t ]
gi (Xi,1 , · · · , Xi,T ) = exp
+
c[X
,
φ
]
.
i,t
i,t
t=1
φi,t
where
µi,t = E[Xi,t | Xi,t−1 ] = F [Xi,t−1 , ui,t , γ]
Vi,t = Var [Xi,t | Xi,t−1 ] = H 2 [Xi,t−1 , ui,t , δ]R 2
υi (t) is a cononical parameter or the link function, where
υi (t) is a function of µi (t) .
φi (t) is a dispersion or a scale parameter.
b[υi (t)], c[z, φi (t)] are a functions taking a different form
depending on the distribution of the Xi (t).
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Estimating the parameters
We use the EM algorithm to estimate the parameters.
At the step p, p = (0, 1, 2, · · · , ), θ(p) = (β (p) , γ (p) , δ (p) )0 , then at
p+1:
s(p+1)
βk
=
n X
T Z
X
Z n
o
s
s
···
u0i,t Dik ,t Σ−1
[y
−
π
]
ik,t
ik,t ik ,t
i=1 t=1
×P(Xi,t | Yik,t , θ(p) )d(Xi ).
γ (p+1)
=
n Z
X
i=1
(3)
)
Z (X
T
[Xi,t − µi,t ] ∂µi,t
···
Vi,t
∂γ
t=1
(p)
×P(Xi,t | Yik ,t , θ
)d(Xi ).
(4)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Estimating the parameters
n
δ (p+1)
=
1X
2
Z
···
"
Z (X
T
−1
i=1
×P(Xi,t | Yik ,t , θ
where
s
Dik,t
=
t=1
(p)
s
∂πik,t
s
∂ηik,t
Vi,t
[Xi,t − µi,t ]2
+
2
Vi,t
)d(Xi ).
#
∂Vi,t
∂δ
)
(5)
s
s
= πik,t
[1 − πik,t
]
Σik,t = cov (Yik,t )
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The AIEKPF algorithm
We can implementing to an individual , for m = 1, · · · , N
1
Initialization (t = 0), draw the states (particles)
m ∼ expf (υ m , φm )
Xi,0
i,0 i,0
m
m
2
generate µm
i,t ∼ expf (υi,t , φi,t ).
3
Update the particles using
h the IEKF algorithm.
i
m
s
m
m
m , and
4
wi,t = q(m | Yik,1:t ) ∝ M πik (ui,t , µi,t ) wi,t−1
m =
normalize weights wi,t
5
6
m
wi,t
PN
.
m
wi,t
m
ςi of
m=1
Resample to obtain the index
particle m’s parent.
Calculate importance weights of particles
m
wi,t
=
m]
p[Yik,t | Xi,t
ςm
p[Yik ,t | µi,ti ]
=
m , um ]
M[πiks (xi,t
i,t
ςm
M[πiks (µi,ti , um
i,t ]
Output: a set of weighted particles (samples) to an
m , w m }N
individual [{Xi,t
i,t m=1 ], i = 1, · · · , n.
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The posterior mode
The posterior distribution of Xt by Bayes’ theorem
( n T q
)
YY Y
1
p(Xt | YT ) =
p(Yik (t) | Xi (t))
p(YT )
i=1 t=1 k=1
( n T
)( n
)
YY
Y
×
gi (Xi (t))
gi (Xi (0))
i=1 t=1
(6)
i=1
p(YT ) does not depend on Xt
( n T q
)
YY Y
p(Xt | YT ) '
p(Yik (t) | Xi (t))
×
i=1 t=1 k=1
( n T
YY
)( n
)
Y
gi (Xi (t))
gi (Xi (0))
i=1 t=1
(7)
i=1
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The posterior mode
The posterior mode smoother is given by
0
0
a ≡ (a0|T
, a1|T
, · · · , aT0 |T )0 = arg max{PL(Xt )},
Xt
q X
ck n
n X
T X
o
X
s
s
PL(Xt ) =
yik,t
log πik,t
+ G1 + G2
(8)
i=1 t=1 k=1 s=1
G1 =
G2 =
n X
Xi,0 υi,0 − b[υi,0 ]
i=1
n X
T X
i=1 t=1
φi,0
+ c[Xi,0 , φi,0 ]
Xi,t υi,t − b[υi,t ]
+ c[Xi,t , φi,t ]
φi,t
We estimate the posterior mode by apply Working
Extended Kalman filtering recursions.
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The Working Extended Kalman Filtering Recursions
Initialization:
ai,0|0 = ai,0 ,
Pi,0|0 = Vi,t .
Prediction
for t = 1, · · · , T
ai,t|t−1 = F (ai,t−1|t−1 , ui,t , γ)
0
Pi,t|t−1 = Ai,t Pi,t−1|t−1 A0i,t + Ti,t R 2 Ti,t
.
Since
A0i,t =
∂F (Xi,t , ui,t , γ)
|Xi,t =ai,t−1|t−1 .
∂Xi,t
Ti,t = H(xi,t−1|t−1 , ui,t , δ)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
The Working Extended Kalman Filtering Recursions
Filtering for t = 1, · · · , T
ai,t|t
Ki,t
Pi,t|t
ei,t − ai,t|t−1 )
= ai,t + Ki,t (Y
−1 −1
0
0
= Pi,t−1|t−1 Bi,t
(Bi,t Pi,t−1|t−1 Bi,t
+ Wi,t
)
= (I − Ki,t Bi,t )Pi,t−1|t−1
where
Bi,t =
∂πi,t
|X =a
.
∂Xi,t i,t i,t|t−1
e
e
with ”working” observation Yf
i,t = (Y1,t , · · · , Yn,t ) where
ei,t
Y
Wi,t
=
n
o0 −1
Di,t
Yi,t − πi, ) + ηi,t ,
0
= Di,t Σ−1
i,t Di,t .
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
plan of simulation
1
2
3
4
Suppose we have (n = 10) individuals, we asked them for
each instant (t) questions (q = 5) for one question has
(ck = 6) categories.
We simulate the latent variables Xi,t by a state equation
(GARCH model ) with noise processes from exponential
family.
We simulate 3 covariate variables ui,t :Age, Sex, Length.
We suppose initial value for parameters βks , and calculate
s =
the probabilities πik,t
s ]
exp[ηik
,t
P ck
j
1+ j=1 exp[ηik ,t ]
if s < ck , where
s
ηik,t
= u0i,t βks + Xi,t ,
5
6
We simulate Yik,t ∼ M(πiks ,t )
We apply the Working Extended Kalman Filtering
Recursions to estimate the latent variables by posterior
mode.
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Sumlation
Figure: Xi,t = ρXi,t + εi,t , εi,t ∼ N(0, 1)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Sumlation
Figure: Xi,t = ρXi,t + εi,t , εi (t) ∼ Student 0 s t(ν = 5)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Sumlation
Figure: Xi,t = ρXi,t + εi,t , εi (t) ∼ exp(λ = 1)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Sumlation
Figure: Xi,t =
q
2
ρ1 + ρ2 Xi,t−1
∗ εi,t , εi,t ∼ N(0, 1)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Sumlation
Figure: Xi,t = ρ1 Xi,t−1 +
q
2
ρ1 + ρ2 Xi,t−1
∗ εi,t , εi,t ∼ N(0, 1)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
Sumlation
2
Figure: Xi,t = {(ρ1 + ρ2 )e−ρ3 Xi,t−1 }Xi,t−1 + εi,t , εi,t ∼ t(ν = 5)
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
References
1
Genshiro Kitagawa (2010)”Introduction to Time Series
modeling”, chapman and Hall/CRC , USA.
2
Ludwig Fahrmeir and Stefan Wagenfeil (1997)”Penalized
likelihood estimation and iterative Kalman smoothing for
non-Gaussian dynamic regression models”. Computational
Statistics and data analysis 24,pp.295-320.
3
Moussedek Bousseboua and Mounir Mesbah”Processus
de Markov longitudinal latent Rasch”, Pub.Inst.
Stat.Univ.Paris, LIV, fasc. 1-2, 2010, 35 Ã 50 .
4
Xi, Y., Peng, H., Kitagawa, G. & Chen, X. (2015)”The
auxiliary iterated extended Kalman particle filter”, Optim
Eng 16, pp:387-407.
Longitudinal multicategorical processes : Generalized state
Longitudinal multicategorical processes : Generalized state space models
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Merci de votre attention
Longitudinal multicategorical processes : Generalized state