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ORIGINAL DISTORTED
Corner handles
Good
prin
/ng qu
ality
Bad prin/n
g qu
ality
We propose a simple hierarchical infinite HMM (iHMM) model, an extension to iHMM with efficient inference scheme. The model can capture dynamics of a sequence in two /mescales and does not suffer from the problems of other related models in terms of implementa/on and /me complexity. We use the model to analyze the dynamics in two /mescales of some synthe/c and real physiological data. We show that the model performs reasonably well compared to a baseline on two physiological datasets.
ABSTRACT MODEL A generaliza/on of iHMM where the transi/on probability is a mixture of:
1. a state-‐dependent transi/on probability distribu/on which resembles the transi/on probability in an iHMM
2. a state-‐independent probability distribu/on The mixture component is sampled from a Bernoulli distribu/on with a parameter that depends on both the hidden state and the observa/on.
REFERENCES
o Time series with mul/ple /mescales appear in domains where there is a hierarchical structure; for instance, in natural language, handwri/ng and mo/on recogni/on.
o iHMM and its variants have been successfully applied to ... Ø Speech recogni/on Ø Document modeling Ø Biology Ø Corporate bond ra/ng
o Their applica/on to /me series with mul/ple /mescales has been limited. The reasons are mainly:
Ø Inefficient inference Ø Complex implementa/on
1Computer Science and Ar/ficial Intelligence Laboratory, MIT 2 Media Lab, MIT 3 Adobe Research
Ardavan Saeedi1, Asma Ghandeharioun2, Ma` Hoffman3
A simple hierarchical infinite HMM with efficient inference
o Defining a Bayesian nonparametric model for /me series with dynamics at two /me scales
o Proposing an efficient stochas/c varia/onal inference scheme
o Applying the model to physiological data and performing reasonably well compared to a baseline.
Katherine A Heller, Yee W Teh, and Dilan Go r̈u ̈r. Infinite hierarchical hidden markov models. In Inter-‐ na:onal Conference on Ar:ficial Intelligence and Sta:s:cs, pages 224–231, 2009. MaChew Johnson and Alan Willsky. Stochas/c varia/onal inference for bayesian /me series models. In Proceedings of the 31st Interna:onal Conference on Machine Learning (ICML-‐14), pages 1854–1862, 2014. Thomas S Stepleton, Zoubin Ghahramani, Geoffrey J Gordon, and Tai S Lee. The block diagonal infinite hidden markov model. In Interna:onal Conference on Ar:ficial Intelligence and Sta:s:cs, pages 552– 559, 2009.
MOTIVATION
CONTRIBUTION
The generaBve descripBon
1. Generate the transi/on probability matrix according to the genera/ve process of iHMM
2. At /me step t, given a hidden state, generate an observa/on from a condi/onal observa/on distribu/on.
3. Sample a segmenta/on variable from a Bernoulli distribu/on with a parameter which depends on both the hidden states and the observa/ons
4. Condi/oned on the segmenta/on variable, either sample the next state from a state-‐dependent distribu/on or ignore the current state and sample from a distribu/on . ⇡0
zt hidden state at time tyt observation at time tst segmentation variable at time tH prior distribution over �F (�zt) the observation distribution
�zt the parameter corresponding to ztGEM(�) the stick-breaking distribution
with parameter �↵ parameter of the DP
!yt observation feature weight
!zt hidden state feature weight
A SIMPLE ILLUSTRATION ON TOY DATASET
o A toy dataset with 15000 data points from 3 different transi/on matrix each with 2 hidden states.
o The goal is to find the points where we have switched from one regime to another one and also the dynamics within each segment.
True segments
Inferred segments
STOCHASTIC VARIATIONAL INFERENCE (SVI)
o Truncated SVI is used for inference; the posterior is approximated with mean field family distribu/on.
o We maximize the marginal likelihood lower bound:
by using stochas/c natural gradient ascent over the global factors and standard mean field updates for the local factors (z and s). o Minibatch of M sequences for upda/ng local factors o Global factors are updated by taking a step of size in
the approximate natural gradient direc/on.
L , Eq
p(z, s,�,!,⇡,�,y)
q(z, s)q(�)q(!)q(⇡)q(�)
�
⇢
VariaBonal factors
o “Direct assignment” trunca/on with trunca/on level K (Johnson & Willsky 2014) for z and s:
If for any to we have and . o Point es/mate for , and : o For , we assume the prior is: Due to conjugacy the op/mal varia/onal factor is in the form of with parameter . o For , we assume the prior is in exponen/al family and
conjugate for the likelihood func/on . Hence,
q(z1:T , s1:T ) = 0z1 zT zt = k k > K
�q(�) = ��⇤(�)
!y !z
q(!y) = �!⇤y(!y) q(!z) = �!⇤
z(!z)
⇡i
p((⇡i1, . . . ,⇡iK ,⇡i,rest)) = Dir(↵�1, . . . ,↵�K ,↵�rest)
⇡i,rest = 1�PK
k=1 ⇡k �rest = 1�PK
k=1 �k
Dir(↵̃i) ↵̃�
f(yt|�)q(�i) / exp{h⌘̃i, t�(�i)i}
SVI update equaBons
o For the expecta/ons with respect to :
o The update for the parameters of the global varia/onal factors:
Where and are expected sufficient sta/s/cs with respect to .
q(z1:T , s1:T )
F (zt, st) , f(yt|�zt)p(st|zt, yt)X
zt�1,st�1
F (zt�1, st�1)p(zt|st�1, zt�1);
B(zt, st) ,X
zt+1,st+1
B(zt+1, st+1)f(yt+1|�zt+1)p(st+1|zt+1, yt+1)p(zt+1|st, zt)
⌘̃i (1� ⇢)⌘̃i + ⇢(⌘i +m.t̃iy)
↵̃i (1� ⇢)↵̃i + ⇢(↵i +m.t̃itrans)
↵̃0 (1� ⇢)↵̃0 + ⇢(↵0 +m.t̃0trans).
t̃itrans t̃iyq(z1:T , s1:T )
�i ⇠H;
� ⇠ GEM(�); ⇡i ⇠ DP(↵�);
z1 ⇠ ⇡0; yt|zt ⇠ F (�zt);
st|zt, yt ⇠ Bern(�(!yyt + !zzt));
zt+1|zt, st ⇠ ⇡1�st0 ⇡st
zt ,
RESULTS
o Electrodermal ac/vity (EDA) refers to changes in electrical proper/es of the skin caused by sudomotor innerva/on. It is an indica/on of physiological or psychological arousal and has been u/lized to objec/vely sleep quality.
o We use two datasets of sizes 12000 and 32000 and split them into sequences of size 1000. In both datasets, we normalize the EDA values and use batch size and heldout size of two. RELATED MODELS
o Infinite hierarchical HMM (Heller et al. 2009) o The block-‐diagonal iHMM (Stepleton et al. 2009) o In contrast, our model is much simpler and easier to
implement inference for. It can also discover transi/on matrices with approximately block-‐diagonal structure; the segmenta/on events provide a mechanism for transi/oning from one group of connected states to another.
0 2000 4000 6000 8000 10000 12000�2.5�2.0�1.5�1.0�0.5
0.00.51.01.52.0
0 2000 4000 6000 8000 10000 12000t
�2.5�2.0�1.5�1.0�0.5
0.00.51.01.52.0
True
segm
ents
Segm
enta/o
n
HDP-‐HM
M
S/cky HD
P-‐HM
M
