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ORIGINAL   DISTORTED  

Corner  handles  

Good

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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

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