Docstoc

HMM

Document Sample
HMM Powered By Docstoc
					Hidden Markov models

       Stat 518
        Sp08
    A Markov chain model
   (Rt | Rt  1,...,R1 ) ~ pRt  1,Rt

January data from Snoqualmie Falls,
Washington, 1948-1983
325 dry and 791 wet days

                    Today wet   Today dry
    Yesterday wet   643 (543)   128 (223)   771
    Yesterday dry   123 (223)   186 (91)    309
                       766         314      1080
            Survival function

S(t) = 1 – Pr(Dry period ≤ t)
Dry period ~ Geom(1/(1-p00))
     1.00



     0.10



     0.01




            0       5          10          15

                Dry period length (days)
          A spatial Markov model

      Three sites, A, B and C, each observing
      0 or 1. Notation: AB = (A=1,B=1,C=0)
      Markov model:
     P(X t  (X A,t ,X B,t ,X C,t )  (i,j,k) | X t  1  (l,m,n),...,X 1 )
            plmn ,ijk


      Great Plains data1949-1984 (Jan-Feb)
        Dry     A        B       C        AB      AC       BC      ABC
Obs     718     1020 1154 957             866     752      728     657
MC      722     942      1076 1031 789            750      727     655
    A hidden weather state

Two-stage model
      Ct Markov chain, c states
  (Rt|Ct,Rt-1,Ct-1,...,C1,R1) = (Rt|Ct)=pt(Ct)
We observe only R1,...,RT.
C clusters similar rainfall patterns. In
atmospheric science called a weather
state
                Likelihood
                      T
 L()              p(ri | ci;)p(ci;)
            cCT     i 1
|C|=3, T=100, |CT| = 5.2 x 1047
Forward algorithm: unravel sum
recursively
  t (j)  Pr(R1,...,Rt ,Ct  j)
          |C|
          t  1 (i)p t (j)pi,j
         i 1
         |C|
 L()    T (j)
         j 1
      Computational algorithm

   Lystig (2001): Write
                                T
L()  Pr(R1,...,RT ;)   Pr(Rt | Rt  1,...,R1;)
                            t 1

       t (j)  Pr(Rt ,Ct  j | Rt  1,...,R1 )
                |C| 
                      t  1 (i)p t (j)pi,j
               
                i 1         t  1
             |C|
       t    t (j)  Pr(Rt | Rt  1,...,R1 )
            j 1
                            T
       ()  logL()   log  t
                           t 1
 Estimating standard errors

The Lystig recursions enable easy
calculation of first and second
derivatives of the log likelihood, which
can be used to estimate standard
errors of maximum likelihood estimates
of .
        Snoqualmie Falls

Two-state hidden model
Pr(rain) = 0.059 (0.036) in state 1
           0.941(0.016) in state 2
     p11  0.674 (0.030)
     ˆ
    p 22  0.858 (0.016)
    ˆ

            1    0.40 0.24 0.71
           
            0.40    1   0.79 0.01 
est corr                         
            0.24 0.79   1   0.16 
                                  
           0.71 0.01 0.16     1 
       Survival function

1.00



0.10

                                      HMM
0.01
                                            MC



       0       5          10          15

           Dry period length (days)
             The spatial case

  MC: 8 states, 56 parameters
  HMM: 2 hidden states (one fairly wet,
  one fairly dry), 8 parameters, rain
  conditionally independent at different
  sites given weather state

      Dry   A    B    C    AB    AC    BC    ABC
Obs   718   1020 1154 957  866   752   728   657
HMM   725   1019 1153 956  862   749   728   657
MC    722   942  1076 1031 789   750   727   655
  Nonstationary transition
       probabilities
Meteorological conditions may affect
transition probabilities

           At-1          At

    Ct-2          Ct-1              Ct

 Rt-2             Rt-1              Rt

     pij (t) 
 log              ij   T At
     1 pij (t)
             
                             j
        A model for
  Western Australia rainfall
1978–1987 (1992) winter (May– Oct)
daily rainfall at 30 stations
Atmospheric variables in model: E-W
gradient in 850 hPa geopotential height,
mean sea level pressure, N-S gradient
in sea-level pressure
Final model has six weather states (BIC
and other diagnostics)
Rain probabilities
Spatial dependence

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:0
posted:4/13/2013
language:Unknown
pages:16