Hidden Markov models Stat 518 Sp08. A Markov chain model January data from Snoqualmie Falls,...
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Transcript of Hidden Markov models Stat 518 Sp08. A Markov chain model January data from Snoqualmie Falls,...
A Markov chain model
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
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(Rt | Rt −1,...,R1) ~ pRt −1,Rt
0 5 10 15
Dry period length (days)
1.00
0.10
0.01
Survival function
S(t) = 1 – Pr(Dry period ≤ t)
Dry period ~ Geom(1/(1-p00))
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:
Great Plains data1949-1984 (Jan-Feb)
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P(Xt ≡ (XA,t ,XB,t ,XC,t ) = (i, j,k) | Xt −1 = (l,m,n),...,X1)
= plmn,ijk
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)=t(Ct)
We observe only R1,...,RT.
C clusters similar rainfall patterns. In atmospheric science called a weather state
Likelihood
|C|=3, T=100, |CT| = 5.2 x 1047
Forward algorithm: unravel sum recursively
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L(θ) = Lc∈CT
∑ p(ri | ci;θ)p(ci;θ)i=1
T∏∑
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α t (j) = Pr(R1,...,Rt ,Ct = j)
= α t −1(i)i=1
|C|∑ πt (j)pi,j
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L(θ) = αTj=1
|C|∑ (j)
Computational algorithm
Lystig (2001): Write
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L(θ) = Pr(R1,...,RT;θ) = Pr(Rt | Rt −1t =1
T∏ ,...,R1;θ)
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λ t (j) = Pr(Rt ,Ct = j | Rt −1,...,R1)
=λt −1(i)πt (j)pi,j
Λ t −1i=1
|C|
∑
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Λt = λt (j)j=1
|C|∑ = Pr(Rt | Rt −1,...,R1)
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l (θ) = logL(θ) = log Λ tt =1
T∑
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
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ˆ p 11 = 0.674 (0.030)
ˆ p 22 = 0.858 (0.016)
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est corr =
1 −0.40 0.24 −0.71
−0.40 1 −0.79 0.01
0.24 −0.79 1 0.16
−0.71 0.01 0.16 1
⎛
⎝
⎜ ⎜ ⎜ ⎜
⎞
⎠
⎟ ⎟ ⎟ ⎟
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
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logpij (t)
1− pij (t)
⎛
⎝ ⎜
⎞
⎠ ⎟ = α ij + βj
TAt
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)