Hidden Markov models

1
Hidden Markov models

• Stat 518
• Sp08

2
A Markov chain model

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

3
(No Transcript)
4
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)
5
A spatial Markov model

• Three sites, A, B and C, each observing 0 or 1.
  Notation AB (A1,B1,C0)
• Markov model
• Great Plains data1949-1984 (Jan-Feb)

6
A hidden weather state

• Two-stage model
• Ct Markov chain, c states
• (RtCt,Rt-1,Ct-1,...,C1,R1) (RtCt)pt(Ct)
• We observe only R1,...,RT.
• C clusters similar rainfall patterns. In
  atmospheric science called a weather state

7
Likelihood

• C3, T100, CT 5.2 x 1047
• Forward algorithm unravel sum recursively

8
Computational algorithm

• Lystig (2001) Write

9
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 q.

10
Snoqualmie Falls

• Two-state hidden model
• Pr(rain) 0.059 (0.036) in state 1
• 0.941(0.016) in state 2

11
Survival function
1.00
0.10
HMM
0.01
MC
0
5
10
15
Dry period length (days)
12
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

13
Nonstationary transition probabilities

• Meteorological conditions may affect transition
  probabilities
• At-1 At
• Ct-2 Ct-1 Ct
• Rt-2 Rt-1 Rt

14
A model for Western Australia rainfall

• 19781987 (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)

15
Rain probabilities
16
Spatial dependence