A Markov chain model for juvenile salmon

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A Markov chain model for juvenile salmon

• E. A. Steel and P. Guttorp (2001) Modeling
  juvenile salmon migration using a simple Markov
  chain. Journal of Agricultural, Biological and
  Environmental Statistics 6 80-88.
• Scientific issue As few as 15 of hatchery
  salmon survive to the first dam.
• Need to understand fish movement and the role of
  covariates, such as river speed
• Data radio tags at 129 yearling chinook in Snake
  River read at 12 receiving stations
• Travel time calculated at each segment (between
  stations). 7 31 observations/segment
• Missing data due to signal strength, antenna
  orientation, tag failure

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The model

• Each fish make 10 decisions per hour (to move 1km
  or to stay)
• It is observed after it has traveled Li km.
• A wait time is defined as a 1-0-0-0...-0-1
  transition. The expected value and variance can
  be computed as a function of the transition
  probabilities.
• Total travel time for a segment is the sum of the
  wait times (independent)

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Estimated parameters
Stretch Obs Length p00 p11
1 31 41 .988 .992
4 16 25 .946 .947
7 21 6 .973 .886
10 20 1 .9996 .932
11 20 7 .9999 .989
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Model intepretation

• Long runs of staying or of moving
• Implication for time spent moving and staying?
• Fish behavior different in different parts of the
  river.
• Confounded with river speed. Length of movement
  can be made depend on average speed. Clearer
  differences between different parts of river,
  higher precision of estimates.

5
Tornado model

• C. Marzban, M. Drton and P. Guttorp (2003) A
  Markov chain model of tornadic activity. Monthly
  Weather Review 131 2941-2953.
• Scientific issue Tornado prediction
• Data 49 years of daily indicators of occurrence
  of a tornado in continental US
• Varies with time of year

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Time-dependent transition probabilities
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Tornado alley
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Regional differences
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Why is it so?

• Frontal systems stay in a region for several
  days, conducive to tornado activity. So then p11
  gt p01.
• In southern Tornado alley frontal systems cease
  around mid-May, decreasing p11, but p01 continues
  to increase for another month due to lots of
  moisture and weak upper atmosphere systems
• SE tornado activity related also to tropical
  storms, so lasts longer, less pronounced peaks

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Quality of forecast
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Precipitation modeling

• J. P. Hughes and P. Guttorp (1994) Incorporating
  spatial dependence and atmospheric data in a
  model of precipitation. Journal of Applied
  Meteorology 33 1503-1515. IPCC SAR.
• Scientific problem Downscaling climate models to
  model regional precipitation

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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)

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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

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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

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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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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

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Rain probabilities
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Blood production in animals

• J. L. Abkowitz, S. N. Catlin, and P. Guttorp
  (1996) Evidence that hematopoiesis may be a
  stochastic process in vivo. Nature Medicine 2
  190-197
• Scientific problem Understanding how stem cells
  for blood production work
• Stem cells are not identifiable except by function

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Hematopoiesis model
Stem cells
Contributing clones
Symmetric division
Specialization
Exhaustion
Asymmetric division
Apoptosis
Observations
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Observation model
Niche hypothesis
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A cat experiment

• Female Safari cats (mix of Geoffroy and domestic)
• Autologous bone marrow transplant
• Smallish number of stem cells replaced
• X-chromosome linked enzyme G6PD
• electrophorically distinguishable
• genetically neutral
• binary marker for phenotype
• tracks contribution of stem cell

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Some data
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A Markov chain Monte Carlo approach

• Want p(?y)
• Marginalize p(?,x0,Ty)
• Outer step (parameter update)
• Draw ? from p(?x0,T,y)
• Gibbs sampler
• Inner step (state update)
• Draw x0,T from p(x0,T?,y)
• RJMCMC
• Non-local updates

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State update moves

• Deletion of randomly chosen event
• Insertion of randomly chosen event
• Shuffle move a randomly chosen event to a new
  time
• Deletion and insertion change state space
  dimension
• Difficulty if too many events between
  observation times

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Insertion of emigration
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Parameter values for different animals
l n a ? N
Cats 1/10 1/13 0-1/50 1/6.7 11.2-22.4k
Mice 1/2.5 1/3.4 1/20 1/6.9 6-16.8k
Rhesus 1/20 (1/6.7)
Baboon 1/30-70 (1/6.7) (11k)
Human 1/23-50 (.71?) (.14?) (1/6.7) (11k)
Independently verified Using different
approaches