Dynamic Models Paul J. Hurtado Mathematical Biosciences Institute (MBI), The Ohio State University...
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Dynamic ModelsPaul J. Hurtado
http://www.pauljhurtado.com/Mathematical Biosciences Institute
(MBI),The Ohio State University
19 May 2014 (Monday p.m.)
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Classic (Linear) vs Dynamic Models
I. Incorporating Stochasticity• Observation/Extrinsic• Process/Intrinsic (both)
II. Common Statistical Endeavors• Parameter estimation• Uncertainty quantification• Diagnostic tests (check assumptions)• Model selection and comparison • Descriptive Statistics• Tests vs. “Brute Force” Approaches?
III. Computer Lab• Resources: (http://www.pauljhurtado.com/docs/nimbios14)• Part I:• Simple regression in R • Uncertainty quantification• Model Selection (AIC,BIC)• Diagnostics
• Part II: Adding Stochasticity• Adding observation noise• Stochastic Simulation Algorithm (Gillespie)
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Dynamic Models and Data
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Real data are “noisy”!
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Source of stochasticity?
Process Noise
The state transitions are stochastic:
N(t+Δt) = f(N(t)) + εt
Y(t) = N(t)
Observation Noise
Observations of states are stochastic:
N(t+Δt) = f(N(t))
Y(t) = N(t) + εt
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Adding StochasticityProcess Noise
Original Model:dN/dt = f(N)
Discretize + noise:N(t+Δt) = N(t) + f(N)Δt
+ εt
Observations/Data:Y(t) = N(t)
Observation Noise
Original Model:dN/dt = f(N)
Simulate + noise:
Observations/Data:Y(t) = N(t) + εt
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Adding StochasticityProcess Noise #1Original Model:
dN/dt = rN
Rederive your model: N(t+Δt) = N(t) +
rbinom(rΔt, N(t))
Observation: Y(t) = N(t)
Process Noise #2Original Model:
dN/dt = bN-mN
Gillespie (SSA): Loop: Δt ~ rexp(1/(bN+mN)) ΔN ~ ±1 w.p. b/(b+m) N(t+Δt) = N(t) + ΔN
Observation: Y(t) = N(t)
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Fitting to data?Process Noise
Stochastic model: N(t+Δt) = N(t) +
rbinom(rΔt, N(t))
…
Max. Likelihood Est.: θ = argminθ –logLik(θ;X)
Observation Noise
Original Model:dN/dt = f(N)
Vary parameters & simulate:
Least-squares (min. SSE):
SSE = Σ(Xi – Ni)2^
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Statistics Application
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Simple ExampleU.S. weight data (2007-2010) by age,
sex:
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Does weight vary by sex?
YES!
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Does weight vary by sex?
0 5 10 15
2040
6080
Age
MeanWeight
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How much? Are we sure? How sure?
Estimation ✓ Uncertainty ✓ Diagnostics ✓
Comparison/Hypothesis Test ✓
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Remarks1. Proper statisticians have provided us
MANY tools for “well behaved” models!
2. Most dynamic models are decidedlynot “well behaved”…
3. We can still ask similar questions and compute similar answers! We just need to 1. Be wary of common pitfalls2. Use brute-force computation
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Robust Regression
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Outliers?
R code:# Load the robust regression packagelibrary(robustbase);…fit=lm(Y~X,data); # Outliers includedrfit=lmrob(Y~X,data); # Outliers excluded…
Robust methods identify “high leverage” data points for down-
weighting or exclusion.
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Pelagic Fish in Lake ErieMechanistic Model: behavior/movement, physiology,
ecology.
Growth Survival
Physical Environment
(Temperature, Dissolved Oxygen)
Sub-lethal consequences: growth (fish mass; w) Direct consequences: survival (# of fish; N)
Movement
# of fishN
fish massw
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Cool Warm
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Cool Warm
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Results: 1987-2005 (Aug-Oct)
Population Size
All Years (1987-2005)
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Robust Regression
Field Observations