Dynamic ModelsPaul J. Hurtado

http://www.pauljhurtado.com/Mathematical Biosciences Institute

(MBI),The Ohio State University

19 May 2014 (Monday p.m.)

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)

Dynamic Models and Data

Real data are “noisy”!

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

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

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)

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^

Statistics Application

Simple ExampleU.S. weight data (2007-2010) by age,

sex:

Does weight vary by sex?

YES!

Does weight vary by sex?

0 5 10 15

2040

6080

Age

MeanWeight

How much? Are we sure? How sure?

Estimation ✓ Uncertainty ✓ Diagnostics ✓

Comparison/Hypothesis Test ✓

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

Robust Regression

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.

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

Cool Warm

Cool Warm

Results: 1987-2005 (Aug-Oct)

Population Size

All Years (1987-2005)

Robust Regression

Field Observations