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ODE Analysis

Analysis of an Ordinary Differential Equation (ODE)

All the figures below are generated using examples/ode/ode.jl.

ODE

$$x''(t) = -\omega^2 * x(t) - k * x'(t)$$

Unknown ODE parameters

  • k
  • $\omega$

Example ODE solution

For model parameters:

  • k = 0.1
  • $\omega$ = 0.2

Local sensitivity analysis

Global sensitivity analysis (using eFAST)

Probabilistic distributions of the prior parameter uncertainties are:

  • k = LogUniform(0.01, 0.1)
  • $\omega$ = Uniform(0.1, 0.3)

Bayesian sensitivity analysis

Observations

Synthetic observations are applied to constrain the ODE parameters

Observation errors are equal for all the sample locations with standard deviation equal to 1 (observation weight = 1 / observation standard deviation = 1 / 1 = 1)

Prior parameter uncertainties

k only

$\omega$ only

Both parameters

The observation data are plotted as a solid black line.

Histograms/scatter plots of Bayesian MCMC results

Posterior parameter uncertainties

Note that now the parameter uncertainties are constrained by the observation data.

k only

$\omega$ only

Both parameters

The observation data are plotted as a solid black line.