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.