Model Fitting: Nuts & Bolts
Fitting a Dynamical System to a PfPR Time Series
In malaria, we want to use mechanistic models to understand patterns in malaria data. For most of the analyses we do, the starting point will be a PfPR time series, \(x(t)\). Our task is thus to fit a dynamical system to a time series. Here, we discuss the nuts-and-bolts of model fitting using SimBA.
Decomposing Pattern
We want to think about the features of a time series in an ordered way:
The time series has a long-term average or mean, \(m\);
The time series has a seasonal pattern, \(S(t);\)
The time series has an inter-annual component, \(T(t);\)
After fitting, we have additive residual error, \(\epsilon(t)\)
Conceptually, at least, we can think of pattern in data as the sum of a time series model giving a signal, \(m S(t) T(t)\) and residual error:
\[x(t) = m S(t) T(t) + \epsilon(t)\] In model fitting, we fit the shapes of functions specifying the signal to minimize the residual error.
Seasonality
In SimBA, we have developed a trigonometry-based function family to model seasonality.
Interannual Variability
We use spline functions with knots and control points.