fitting.qmd

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title: "Model Fitting: Nuts & Bolts"
subtitle: "Fitting a Dynamical System to a *Pf*PR Time Series"
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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 *Pf*PR 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.

# Initial Conditions

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