Plotting Fit Results

Given a dataset type or DataModel, there are pre-defined recipes for the Plots.jl package so that they can be visualized via:

DM = DataModel(DataSet(1:3, [4,5,6.5], [0.5,0.45,0.6]), (x,p)->(p[1]+p[2])*x + exp(p[1]-p[2]), [1.3, 0.2])
plot(DM, MLE(DM))

where the best fit corresponding to the MLE parameters MLE(DM) is drawn in by default. Alternatively, other parameter values may optionally be specified in the second argument.

By default, a linearized (i.e. approximate) confidence band is computed around the prediction via Gaussian error propagation of the inverse Fisher information, which provides a lower bound on the parameter uncertainty by the Cramér-Rao theorem.

A desired (vector of) confidence level(s) in units of $\sigma$ can be specified via the Confnum keyword for more control over the bands, with Confnum=0 disabling the bands.

plot(DM; Confnum=[1,2])

These linearized pointwise confidence bands only take into account the uncertainties in the fit parameters and therefore quantify pointwise at every input x (again, approximately for non-linearly parametrized models!) and quantify the interval in which the model prediction of the true parameters is estimated to lie with the specified confidence, given the observed data.

However, depending on the uncertainty associated with the measurement process, future recorded data points can lie far outside the confidence bands. In contrast, to assess the range in which future validation measurements are likely to land, one can draw bands where the data uncertainty is added on top of the prediction uncertainty due to the parameters. This can be achieved via the Validation=true keyword in the plot method, where the Confnum can be controlled as before:

plot(DM; Confnum=[1,2], Validation=true)

In the literature, these are often instead referred to as "prediction bands" instead of validation bands. However, this terminology is somewhat prone to confusion in my humble opinion which is why I personally prefer to avoid it.

The linearized confidence and validation bands are computed via the VariancePropagation and ValidationPropagation methods respectively.

For DataModels or simply datasets which have multiple different components (i.e. ydim > 1), the Symbol :Individual may be appended as the last argument to split the components into separate plots:

DS122 = DataSet([1,2,3],[2,1,4,2,6.8,3.5],[0.5,0.5,0.45,0.45,0.55,0.6], (3,1,2))
DM122 = DataModel(DS122, (x,p)-> [p[1]*x, p[2]*x])
plot(DM122, MLE(DM122), :Individual; Confnum=1, Validation=false)

as compared with plotting both in one:

plot(DM122, MLE(DM122); Confnum=1, Validation=false)

Residuals

PlotQuantiles
ResidualVsFitted
ResidualPlot

Plotting Parameters

ParameterPlot
ParameterSavingCallback
TracePlot