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D2D Software / Computation of integration-based prediction bands

Here, a basic introduction to the integration-based computation of prediction bands (PBs) on the model dynamics will be given. The example in /Examples/ABC_toyModel was added in the revision of 07/22/2015.

The integration framework is located in the folder /PPL. It allows to compute prediction bands on the whole time-course of model dynamics of the model components (internal states) as well as for the observations. In addition, new observations like ratios or sums of the internal states or observations can be added to the model and their prediction band can be calculated. At the beginning, a validation profile likelihood is computed (see Likelihood based observability analysis and confidence intervals for predictions of dynamic models. ). Thereof, the auxiliary data point and parameter set to a specified confidence level is taken as starting point for the integration. Based on the penalised likelihood-function, a system of ODEs is obtained through the implicit function theorem, which is executed to establish the prediction band.

To run the ABC toy model in the Examples folder, at first the D2D framework is initialised and the model and data is loaded, followed by a compilation of them.

arInit

arLoadModel('ABC_model');
arLoadData('ABC_data_BCobs');

arCompileAll();

Further, the standard deviation of 0.1 which was taken for the data simulation is fixed for the optimisation procedure, which is then performed:

ar.config.fiterrors = -1;
arSetPars('sd_B_au',[],2);
arSetPars('sd_C_au',[],2);

arFit();

Subsequently, to obtain the prediction bands on all three states A, B and C, the function doPPL is used. It takes the desired model as first input and the condition or data, dependent on the fifth flag (takeY). Also, the desired internal states for which PBs should be computed have to be stated as third input (in this case 1:3). In addition, a vector t can be added for which the complete validation profile should be calculated. In the final plot, black stars indicate the alpha thresholds of the hereby obtained prediction intervals. The first time point stated will be used as starting point of the integration.

The doPPL statement in the Setup.m of the ABC toy example is given by

doPPL(1,1,1:3,linspace(0,100,11),0,1,0.25);

where the initial correction factor and the integrator step size (0.25) is added.

As the normal arPlot2-function plots the data points, best fit and the corresponding (estimated) error of the data points, the plotting of PBs has to be stated in the config struct:

ar.config.ploterrors = -1;
arPlot2

The plot then shows the data points as dots, the best fit as solid line and the PBs as shaded area, together with the thresholds of the confidence intervals calculated via validation profile likelihood for the stated time points:

ABC.png

Prediction bands in a Systems Biology application can be computed by commenting out the last line in the Setup.m file within the Swameye example at /Examples/PPL, or including an appropriate line in the other examples of the D2D framework.

Updated