I am looking for advice how to show the changes in a dynamical model system based on equations. I want to carry out a sensitivity analysis and want to show how strong the impact of a percentage change in one parameter is displayed in the system itself. Can anyone recommend some literature or how you did it by yourself?
Dear Marie-Charlott Petersdorf
You should first learn Matlab software programming which is a technical language for computer-based computation and simulation .
It can give you plot illustrations for any nonlinear system.
Assume you want to study mass-spring-damper system, with a second order ordinary differential equation:
my''+cy'+ky=0,
then you can use Matlab to simulate this ODE numerically, and get the plot of y(t) and y'(t) versus time, starting with any initial condition y(0), y'(0).
Matlab also can give you the plot illustrations and display the change in plots even for a range of variation in one, two or more parameters as m, c, or, k. Then you can quantize these variations as prozentual, and see their impact on dynamical model system, as a sensitivity analysis. You can also quantize y_maximum, or y'_maximum for a range of m, c, and, k, and even illustrate them in Matlab or Excel with a simple plot, or tabulated data, or so on.
Please also take a look at:
https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html
I developed a model of seed dispersal based on equations that depicted interactions between birds and forests on a grid. I also did sensitivity testing as part of the model. I've attached my paper. Does 'prozentual' mean in % terms? (I think it does) :)
Saeb AmirAhmadi Chomachar Thanks, but that is not the topic my question is adressed to. I am not asking for the method to plot something but the idea to keep a clear and focused plot on the main idea what I want to show in as few figures as possible. If I just plot the graphes of the equations itself, I don't have a quantitative scale to express the impact of one parameter for the whole system.
So far, I thougt about showing the trajectorie of the species in the system but I am not sure how to quantify the system changes.
Dear Marie-Charlott Petersdorf
If you have plotted the ODE, with probably Matlab, then the computer has rendered you the array of each ODE variable in a data format.
For example, if you simulate the ODE; y'=-c*y, then Matlab will save the digital data as an array in its memory. To access the data, you need only to command y(:,2).
So, for example if you want to investigate the effect of damping parameter (c) on over-shoot value, you write a loop as:
For i=1:N
c(i)=2*i,
end.
Then you should write an MFile in Matlab to run the ODE for each c=c(i), and then write a code to get: y_max=max(y(:,2)), for each c=c(i).
This is a bit glitchy, but after some tries you mange to get the values.
I have attached an example from Ogata control engineering book, about the Matlab syntax required for this purpose. You do not need to know more about control engineering. Only notice the computer code with a programming eye.
Then you may plot y_max versus c, tabulate it, chart it, or any other math operations to correlate the c data with the y_max data. For example, you can find the standard deviation, mean value or any other statistic parameter, even using some sensitivity function definitions, or error indices as IAE:
http://www.online-courses.vissim.us/Strathclyde/measures_of_controlled_system_pe.htm#:~:text=IAE%20integrates%20the%20absolute%20error,by%20the%20time%20over%20time.
or any other performance or sensitivity index.
Three typical methods are introduced in Article Sensitivity Analysis and Joint Estimation of Parameters and ...
Maybe you can consider the recursive least squares algorithm (RLS) with forgetting factor (RLS-FF). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For the RLS-FF algorithm, acquired data is weighted according to its age, with increased weight given to the most recent data. The correlation parameters are updated gradually.
Application example ― I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data:
Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...
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