Parameter estimation - How to input experiment data?

Hi All,

I want to perform parameter estimation of the following ode model.

5 parameters have to be estimated after performing a least square error minimization of the cost function. The input that I have is steady state multi -experiment values of y1 to y10.

I found an example here that illustrated how to input time series experiment data. Could someone explain how to input staedy state multi-experiment data?

[dpb]

Start here (link) Boundary Value Problems

[Deepa Maheshvare]

Most of the examples have been shown for second order differential equations. The system of equations that I have is first order. I am not sure if it's applicable for my system.

[dpb]

Well, since higher order DEs are reduced to a system of first order ones, it seems the first step of your work is already done... :)

Just write the necessary routine that returns the vector of dy/dx for each component in the system; the only difference is you don't have the higher-order terms that are present in the examples. TMW did that to illustrate how to reduce higher-order systems to first-order.

There are also other examples of passing extra parameters in the ODE documentation as well.

[Deepa Maheshvare]

Thanks a lot for the explanation. In the documentation it is mentioned that " The boundary conditions specify a relationship between the values of the solution at two or more locations in the interval of integration"

The data that I have is steady state value from different experimental studies and not at different time instants.

I would like to know if it is still appropriate to solve my system as a bvp

[dpb]

Why not? Simply enter a time for the ending value that is far enough out for the system to have reached SS...I don't recall if there's a special case to let you use inf for that purpose or not; see what the documentation says and experiment...

[Deepa Maheshvare]

Hi,

Like you suggested I'm entering a time values that's long enough for the system to reach steady-state.

I'm following the example given here

https://in.mathworks.com/help/matlab/math/solve-bvp-with-unknown-parameter.html

I'm not sure how the residual is calculated at the boundary (t=2000s)using the function 'mat4bc' given in the above link.

It is explained that ,

"To calculate the residual values, you need to put the boundary conditions into the form g(x,y)=0. "

I'm facing challenge in implementing the above form . However, I have defined afunction that computes the residual,

function residual = LQminimization(initialConditions,tSpan,y_SS)
y0 = initialConditions;
[t,y] = ode15s(@myModel,tSpan,y0);
residual= (y[:,end] - y_SS).^2;

I have reached till this point.I'm not sure how to proceed with estimating parameters from here.

Any suggestions?

[dpb]

The mat4bc function accepts the arguments ya,yb which is the shorthand nomenclature for the y of the ode function at [a,b] of the interval.

In the example (and for any SS solution), the BC for the derivative is zero which is what is yb(2) returns.

The example has three BCs so the function returns three values; if your case has additional, you simply set the other terms of ya,yb appropriately. However, you have to be certain the system is not over-constrained by more BCs than are DOF to match--iow, must be realizable.

[Deepa Maheshvare]

Thanks a lot for the response. In my case, there are 10 variables. Basically, the

arguments ya and yb are column vectors of 10 rows each.

I'm not sure if I follow this,

"you have to be certain the system is not over-constrained by more BCs than are DOF to match--iow, must be realizable".Could you please explain?

[dpb]

Just like you couldn't fit a quadratic equation to only two points, you can't try to force more conditions onto the solution of your set of PDEs than can be achieved.

Just an observation on a elementary fact; it's sometimes not so easy to tell with such a system of equations what isn't feasible when setting constraints unless there is a closed-form solution that you can examine.