# Formulation of the least-squares parameter estimation problem

I have a system of 10 ordinary differential equations of the form,

$$\frac{dy_1}{dt} = f1(V1,k1,y1,y2)\\ \vdots \\ \frac{dy_{10}}{dt} = f_{10}(V_{10},k_{10},y_{9},y_{10})$$

I want to estimate the parameters $$V_1 ,V_6,V_7,V_{10}$$ using a global fit.

From the approaches suggested in the literature, I understand least-squares error minimization is commonly used. However, I'm not able to understand how optimization problem is actually formulated.

Cost function = $$\Sigma_{i=1}^{10} (y_i^\text{experiment} - y_i^\text{model})^2$$

Where, $$y_i^\text{experiment}$$ is the steady-state value obtained from experiments and not the time series data of $$y_i$$. Could someone explain how $$y_i^\text{model}$$ is expressed in terms of the parameters that are to be estimated?

Is is differential equation,(say) $$\frac{dy_1}{dt} = f1(V1,k1,y1,y2)$$

expanded using Taylor polynomial to find $$y_1^\text{model}$$?

Could someone provide an example?

• Typically it's necessary to use a numerical method (e.g. a Runge-Kutta method) to solve the ODE initial value problem. Derivatives of the solution with respect to the parameters are then computed by finite difference approximations. – Brian Borchers Apr 7 at 17:41
• In what form do you have the system of equations? Do you have dy_n/dt at several times for some initial conditions? Do you have y_n(t)? – Jacob Panikulam Apr 7 at 20:57
• @BrianBorchers Thanks a lot for your response. I'm using stiff- equation solver.I'm facing challenge in understanding how the optimization problem is solved to estimate the unknown parameters – Natasha Apr 8 at 3:10
• @JacobPanikulam I have multiple steady- state values of $y_1$ to $y_{10}$ obtained from experiments – Natasha Apr 8 at 3:13
• You might find following link useful. The way you state the problem is a classical optimization problem subject to a ODE as constraint. There are two ways of solving this, the forward approach Brian Borchers mentioned or the adjoint method linked here. – Bort Apr 10 at 11:15

You can think of the ODE as a constraint between the parameters $$V$$ and the observed variable $$y$$. To enforce constraints in optimization problems, you can introduce a Lagrange multiplier, which we'll call $$\lambda$$. In that case, the Lagrangian for your problem is
$$L(y, \lambda, V) = \sum_i|y(t_i) - y_i^\text{experiment}|^2 + \int_0^T\lambda\cdot (\dot y - f(y, V))dt.$$