How to find models to estimate parameters

Question

How can I find models for estimating the parameters $w$, $c_1$, $c_2$ and $c_3$ in the function: $y(t)=c_1\sin(wt)+c_2\cos(wt)+c_3$

Answer 1

You have $n$ data points $(t_i,y_i)$ and you want to fit the model $$y=c_1\sin(wt)+c_2\cos(wt)+c_3$$ which is nonlinear because of $w$.

Suppose that $w$ is given a value; the problem becomes linear and a str=andard multilinear regression would easily provide $c_1,c_2,c_3$ which depend on the value selected for $w$. For this value, you can recompute $$SSQ(w)=\sum_{i=1}^n\left(c_1\sin(wt_i)+c_2\cos(wt_i)+c_3-y_i\right)^2$$ Try ddifferent values of $w$ until you see (graph $SSQ(w)$ a a function of $w$) an area where there is a minimum.

Now, you have all required elements to start a full nonlinear regression.

I must mention here that JJacquelin, an MSE user, proposed a method which does not require any initial guess and any nonlinear regression. This is explained in the paper "Régressions et équations intégrales" (starting on page $21$) published on Scribd. The paper is in French but this will not make any problem to you.

Edit

Using the $15$ data points used by JJacquelin (page 23 of his paper), the method I described gives the following results $$\left( \begin{array}{cc} w & SSQ(w) \\ 0.5 & 8.3687 \\ 1.0 & 7.2649 \\ 1.5 & 3.8688 \\ 2.0 & 0.3261 \\ 2.5 & 3.2727 \\ 3.0 & 8.7657 \end{array} \right)$$

For the best point, corresponding to $w=2$, we have from the multilinear regression $c_1=1.28306$, $c_2=-0.573569$, $c_3= -0.397904$. Staring with these estimates, the nonlinear regression leads to $c_1=1.28934$, $c_2=-0.571687$, $c_3= -0.390698$, $w=1.98131$ to which corresponds $SSQ=0.320353$. You can compare with the results of the non-iterative method (page $32$ of the paper - curve $3$). This is indeed a very good method.