Good morning, I have the a set of data $(\sigma,D,\alpha_0)_i$, $i=1...n$ data.

I want to determine two parameters $K_{IC}$, $C_f$ in the **basic equation** given as

$K_{IC} = \sigma \sqrt{D} k_0(\alpha_0+C_f/D)$.

$k_0$ is a 4th order polynomial function with known coefficients. For instance $k_0(x) = p_4 x^4 + p_3 x^3 + p_2x^2 + p_1x + p_0 $.

I am using least squares error minimization to solve for the unknown $K_{IC}$ and $C_f$.

The choice of the residual function $r(K_{IC},C_f)$ is my question of interest. **The residual function can be obtained from the basic equation.** For instance, the residual for $i^{th}$ data set $(\sigma,D,\alpha_0)_i$ can be given as
$r(\sigma,D,\alpha_0)_i = K_{IC} -\sigma^i \sqrt{D^i} k_0(\alpha_0^i+C_f/D^i) $.

The net residual error is given by summing the squares of the residuals for arbitrary $K_{IC},C_f$: $ SSE= \sum\limits_{i=1}^{n} [ r_i ]^2$.

This residual can be evaluated at various $K_{IC}$ and $C_f$ and the minimum found and the corresponding $K_{IC}$, $C_f$ are chosen as the best fits.

Similar, by squaring of LHS and RHS of the **basic equation** and rearranging, we can get another residual of the form say

$r(\sigma,D,\alpha_0)_i = \frac{1}{(\sigma^i)^2 D^i} -\frac{(k_0(\alpha_0^i+C_f/D^i))^2}{K_{IC}^2} $

and another ($K_{IC}$, $C_f$) value can be found minimizing the sum of squares of the residual$\sum\limits_{i=1}^{n} [ r_i ]^2$.

There can be infinite choice of residual functions(obtained by recasting the basic equation obtained from the physics) and corresponding sets of $K_{IC},C_f$ values.

For instance:

$r(\sigma,D,\alpha_0)_i = {(\sigma^i)^2 D^i} -\frac{K_{IC}^2} {(k_0(\alpha_0^i+C_f/D^i))^2}$

or

$r(\sigma,D,\alpha_0)_i = \frac{1}{K_{IC}} -\frac{1}{\sigma^i \sqrt{D^i} k_0(\alpha_0^i+C_f/D^i)} $.

**Questions**

How is one to evaluate which of the residuals is the best one?

How can one bring statistics (say linear regression fitting like $y=\beta_0 + \beta_1 x$, the coefficient of variation of $\beta_0$, $\beta_1$ etc) for this problem?

Are concepts like conditioning number used in linear algebra etc applic able for this problem?

Any specific suggestions are welcome.

Thanking you, Sincerely, Gama.