# Choice of residual function for least squares error minimization

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.

• (1) This question is likely better suited for stats.stackexchange.com (2) How the residual is distributed depends on the specifics of your data and the processes that generate it. I would suggest thinking carefully about what phenomenon you are trying to model and using your understanding of that to generate a hypothesis on your residuals. – Neal Nov 11 '18 at 4:56
• Hi Neal, The basic governing equation is known. The residual is obtained from the basic equation. I would want some information regarding conditioning number etc... if applicable. – gama Nov 11 '18 at 5:22
• Thank you for your suggestion, Posted this question in stats.stackexchange also. stats.stackexchange.com/questions/376412/… – gama Nov 11 '18 at 5:37