So I'm working on a non-linear least squares optimization problem, and I'm wondering if any of my optimization parameters might be redundant on the basis that they might be considered factors of other parameters.
To demonstrate what I mean, I originally tried to fit the following curve to some data: $$f(x)=\beta_1\left(1-\left(1-e^{\beta_2\left(x-\beta_3\right)}\right)^2\right).$$ When this failed to reduce the sum of squared residuals to an acceptable value, I thought that introducing more parameters might help: $$f(x)=\beta_1\left(\beta_2-\left(\beta_3-\beta_4\beta_5^{\beta_6\left(x-\beta_7)\right)}\right)^2\right)=\beta_1\left(\beta_2-\left(\beta_3-\beta_4e^{\beta_6\left(x-\beta_7)\right)\ln\beta_5}\right)^2\right)$$ Expanding the above expression yields $$ f(x)=\beta_1\beta_2-\left(\beta_3\sqrt{\beta_1}-\beta_4\sqrt{\beta_1}e^{\beta_6\left(x-\beta_7\right)\ln\beta_5}\right)^2.$$ Isn't it correct that for $\beta_1\beta_2$, $\beta_3\sqrt{\beta_1}$,$\beta_4\sqrt{\beta_1}$, and $\beta_6\ln\beta_5$, each could be expressed by an individual parameter (e.g. $\beta_1^*$, $\beta_2^*$, $\beta_3^*$, and $\beta_4^*$, respectively)? If this is correct, I should be able to determine parameters for the function $$f(x)=\beta_1^*-\left(\beta_2^*-\beta_3^*e^{\beta_4^*\left(x-\beta_5^*\right)}\right)^2,$$ where $\beta_5^*:=\beta_7$, that would render it equivalent to the function $$f(x)=\beta_1\left(\beta_2-\left(\beta_3-\beta_4e^{\beta_6\left(x-\beta_7)\right)\ln\beta_5}\right)^2\right).$$ This would also mean that two of the parameters in my second equation are redundant. What are your opinions of this, StackExchange? Furthermore, does it look like any of my other parameters could be redundant?