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I encountered several regression problems in the field of engineering that are formulated similar to the following form (I will ommit the error terms):

$$ y=a_0+a_1(x_1-c_{ref})+a_2\ln(x_2/d_{ref})\,, $$

where $y$ is the dependent variable; $x_1$ and $x_2$ are independent variables; $a_0$, $a_1$ and $a_2$ are unknown coefficients need to be estimated. What I have difficulties in the kind of regression problems is the interpretation of the two reference parameters $c_{ref}$ and $d_{ref}\,$. In papers, these two reference levels are usually given (i.e., they are preset values and not estimated together with $a_0$, $a_1$ and $a_2$) without any explanations on why they are considered and how their values are chosen...

In my view, all these preset reference parameters can be merged with the constant term $a_0$ into a new constant, say $\tilde{a}_0$, i.e.,

$$\tilde{a}_0=a_0-a_1c_{ref}-a_2\ln(d_{ref})$$

and the original regression problem becomes to

$$y=\tilde{a}_0+a_1x_1+a_2\ln(x_2)\,.$$

Therefore, I do not understand why these reference parameters are involved in the very first place, except that they represent some important information. For this reason, I wonder if anyone dealt with this kind regression problem and can share some ideas on

  1. Why are these reference parameters needed? What roles do these parameters play in a regression problem? How can I interpret them in terms of the corresponding independent variables?

  2. How can I determine their values in advance? Is there any formal way to decide their values?

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  • $\begingroup$ I have an idea on Cref. If I am regressing data on the years 1971, 1972, 1973, etc. but what I really want is years since 1970 I would use (year - 1970) as my regression variable, so that the regression is instead on 1, 2, 3, etc. This is common in some types of regression. $\endgroup$ – James Phillips Aug 18 '18 at 18:24
  • $\begingroup$ Dref might be a scaling factor for unit conversion, for example converting data in units of meters to data in in units of kilometers. $\endgroup$ – James Phillips Aug 18 '18 at 18:31
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You could, but the way these are written looks like pretty standard general-purpose engineering equations that might have different known values for ther _ref parameters for, say, different materials, temperatures, etc.

Suppose every diode has a voltage at which it starts conducting. If you wanted to look at the apparent resistance of a conducting diode, you would put a range of voltages across it above cref and measure the current to get a linear fit. One model of diode might conduct when 0.9 volts is reached, and another when 1.2V is reached. so that cref means that you would subtract off that voltage since it is needed just to get past turned-on. Now that term starts at 0 for any diode.

Same for the second term. Taking the ln implies there is a belief that term is of the form $e^{x_2/d_{ref}}$. Maybe this is the current in a wire attached to a large battery with an inductor in it. The $X_2$ would be time, and $d_{ref}$ would be a known quantity for different-sized inductors.

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