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I am currently evaluating the measurement of the thermal impedance of different semiconductor devices. In order to properly evaluate these measurements, I have to determine the derivative of the thermal-impedance function $Z_{th}(a)$, where $a = \ln(t)$. $Z_{th}(a)$ is a continuous function, but due to the way the measurement works, I only have a set of data points $\{(a_1, Z_1), \dots, (a_{100}, Z_{100})\}$. Because of this, the derivative is estimated using linear regressions over small subsets of the data points (e.g. 9 of the 100 data points at a time). So I would calculate slope using a linear regression of the points $a_1$ to $a_9$, then the next slope using the points $a_2$ to $a_{10}$ and so on.

Now I am faced with the following question: When I perform a linear regression over a small set of points $\{(x_1, y_1), \dots, (x_9, y_9)\}$ that belong to a nonlinear function, I get one resulting slope. How do I find the point x where this resulting slope is closest to the slope of the nonlinear function? Should I just assume it is closest in the middle of my regression intervall? Would "middle" mean $x = 1/2 \ (x_1 + x_9)$, or would it be the average of all my x values (since the points x might not be equidistant)?

My intuition would be the latter, but I am wondering if there is a better way to do this.

idkwiad
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  • Because the descriptions are vague, I don't trust that I understand your question, but it seems to be concerned with estimating the derivative of a regression function. A general way to accomplish that is described in my answer at https://stats.stackexchange.com/a/216216/919. Do you think you could describe your data specifically and narrow the focus of this post to one issue alone? – whuber Sep 08 '21 at 16:14
  • What do you mean by *the slope of the nonlinear function?* Nonlinear means that its slope is varying! – kjetil b halvorsen Sep 13 '21 at 03:22
  • @kjetilbhalvorsen That is of course correct. The problem is that I have of a set data points and have to estimate the derivative, so I can only perform some kind of regression. This regression is done repeatedly over the same interval, but with varied starting point and end point (I edited the question for clarity). Over every regression interval, the slope of the original function will vary, since the function is nonlinear. That is why I want to know my "best bet" for the value x, where "best bet" means the slope of regression is as close to the slope of the function as possible. – idkwiad Sep 13 '21 at 06:49

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