# Linear Regression Filter Properties

I have a filter to detect when a slow moving signal (oven temperature) reaches steady state. I am using a linear regression of a moving window and looking at the slope. Based on the equations for linear regression including $\sum xy$, $\sum x^2$ and $(\sum x)^2$ I am thinking this is a non linear filter, is this true? Is it still true to say this is a FIR filter though?

Also if I am trying to get an estimate for reaching steady state is there a better filter, or can similar quality be achieved with a linear filter given Gaussian noise? Basically I am looking for a good theoretical analysis of this type of filter to way its pros and cons?

This is a linear filter because the signal values ($y_i$ in your notation) enter the equation linearly. It can be implemented as an FIR filter, and then it's called "Slope Filtering". The method was described in the IEEE Magazine on Signal Processing ("DSP Tips & Tricks"), Nov. 2008. You can read the article here.