I have a discrete-time signal (with noise) and I want to determine the best non-increasing function that represents it.
Of course it would be useful if the obtained function were defined in terms of a continuous variable of time, but to me this sounds too general. I think it is not a good idea to specify a particular equation with coefficients to be determined, simply because in so doing I am imposing a too restrictive space of test functions, thus possibly causing "underfitting". For example, I could impose an exponential function like $f(t) = c \, \exp(-\lambda t)$, where $c$ and $\lambda$ are parameters to be determined, but what if the given signal has another form?
For a discrete signal I think it is better try to generate a discrete approximation as well. In this case, the "fit" would be a non-increasing discrete function that is the best representation of the given general signal. Does anybody know a method to do that?
One could use either the raw signal or then a low-pass filter firstly. As a remark I would like to say that polynomial interpolation simply does not work because of the noise.