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I'm new to system ID so be gentle.

I have a modeled signal of streamflow that I would like to map to an observed signal of stream height. This lends itself nicely to a simple impulse-response kind of relationship.

Problem is that the streamflow is modeled. So sometimes the magnitude of the impulse does not really match the magnitude of the response (e.g. big impulse, small response followed by small impulse, big response later in the time series). Even worse, sometimes I have false positives (modeled impulse, no response) and false negatives (no impulse, but there is a response). I have hundreds of sites I'm working with, and it's tedious to go through each one by hand and eyeball whether there's a good input-output relationship.

So my question is: Is there a way to compare the impulse signal to the response signal to determine the "quality" of the relationship? That way I can start working with the "good" sites, and also try to characterize what makes a site good vs bad based on other factors. Is cross-correlation the way to look at it? Or is there a better way to look at this?

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I don't yet see the connection of your input with "impulses" in that if I understand correctly you are looking at the settled state from prior changes rather than dynamic responses. What you are asking for I believe is a statistical correlation between "Stream Flow" and "Stream Height"; given a stream flow (in the steady state), what is the "settled" stream height that would be sustained by that flow. I would argue that to be a correlation problem to the degree you are looking for a linear relationship between the two. We have other posts that detail how to compute the correlation coefficient, if you need further info on that you could look here:

Noise detection

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  • $\begingroup$ Actually what I am looking for is dynamic responses. The basic question I'm trying to answer is: given an impulse from the modeled flow, what will the expected change in height be? The problem is that the flow is modeled, so sometimes you get an input when there is no output, and vice-versa. What I want to try and determine is, purely by looking at the two signals, is there a metric that can be used to assess, prior to performing any system ID, whether the system could even be modeled effectively by a transfer function or something like that. $\endgroup$ – kjfries27 Mar 29 '17 at 0:57
  • $\begingroup$ @kjfries27 I am just seeing your comment now: an immediate test is to see if the system is linear by comparing the input and output frequency content. If the system is non-linear, it cannot be modeled by a simple transfer function. If non-linear there will be additional frequencies of significant strength at the output that are not present at the input. The total power of these additional frequencies (when tested with a single frequency tone at the input) is the total harmonic distortion and from that you can assess the extent of linearity for the system. $\endgroup$ – Dan Boschen May 27 '17 at 12:20
  • $\begingroup$ Other tests are for time invariance- for this I suggest comparing impulse responses with the same input tested over different time intervals. If the system is linear and time invariant it can be modeled with a transfer function $\endgroup$ – Dan Boschen May 27 '17 at 12:22

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