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I have two sets of responses from two different sensors. In each set, the first column is distance measured in feet, and the second column is the response of the sensor. Sensor A has response values in the 10-20 range, with very low variance, and Sensor B has responses in the 50-1000 range, with higher variance, over and beyond the fact that the values are of another order of magnitude. Another important issue is that the sensors fire at different, irregular rates, so the sampling rates do not match up between the sensors.

I would like to combine the data from the two sensors into one plot that reflects the confidence I have that something was sensed, based on the two responses. I am not trying to prove correlation between the two sensors; I expect them to be highly correlated. What sort of tools should I use to explore this data?

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    $\begingroup$ Do you have any reference data for each sample? ie how do you determine whether or not something is being correctedly sensed or not? To compare the data on the same axis I suggest you need to interpolate both to a reasonable sampling rate, you can then normalize each time series individually to have zero mean and unity standard deviation $\endgroup$ – BGreene Sep 20 '12 at 18:00
  • $\begingroup$ I do not have any reference data. I'm doing a blind study. Sampling an interpolation, normalizing and then comparing the two resultant series sounds like a good idea. Thanks! $\endgroup$ – cjohnson318 Sep 21 '12 at 15:10
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    $\begingroup$ Could you please tell us what is being sensed and the meaning of the sensor values? Otherwise, we can only guess what the relationship between the two sets of responses is supposed to be, which makes it difficult to suggest any meaningful or useful procedure. $\endgroup$ – whuber Sep 21 '12 at 15:23
  • $\begingroup$ The two sensors are the OpenCV template matching function, and the SURF algorithm. I am trying to detect certain shapes in an image. $\endgroup$ – cjohnson318 Sep 24 '12 at 21:09
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    $\begingroup$ A simple Kalman filter is very good at fusing two or more information sources knowing measure (and process) means and standard deviations. $\endgroup$ – Vladislavs Dovgalecs Apr 18 '15 at 2:23
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You may want to have a look on Dempster-Shafer theory. This framework has been investigated for merging information coming from sensors, for instance you can have a look at the following papers:

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    $\begingroup$ I never resolved the problem, but I think this answer is definitely the most helpful. $\endgroup$ – cjohnson318 Aug 17 '15 at 17:03
  • $\begingroup$ Dezert-Smarandache theory is used for sensor fusion, try that. $\endgroup$ – E. Douglas Jensen Aug 9 '18 at 15:46
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Being sensed is a binary outcome. The sensors' distance from target is a covariate that determine the outcomes. Logistic regression is one way to model this. The model would look like a smooth 2-dimensional surface over the x-y plane where x and y represent the respective sensor's distance from the target, with the height being the logit of p where p is the probability that the target is sensed (I am assuming that the definition of target being detected is that at least one of the sensors detects it).

There are other possibilities of modeling a binary outcome, probit analysis is one alternative for example. The choice will depend on what modeling assumptions you think are reasonable.

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    $\begingroup$ I'm a little lost. Being sensed is a binary outcome, but the response of the sensors is continuous, and I do not have an a priori threshold. I'm not sure I am comfortable interpolating a two dimensional surface from two one dimensional data sets. I think I was more curious if there were existing models for fusing data from one dimensional functions. $\endgroup$ – cjohnson318 Sep 21 '12 at 15:17
  • $\begingroup$ Your question is a little vague and perhaps I should not have assumed that this was a detection problem. The way I interpreted your question was that the sensors are trying to detect a signal that is a target. But of course there is background noise. The output from the sensor is a positive number that I assumed represents an evaluation of the signal. The signal should be above a certain level to be classified as a target because values below that level could just be the background noise. This makes the outcome of interest target detection which is binary. $\endgroup$ – Michael R. Chernick Sep 21 '12 at 15:29
  • $\begingroup$ Of course the outcome depends on distance as signal waves have amplitude degrading (such as with an inverse square law) as the distance to the target increases. So if detection is the response the two sensors would have different logistic regression curves because of their different characteristics. $\endgroup$ – Michael R. Chernick Sep 21 '12 at 15:35
  • $\begingroup$ If this is not your problem maybe you could explain it better so that we can understand. It sounded like you were looking for an improved detection algorithm based on the output from the two sensors. a decision rule could then be based on the points in the positive quadrant of the plane. You would use training data to construct a classifier apply the classifier to test data on targets to get binary outcomes (successful detection, failed detection). If this is what you want I can modify my answer to add some of this detail. $\endgroup$ – Michael R. Chernick Sep 21 '12 at 15:44
  • $\begingroup$ If I am totally off base I will delete my answer and hope that you will clarify your objective so that me or someone else can give you am amswer. $\endgroup$ – Michael R. Chernick Sep 21 '12 at 15:44

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