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We are using resp. manufacturing a certain device which measures, among others, the humidity of liquids. I got asked by a colleague if it makes sense to use two sensors instead of one (which a competitor is apparently doing). Mainly, in a statistical manner.

My ideas are: Redundancy: In a technical manner, just a backup. Or to use the one that provides more meaningful measurements (but I guess this is not possible because how can you know which is (more) true?).

Statistically: Using the mean of both. Maybe even with a sensor-specified weight (though I wouldn't). One could define a kind of deviation band in which the sensors must be within. In case their signals differ too much, one knows that something is going on.

Are there some facts which become improved by going from one sensor to two? I'd say, e.g., the confidence is increasing but I don't know how to underline this (statistically). Finally, I'd simply say: The more, the better. Unfortunately, this isn't solid argument.

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    $\begingroup$ If you do use two different sensors throughout, then you can it make part of your analysis to see, (a) if the two are consistently different, (b) if one has greater variation than the other, (c) if one seems less variable at low (or high) humidity than the other. // If you made two measurements throughout--even with the same kind of sensor, that would give you somewhat better power to detect differences in humidity. // In spite of trouble and expense, I see no downside to double measurements. $\endgroup$ – BruceET Jul 9 at 7:01
  • $\begingroup$ Thanks a lot! The first aspect aims more at a generel analysis about the sensors themselves, I'd say. For the second, could you please provide an elaborated explanation? In which way is the power better? How can I make us out of it? $\endgroup$ – Ben Jul 9 at 7:24
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    $\begingroup$ Hard to be specific without more detail on what the simplest design would be. Maybe you have five humidity levels and four specimens per level. Then you'd have a nested ANOVA: specimens within levels. If each specimen were measured twice you would have more information about variability (hence better power to distinguish among levels) than with only one measurement per specimen. // Even if both types of sensor turn out to be essentially the same, you'd have some power advantage. // Also, @Skander (+1) makes some good points, which I hope you will consider. $\endgroup$ – BruceET Jul 9 at 8:24
  • $\begingroup$ When I understood it correctly it is basically one specimen as it is meant to monitor a larger device. So, it will be continous signal. Should have mentioned that earlier, sorry! $\endgroup$ – Ben Jul 9 at 9:23
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Intuitively, if the noise and measurement errors from your sensors are independent and have zero mean, then using multiple sensors will improve the signal to noise ratio because the noise and errors will tend to cancel each other out as you increase the number of sensors.

There is an interesting parallel to be made with the concept of ensembling in machine learning: Averaging the output of multiple models will improve accuracy for similar reasons (They reduce the variance that is inherent in a single model).

So you can check the math underlying ensemble prediction methods to figure out a similar mathematical justification to using multiple sensors. Look for example at how Random Forests work as ensembles of high variance decision trees.

But, but...this only works if the noise and errors are independent and zero mean. Often times in lab and industrial instrumentation situations, the noise and errors have some systemic environmental cause or are due to flaws in the measurement process which is introducing a non-zero mean bias to your model, so a more in-depth analysis is necessary than simply averaging the output of multiple censors.

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  • $\begingroup$ Thanks a lot! How exactly do noise and ratio cancel each other? Just conceptually or can this mathematically be taken up? $\endgroup$ – Ben Jul 9 at 7:58
  • $\begingroup$ To clarify: The signal to noise ratio, also referred to as SNR, $\frac{signal}{noise}$ is improved, that is increased, because the noise and measurement errors are reduced. I am using noise in a confusing way because on one hand I'm using measurement error and noise as two separate quantities and then I am combining them into one noise in the term SNR. So clarify $SNR = \frac{signal}{totalnoise}$ and $totalnoise = measurementerror + environmentnoise$. $\endgroup$ – Skander H. Jul 9 at 8:11
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    $\begingroup$ To explain why they cancel out: Since they are zero mean and they have the same variance, the chances on average that the measured value is less than the true value are the same as the chances on average that the measured value is more than the true value. Negative and positive then simply cancel each other out. $\endgroup$ – Skander H. Jul 9 at 8:14
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    $\begingroup$ Yes. The more signals the better. In theory, if you average a large enough number of signals the noise goes to 0. $\endgroup$ – Skander H. Jul 9 at 9:31
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    $\begingroup$ Good answer. This also relates to the concept of the standard error of the mean. Each individual measurement is drawn from some distribution which has some amount of noise around the true value, which can be described by the standard deviation or variance. This measurement variability does not change with how many samples you take. The mean of multiple measurements has its own distribution which should be centered on the true value, however, its variance decreases with the sample size. As you take more measurements, the standard error of the mean shrinks. $\endgroup$ – Nuclear Wang Jul 9 at 18:35
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Sensor fusion via a Kalman filter takes 2 (or more) separate measurements, combines them within the filter calculations, and outputs a smoother/more accurate measurement of the underlying.

Some good stuff online about this, e.g. https://simondlevy.academic.wlu.edu/kalman-tutorial/the-extended-kalman-filter-an-interactive-tutorial-for-non-experts-part-14/

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  • $\begingroup$ Coincidentally, I was actually reading up Kalman filters (again) :) Thank you! $\endgroup$ – Ben Jul 9 at 7:57
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    $\begingroup$ Worth pointing out that Kalman filters fully support 2 sensors with complimenting accuracy and precision properties (aka, GPS and barometer for altitude) $\endgroup$ – Michael Jul 9 at 19:22
  • $\begingroup$ Thanks, what exactly does this mean? That I need at least 2 sensors for Kalman filter? $\endgroup$ – Ben Jul 10 at 4:47

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