I am using two different methods to measure velocity. (Here, I’m measuring velocity in one dimension, with velocity in one direction counted as positive and velocity in the other direction counted as negative.) One method (‘Method 1’) measures low velocities (and no velocity) accurately, but it cannot measure velocities that are large in magnitude (in either the positive or the negative direction) accurately. The other method (‘Method 2’) measures velocities that are large in magnitude accurately but does not measure low velocities accurately.
I want to integrate these two different methods to generate velocity values that are accurate across all velocities. These integrated values will weigh method 1 values more heavily if the method 1 values are low, and they will weigh method 2 values more heavily if the method 2 values are large in magnitude. If the method 1 value is 0, its weight would be 1 and the method 2 value wouldn’t be counted at all. If the method 2 value is infinite (which obviously isn't possible, but I'm just mentioning the case here to help to illustrate my point), its weight would be 1 and the method 1 value wouldn't be counted at all.
What is the best way to integrate both of these methods?
The best solution I have at the moment is to transform both method 1 and method 2 values separately so they're on a finite scale. Since I want to base one of my weighing parameters off of infinite values, I need to make both positive and negative infinity finite values. By transforming them with the logistic function or some other sigmoid function that is continuous, differentiable, and has asymptotes at 0 and 1, values that were negative infinity become 0, values that were 0 become 0.5, and values that were positive infinity become 1. Now, with both method 1 and method 2 values on the same finite scale, I can apply the weights: method 1 would have a weight of 1 if its transformed value is 0.5 and a weight of 0 if its transformed value is either 0 or 1, and method 2 would have a weight of 1 if its transformed value is either 0 or 1 and a weight of 0 if its transformed value is 0.5. I'm not sure which sigmoid function is most appropriate or how quickly it should approach the asymptotes - is there a way to determine these things using the data?
I also asked this same question here.
I would approach this problem by seeing the weight function as curves. There will be one velocity where both the measurements will give the same value, and this will be the accurate value of the velocity.
As infinity is an impractical limit, a linear curve for weights may not make sense as it will bias the low velocity measurements more in the practical range. I would suggest a cubic or quartic function depending on your working range of velocities. The coefficients of the cubic curve will be dependent on the velocity of equal measurement. Please refer to the following illustration of the idea.
The dotted lines show the linear weights and the advantages of cubic or higher degree weight is clear from the picture.
Hope this helps