Two measurements from different sources - how to combine?

[9988776655]

I can get a measurement of my distance between points (x1,y1,z1 and x2,y2,z2) by analysing position data from a GPS/barometer system, which has a standard deviation of about 2m for x y z positions. I can also analyse data from a system that provides velocity with a standard deviation of about 0.5 m/s for x y z velocities. I integrated the magnitude of the velocity values, obtained position and used this to find distance. Given that I have 2 values for distance, how should I combine them to report the best estimate?

I have not repeated the experiment. The object moved while the two devices outputted their measurements.

 

[mfb]

how should I combine them to report the best estimate?

With a weighted average, where the weights are the inverse variances.

If your two points are not far away (both in space and time), GPS will give the difference much more accurate than the positions on their own because the uncertainties are highly correlated.

 

[9988776655]

I don't know the standard deviation of the distance that I obtained after integrating velocity. I only know that the standard deviation for the velocity was 0.5 m/s. Would the standard deviation of the distance be the standard deviation of the velocity multiplied by the sample period? I numerically integrated the magnitude of velocity:
d(i) = d(i-1) + v(i)*dt
Where d is the distance, v is the the velocity and dt is he sample period. i is the sample index.
Is the standard deviation of the distance (obtained by finding the difference in two consecutive GPS positions) be equal to the standard deviation of the position given by the GPS?

 

[Stephen Tashi]

How much time do you have to invest in this mathematical question? For example, it might be that you are writing a report than very few people will read carefully and those that do read it won't take any significant actions based on the techncial details. . Or it might be that the object is a sunken treasure and estimating cost of a multi-million dollar expedition to find it depends crucially on the "uncertainty" in the object's location.

I can get a measurement of my distance between points (x1,y1,z1 and x2,y2,z2) by analysing position data from a GPS/barometer system, which has a standard deviation of about 2m for x y z positions.

For a routine report, you could assume the errors are independent. For a sunken treasure, there might be papers written on how (x,yz) errors in GPS/barometer systems are correlated and those should be consulted

I can also analyse data from a system that provides velocity with a standard deviation of about 0.5 m/s for x y z velocities.

For a routine report, we could assume the velocity measurements are indpendent and identically distributed. For sunken treasure, we should look at how your x,y,z velocity data was generated. Often such data doesn't not come directly from 3 sensors, each measuring a particular coordinate direction. Instead it may come from other less direct measurements that are run through some algorithm to produce (x,y,z) data.

I integrated the magnitude of the velocity values, obtained position and used this to find distance.

It will be important to know how many velocity values were used in the integration. One would expect the estimate of total distance to be more subject to error when more measurements are involved. There are different ways to estimate integrals from discrete data. Some involve fitting polynomials to intervals of the data and then integrating the polynomials. What method did you use?

Given that I have 2 values for distance, how should I combine them to report the best estimate?

Unfortunately, there is no definition for "best estimate" in mathematical statistics. There are definitions for technical things like "minimum variance unbiased estimator". For a routine report, you can try to find the "minimum variance unbiased estimator" if it exists.

The meaning of "the distance" needs to be confirmed. Suppose the object starts its trip at the origin of the coordinate system. Are you interested in the standard deviation of the distance between the origin and the estimated final position ( that one distance, not the values of the x,y,z coordinates)? Or are you interested in the distance between the final estimated position and the actual final position of the object?

For sunken treasure, you need to consider the cost of making errors in estimates. For example, thinking of a flat earth, the (x,y) position of the treasure might be more critical to get right than the z of it, if the z is always the bottom of the ocean at (x,y).

 

[mfb]

I don't know the standard deviation of the distance that I obtained after integrating velocity. I only know that the standard deviation for the velocity was 0.5 m/s. Would the standard deviation of the distance be the standard deviation of the velocity multiplied by the sample period?

Only if you know the velocity is constant. Otherwise the uncertainty will grow slower than linearly.
What are typical velocities and distances in your setup? It looks like the velocity measurement will be very imprecise compared to the GPS data, then you can simply ignore it.

Is the standard deviation of the distance (obtained by finding the difference in two consecutive GPS positions) be equal to the standard deviation of the position given by the GPS?

It would be sqrt(2) times this value of the uncertainties would be uncorrelated, otherwise it will be smaller (differential GPS can get precisions of a centimeter). It would help to get some rough size of the system to see if that is a good assumption.