Linear regression considering standard deviation of sample points?

Question

I am trying to do many linear regressions on series of data points where each data point is associated with a standard deviation.

For simplicity sake, let's say that each series has 3 points (of mean measurements), each of which has its own standard deviation.

Series 1, points A, B and C. Points given in (XX-coordinate,YY-coordinate,SD)
A= (1,2,0.5)
B= (2,3,0.75)
C= (3,2,0.4)

I am trying to find a way to do a linear fit to the data points and consider their standard deviation to help determine the confidence that I have on each point (larger SDs meaning smaller confidence in that mean value).

I was thinking about three possible ways to do this, but I'm looking into a different approach:

1. Randomize the list of data. Say, for each series I would randomly choose a list of points that would vary around the mean value within the SD. This would, for instance, generate 1000 datasets for each series. I would then do the fittings to each dataset and to each series.

2. I would assign weights to each of the points that would be inversely proportional to the SDs. Those points where a higher SD is observed would have a smaller weight. However, in some series I have very big SDs for all 3 points, meaning that I am essentially giving each point the same weight anyway.

3. To each of the datapoints, assign the mean+SD and mean-SD as 2 separate points. For instance, in A= (1,2,0.5) I would have points (1,2),(1,1.5),(1,2.5). Then, do the regression to the 9 points instead of the three initial values.

How would you approach this issue?

Thanks

Answer 1

Lookup weighted least squares and generalized least squares. The former has some predefined weights, e.g. reciprocal of the point's standard deviation can be used. The latter can estimate the weights itself.