binbagsss said:Thanks for the reply. ok so when its eg Ʃwixi - so this means the product of each x value and its weighting, summed up.
So how would you include the y weightings ?
haruspex said:I don't understand what it is that you don't understand. The formula you posted in the OP showed how include weightings on both x and y, and so (in a different format) does the link I posted. They're not separate weightings. Each weight is associated with a datapoint, both x and y.
haruspex said:If there are n datapoints, there are n weights. Weight w1 is used for both x1 and y1. Ʃwi is the sum of the n weights.
I doubt weighting in that way will have any effect.binbagsss said:Basically for my data all the x values have the same weighting, and all the y values have the same weighting
binbagsss is proposing to apply just two weights, one to all the x terms and another to all the y. I think this is based on the knowledge that one set of values is known more accurately than the other. It's not apparent what one would do with the Ʃwi term in that case, but even if that could be sensibly handled I believe such a weighting would have no effect, or if it had an effect not a desirable one. Can you corroborate that?rcgldr said:What the weighting does is effectively multiply the number of occurances of each sample point, including the ability to multiply by a non-integer, such as a weight factor of 1.5 which would mean that a sample point would be treated at 1.5 sample points in the formula, while a weight factor of 0.3 would mean that a sample point would be treated as 0.3 sample points. The "total" number of weighted sample points would be the sum of all the weighting factors.
If you want to use the same formula for non-weighted points, just set all the weighting factors to 1.0.
I reach the opposite conclusion. The weights should be related to the absolute uncertainties, not relative uncertainties.rcgldr said:If the x and y uncertainties are fixed instead of relative to the magnitude of x and y, you could base the weighting factors on the ratio of (magnitude of a data point) / (magnitude of uncertainty of that data point). If the uncertainties are relative to the magnitude of x and y, I don't see any point in using weighting factors.
rcgldr said:If the x and y uncertainties are fixed instead of relative to the magnitude of x and y, you could base the weighting factors on the ratio of (magnitude of a data point) / (magnitude of uncertainty of that data point).
haruspex said:I reach the opposite conclusion. The weights should be related to the absolute uncertainties, not relative uncertainties.
Suppose the plan is to fit some curve to the data then use this as the basis for predicting y values given x values in the future. If the cost function of an error it that prediction is related to the relative error (error/value) then you want the curve to be particularly accurate at small values. This means you want to give data in that part of the curve high credence and stop the curve becoming distorted by errors elsewhere in the data. This argues against basing the weighting on relative error.rcgldr said:An error of ± .001 has more effect on a data value of .01 than it does on a data value of 100, but I'm not sure if or how this should be taken into account.
haruspex said:As I understand it now, there is a known error range in both x and y values, and these can be different at each datapoint. The larger the error range, the smaller the weight to be allocated. Since only one weight can be assigned to each datapoint, it must represent both the x and y uncertainties. A root-mean-square combination sounds reasonable.
The weighted least squares formula is a statistical method used to minimize the sum of squared errors in a regression model. It takes into account the variability or weight of each data point, giving more weight to data points with lower variability and less weight to those with higher variability.
The weighted least squares formula is commonly used in regression analysis when the data points have unequal variances or when there is heteroscedasticity present in the data. It is also used in cases where there is a large amount of noise in the data.
The ordinary least squares method assumes that all data points have equal variance while the weighted least squares takes into account the variability of each data point. This makes the weighted least squares more robust and accurate in situations where there is unequal variance in the data.
The weighted least squares formula is advantageous because it gives more weight to data points with lower variability, which leads to a more accurate regression model. It also accounts for heteroscedasticity, a common issue in real-world data, and provides more reliable estimates of the regression parameters.
The weight for each data point is typically determined by the inverse of the variance of that data point. This means that data points with lower variability will have higher weights, while those with higher variability will have lower weights. In some cases, the weights can also be determined through expert judgment or by using a specific weighting function.