I am estimating parameters on a dataset that would, for the most part, result in a weakly constrained solution. The dataset however also contains a few more data points that make the solution well-conditioned. The issue is illustrated in an example of a simple line fit in the sketch below.
While the few constraining points may actually be good/inliers the obtained solution clearly still heavily depends on these few data points.
I would now like to have a direct measure that can tell me the relative amount of evidence that contributed to the 'well-constrainedness' of a solution. (In the illustration below that number would indicate a low amount of evidence).
Does such a direct measure exist or would I have to go through sampling/clustering/segmenting the dataset?
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As pointed out in the previous answer, the concept you are looking for is sensitivity.
You take the formula that computes the parameters of your regression line and compute the derivatives with respect to your point positions. These should be large for the three isolated points, indicating that moving them will have big influence.
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$\begingroup$ Is there a direct way to compute sensitivity then? What you suggest amounts to computing d_solution / d_point for each data point which is a lot of work, especially for higher dimensional parameter spaces... $\endgroup$– rsp1984Jul 21 '13 at 15:23
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You want to look for the concept of "parameter sensitivity" in parameter estimates.
Another direction worth looking into is Bayesian inversion that gives you either a very narrow or a broad probability distribution in parameter space. There are many good introductions. My (partial, biased) favorite is number 35 here: http://www.math.tamu.edu/~bangerth/publications.html#x-reviewed
answered Jul 21 '13 at 2:11