Add plot recipe for fit result #180
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We could maybe do this. I'm not sure I understand the prediction part. |
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The confidence interval is likely to contain the true mean. The prediction interval is likely to contain any new observation. |
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Sorry about the low quality illustration, here's a better showcase of the feature, generated by the test code. (Note that the first two series in every subplot are generated by separate code calls, this PR adds the rest)
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| function margin_error(model::Function, x, fit::LsqFitResult, alpha=0.05; purpose=:confidence) | ||
| g = p -> ForwardDiff.gradient(p -> model(x, p), fit.param) | ||
| c = g(fit.param)' * estimate_covar(fit) * g(fit.param) | ||
| if purpose === :prediction |
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I see this was in the original formulation on stackexchange, but where does the 1 come from?
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My statistics textbook has a derivation for this, but it's sadly in Swedish. I'll see later today if I can find a good reference. Essentially the difference between prediction and confidence coincidentally comes down to 1 dof.
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I see. You can post the picture of the Swedish text, I'll be able to read it.
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Here's the derivation in Seber, G.A.F & Wild, C.J 1989: Statistical inference (My textbook only had it for the linear case). My statistics isn't quite strong enough that I could convert this directly to working code, but I think it contains the source of the 1.
Intuitively, your next value is likely to land in an interval that is the confidence interval plus one "standard deviation" to either side, because that's where it's likely to land given the position of the true mean is inside the confidence interval. I hope this makes sense, otherwise it might be an idea to hold off on this part of the PR.
Yeah, I like it. Just want to understand the prediction part first, I've never used that myself :) |
(might close #139)
Implements a plot recipe for fit results. Uses the delta method (as described here) to form confidence and prediction bands.
My statistics is shaky, it's been a couple of years, so this might be a terrible way to do this. Which is why I'm asking for input before I go through the trouble of writing docs and tests.
Is this desirable?
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