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Forecasting solutions to a differential equation is hard, especially when there are infinitely many solutions. It is a matter of keeping constants up to date in

Forecasting solutions to a differential equation is hard, especially when there are infinitely many solutions. It is a matter of keeping constants up to date in light of new information. If those constants are wrong, the whole model is essentially useless.

jkqwzsoo 7y ago
> It is a matter of keeping constants up to date in light of new information. I'd argue it's a matter of calculating confidence intervals for all of your fitted parameters and displaying joint confidence intervals if they display a large degree of interdependence/nonlinearity, as well as showing the fitted residuals beneath the main plot so it's easily apparent what issues the fitted function might have, such as high leverage, or overfitting. The prediction made will be from the best-fit parameters, but the contributions of other possible models can be used to infer a distribution of potential outcomes. A parameter sensitivity analysis wouldn't hurt either. Then you'll immediately know if you're looking at (or about to publish) junk.
LolWolf 7y ago
> Forecasting solutions to a differential equation is hard, especially when there are infinitely many solutions. Not sure what you mean by infinitely many solutions, but this is not true in general. A silly example is y'(t) = Ct, where, for most distributions, you would only need a few points to get both C and the initial condition to reasonably high accuracy (~O(sqrt(n))). More complicated examples exist that have much more interesting dynamics, but their general trajectories are just not as sensitive/chaotic (w.r.t. the initial parameters). > If those constants are wrong, the whole model is essentially useless. I think what makes this hard is not if they're wrong, but rather, being even just a tiny bit wrong makes the whole future prediction change drastically. In other words, two possible inferred parameters which are statistically indistinguishable given our current observations will yield incredibly different outcomes under many of these models. > It is a matter of keeping constants up to date in light of new information. Indeed! :)

jkqwzsoo 7y ago

> It is a matter of keeping constants up to date in light of new information. I'd argue it's a matter of calculating confidence intervals for all of your fitted parameters and displaying joint confidence intervals if they display a large degree of interdependence/nonlinearity, as well as showing the fitted residuals beneath the main plot so it's easily apparent what issues the fitted function might have, such as high leverage, or overfitting. The prediction made will be from the best-fit parameters, but the contributions of other possible models can be used to infer a distribution of potential outcomes. A parameter sensitivity analysis wouldn't hurt either. Then you'll immediately know if you're looking at (or about to publish) junk.

LolWolf 7y ago

> Forecasting solutions to a differential equation is hard, especially when there are infinitely many solutions. Not sure what you mean by infinitely many solutions, but this is not true in general. A silly example is y'(t) = Ct, where, for most distributions, you would only need a few points to get both C and the initial condition to reasonably high accuracy (~O(sqrt(n))). More complicated examples exist that have much more interesting dynamics, but their general trajectories are just not as sensitive/chaotic (w.r.t. the initial parameters). > If those constants are wrong, the whole model is essentially useless. I think what makes this hard is not if they're wrong, but rather, being even just a tiny bit wrong makes the whole future prediction change drastically. In other words, two possible inferred parameters which are statistically indistinguishable given our current observations will yield incredibly different outcomes under many of these models. > It is a matter of keeping constants up to date in light of new information. Indeed! :)