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I am working on modeling psychiatric disorders, where multivariate time series data - which I think is necessary to do any interesting modeling - is only broadly available recently, due to the availability of surveys via mobile devices.

To get into the dynamic systems modeling literature, I would like to look into a biological mechanism, which is well-modeled, by which I mean that the error/misfit is negligible for many applications. Basically, I am looking for a (historical) example, where I can follow the steps that were taken to get from the observed (maybe experimental) data to a good model, with the hope that I can learn some lessons for my own work. Ideally, it would be an example with about 5 < p < 20 variables.

Any hints and suggestions are greatly appreciated and my apologies if this is a too trivial question.

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    $\begingroup$ Assuming you are talking about analytical modelling and not about statistical modelling, you can learn about modelling in general by learning the basics of math. This book from Otto and Day talks about analytical modelling in the fields of evolution and ecology. $\endgroup$ – Remi.b Feb 12 '17 at 16:35
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    $\begingroup$ while @Remi.b has a good point, I would counter from own experience that getting hands dirty with data is the best learning method for me. there are thousands of examples of biologically interesting mechanisms with data and models, but it would be very helpful to know more about what you want to do. Does it need to be time series data? Or psychiatric data? There is for instance really great data and models on molecular networks running plant circadian clocks from Andrew Millar's group (onlinelibrary.wiley.com/doi/10.15252/msb.20145766/abstract). $\endgroup$ – Maximilian Press Feb 12 '17 at 22:52
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I would point out that, assuming (as the other commenter noted) you are interested in analytic rather than statistical modelling (which seems a little unlikely to me given your area), most mathematical models are derived from first principles (often from physics) or other assumptions. This suggests (to me) that looking at models of totally unrelated systems is not so useful; the whole idea of such models is to formalize your domain knowledge!

But I'll give some examples anyway.

For example, consider biophysical models of neurons or continuum mechanical models of cardiac tissue, both of which have rich histories of mathematical models developed over decades of work, based on both theoretical and empirical considerations (albeit largely from physics). Perhaps slightly more related (or at least empirically driven) are the advances in systems biology, which tend to derive from some underlying laws of chemical kinetics, but utilize enormous amounts of (multivariate, temporal) data in fitting (often non-linear stochastic) systems of differential equations. Methods for computational protein structure prediction tend to use a mix of ad hoc rules (for computational time-cost reasons) and physics-based considerations in their models. Evolutionary biology and population dynamics also have wonderful models (see the book from the commenter above).

My guess, given that your question does not have much detail, is that you would prefer statistical, i.e. probabilistic graphical or stochastic process, models instead.

Let's suppose you have a bunch of variables over time (e.g. the actions or status of a person) but you want to estimate an unobservable quantity (e.g. say, the degree to which they are insane [not my field, sorry]). One simple probabilistic model is the hidden Markov model. It has many examples of applications in sequence biology. It also lets you figure out, in a way, which variables are important, and let's you tune certain parameters (e.g. it's order, which here describes how much the person's state at time $t$ affects their state at time $t+\Delta t$). In terms of predictive modelling power, though, the state of the art is in deep learning models for time series.

Anyway, I'd consider what your goal is more carefully. If you indeed have a strong idea of an analytic model (motivated by some theory you know), then by all means formalize it. However, if you want the data itself to suggest a model for you, then use a statistical approach.

Note: apologies if this answer was too trivial, I was not sure what you did and did not know.

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