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Suppose I have a machine that has some output behavior y over some independent variable x. I set up a predictive model to predict the value of y for an arbitrary x and it's working well.

Then I get a second machine which is a lot like the first. I build a model for this as well. The models are completely independent. It works fine, too. When you examine the data, you see that the machines behave in pretty similar ways -- maybe they both have a linear relationship, but a slightly different slope. Probably there's a physical variation between the machines.

Maybe my third machine is similar, but I have almost no operating data for it so I can't make a very good model. I know it will behave similarly to the other machines and I want to exploit this knowledge.

What sort of options exist for modeling $n$ related items as a "fleet"?

Here are some naive ideas:

  • Treat all data as one big dataset, i.e. fit one 'average' model for all machines
  • As above but add as many additional features as possible that might capture variations across machines
  • Cluster the machines into k groups and fit k models
  • Use a more sophisticated method which is designed for this type of problem (what options exist?)

If the input data is time-series, does that change anything?

Also, my company is calling this "fleet analytics", is there another name for this type of problem?

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The classical approach to this kind of problem is a multilevel model. The basic idea is that the parameters of each cluster (machines, in your case) are treated as random variables themselves, coming from one or more higher-order distributions that are common to all the clusters. The practical consequence of using a multilevel model, compared to fitting a separate model for each cluster, is that parameters sharing a common distribution are estimated as being closer together in value.

There's a rich frequentist toolbox for multilevel models, but they also work well with a Bayesian approach; the distributions of the random effects work just like prior distributions.

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