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Linear regression would not be a sensible approach for the data in that paper, since the relationship between X and Y is does not linear (conditional on knowing the clustering, there doesnt seem to be any relationship between X and Y at all, and any 'linearity' you find is going to be a spurious result of marginalising over the cluster allocations). You ...

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After your update, I can say by eye that your model seems to separate low and high Y well. The Ys are clustered. The fact that you observe that also the Xs are clustered in a similar way means that they are capturing something related to the Ys. You can estimate the predictive power by the mean squared error between the X and Y. Also, you can binarize the ...

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I'm interested in why you think your proposed approach is costly? Surely, the business would not be surprised that customers game the system. To me, this sounds like the only way to solve the problem (barring cases where you might have prior information). I don't think a classifier is the right approach here. You would probably want risk estimates (i.e. ...

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Consolidating the comments into an answer, this is a linear system. Each shipment has some number cartons which contain some number of PINs. We don't seem to be interested in the cartons themselves, so we can ignore that detail and represent the source data as an $n \times p$ matrix $A$ where there are $p$ products (PINs) and $n$ shipments. Likewise, there ...

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I know I'm late to the party, but: the theory behind the data imbalance problem has been beautifully worked out by Sugiyama (2000) and a huge number of highly cited papers following that, under the keyword "covariate shift adaptation". There is also a whole book devoted to this subject by Sugiyama / Kawanabe from 2012, called "Machine Learning ...

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