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I have sets of projected and actual time series, and given a new set of projections, I'd like to predict the actual series. I'm not trying to extend a series into the future, I'm trying to predict an entire set of series given past correspondences plus a "hint." Does this problem have a name and/or standard solutions?

A little more concretely: I have projections of resource usage over time. For each customer, say there are resources A, B, and C, and the projected usage over 6 months:

A: 1 1 2 3 3 3
B: 0 0 1 1 2 4
C: 2 2 1 0 0 0

Later, the actual measured usage comes in for each resource for each month, and it has the same shape but of course the numbers may be different (and are not integral in general). I have say ten resources for each of tens of customers over tens of months. The resources are not all the same for all customers, but they are drawn from the same universe of tens of resources. The hypotheses are that a) projected usage informs actual, b) usage between customers is similar, and c) usage between resources is similar.

I'd like to predict a customer's actual usage (the whole series for all resources), based on their projected usage and all past projected and actual usages. To be clear, I am not trying to predict the next value in a sequence; I'm trying to predict an entire set of sequences. I have more information than a traditional autoregressive or HMM situation, seems like I should use it. I could try supervised learning, but I'm curious if there's something specifically for timeseries. What's the best way to approach this problem?

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I'm not trying to extend a series into the future, I'm trying to predict an entire set of series given past correspondences plus a "hint." Does this problem have a name and/or standard solutions?

Yes. This is called "nowcasting".

The BSTS and CausalImpact packages in R can perform this type of analysis, although I've only used them for one series at a time (i.e one time series + the set of additional variables, in your case that would be one customer + the resources).

See this paper for details.

More generally, this type of problem can be modeled using state space models and dynamic linear models.

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