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I am interested in making a time-dependent prediction model. The set up looks something like this:

Given weather data from the days Monday-Saturday I would like to predict the high point on Sunday. Now, this unfeasible since we only have a small number of data points. But I do have Mon-Sat weather data from $M$ different locations, say in California, where $M>300,000$. Part from the feature set includes location data. So I would like my model to have as input Mon-Sat weather data from any location in California, for example, Santa Monica, and as output a prediction of the high point. Given that I have $M$ locations to train on, I would like a model that takes into account areas that are similar to Santa Monica (other beaches and locations that are within a certain radius of Santa Monica) and ignores other locations like mountain/inland areas.

I have been searching for a model that fits this description but I haven't stumbled upon anything. Or does this training method simply not make sense? I have never done any time dependent models, so I wouldn't know how feasible this problem is. Thanks in advance for the help.

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Your description "I would like a model that takes into account areas that are similar to Santa Monica ... and ignores other locations" is a good summary of the k-NN family of approaches.

In the context of meteorology, relationships between climate at different locations are known as teleconnections (e.g. EOF maps).

However your problem also has a shorter-term weather component. For example "time dependent" would include things like advection of pressure systems. Weather forecasting incorporates these effects in great detail via sophisticated physics-based models that assimilate data using techniques such as Ensemble Kalman filtering.

Probably this is more than you would need, but simpler approaches could be possible. For example, advection could induce time-lagged correlations between nearby locations. However which locations are (tele-)connected, and the sign of the lag, would depend on which way the wind is blowing (i.e. the upwind direction). Now if you have a west-coast US location, and no data over the Pacific, this may be less relevant (given common circulation patterns).

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