Say I have a number of time series, e.g. a number of temperature records from various stations in a region. I want to get a single temperature record for the whole region with which I could describe aspects of the regional climate. The intuitive approach might be to simply take the average of all stations at each timestep, but my statistical spider-sense (which I'm definitely not well in touch with yet) tells me that this might not be so easy. In particular, I imagine that averaging over the entire region will remove some of the interesting temperature extremes, and I might have problems with dependence between close stations.

What other problems might I face if I tried a strategy like this, and are there ways to overcome them, or more sensible methods of combining this kind of data?

Note: Answers can be more general than the spatial example I've provided.

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    $\begingroup$ The problem might be in your conflict between "a single temperature record for the whole region" and any interest you have in intra-region variation. A solution might involve some way of reconciling these two issues eg partitioning variance into intra- and inter- region components. $\endgroup$ Apr 5 '12 at 10:38
  • $\begingroup$ @PeterEllis, yeah, I was vaguely thinking of that. For the purposes of the question, let's assume I don't care about intraregional spatial variability. $\endgroup$
    – naught101
    Apr 5 '12 at 10:56
  • $\begingroup$ in that case, I think the main thing you have to worry about is the dependence between close stations. Find a way to weight down observations that effectively duplicate the station next door, and you should be ok. $\endgroup$ Apr 5 '12 at 11:16
  • $\begingroup$ @PeterEllis: ok, but there might not be a reasonable physical way to do that - Closeness of stations doesn't necessarily mean that they are more dependent - ie. two close stations on the opposite sides of a mountain range might be less similar than two distant stations on a broad plain. Is there a reliable way to define dependence statistically? Covariance, I suppose... There are still likely to be less peaks in the resultant series (I guess that reflects the physical situation though - temp changes over a broad region are likely to be slower and steadier than at a single place). $\endgroup$
    – naught101
    Apr 5 '12 at 11:31
  • $\begingroup$ @naught, regarding the spatial aspect of your question, how are your regions defined? In your comment, you mention that two close stations on opposite sides of a mountain could be different from two distant stations on a broad plain. Have you considered re-defining the station regions based upon proximity and similarity for your analysis? They wouldn't have to necessarily match up to conventional regional boundaries. Instead they could become an analytical overlay that could be plotted over a traditional map. $\endgroup$
    – dav
    Apr 5 '12 at 11:43

First, I'd like to say that I would be adding a comment, but I can't do that yet (rep), but I like the question and wanted to participate, so here's an "answer". Also, I see that this is old, but it's interesting.

First, would it be possible to use a dimension-reduction technique, like PCA, to condense the time series? If the first eigenvalue is large, maybe that means that your use of the eigenvector would represent most of the dynamics.

Second, and more generally, what is your desired use of the time series? Not knowing much else, I would guess that the temperatures could vary quite a bit. E.g., if some temperature records are near cities, you could get a "heat island" type effect. Or perhaps a small change in lateral distance happens to yield a large change in vertical distance--- one location could be at sea level and right on the ocean, and another not "too far away", but at a kilometer in elevation. Those would definitely have different temperatures!

These are just some thoughts. Maybe someone else could jump in and give a better answer.

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    $\begingroup$ Good point. To be honest, I can't remember what the context around this question was, and I have a feeling that my comments were misleading. I was specifically interested in not losing variability common to all stations, but out of phase. Think of stations all over a continent, and a cold front passing across. A simple spatial average might basically remove the cold front, which is not really a good thing, as each station would exhibit it strongly, but at different times. Probably running some kind of PCA on each station and then averaging the results somehow might be a way to get around that. $\endgroup$
    – naught101
    May 6 '13 at 6:26
  • $\begingroup$ O wow, ok, so if you're trying to characterize the trend in the time series over the region, maybe you should make the time series stationary, and each with a mean of 0. You could also try to remove the daily cycle from each one (or just take daily averages). Then you'd be left with lower-frequency changes in temperature, each centered around a mean of 0. Once you have that, then maybe you could condense the centered+stationary time series using some dimension reduction technique, such as PCA. I'm glad you talked about the context of the question a bit, b/c that really helps. Good stuff! $\endgroup$
    – rbatt
    May 6 '13 at 11:20

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