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I have a bunch of time series where the data has a natural (known) cycle, for example daily or annual (or both). Here is an example (this is 6 years worth of temperature data sampled hourly):

enter image description here

Source data: http://pastebin.com/GESpGqVu

How can this data be compared? How can we check (for example) whether the first three years are significantly different from the last three years? That a particular month is significantly different from that month in all other years? That a particular day is unusual? (compared to both the same day in other years and adjacent days in the same year)

I would like a test for change vs no change, something that works as the cyclical data equivalent to a test for stationarity (unit root test, Dickey–Fuller, KPSS, etc).

I would also like some kind of descriptive statistics that works in this case. It is good to know that there is some change, but what is it? :)

I am mainly interested in detecting trends (eg: the summer-winter swing is increasing over time) and outliers (eg: a particular month is unusually cold) while taking all the available information about cyclical variation into account.

Some of the answers to other questions about time series comparison suggest doing ARMA/ARIMA etc on the two series (or parts of a series) that one is interested in comparing, and then an F test to see if the model parameters change. This may address the first part of my question (have the series changed), but it does not address the second part (how to explain in human-understandable terms how they have changed). I am also interested in any other (not autoregressive model comparison) standard or particularly good approaches to this.

See also (related questions, no full answer):

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    $\begingroup$ I believe that once you read through more than one of David Reilly's answers you will have a pretty full account of all the (many) questions you ask here. Could you point specifically to anything that is not addressed? $\endgroup$ – whuber Jul 5 '14 at 21:01
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    $\begingroup$ The first example here videolectures.net/mlss09uk_rasmussen_gp looks similar to your data, although with an evident trend. These are called Gaussian Processes, a semi-parametric technique that may apply to your case. $\endgroup$ – carlosayam Jul 8 '14 at 7:34

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