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Hi all,

I have a basic question about moments of time series (say, the average temperature at some place)

is there a mathematically rigorous way to define time-dependent moments of a time series? E.g. I've heard the claim that "since 1987 the average temperature of place X has become one degree higher" but I've got some problems with comparing the time average before the year $T$ to the time average after the year $T$, because I guess it depends on $T$. (in the case of the claim I suppose that $T=1987$ yields the maximal temperature difference)

I've read that for climate models one can use the ensemble mean. One can run an ensemble of models (and assume ergodicity to relate it to the time average) and take moments with respect to this ensemble. In this case one can notice that the ensemble moments exhibit time-dependent behaviour.

But I'm wondering if something similar is possible for just one time-series. Is there a mathematically rigorous and meaningful way to examine if some moments of a certain time series are time-dependent?

## Comments

There isn't any objectively "best" way of doing this with a single time series. What people typically do is smooth the time series and compute differences between the smoothed values. Simple moving averages are common among climatologists; statisticians often use lowess. You can get uncertainties this way if you use a smoothing method that gives them (e.g., smoothing spline or Gaussian process). But even this doesn't solve the "base year" problem, which is that differences are respect to some time (T), which may be arbitrarily chosen. If you care, you can compute the differences for a range of T and examine the dependence.

`There isn't any objectively "best" way of doing this with a single time series. What people typically do is smooth the time series and compute differences between the smoothed values. Simple moving averages are common among climatologists; statisticians often use lowess. You can get uncertainties this way if you use a smoothing method that gives them (e.g., smoothing spline or Gaussian process). But even this doesn't solve the "base year" problem, which is that differences are respect to some time (T), which may be arbitrarily chosen. If you care, you can compute the differences for a range of T and examine the dependence.`

Thanks for the explanation. I didn't know lowess (nor loess) before you had mentioned it.

`Thanks for the explanation. I didn't know lowess (nor loess) before you had mentioned it.`

for time series Sage has:

and

`for time series Sage has: > central_moment(k) > Return the k-th central moment of self, which is just the mean of the k-th powers of the differences self[i] - mu, where mu is the mean of self. Note that the central moment is different from the moment (here), since the mean is not and > moment(k) > Return the k-th moment of self, which is just the mean of the k-th powers of the elements of self.`