Irregular events within an otherwise cyclic time series

(I have asked the same question on math.stackexchange, but I figured that physicists might actually be more likely to have encountered the same problem before.)

I am considering a time series with a fairly cyclical baseline behaviour. By averaging over multiple periods, one might obtain a good prediction for it.

However, there are a bunch of events that occur irregularly. (Even though they usually occur once during each period and are loosely correlated.) Each of these events has a strong and characteristical, but temporally limited impact on my time series.

I am looking for a forecasting method that can handle both the stationary, cyclic background and the spontanous but typical offsets. Does anybody have a suggestion for me?

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Cross-posted from math.stackexchange.com/q/306657/11127 – Qmechanic Feb 18 at 13:40
Perhaps stats.stackexchange.com is a better home? – Qmechanic Feb 18 at 13:43
I am personally puzzled by the question. The description of when the irregular events occur seems vague to me - yet it's clear that the answer you're looking for crucially depends on the precise rules "when" i.e. on the question. Also, if the irregular events occur randomly, it's clear that only statistical features of the time series (those obtained by collecting/averaging many periods etc.) may be predicted. But when it comes to those, you haven't specified what you really want to be predicted. To summarize, you want the folks to give you a nice answer but you haven't asked a nice question. – Luboš Motl Feb 18 at 13:50
@LubošMotl I will try to express myself better: There is a basal component in the time series, of which the expectation value and variance is a function of the time that has passed since the beginning of the period. And there are other events that are correlated and who have a coherent, strong and short-term effect on the measured values. Imagine a light sensor: The effect through sunshine can be predicted rather well if we know the time (since midnight), but sometimes a group of people in cars drives by, which causes coherent, correlated increases in light intensity. – Konstantin Feb 18 at 14:31