Extreme value theory for detrended series I'm reading "An Introduction to Statistical Modeling of Extreme Values" by Stuart Coles, and using the pyextremes package for exploring the data which is time to return (in days). After detrend (as suggested in the book) my data I've got the model as 
                           Univariate Extreme Value Analysis                            
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                                      Source Data                                       
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Data label:                          None      Size:                                 781
Start:                          July 2020      End:                       September 2022
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                                     Extreme Values                                     
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Count:                                 17      Extraction method:                    POT
Type:                                high      Threshold:                             15
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                                         Model                                          
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Model:                                MLE      Distribution:                       expon
Log-likelihood:                   -36.168      AIC:                               74.603
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Free parameters:              scale=3.088      Fixed parameters:             floc=15.000
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Usually, with you detrend your series and make a prediction, the value would be something like $\hat{y} = \theta + \text{trend}$ where $\theta$ is the prediction. My results are

so, for once every 10 years a person will return after 28 days from the last visit. But this is for the detrended series. How can I interpret this result for the original series?
 A: I'm puzzled by your expression of $\hat{y} = \theta + \text{trend}$, I don't think this is the standard expression of a time-series. $\theta$ we judge to be a constant, but the "fixed part" is more nuanced. TS, being a regression, comprises a fixed part and the random part. The "trend" is the fixed part. The "trend" may be exogenous covariates (other "x's") or lagged effects of $Y$, and the random part constructs uncertainty bounds, makes inference, and flags outliers. In other words, the "trend" part should give the point-prediction (the red line in the upperleft quadrant does just that - we seem to assume a linear effect with time). The random part is a complex expression with autoregressive effects. Detrended time series should moreover be likened to residuals than to predictions. As such, the marginal uncertainty exhibits heteroscedasticity. In particular, the bounds seem to reflect an AR-1 autoregressive effect, a somewhat standard TS model. For a particular observation to be "extreme" it would lie beyond the bounds defined by the autoregressive funnel in said plot.
The predictions shown in the last panel do NOT appear detrended. The upper left quadrant shows precisely 1 time point labeled 1 where the red line seems to have a Y level of about 21: an exact correspondance. Then time points at 2 up to 10 - far far beyond the range of data - are shown. The uncertainty interval width magnifies accordingly.
Note the term "extreme value" is well defined yet Coles seems to adopt the term for a time-series specific application. In particular, the presence of an autoregressive effect means that the plot of response versus time shows a heteroscedastic funnel. And what we mean by "extreme value" may be considered to simply be an outlier. Note, outlier is not well defined, and so we may generally mean points that do not fit with trend in spite of accounting for error.
