For example, in hydrology, for a monthly flow series, how can I determine the possibility that one point indicates an extreme high (or low) flow event, and the duration of that event?
Hi Yang,
You may want to consider the paper attached. Here's its Abstract:
"In this article we consider the problem of detecting unusual values or outliers from time series data where the process by which the data are created is difficult to model. The main consideration is the fact that data closer in time are more correlated to each other than those farther apart. We propose two variations of a method that uses the median from a neighborhood of a data point anda threshold value to compare the difference between the median and the observed data value. Both variations of the method are fast and can be used for data streams that occur in quick succession such as sensor data on an airplane."
Best wishes
One-point extreme events are often handled with extreme events distributions or outlier analyais. In case of, let's say, "extreme periods" (i.e. more than one extreme value close to the other) as it seams since you are considering duration, you probably should consider the change-point detection analysis. This approach is not that popular even if it is widely applied in the homogeneity assessment and in the homogenization procedures adopted in climatology.
A broade literature exists on this topic. To have a look on the part concerning climate go to: www.homogenisation.org , the site provides a nearly complete list of papers on the topic (theory, applications and so forth). Obviously the same techniques adopted in climatology can be applied to other tipes of data series, provided that they comply the conditions the specific method is suited to handle.
Best wishes
Hi Yang
As you can see there are several solutions. It depends on the meaning of Extreme for you. I mean, it depends on the final target of the analysis. E.g. if you are dealing with floods, you are not interested on the actual duration of the rainfall itself but on the time scale of the basin response. I attach here a paper where a method for identification of extreme rainfall events is proposed for climatological studies. It is based on statistical criteria. Hope that helps
I agree with Michelle Rienzner on the use of extreme value distributions as a useful tool. However, there are a number of traps for the unwary! Firstly, the extreme value theorem only applies in similar circumstances to which the Central Limit Theorem applies: independent, iid observations and stationarity for example. This can be approximately achieved by regressing out any known effects, to achieve a residual time-series, and fitting that with a Weibull distribution (for extreme high values). An appropriate quantile-quantile plot will reveal outliers as points that plot off the line. Like the CLT, this analysis applies to any distribution of extreme values. Note that the coefficients of the above line (slope, intercept) are effectively proportional to the coefficients of relevant Weibull distribution.
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