# What is the probability of one peak over threshold (POT) event in a given year?

I have got a dataset of daily rainfall data (mm) from different hydrological gauges. The observed time period is not fixed, and therefore the gauges' time series have different lengths (i.e. 10, 23, 50, 110 years) I selected a threshold u = 99th percentile and I would like to know:

What is the probability of a peak over threshold event happening in a given year (365 days)?

With annual maximum (AMAX) data p = 1/365 as per AMAX definition, i.e. one event per year but with POT data I can theoretically have one POT event per day.

Should I count the number of POT events in that particular year and divide them by 365? e.g. 40 POT events in a year, probability of one POT event in the same year p = 40/365 = 0.11

Is this right? If not, how would you approach the question? Thanks

• Are these "given years" within the range of your dataset or are they outside the data range?
– whuber
Apr 10 '17 at 18:40
• actually my dataset of gauges has no fixed time period length, meaning that I have time series of e.g. 10, 30, 55, 100 years mixed. thanks for your help Apr 10 '17 at 20:55
• It makes a difference whether you are estimating the probability for one of the years comprised by your data or for a year not included in your data.
– whuber
Apr 10 '17 at 21:15
• I need the probability for a year WITHIN my data Apr 10 '17 at 22:24
• That is of fundamental importance! It might (should) completely change the answers you get. Please make that clear within your question.
– whuber
Apr 11 '17 at 16:22

If I understand your question correctly, you want P(1+ events in a given year).

I'm going to make this tenable with the data provided by assuming rainfall is for any given day is an independent event from any other day. Clearly not the case in nature, but perhaps close enough).

For this type of problem it is almost always easier to find it this way: P(1+ event) = 1 - P(0 events)

The probability of no events in 365 days is $P_0^{365}$

If you've set your threshold to the 99th percentile, P_0 = 0.99

P(1+ event) = 1 - P(0 events) = $1 - P_0^{365} = 1 - 0.99^{365} \approx 0.9745$

Although any event is unlikely (only 1% for over the 99th percentile), the odds of 365 consecutive days without is very small. Over 97% of all years will have at least one day with such an event.

• A lurking issue in this context is that the threshold itself is an estimate--a 99th percentile of a 40-year period of observations--and therefore you also need to account for its uncertainty in your calculation. This question appears to be closely related to questions about tolerance limits.
– whuber
Apr 10 '17 at 18:24
• Excellent point! With the large number of data points, hopefully this can be estimated pretty accurately. Apr 10 '17 at 18:52
• Yes, one would hope that. We can check by computing confidence limits for the 99th percentile. The UCL, e.g., will be around the 99.15 percentile of the data if all 40*365 days were independent, but they were not. As an approximation we might suppose each year represents, say, 50 quasi-independent precipitation periods. If that's the case, then the 99.4 percentile of the data will serve. Since we're peering into the upper extremes, we should expect there to be substantial differences between the 99th percentile and the 99.4 percentile, suggesting this uncertainty should not be neglected.
– whuber
Apr 10 '17 at 19:23
• Agreed, depending upon the distribution, it could certainly be material. Apr 10 '17 at 19:57