Hi all,
I have a question on how to use/interpret Gumbel's law. Suppose I have 6 years of data on total yearly rainfall on a particular region, from 3 sites (mm/year). Say
Site 1: 541, 781, 978, 1421, 1017, 1267
Site 2: 1169, 1283, 1209, 1211, 761, 1076
Site 3: 998, 1219, 630, 798, 662, 989
Recall these are yearly values, not daily or monthly maxima. If I group the data per site, I get a sample of maximum values: x1 = {1421, 1283, 1219}. But if I group the data per year I get a different sample: x2 = {1169, 1283, 1209, 1421, 1017, 1267}. The two samples yield different Gumbel parameters y = 1/0.78 sd(x) * [max(x) - mean(x) + 0.45*sd(x)], namely
y1 = 1.984392
y2 = 2.422774
These lead to different probability and recurrence periods:
y1: Prob[ yearly rainfall in region > 1421] = 0.128 (1 in ~8 years)
y2: Prob[ yearly rainfall in region > 1421] = 0.0849 (1 in ~12 years)
Which one should I use for prediction of future rainfall, or are they both correct and it's just a matter of interpretation of the result? You can assume all data points are drawn from the same underlying normal distribution (which they actually are, with mean 1000, sd 200).
Thanks!
Well,
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Regards
Could you please explain a little more what is your data and how did you build your groups, I mean it is not clear what is site, what are the years and what are the maximums. Despite this, there are a couple of things to be said about your analysis.
1. Usually, you do not mix data from different sites because strictly speaking, each site represents a sampling location with its own statistical characteristics. Yes, in practice, if the sites are very close to each other (a few meters, maybe up to 100 meters or so), you could assume that you measure the same rainfall. But even then, you need to consider if both sites are under the same conditions, for example if they are both free of shadow from any structure or tree.
2. It is obvious that if you build a set with maximum values, this will yield different statistics than another set built under different criteria; in your case, grouped by time, no matter what the underlying distribution is. You can look at it in another way namely that by taking maxima, you have created a biased sample.
3. If for any reason you are working with artifically generated data (random number generator), I would say that also the number of values you have is too small to expect any "convergence".
Both the analysis and its interpretation depend on the question you are trying to answer and the data that you have actually used for it. But my first guess, is that actually non of your sets delivers you the expected answer. If you write some more details about what you are doing I might be able to help.
Finally, one word of caution: with this kind of analysis you are not doing any kind of prediction for the future. A more proper interpretation of this results would be to say something like, "under asumption of steady conditions (no change in time), a rainfall of XXXX ammount has an annual exceedance probability of YYYY.
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