I have some time series that is a list of positive numbers $x_1, x_2 ... x_n$. These numbers are ordered in time, and equally spaced, and the order between the number is important.
Now, I would like to identify some consecutive numbers in that time series that are unusually high. For example, if I have this series: 2,1,1,2,1,1,2,1,3,5,6,7,3,2,1, I would like to find that 5,6,7 are some consecutive numbers that are unusually high in that time series. These consecutive numbers represent something unusual.
Thus, my questions is what kind of mathematical models can be used to identify some consecutive numbers that are usually high in a time series? Some simple idea could be to just look at the average and look for numbers that are higher than the average + standard deviation. But it sounds a bit simple. Is there some other, perhaps better approaches to this problem?
About my data, I do not make any assumptions except that this is a time series of positive numbers. I also do not know the distribution of the data.
Note: I have previously asked a similar question on StackExchange. But this question is different. I want to identify an interval of consecutive values (in that question) rather than a single value (in my previous question).