Outlier detection under interval and fuzzy uncertainty: algorithmic solvability and computational complexity

In many application areas, it is important to detect outliers. Traditional engineering approach to outlier detection is that we start with some "normal" values x₁,..., x_n, compute the sample average E, the sample standard variation σ, and then mark a value x as an outlier if x is outside the k₀-sigma interval [E-k₀·σ, E+k₀·σ] (for some pre-selected parameter k₀). In real life, we often have only interval ranges [x_i, x~_i] for the normal values x₁,...,x_n. In this case, we only have intervals of possible values for the bounds E-k₀·σ and E+k₀·σ. We can therefore identify outliers as values that are outside all k₀-sigma intervals. In this paper, we analyze the computational complexity of these outlier detection problems, and provide efficient algorithms that solve some of these problems (under reasonable conditions). We also provide algorithms that estimate the degree of "outlier-ness" of a given value x-measured as the largest value k₀ for which x is outside the corresponding k₀-sigma interval.