Outlier Detection under Interval Uncertainty: Algorithmic Solvability and Computational Complexity

Abstract

In many application areas, it is important to detect outliers. Traditional engineering approach to outlier detection is that we start with some ”normal” values x ₁,...,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 k ₀-sigma interval [E − k ₀·σ,E + k ₀·σ] (for some pre-selected parameter k ₀). In real life, we often have only interval ranges $[{\underline x}_i,{\overline x}_i]$ for the normal values x ₁,...,x _n . In this case, we only have intervals of possible values for the bounds E-k ₀·σ and E+k ₀·σ. We can therefore identify outliers as values that are outside all k ₀-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).