Detecting Outliers under Interval Uncertainty: A New Algorithm Based on Constraint Satisfaction

In many application areas, it is important to detect outliers. The traditional engineering approach to outlier detection is that we start with some “normal” values x1, . . . , xn, compute the sample average E, the sample standard deviation σ, and then mark a value x as an outlier if x is outside the k0sigma interval [E − k0 · σ,E + k0 · σ] (for some pre-selected parameter k0). In real life, we often have only interval ranges [xi, xi] for the normal values x1, . . . , xn. In this case, we only have intervals of possible values for the bounds L def = E − k0 · σ and U def = E + k0 · σ. We can therefore identify outliers as values that are outside all k0-sigma intervals, i.e., values which are outside the interval [L, U ]. In general, the problem of computing L and U is NP-hard; a polynomial-time algorithm is known for the case when the measurements are sufficiently accurate, i.e., when “narrowed” intervals [ x̃i − 1 + α 2 n ·∆i, x̃i + 1 + α 2 n ·∆i ] – where α = 1/k0 and ∆i def = (xi − xi)/2 is the interval’s half-width – do not intersect with each other. In this paper, we use constraint satisfaction to show that we can efficiently compute L and U under a weaker (and more general) condition that neither of the narrowed intervals is a proper subinterval of another narrowed interval.