Assessing the severity of missing data problems with the interval discrete Fourier transform algorithm

The interval discrete Fourier transform (DFT) algorithm can propagate in polynomial time signals carrying interval uncertainty. By computing the exact theoretical bounds on signal with missing data, the algorithm can be used to assess the worst-case scenario in terms of maximum or minimum power, and to provide insights into the amplitude spectrum bands of the transformed signal. The uncertainty width of the spectrum bands can also be interpreted as an indicator of the quality of the reconstructed signal. This strategy must however, assume upper and lower values for the missing data present in the signal. While this may seem arbitrary, there are a number of existing techniques that can be used to obtain reliable bounds in the time domain, for example Kriging regressor or interval predictor models. Alternative heuristic strategies based on variable (as opposed to fixed) bounds can also be explored, thanks to the flexibility and efficiency of the interval DFT algorithm. This is illustrated by means of numerical examples and sensitivity analyses.