Discrete orthogonal polynomials: anomalies of time series and boundary effects of polynomial filters

Abstract: We describe a new result in the classical theory of univariate discrete orthogonal polynomials: extremely fast decay of their values near the interval boundary for polynomials of sufficiently high degree. This effect dramatically differs from the behavior of much more popular in mathematical curricula continuous orthogonal polynomials.

The practical importance of this new result for the theory of discrete polynomial filters (widely applied for detection of anomalies of time series of measurements) is demonstrated on the practical example of detection of outliers and small discontinuities in the publicly available GPS and GLONASS trajectories.

Discrete polynomial filters, on one hand, can detect very small anomalies in sparse time series (with amplitude of order 10^(-11) relative to the typical values of the time series). On the other hand our general result limits sensitivity of polynomial filters near the boundary of the time series. The main problem in practical applications of the discussed method is numerical instability of construction of the discrete orthogonal polynomials of high degree.

Zoom link for the talk: us04web.zoom.us/j/73902155099?pwd=ZnhXVUtIbUhPNmk4MFJ2dGpLNllZUT09

machine learningmathematical softwaremathematical physicsdynamical systemsnumerical analysisdata analysis, statistics and probability

Audience: researchers in the topic

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Mathematical models and integration methods