High degree least squares polynomial fitting using discrete orthogonal polynomials

© Siberian Federal University, 2017

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Getting Started

This repository contains the code used in: S.P.Tsarev, A.A.Kytmanov, Discrete orthogonal polynomials as a tool for detection of small anomalies of time series: a case study of GPS final orbits: https://arxiv.org/abs/2004.00414

All algorithms for best least squares polynomial approximation in this repository were implemented in Julia programming language.

The Julia scripts robustly approximate a (probably vector-valued) time series with a very large number of points (thousands) with polynomials of large degrees (up to a few hundred) in the sense of the best RMS (best least squares) approximation and find the approximation residue. You can approximate your dataset by polynomials on the entire data (or a part thereof) available in the file, or alternatively approximate the data using a sliding window (possibly with overlapping).

Details on script parameters and the obtained results are given as comments inside the scripts themselves.

The scripts come in 4 different versions:

• for calculation with the standard (8-byte) floating point type Float64 for time series with constant time step (equidistant argument lattice);
• for calculation with the standard (8-byte) floating point type Float64 for time series with (slightly) non-constant time step (nonequidistant argument lattice);
• for calculation with the arbitrary precision floating point type BigFloat for time series with constant time step (equidistant argument lattice);
• for calculation with the arbitrary precision floating point type BigFloat for time series with (slightly) non-constant time step (nonequidistant argument lattice);

When you are not sure if your problem may involve fast error accumulation use BigFloat versions, but they are slow and more memory-consuming! Usually (for data up to 10,000 points) and approximation up to degree 200 you will be safe with Float64 versions. We found that using a modest size sliding window on the big time series may be more informative and much faster than approximation of the complete set of data: the log file result_max_resid.txt will give a concise report on approximation results for every window.

The Julia scripts for processing time series data (with English comments) can be found in subdirectory en.

Complete examples of running the scripts with the log files and results can be found in subdirectory test.

The Julia scripts with comments in Russian (otherwise absolutely identical to the English versions) as well as the Legal disclaimer (ru).txt and SFU OPEN LICENSE 2020 (ru).docx from Siberian Federal University can be found in subdirectory ru.

Installing

You need only a Julia installation to run the code. Please note that all version require the file service_functions-en.jl (or service_functions-ru.jl for Russian language versions) where some utility functions are collected.

Authors

Sergey P. Tsarev, Alexander S. Pustoshilov

Please write to [email protected] if you have questions, comments or suggestions.

Cite the paper S.P.Tsarev, A.A.Kytmanov, Discrete orthogonal polynomials as a tool for detection of small anomalies of time series: a case study of GPS final orbits: https://arxiv.org/abs/2004.00414 if you need a publication reference.