The Application Of Manifold Learning To Noise Reduction In Chaotic Time Series

With phenomena of chaos acknowledged more often in every field，chaotic time seriesnoise reduction has been attracting much more attention since20century. Due to the charactersof chaotic time series, pseudo random and broad band in both the time and frequency domain,the traditional linear noise reduction method which is based on power spectral analysis is nolonger suitable for the chaotic time series. With the aid of analyzing the methods of nonlineartime series noise reduction based on phase space reconstruction, the thesis proposes the frame ofthis kind of noise reduction methods: we can get high dimensional noisy trajectory matrix fromone dimensional noisy time series via phase space reconstruction, then obtain the much cleanertrajectory matrix from the original one by principal manifold learning and principal manifoldreconstruction, finally map the much cleaner trajectory matrix back to one dimensional cleanersignal. The main tasks of the thesis are as follow:(1) We propose the uniform frame of the nonlinear noise reduction based on phase spacereconstruction. What is more, we implement principal manifold learning by nonlineardimensional reduction, while we reconstruct principal manifold through local polynomial fitting.Finally, we prove that the algorithms based on principal component analysis agree with theframe.(2) Acoording to the uniform frame of the noise reduction algorithm based on phase spacereconstruction, the key improvement of the kind of noise reduction algorithm fouces on themanifold learning algorithm. The paper studies linear dimensionality reduction algorithms (PCAand MDS) and nonlinear dimensionality reduction algorithms(LTSA) on the basis of lineardimensionality reduction algorithms in detail.(3) We propose the criterion for choosing the optimal embedding dimension and time lag forglobal singular value decomposition noise reduction algorithm based on principal componentanalysis.The project is supported by the National Science Foundation called the extraction andapplication of the non-stationary signal’s fractal character based on image processing (NO.51105284).