Symplectic geometry spectrum regression for prediction of noisy time series

We present the symplectic geometry spectrum regression (SGSR) technique as well as a regularized method based on SGSR for prediction of nonlinear time series. The main tool of analysis is the symplectic geometry spectrum analysis, which decomposes a time series into the sum of a small number of independent and interpretable components. The key to successful regularization is to damp higher order symplectic geometry spectrum components. The effectiveness of SGSR and its superiority over local approximation using ordinary least squares are demonstrated through prediction of two noisy synthetic chaotic time series (Lorenz and Rössler series), and then tested for prediction of three real-world data sets (Mississippi River flow data and electromyographic and mechanomyographic signal recorded from human body).

Figure 1

Prediction with SGSR regularization technique and OLS for data generated from the Lorenz system corrupted with $2\%$ (a) and $5\%$ (b) NLs.

Figure 2

Prediction with SGSR regularization technique and OLS for data generated from the Rössler system corrupted with $5\%$ (a) and $10\%$ (b) NLs.

Figure 3

The effect of nearest neighbors for 20-step prediction of Lorenz series with $2\%$ (a) and $5\%$ (b) NLs.

Figure 4

The effect of nearest neighbors for 20-step prediction of Rössler series with $5\%$ (a) and $10\%$ (b) NLs.

Figure 5

Mississippi River flow over year 2016 and its 30-step prediction values by SGSR and OLS: direct time series plot (a) and receiver operating characteristic curves (b).

Figure 6

The Mississippi River flow prediction NMSEs with lead time from 1 to 30 days for SGSR and OLS.

Figure 7

A 500-point EMG segment and its 32-step prediction by SGSR and OLS: direct time series plot (a) and scatter plot (b).

Figure 8

A 500-point MMG segment and its 32-step prediction by SGSR and OLS: direct time series plot (a) and scatter plot (b).