Nonstationary time series prediction combined with slow feature analysis

using the Slow Feature Analysis (SFA) approach, then introducing the driving force into a predictive model to predict non-stationary time series. In essence, the main idea of the technique is to consider the driving forces as state variables and incorporate them into the prediction model. To test the method, experiments using a modified logistic time series and winter ozone data in Arosa, Switzerland, were conducted. The results 10

SFA is a method that extracts slowly varying driving forces from a quickly varying (2) 74 where H(t) is an N k × matrix and k = m + m (m + 1)/2.

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To simplify (2) as ( (4) 81 Here, k y  is first-order derivative, calculated by ) 3) Normalize the expanded signal H(t), by an affine transformation to generate H'(t) 83 with zero mean and uni t covariance matrix: 4) By means of the Schmidt algorithm, the function space (5) is orthogonalized as 98 Where c is a given constant and )} ( { 1 t y is the output signal of the slowest driving 99 force obtained by equation (7).

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In this study, the SFA was tested on a logistic map To test the ability to construct the driving force from this modified logistic map, we Here, m 1 and m 2 are the given embedding dimensions for respectively, and N = n − (max (m 1 , m 2 ) − 1)τ is the number of phase points on the 126 trajectory.

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Based on this trajectory, a predictive model to predict the future state of the system 128 can be established as: Where p is the prediction time step (considered as 1 in the present study), ) (t ε is the 131 fitting error, and fˆis assumed to be a quadratic polynomial in this study. The Takens 132 embedding theorem is only appropriate for an autonomous dynamical system, 133 therefore we followed the method of Stark (1999) to embed the driving forces in the 134 same state space for a nonstationary system. The next task is to find the cost function (12 ) when it reaches its minimum value. For more 136 details, refer to the studies of Farmer and Sidorowich (1987) and Casdagli (1989).

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3 Experime nts 138 We applied the prediction technique described above to perform some prediction 139 expe riments using several given non-stationary time series. The experiment presented 140 in Section 3.1 was performed with data from the modified logistic model given abo ve.

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The prediction experiments were based on 5000 data points from the above verified 143 logistic map (8) with the assumed driving force (9). The first 4800 data points were driving force } { 1 y when the embedd ing dimens ion was chosen as 3, 5,7,9,11,174 respectively (shown in Figure 3). Note that the result did not change significantly with 175 different embedding dimension values. 176 We established a pr ediction mod el for winter ozone data by incorporating the The experimental results for this case are listed in Table 1, also shown in Figure 4 186 and Figure 5. From Table 1, it can be seen that all RMSE values given by the forcing 187 mod el were much lower than those by the stationary mod el. Figure  In this study, we first constructed the driving forces of a time series based on the SFA 207 approach, and then the driving forces were introduced into a predictive mode l. By  The true and constructed driving force.

Figure 2
The comparison of prediction skills between models combined with or 313 without driving force.

Figure 3
The slowest driving force with different embedding dimension for total 315 ozone data.
316 Figure 4 The comparison of prediction skills between models combined with or 317 without driving force. Errors (Dobson Units) at prediction steps with or without forcing input. 349