*2.3. Error Forecasting Model Based on Chaotic Time Series*

Chaos is a quasi-stochastic irregular motion possibly appearing in deterministic nonlinear dynamic systems [38]. Since various nonlinear systems exhibit chaotic features, chaos theory is widely used in nonlinear system analysis to detect deterministic relationships hidden behind random-looking phenomena, and has been increasingly used in time series analysis [30,31]. According to the delay coordinate embedding technique, the underlying dynamical system can be faithfully reconstructed from stochastic time series under fairly general conditions [39]. Therefore, a one-to-one correspondence can be established between the reconstructed and the true but unknown dynamical systems [40].

Given a scalar time series of model errors *e* = (*e*1, *e*2, ... , *eN*) with time step Δ*t*, and *N* is the number of elements in the time series, the element *ei* (*i* = 1, ... , *N*) is computed by Equation (1). According to the procedure of phase space reconstruction, the scalar time series *e* is transformed in phase space as follows:

$$\begin{aligned} \mathbf{E}\_1 &= \begin{pmatrix} \varepsilon\_{1\prime} & \varepsilon\_{1+\tau\prime} & \varepsilon\_{1+2\tau\prime} & \dots \end{pmatrix} \\ \mathbf{E}\_2 &= \begin{pmatrix} \varepsilon\_{2\prime} & \varepsilon\_{2+\tau\prime} & \varepsilon\_{2+2\tau\prime} & \dots \end{pmatrix} \\ &\dots \\ \mathbf{E}\_M &= \begin{pmatrix} \varepsilon\_{M\prime} & \varepsilon\_{M+\tau\prime} & \varepsilon\_{M+2\tau\prime} & \dots \end{pmatrix} \end{aligned} \tag{7}$$

where τ is the delay time, it could be several times of Δ*t*; **E***<sup>i</sup>* (*i* = 1, ... , *M*) is a chaotic vector in the phase space, *m* is the embedding dimension of the phase space, *M* = *N*–(*m*–1)τ is the number of phase point.

Takens [39] has proved that the chaotic attractor of a time series would be revealed in the phase space if the parameters τ and *m* are properly selected. The dimension parameter *m* is usually larger than three, to entirely reveal the underlying information of the time series [31]. Among existing methods for determining parameters τ and *m*, the coupled-cluster (C-C) method [41] is used in this study.

In the case of chaotic systems, the Lyapunov exponent (λ) gives a system the sensitivity to initial conditions and determines the total predictability of the system, and a positive λ indicates the system is chaotic [42]. Therefore, the reconstructed time series (**E**1, **E**2, ... , **E***M*) is tested for the chaotic signature through the maximum Lyapunov exponent which is evaluated by Wolf's algorithm [43].

In the phase space of a chaotic system, the dynamic information could be interpreted in the form of *m*-dimensional mapping as [30]:

$$\mathbf{E}\_{M+1} = f(\mathbf{E}\_M) \tag{8}$$

where **E***<sup>M</sup>* is the state at current time, **E***M*+<sup>1</sup> = (*eM*+1, *eM*<sup>+</sup>1+τ, *eM*<sup>+</sup>1+2τ, ... , *eM*<sup>+</sup>1+(*m*−1)τ) is the state at future time. Note that, the last element *eM*<sup>+</sup>1+(*m*−1)<sup>τ</sup> of **E***M*+<sup>1</sup> is exactly the next element *ek*<sup>+</sup><sup>1</sup> of the error series *e* which needs to be forecasted. Therefore, the phase point **E***i*(*i* = 1, 2, ... , *M)* further evolves into **E***i*<sup>+</sup>1, and there is a determinism mapping function between *ei*<sup>+</sup>1+(*m*−1)<sup>τ</sup> (i.e., the last element of **E***i*<sup>+</sup>1) and **E***<sup>i</sup>* as follows:

$$x\_{i+1+(m-1)\tau} = f(\mathbf{E}\_i) = f(\mathbf{e}\_1, \mathbf{e}\_{1+\tau\prime}, \mathbf{e}\_{1+2\tau\prime}, \dots, \mathbf{e}\_{1+(m-1)\tau}) \tag{9}$$

According to the properties shown in Equations (8) and (9), the chaotic time series can be utilized for prediction, and then the LSSVM approach described in Section 2.3 can be used to establish the nonlinear functions in Equation (9). The model input data and output data for LSSVM training are shown as follows:

$$\mathbf{X}\_{\text{error}} = \begin{vmatrix} \mathbf{E}\_1 \\ \mathbf{E}\_2 \\ \mathbf{E}\_{M-1} \end{vmatrix}; \ \mathbf{Y}\_{\text{error}} = \begin{vmatrix} \mathcal{e}\_{2+(m-1)\tau} \\ \mathcal{e}\_{3+(m-1)\tau} \\ \mathcal{e}\_{M+(m-1)\tau} \end{vmatrix} \tag{10}$$

where **X**error is the input data with the dimension of (*M*–1) × *m*, **Y**error is the output data with the dimension of (*M*–1) × 1.

Note that, due to *M* = *N*–(*m*–1)τ, the last element of **Y**error is actually *eN*, in other words, the last element of the error time series *e*. After the nonlinear function of Equation (9) is established by LSSVM, one can predict the future element of *e* at next time step through *eN*<sup>+</sup><sup>1</sup> = *f*(**E***M*).
