High accuracy behavior prediction of non-linear dynamic system with signal separation and support vector regression

Abstract

The behavior prediction of non-linear dynamic system is a challenging problem, especially when the system includes many independent subsystems. The observations from the complex dynamic system are the result of the interaction of multiple dynamic subsystems, which results in a loss of predictability. In this paper, signal separation technique is adopted to separate the observations of complex non-linear dynamic system in order to improve its predictability. And then local support vector regression technique is used to model the separated observations and make prediction. Finally, the prediction results are remixed as the original observation prediction or the behavior prediction of the complex non-linear dynamic system. The experimental results show that the proposed method improves the prediction accuracy substantially, which proves that signal separation can improve the predictability of complex non-linear dynamic system.

Keywords

Non-linear dynamic system signal separation support vector regression

1. Introduction

A time series is a series of observations from some dynamic system. In many situations, people always want to know the mechanism of the under-observed dynamic system and try to predict the system evolution in the future. The equivalent dynamic system needs to be reconstructed from the observed time series. Nevertheless, the observed time series are always noisy, non-stationary or even non-linear, which makes the equivalent dynamic system modeling quite difficultly. So, time series forecasting is a challenging statistical signal processing problem and has been applied in many fields such as network traffic demand prediction, medical diagnosis, air pollution forecasting, machine condition monitoring, financial investments, production planning, sales forecasting, channel prediction and so on [1, 2, 3, 4, 5].

Nowadays, many methods have been proposed to deal with the modeling and prediction of the time series. Box and Jenkins proposed the widely used ARMA model to model the time series and make prediction. Unfortunately, the ARMA model can work well only for stationary time series. Non-linear modeling methods have been proposed to model the non-stationary or non-linear time series and make prediction. Lee et al. proposed RBF neural network as a global model for prediction [6]. Wu et al. proposed support vector regression (SVR) as a global model for time series forecasting [7]. The global prediction model works well only when the non-linear time series varies slowly. Non-linear time series in the real world varies complicatedly and has strong self-similarity. Many local modeling techniques have been proposed for non-linear time series prediction, such as local wavelet neural network, FIR neural network, mixture expert models, support vector machine experts and so on [8, 9, 10, 11]. However, the prediction performance of local modeling is still not good enough when the observed time series is mixed with many non-linear time series generated by independent non-linear dynamic sub-systems.

In this paper, the signal separation technique and local prediction modeling are combined to do behavior prediction of complex non-linear dynamic system with many independent subsystems. Firstly, signal separation algorithm such as FastICA [12] is used to improve the predictability of observations. Secondly, the local support vector regression is trained with every separated independent series. Finally, the prediction results of every separated series with its local support vector regression are merged with mixed matrix obtained in the separation stage in order to predict the original observations.

2. Technical background

2.1 Support vector regression

The SVR [11, 13] is a linear regression in a feature space. The feature space is introduced by a function which maps the input space to the high dimensional feature space, i.e. $\phi:R^{m}\to F$ such that the non-linear function under approximation in the input space becomes a linear function in the feature space. Considering a linear function in the input space, i.e. $f(\textbf{x})=\left\langle{\textbf{w},\textbf{x}}\right\rangle+b$ , where w is the function coefficient vector, b is a real constant and $\left\langle{\cdot,\cdot}\right\rangle$ denotes the inner product on the input space. Let $\hat{f}$ be the estimation of a function $f$ from $N$ samples $DN=\{\textbf{z}_{i},y_{i}\}_{i=1}^{N}$ within a finite accuracy and a loss function is needed. A loss function known as a $\varepsilon$ -insensitive function, as defined in Eq. (1), is usually used to evaluate the robustness of SVR.

$\displaystyle\left|\zeta\right|_{\varepsilon}=\left\{{\begin{array}[]{ll}0&% \textit{if}\left|\zeta\right|<\varepsilon\\ \left|\zeta\right|-\varepsilon&\textit{otherwise}\\ \end{array}}\right.$ (1)

It allows errors to occur within the interval $\left[{+\varepsilon,-\varepsilon}\right]$ but no errors are allowed outside this interval. The $\varepsilon$ -insensitive function corresponds with a probability distribution of the noise, as shown in the following:

$\displaystyle P(\zeta)=\frac{1}{2(1+\varepsilon)}\exp(-|\zeta|_{\varepsilon})$ (2)

where $\zeta=\left({y-f(\textbf{x})}\right)$ .

The cost function penalizes the large values of w by using $||$ w $||^{2}$ so that the estimated function $\hat{f}$ is a smooth function.

The cost function of SVR is

$\displaystyle\min\frac{1}{2}||\textbf{w}||^{2}$ (3) $\displaystyle s.t.\left\{{\begin{array}[]{l}y_{i}-<\textbf{w},\textbf{z}_{i}>-% b\leqslant\varepsilon\\ <\textbf{w},\textbf{z}_{i}>+b-y_{i}\leqslant\varepsilon\\ \end{array}}\right.$

This constrained optimization problem can be solved by applying the Lagrange technique [14]. Therefore, we have

$\displaystyle L_{p}=\frac{1}{2}||\textbf{w}||^{2}-\sum\limits_{i=1}^{N}{\alpha% _{i}\left({\varepsilon-y_{i}+\left\langle{\textbf{w}_{i},\textbf{z}_{i}}\right% \rangle+b}\right)}-\sum\limits_{i=1}^{N}{\alpha_{i}^{\ast}\left({\varepsilon+y% _{i}-\left\langle{\textbf{w}_{i},\textbf{z}_{i}}\right\rangle-b}\right)}$ (4)

where $\alpha_{i}$ , $\alpha_{i}^{\ast}$ are Lagrange parameters. Equation (4) is known as the primary objective function. The corresponding dual objective is

$\displaystyle\max-\varepsilon\sum\limits_{i=1}^{N}{\left({\alpha_{i}+\alpha_{i% }^{\ast}}\right)}+\sum\limits_{i=1}^{N}{y_{i}\left({\alpha_{i}+\alpha_{i}^{% \ast}}\right)}$ $\displaystyle\qquad-\frac{1}{2}\sum\limits_{i,j=1}^{N}{\left({\alpha_{i}-% \alpha_{i}^{\ast}}\right)\left({\alpha_{j}-\alpha_{j}^{\ast}}\right)\left% \langle{\textbf{z}_{i},\textbf{z}_{j}}\right\rangle}$ (5) $\displaystyle s.t.\left\{{\begin{array}[]{l}\sum\nolimits_{i=1}^{N}{\left({% \alpha_{i}-\alpha_{i}^{\ast}}\right)}=0\\ \alpha_{i},\alpha_{i}^{\ast}>0\\ \end{array}}\right.$

Equation (2.1) represents a standard constrained quadratic programming problem, which can be solved by using optimization toolbox in our simulation study. The solution of the dual objective function is equal to the solution of the primary objective function. For non-linear cases, the dot product $\left\langle{\textbf{z}_{i},\textbf{z}_{j}}\right\rangle$ in Eq. (2.1) becomes a kernel $\left\langle{\phi(\textbf{z}_{i}),\phi(\textbf{z}_{j})}\right\rangle$ $=K(\textbf{z}_{i},\textbf{z}_{j})$ . The prediction is achieved by

$\hat{y}=\hat{f}(\textbf{x})=\sum\limits_{i=1}^{N}{\left({\alpha_{i}-\alpha_{i}% ^{\ast}}\right)K\left({\textbf{x},\textbf{z}_{i}}\right)}$ (6)

where $K\left({\textbf{x},\textbf{z}_{i}}\right)$ is known as a kernel function, such as the Gaussian kernel function:

$K\left({\textbf{x},\textbf{z}_{i}}\right)=\exp\left({\frac{-||\textbf{x}-% \textbf{z}_{i}||^{2}}{2\sigma^{2}}}\right)$ (7)

where $\sigma$ is the kernel parameter of variance.

2.2 The FastICA algorithm

Independent component analysis (ICA) of observed n-dimensional random vectors ${\bm{X}}$ ( $\textbf{X}=(x_{1},x_{2},\ldots,x_{n})^{T})$ is defined as the process of finding an m-dimensional linear transform $S={\bm{W}}{\bm{X}}(\textbf{S}=(s_{1},s_{2},\ldots,s_{m})^{T})$ such that the separation matrix ${\bm{W}}$ is a linear transformation and and the hidden components $s_{i}$ are as independent as possible. Usually, the dimension of ${\bm{S}}$ is less than that of ${\bm{X}}$ ( $m\leqslant n)$ . Only the case of $m=n$ is considered in this paper. Here, the matrix $W$ is assumed to be invertible. FastICA [12] is just such an algorithm, which is based on a fixed-point iteration scheme for finding a maximum nongaussianity of ${\bm{W}}{\bm{X}}$ . There are many different measures for nongaussianity, such as kurtosis and negentropy. In this paper, the kurtosis is used as the maximum nongaussianity measure (i.e., the score function), which is defined for a zero-mean random variable $s$ as [15]

$\textit{kurt(s)}=E\left\{{s^{4}}\right\}-3\left({E\left\{s\right\}^{2}}\right)% ^{2}$ (8)

Using the natural gradient method under the constraint $\left\|{w_{j}}\right\|_{2}=1$ (where $w_{j}$ is the $j$ th column of ${\bm{W}}$ ), the updating rule of $w_{j}$ is written as

$\displaystyle w_{j}(t+1)=w_{j}(t)+\mu(t)\left\{{v_{i}(t)\left[{\left({w_{j}(t)% }\right)^{T}v_{i}(t)}\right]}\right.^{3}\left.{-3\left\|{w_{j}(t)}\right\|^{2}% w_{j}(t)+f\left({\left\|{w_{j}(t)}\right\|^{2}}\right)w_{j}(t)}\right\}$ (9)

where $v_{i}$ is the $i$ th row of whitened results of observed vectors ${\bm{X}}$ , $\left({w_{j}(t)}\right)^{T}$ is the transpose of $w_{j}(t)$ , $\mu(t)$ is the learning rate, and $f(\cdot)$ is a penalty term due to the constraint $\left\|{w_{j}}\right\|=1$ . The learning rule will stop at a fixed point for which $\left\|{w_{j}(t)-w_{j}(t-1)}\right\|$ is sufficiently close to zero. The linear combination ${\bm{W}}{\bm{X}}$ will be one of the required independent components in accordance with the formula ${\bm{S}}={\bm{W}}{\bm{X}}$ .

3. High accuracy behavior prediction of non-linear dynamic system with signal separation and support vector regression

Improving the behavior prediction performance of complex non-linear dynamic system is a challenging problem. The behavior prediction performance of complex non-linear dynamic system with several independent subsystems is worse than the prediction performance of non-linear dynamic systems with single subsystem. The observations of complex dynamic system are the mixtures of several observations from different non-linear dynamic subsystems. Some useful information for prediction is lost when mixing happens. Two main problems should be resolved to improve the prediction performance of the complex non-linear dynamic system. One problem is whether signal separation can improve prediction performance or not. The other problem is how much linear or non-linear mixing affects the prediction performance. This paper focuses on the first problem and linear instantaneous mixing is assumed for simplicity. The second problem will be considered in the future work. Linear instantaneous mixing assumption means that the channels between the state variables of dynamic subsystems and the observing sensors are linear and non-memory. Let $x_{1}(t)$ and $x_{2}(t)$ be one state variable of two independent non-linear dynamic systems respectively, and a complex dynamic system consists of these two systems. Let one observed time series of the complex dynamic system be $\alpha x_{1}(t)+\beta x_{2}(t)$ . According to the principles of dynamic system, the characteristics measure of non-linear dynamic system can be the fractal dimension or box dimension [16]. The fractal dimensions of the $x_{1}(t)$ , $x_{2}(t)$ and $\alpha x_{1}(t)+\beta x_{2}(t)$ can be represented as $\dim\left\{{x_{1}(t)}\right\}$ , $\dim\left\{{x_{2}(t)}\right\}$ and $\dim\left\{{\alpha x_{1}(t)+\beta x_{2}(t)}\right\}$ respectively. It will be found that $\max\left\{{\dim\left\{{x_{1}(t)}\right\},\dim\left\{{x_{2}(t)}\right\}}\right% \}\geqslant\dim\left\{{\alpha x_{1}(t)+\beta x_{2}(t)}\right\}$ . This means that the information for prediction is lost when the mixing procedure happens. If the mixed observation can be separated at first, the prediction performance may be improved.

In this paper, a novel behavior prediction method for complex non-linear dynamic system is proposed. Let a complex non-linear dynamic system be composed of several independent subsystems. Several sensors are used to investigate the state of the complex system. The output of every sensor contains the information of all subsystems. As mentioned earlier, it is assumed that the output of every sensor is linear instantaneous combination of the states of all subsystems. Let $n$ be the number of sensors and let $X(t)=\left[{x_{1}(t),x_{2}(t),...,x_{n}(t)}\right]^{T}$ be the observed states (a vector) of the complex system at time $t$ . After the observation series are separated, local support vector regression for its robust fitting or modeling performance is used to model the behavior of every separated time series. Every separated time series can be considered as the output of some dynamic subsystem. The prediction results of every separated component are remixed with mixing matrix as the prediction result of the complex non-linear dynamic system. In summary, the proposed method can be described as follows:

Step 1:
At time $T_{p}$ , the observed data matrix ${\bm{X}}=\left[{{\bm{X}}(1),{\bm{X}}(2),\ldots,{\bm{X}}(T_{p})}\right]$ is separated with FastICA algorithm. The separation matrix W and the separated signals ${\bm{S}}=\left[{{\bm{S}}(\ref{eq1}),{\bm{S}}(\ref{eq2}),\ldots,{\bm{S}}(T_{p})% }\right]$ are obtained, where ${\bm{S}}(t)=\left[{s_{1}(t),s_{2}(t),\ldots,s_{n}(t)}\right]^{T}$ is independent signal vector at time $t$ .
Step 2:
Every separated signal $s_{i}(t)$ is segmented into $\left\langle{{\bm{S}}_{i}(j),y_{i}(j)}\right\rangle$ with slide windows, where ${\bm{S}}_{i}(j)=\left[{s_{i}(j),s_{i}(j+1)\ldots,s_{i}(j+L-1)}\right]^{T}$ , $y_{i}(j)=s_{i}(j+L)$ and $L$ is the length of slide window. The length of slide windows should be long enough to get sufficient information about the intrinsic dynamic subsystem. The reason why the embedding dimension method [2, 3, 4] is not adopted to segment the separated signal is that the embedding dimension method cannot extract the sufficient information for prediction. The prediction performance with embedding dimension method is worse than the prediction performance with slide windows [17].
Step 3:
Feature extraction is done on the segmented samples with wavelet and PCA in order to avoid over-fitting and to improve efficiency of training.
Step 4:
An $\varepsilon-SVR$ model is trained for every independent source signal with first $T_{p}-L$ samples.
Step 5:
Do prediction with the well trained $\varepsilon-SVR$ model for the sample $S_{i}(T_{p}-L+1)$ and get the prediction value $\hat{y}_{i}(T_{p}-L+1)$ .
Step 6:
Get the prediction of observations by ${\hat{\bm{X}}}(T_{p}+1)={\rm{\bf M}}{\hat{\bm{Y}}}(T_{p}-L+1)$ , where M is the mixing matrix and the ${\hat{\bm{Y}}}(T_{p}-L+1)=\left[{\hat{y}_{1}(T_{p}-L+1),\ldots,\hat{y}_{n}(T_{% p}-L+1)}\right]^{T}$ . Mixing matrix M is inverse of the separation matrix W, i.e. MW = I, where I is an identity matrix.
Step 7:
After the observations are obtained at time $T_{p}+1$ , let $T_{p}=T_{p}+1$ and go to the Step 1.

4. Experimental results

Three real-world datasets are used as the outputs of three independent non-linear dynamic subsystems. The datasets are Darwin Sea Level Pressure (DSLP), Santa Fe Laser (SFL) and Electricity demand (ED) [18]. These time series are non-linear and non-stationary. Five mixed signals are used to do experiments with different mixing matrix to verify the performance of the proposed method. The first mixed signals are

$\textit{MixedSignal\_1}=\left[{\begin{array}[]{ll}\textit{First}&\textit{% Component}\\ \textit{Second}&\textit{Component}\\ \end{array}}\right]=\left[{{\begin{array}[]{*{20}c}{0.4}\hfill&{0.6}\hfill\\ {0.6}\hfill&{0.4}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{\textit{signal}_{\textit{% DSLP}}}\\ {\textit{signal}_{\textit{SFL}}}\\ \end{array}}}\right]$ (10)

The second mixed signals are

$\textit{MixedSignal\_2}=\left[{\begin{array}[]{ll}\textit{First}&\textit{% Component}\\ \textit{Second}&\textit{Component}\\ \end{array}}\right]=\left[{{\begin{array}[]{*{20}c}{0.3}\hfill&{0.7}\hfill\\ {0.7}\hfill&{0.3}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{\textit{signal}_{\textit{% DSLP}}}\hfill\\ {\textit{signal}_{\textit{SFL}}}\hfill\\ \end{array}}}\right]$ (11)

The third mixed signals are

$\textit{MixedSignal\_3}=\left[{\begin{array}[]{ll}\textit{First}&\textit{% Component}\\ \textit{Second}&\textit{Component}\\ \end{array}}\right]=\left[{{\begin{array}[]{*{20}c}{0.2}\hfill&{0.8}\hfill\\ {0.5}\hfill&{0.5}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{\textit{signal}_{\textit{% DSLP}}}\hfill\\ {\textit{signal}_{\textit{SFL}}}\hfill\\ \end{array}}}\right]$ (12)

The fourth mixed signals are

$\textit{MixedSignal\_4}=\left[{\begin{array}[]{ll}\textit{First}&\textit{% Component}\\ \textit{Second}&\textit{Component}\\ \end{array}}\right]=\left[{{\begin{array}[]{*{20}c}{0.6}\hfill&{0.4}\hfill\\ {0.4}\hfill&{0.6}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{\textit{signal}_{\textit{% DSLP}}}\hfill\\ {\textit{signal}_{\textit{ED}}}\hfill\\ \end{array}}}\right]$ (13)

The fifth mixed signals are

$\textit{MixedSignal\_5}=\left[{\begin{array}[]{ll}\textit{First}&\textit{% Component}\\ \textit{Second}&\textit{Component}\\ \end{array}}\right]=\left[{{\begin{array}[]{*{20}c}{0.3}\hfill&{0.7}\hfill\\ {0.7}\hfill&{0.3}\hfill\\ \end{array}}}\right]\left[{{\begin{array}[]{*{20}c}{\textit{signal}_{\textit{% DSLP}}}\hfill\\ {\textit{signal}_{\textit{ED}}}\hfill\\ \end{array}}}\right]$ (14)

In the experiments, the last 190 data points of each time series in the mixed signals are used for doing prediction and others are used for training. Figure 1 shows us the performance comparison of the SVR prediction with separation or without separation of the second component in the second mixed signals. From Fig. 1, it can be found that the predicted series with the proposed method is closer to the real series than the predicted series without separation. That is to say, the SVR prediction with separation can get higher prediction accuracy and improve the predictability of the complex time series mixed by several series. Figure 2 shows us the prediction error evolution of the time series prediction with separation or without separation of the second component in the second mixed signals. From Fig. 2, it can be found that the prediction error with signal separation is significantly lower than the prediction error without signal separation. This also proves that the proposed prediction method can get much higher prediction accuracy and improve the predictability of the complex time series. The reason why the prediction performance of the proposed method is better is that the separated time series are much simpler and easier to be modeled than the original series. Higher prediction accuracy can be obtained for every separated time series. Higher prediction accuracy of original series can be expected with signal separation. In other words, some useful information for prediction is improved by signal separation.

Figure 1.

Prediction performance comparison of the time series prediction with separation and without separation.

Figure 2.

Error evolutions of the time series prediction with separation and without separation.

The prediction performance can be quantified by the Normalized Root Mean Square Error (NRMSE) and Mean Absolute Error (MAE) [17]. Table 1 shows us the MAE and NMRSE comparison of SVR prediction with or without separation. From Table 1, it can be found that both the MAE and NMRSE of the prediction with signal separation are lower than those of the prediction without signal separation. That is to say, the prediction accuracy with signal separation is much higher than the prediction accuracy without signal separation. These results verify that signal separation can improve the predictability of complex time series mixed with several time series substantially.

Table 1

The prediction performance comparison of SVR prediction with or without signal separation

		MAE		NMRSE
		First component	Second component	First component	Second component
Mixed signal 1	With separation	0.05183	0.04323	0.03349	0.03304
	Without separation	0.05924	0.05249	0.04477	0.05004
Mixed signal 2	With separation	0.04597	0.02941	0.02070	0.01477
	Without separation	0.06014	0.05253	0.03727	0.05268
Mixed signal 3	With separation	0.05230	0.03755	0.02135	0.02012
	Without separation	0.05724	0.05721	0.02521	0.05196
Mixed signal 4	With separation	0.07495	0.06012	0.08099	0.03679
	Without separation	0.08698	0.07531	0.10634	0.05681
Mixed signal 5	With separation	0.05409	0.08352	0.02506	0.10738
	Without separation	0.06724	0.09346	0.03679	0.12866

5. Conclusions

The prediction of the complex time series from complicated dynamic system is a challenging problem and still not resolved well, especially when the observed dynamic system includes several non-linear dynamic subsystems. In this paper, the signal separation technique and local prediction modeling are combined to do behavior prediction of complex non-linear dynamic system with multiple non-linear dynamic sub-systems. Firstly, signal separation algorithm is used to separate the observations in order to improve the predictability. Secondly, local support vector regression is trained with each separated independent component series. Finally, the prediction of the original observations is obtained by merging the prediction results of the separated independent component series with its local support vector regression. Experimental results show that prediction performance of the proposed method is much better than that of the prediction method without signal separation. So the proposed method is more qualified for behavior prediction of complex non-linear dynamic system.

Footnotes

Acknowledgments

This work is supported by the NSFC Breeding Program in Shantou University.

References

Povinelli

R.J.

and Feng

, A new temporal pattern identification method for characterization and prediction of complex time series events, IEEE Transactions on Knowledge and Data Engineering 15(2) (2003), 339–352.

Zhou

Song

and Liu

, Medium term load forecasting for chaotic neural networks based on the Euclidean distance, Journal of Computational Methods in Sciences and Engineering 16(3) (2016), 493–503.

Rafsanjani

M.K.

and Samareh

, Chaotic time series prediction by artificial neural networks, Journal of Computational Methods in Sciences and Engineering 16(3) (2016), 599–615.

Potter

Venayagamoorthy

G.K.

and Kosbar

, RNN based MIMO channel prediction, Signal Processing 90 (2010), 440–450.

Gorriz

J.M.

et al., Time series prediction using ICA algorithms, IEEE International Workshop on Intelligent Data Acquisition & Advanced Computing Systems: Technology & Applications (5) (2003), 226–230.

Lee

et al., Noisy time series prediction using M-estimator based robust radial basis function neural networks with growing and pruning techniques, Expert Systems with Applications 36(3) (2009), 4717–4724.

et al., Travel-time prediction with support vector regression, IEEE Transaction on Intelligent Transportation Systems 5(4) (2004), 276–220.

Chen

et al., Time-series prediction using a local linear wavelet neural network, Neurocomputing 69 (2006), 449–465.

et al., An improved time series prediction by applying the layer-by-layer learning method to fir neural networks, Neural Networks 10(9) (1997), 1717–1729.

10.

Milidiu

et al., Time series forecasting through wavelets transformation and a mixture expert models, Neurocomputing 28 (1999), 145–146.

11.

Cao

, Support vector machines experts for time series forecasting, Neurocomputing 51 (2003), 321–339.

12.

Hyvarinen

Karhunen

and Oja

, Independent component analysis. Beijing: Publishing House of Electronics Industry, 2014.

13.

Lau

K.W.

and Wu

Q.H.

, Local prediction of nonlinear time series using support vector regression, Pattern Recognition 41 (2008), 1539–1547.

14.

Simmons

G.F.

. Differential Equations, New York: McGraw-Hill, 1972.

15.

Chiu

S.H.

and Lu

C.P.

, Noise separation of the yarn tension signal on twister using FastICA, Mechanical Systems and Signal Processing 19 (2005), 1326–1336.

16.

Robinson

R.C.

, An Introduction to Dynamical Systems: Continuous and Discreet. Prentice Hall, 2004.

17.

Wang

Zhang

and Xu

H.C.

, Local prediction of Complex Time Series based on Support Vector Machine and Differential Evolution algorithm, The Second International Symposium on Computational Intelligence and Design 2 (2009), 425–428.

18.

Time Series Prediction group [EB/OL], http://www.cis.hut.fi/projects/tsp/?page=Timeseries, 2006-05-08.