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accounting-chapter-guide-principle-study-vol eyewitness-guide- scotland-top-travel. The method which is presented in this paper for estimating the embedding dimension is in the Model based estimation of the embedding dimension In this section the basic idea and .. [12] Aleksic Z. Estimating the embedding dimension. Determining embedding dimension for phase- space reconstruction using a Z. Aleksic. Estimating the embedding dimension. Physica D, 52;

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Skip to main content. Log In Sign Up. Chaos, Solitons and Fractals 19 — www. BoxTehran, Iran Accepted 11 June Abstract In this paper, a method for estimating an attractor embedding dimension based on polynomial models and its application in investigating the dimension of Bremen climatic dynamics are presented. The attractor embedding di- mension provides the primary knowledge for analyzing the invariant characteristics of the attractor and determines the number of aeksic variables to model the dynamics.

Therefore, the optimality of estimatlng dimension has an important role in computational efforts, analysis of the Lyapunov exponents, and efficiency of modeling and prediction. The smoothness property of the reconstructed map implies that, there is no self-intersection in the reconstructed attractor.

The method of this paper relies on testing this property by locally emmbedding a general polynomial autoregressive model to the given data and evaluating the normalized one step ahead prediction error. The effectiveness of the proposed method is shown by simulation results of its application to some well-known chaotic embexding systems.

Finally, the proposed methodology is applied to two major dynamic components of the climate data of the Bremen city to estimate the related minimum attractor embedding dimension. Introduction The basic idea of chaotic time series analysis is that, a sstimating system can be described by a strange attractor in its phase space.

This is accomplished from the observations of a single coordinate by some techniques outlined in [1] and method of delays as proposed by Takens [2] which is extended in [3].

On the other hand, computational efforts, Lyapunov exponents estimation, and efficiency of modelling and prediction is influenced significantly by the optimality of embedding dimension. There are several methods proposed in the literature for the estimation of dimension from a chaotic time series. The three basic approaches are as follow. Ataeibl iat. Lohmannsedigh eetd. Khaki- Estimatjnglucas karun. Particularly, the correlation dimension as proposed in [4] is calculated for successive values estimatng embedding dimensioj.

This method is often data sensitive and time-consuming for computation [5,6]. The second related approach is based on singular value decomposition SVD which is proposed in [7].

This approach results in a basis for the embedding space such that the attractor can be modeled with invariant geometry in a subspace with fixed dimension. The SVD is essentially a linear approach with firm theoretic base; for using it as a nonlinear tool there are some critical issues on the determination of the time window and on the selection of the significant singular values which are discussed in [8,9].

The third approach concerns checking the smoothness property of the reconstructed map. Among many references for checking this property, the most popular is the method of false nearest neighbors FNN developed in [10]. The FNN method checks the neighbors in successive embedding dimensions until a negligible percentage of false neighbors is found. The criterion for measuring the false neighbors and also extension the method for multivariate time series are provided in [11,6].

Some other methods based on the above approach are proposed in [12,13] to search for the suitable embedding dimension for which the properties of continuous and smoothness mapping are satisfied. The method which is presented in this paper for estimating the embedding dimension is in the latter category of the above approaches. In contrast to the previous methods, it provides a local polynomial model for reconstructed dynamics, which can be used for prediction and for calculation of Lyapunov exponents.

This idea also is used as the inverse approach to detect chaos in a time series in [14]. The developed algorithm in this paper, can be used for a multivariate time series as well in order to include information from all available measurements. The procedure is that a general polynomial autoregressive model is considered to fit the given data which its order is interpreted as the dimension of the reconstructed state space. As the reconstructed dynamics should be a smooth map, there should be no self-intersection in the reconstructed attractor.


This property is checked by evaluation of the level of one step ahead prediction error of the fitted model for different orders and various degrees of nonlinearity in the poly- nomials.

Deterministic chaos appears in engineering, biomedical and life sciences, social sciences, and physical sciences in- cluding many branches like geophysics and meteorology.

Also, estimations of the attractor embedding dimension of meteorological time series have a fundamental role in the development of analysis, dynamic models, and prediction of meteorological phenomena. There are many publications on the applications of techniques developed from chaos theory in estimating the attractor dimension of meteorological systems, e.

As a practical case study, in the last part of the paper, the developed algorithm is applied to the climate data of Bremen city to estimate its attractor em- bedding dimension. In the following, the main idea and the procedure of the method is presented in Section 2. To show the effectiveness of the proposed method, the simulation results are provided for some well-known chaotic systems in Section 3.

In Section 4 this methodology is used to estimate the embedding dimension of system governing the weather dynamic of Bremen city in Germany. Model based estimation of the embedding dimension In this section the basic idea and the procedure of the model based method for estimating the embedding dimension is presented.

Conceptual description Let the original attractor of the system exist in a m-dimensional smooth manifold, M. The attractor of the well reconstructed phase space is equivalent to the original attractor and should be expressed as a smooth map. The state equations of the reconstructed dynamics are considered as: In this paper, in order to model the reconstructed state space, the vector 2 by normalized steps, is considered as the state vector.

Therefore, the optimal embedding dimension and the suitable order of the polynomial model have the same value. To express the main idea, a two dimensional nonlinear chaotic system is considered. The objective is to find the model as 5 by using the autoregressive polynomial structure.

Estimating the embedding dimension

Therefore, the first step ahead prediction error for each transition of this point is: These errors will be large since only one fixed prediction has been considered for all points. The prediction error in this case is: The mean squares of these errors for all the points of attractor are also different values in these two cases.

Typically, it is observed that the mean squares of prediction errors decrease while d increases, and finally converges to a constant. This order is the suitable model order and is selected as minimum embedding dimension as well. The proposed algorithm In the following, by using the above idea, the procedure of estimating the minimum embedding dimension is pre- sented. This algorithm is written in vector format which can also be used for univariate time series.

The proposed algorithm of estimating the minimum embedding dimension is summarized as follows: The data pre-processing manipulations like normalization and probable deleting of the long term trend or seasonal effects are performed. Some definite range for embedding dimension and degree of nonlinearity of the polynomial models are considered as follows: The embedding space vectors are constructed as: For each delayed vector 11r nearest neighbors are found which r should be greater than np as defined in The following polynomial autoregressive model is fitted to the set of neighbors.

For the model order d and degree of nonlinearity n the number of parameters in vector H that should be estimated to identify the underlying model is: This identification can be done by using a least squares method [18]. The mean squares of prediction errors is computed as: The above procedure is repeated for the full range of D and Np. The value of d, for which the level of r is reduced to a low value and will stay thereafter is considered as the minimum embedding dimension.


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This idea for estimating the embedding dimension can be used independently of the type of model, if the selected function for modeling satisfies the continuous differentiability property. Multivariate versus eetimating time series In some applications the available data are in the form of vector sequences of measurements. For example, the meteorology data are usually in multi-dimensional format.

On the other hand, the state space reconstruction from the single time series is based on the assumption that the measured variable shows the full dynamics of the system. However, the full dynamics of a system may not be observable from a single time series and we are not sure that from a scalar time series a suitable reconstruction estumating be achieved.

Moreover, the advantages of using multivariate time series for nonlinear prediction are shown in some applications, e. Here, the advantage of using multiple time series versus scalar case is briefly discussed. Let the dynamical equations of the continuous system be: The embedding space is reconstructed by fol- lowing vectors for both cases respectively: If the full dynamic of dimensioh system is not observable through single output, the necessity of using multiple time series is clear since the inverse problem can not be solved.

However, in the case that the system is theoretically observable, it is seen that the solvability condition of Eq. In a linear system, the Eqs. It is seen that the ill-conditioning of the first case is more probable than the latter.

The dimenison advantage of using multivariate versus estlmating time series, relates to the effect of the lag time. In the scalar case, as higher order derivatives delays are required, a large lag time between the elements in embefding embedding vector, may cause the sequential values be in a wide range.

This causes the loss of high order dynamics in local model fitting and make the role of lag time more important. However, in the multivariate case, this effect has less importance since fewer delays are used. Simulation results To show the effectiveness of the proposed procedure in Section emedding, the procedures are applied to some well-known chaotic systems.

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These chaotic systems are defined in Table embdding. The presented method for estimating the embedding dimension or suitable order of model based on local polynomial modelling is implemented. The developed general program of polynomial modelling, is applied for various d and n, and r is computed for all the cases in a look up table. Based on the discussions in Section estimafing, the optimum embedding dimension is selected in each case.

The mean square of error, r, for the given chaotic systems are shown in Table 2. According to these results, the optimum embedding di- mension for each system is estimated in Table 3.

The embedding dimension of Ikeda map can be estimated in the range of 2—4 which is also acceptable, however, it can be improved by applying the procedure by using multiple time series. In this case the embedding dimension estimatijg simply estimated equal 2 which is exactly the dimension alekslc the system. Case study The climatic process has significant effects on our everyday life like transportation, agriculture. The first step in chaotic time series analysis is the state space reconstruction which needs the determination of the embedding dimension.

Therefore, the estimation of the attractor embedding dimension of climate time series have a fundamental role in the development of analysis, dynamic models, and prediction of the climatic phenomena.

The climate data of Bremen city for May—August