Mutual Information for Quaternion Time Series
Mutual Information method is a widely used method for estimation of time delay value in the process of time delay embedding. It’s designed for a univariate scalar time series. In the real systems often many outputs of investigated system are available. In
- PDF / 242,302 Bytes
- 9 Pages / 439.37 x 666.142 pts Page_size
- 59 Downloads / 226 Views
ion
The time delay embedding is heavily explored area of nonlinear time series analysis and nonlinear dynamical systems areas. Usually the most common application of this method is when we want to discover the dynamics of underlying system from the univariate scalar time series created from the measurements of one of the investigated system’s outputs. Taken’s Embedding Theorem [18] implies, that one can reconstruct an equivalent dynamics from univariate time series using it’s time delays. To carry out the embedding procedure, which should result in the reconstructed attractor in the output, two parameters need to be estimated: time delay and embedding dimension. The time delay is a integer value describing which samples from the investigated time series we need to incorporate to time-lagged embedding vector reconstructing the underlying phase space. There are few approaches of estimation of time delay value Td [9]. One group of methods is series correlation approaches (autocorrelation, mutual information [8] or high order correlations [2]). Second grop are approaches of phase space extension (fillfactor [5], wavering product [4] or average displacement [17]). There are available also multiple c IFIP International Federation for Information Processing 2016 Published by Springer International Publishing Switzerland 2016. All Rights Reserved K. Saeed and W. Homenda (Eds.): CISIM 2016, LNCS 9842, pp. 453–461, 2016. DOI: 10.1007/978-3-319-45378-1 40
454
M. Pi´ orek
autocorrelation and non-bias multiple autocorrelation methods [11]. The embedding dimension is an equivalent of the real underlying phase space dimension. It could be estimated using the false nearest neighbors method [3] or it’s extension - Cao’s method [7]. Another methods are also the saturation of system invariants method [1] or neural network approaches [13]. The popularity of the univariate embedding may be caused by the fact, that according to the embedding theorem, for recovering dynamics only a univariate time series is needed. In fact, often many time series measured in the output of the test process are available. Since in multivariate case more data is available, it helps to establish more accurate embedding - in the sense of further predictions or in the presence of data noise. However, it brings a new dilemma: which quantities from multivariate time series to use and whether is better to use constant or non-constant embedding parameters for all quantities selected to embedding vector [6]. The problem of multivariate time series embedding can be seen in terms of suitable conditioned embedding of the considered set of time series [19]. In the related work there are two approaches of multivariate time series embedding: Uniform embedding and Non-uniform embedding. The uniform embedding scheme is more popular approach and assumes that embedding parameters: the time delay and embedding dimension are selected a priori and separately for each time series. The non-uniform embedding is based on the progressive selection of time delayed values from a s
Data Loading...
