Methods of utilizing independent component analysis (ICA) give little guidance about practical considerations for separating singlechannel realworld data, in which most of them are nonlinear, nonstationary, and even chaotic in many fields. To solve this problem, a threestep method is provided in this paper. In the first step, the measured signal which is assumed to be piecewise higher order stationary time series is introduced and divided into a series of higher order stationary segments by applying a modified segmentation algorithm. Then the state space is reconstructed and the singlechannel signal is transformed into a pseudo multiple input multiple output (MIMO) mode using a method of nonlinear analysis based on the high order statistics (HOS). In the last step, ICA is performed on the pseudo MIMO data to decompose the single channel recording into its underlying independent components (ICs) and the interested ICs are then extracted. Finally, the effectiveness and excellence of the higher order singlechannel ICA (SCICA) method are validated with measured data throughout experiments. Also, the proposed method in this paper is proved to be more robust under different SNR and/or embedding dimension via explicit formulae and simulations.
As one of the most attractive solutions for the blind source separation (BSS) problem, independent component analysis (ICA) has a strong practical background and wide applications in multiway data analysis such as biomedicine [
Generally, the number of sensors must be no less than that of the sources to acquire information to support the BSS work. Often in real cases, however, one has just a single measure of a certain specific physical variable, from which information on the underlying source mechanism has to be derived. In this case, the topic faced by the researchers is very important and difficult, that is, the extraction of characteristics from single experimental series, because of the lack of prior information. The method called SCBSS is proposed to exact the independent feature by using only one transducer. The methods employ ICA [
In a dynamical embedding framework, the measured data can be assumed to be generated by the nonlinear interaction of just a few degrees of freedom, with additive noise, and suggests the existence of an unobservable deterministic generator of the observed data. Obviously, in this case the reconstructed phase space (RPS) can be used to uncover as much information as possible about the underlying generators based only on the measured data [
In order to find a solution to the aforementioned problems, a modified method based on HOS is developed in this paper. Section
Generally, the observed singlechannel signal
When the actual data is treated as a nonlinear time series with additive noise which is generated by the nonlinear interaction of just a few degrees of freedom, we can use the SCICA algorithm to solve the SCBSS problem. RPS is the first and foremost step, when the dynamic system theory is utilized to analyze a nonlinear time series. In [
Any approach to state space reconstruction uses the information in delay coordinates as a starting point. Obviously, Takens’ theorem allows us to reconstruct the unknown dynamical system that generates the measured time series by reconstructing a new state space based on the successive observations of the time series. It is indicated that the RPS of the nonlinear time series is the essential projection of the strange attractor on the axis of the space spanned by delay vectors. Therefore, each time series constructed by each delay vector can be regarded as a mixture of source signals. As shown in [
SCICA could separate a singlechannel time series successfully if and only if this method satisfies the following conditions [
The measured signal is stationary.
The phase state can be reconstructed perfectly.
Each time series constructed by RPS could be considered as a singlechannel instantaneous linear mixture (SCILM) of source signals.
All the independent random processes must be bandlimited with disjoint spectral support.
Unfortunately, SCICA algorithms cannot be used directly for sources separation or extraction while the signal is nonstationary. Therefore, a nontrivial structure with nonstationarity of the actual signal with variable statistical property such as the mean and the variance is expected. The problem addressed in this paper is to segment a nonstationary time series, which consist of many segments with different statistical property, in such a way as to maximize the differences in the statistical property between adjacent segments. The BG algorithm in [
Takens’ theorem [
Assuming instantaneous linear mixing of the sources at the sensors, ICA performs a blind separation of statistical independent sources with techniques involving higherorder statistics. However, RPS, which reconstructs the nonlinear time series in the state phase based on the delay coordinates, is essentially a nonlinear transform and cannot change singlechannel data into multiple instantaneous linear mixture. Therefore, SSA [
In this section, the actual data is assumed to be a stochastic process
where
Furthermore, in this paper the actual data is considered as nonstationary signal, which is composed of many zeromean, quasistationary (up to the fourthorder) segments with different higherorder statistical properties. Then the different statistical properties will be selected to segment the time series into several subsets by means of the BG algorithm [
Considering a zeromean, quasistationary (up to the fourthorder) subset of the actual data, which is generated by a nonlinear dynamical system, the information about the underlying generators is uncovered by employing RPSbased method. Using the mean time between peaks (MTBP) as the time window, the reconstructed parameters
Bloch diagram of SCICA based on HOS for a nonlinear time series.
Since SCICA does not work to the nonstationary signals which are the property indeed of the measured singlechannel actual signals, a modified BG algorithm is necessary for developing HOSbased methods. The BG algorithm is designed to characterize the stationary durations of human heart beat time series in [
The changepoints with different segment length

Changepoint 













(a) An artificial time series
The original algorithm is modified as follows: After selecting a larger value as the minimum segment length, a sliding pointer is moved from left to right along the signal. At each position of the pointer, the time slots are computed as
The zeromean, quasistationary (up to the fourthorder) segments prepared for SCICA could be exacted by a modified BG algorithm, as discussed above. Unfortunately, SCICA cannot be directly used to recover all sources from the recorded mixtures, in particular scenarios that the useful sources cannot be perfectly reconstructed into a phase state and the mixing segment cannot be considered as a SCILM. Applied to the actual data, therefore the reconstruction parameters can be selected by a new method to ensure validity of the RPS, and the reconstructed phase state in the delay coordinate system can be transformed into a multipath instantaneous linear mixture, which can be solved by ICA.
The performance of RPS depends on two parameters, namely, the selection of the embedding dimension
Paper [
For some purposes, such as reducing the dimension of the reconstruction, it may be desirable to make a further coordinate transformation
Thus by applying the highorder cumulants theory, a modified method is provided. A fourthorder cumulant is defined as
Then, the elements from the 4thorder cumulants function are selected as the elements of
Finally, the SVD method is applied to
However, the different fourthorder cumulants slices functions of
Singular values of the measured signal processed by the different
Obviously, the noise amplification depends on
Then we show that selecting the optimal slice is equivalent to minimizing the noise amplification
As shown in the following experiments, there are several general motivations behind the use of the HOSbased method in signal processing.
(1) This method is used to distinguish the feature signal from the Gaussian noise. Clearly, consider signal
Singular values of the measured signal: (a) SNR = 0 dB,
(2) As shown in Figure
Singular values of the measured signal: (a) HOSbased method; (b) autocorrelationbased method.
(3) The
Singular values of the chaotic time series for the Lorenz maps: (a) HOSbased method; (b) autocorrelationbased method.
The HOSbased method is used to process the trajectory matrix
(1) Calculate the fourthorder correlation function
(2) Apply SVD on the Toeplitz matrix
(3) Project the trajectory matrix
(4) Reconstruct the original series
Then, ICA [
In essence, ICA must find a separating or demixing matrix
Performing ICA on the matrix processed by HOSbased method
Based on the previous sections, we can introduce a modified SCICA algorithm by the following implementation:
Detect and segment the measured signal into several highorder stationary subsets (with the modified BG segmentation algorithm) in time domain.
Determine the window
Perform the HOSbased coordinate transformation to change the trajectory matrix
Separate the source signals with ICA.
A singlechannel observation of two sources is taken as an example. The sampling frequency
FFT spectrum of the singlechannel observation of two sources, which can be easily mistaken for three sources of three sinusoidal signals,
Singular spectrum of time series by means of autocorrelationbased method.
Singular spectrum of time series by means of HOSbased method.
FFT spectra of two principal signals after HOSbased coordinate transformation.
FFT spectra of two principal ICs after ICA operation.
FFT spectra of the other ICs after ICA operation.
Overall, this paper presents an approach for exacting information from single realworld data. The idea is firstly to segment the measured signal and then to form a pseudoMIMO system by means of decomposing the observed segment into several signals using a representation method based on higherorder statistics (HOS). Finally the fixedpoint FastICA algorithm is applied to estimate the source signals (independent components). The simulations show that the method is successful in isolating components from the singlechannel data. Also, the methods based on fourthorder cumulants are more robust than those based on autocorrelation as the properties of the reconstruction are changed. Compared with the autocorrelation method, HOSbased SCICA is better for low SNR. At this stage, HOS will be more sensitive to the number of the used samples. This is a problem for all the HOSbased method when they are applied to the actual data, but it is not a problem for this method. Moreover, since the available techniques used in this paper can process the singlechannel signal without depending on a priori, the method is a very powerful method that can isolate feature components in the actual data.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (61201282) and the Fundamental Research Funds for the Central Universities of China (no. ZYGX2013J016).