پیاده سازی ماشین آلات رگرسیون بردار پشتیبانی برای پردازش اثرات تغییر شکل نهایی هیلبرت، هوانگ

The end effects of Hilbert–Huang transform are represented in two aspects. On the one hand, the end effects occur when the signal is decomposed by empirical mode decomposition (EMD) method. On the other hand, the end effects occur again while the Hilbert transforms are applied to the intrinsic mode functions (IMFs). To restrain the end effects of Hilbert–Huang transform, the support vector regression machines are used to predict the signals before the signal is decomposed by EMD method, thus the end effects could be restrained effectively and the IMFs with certain physical sense could be obtained. For the same purpose, the support vector regression machines are used again to predict the IMFs before the Hilbert transform of the IMFs, thus the accurate instantaneous frequencies and amplitudes could be obtained and the corresponding Hilbert spectrum with physical sense could be acquired. The analysis results from the simulation and experimental signals demonstrate that the end effects of Hilbert–Huang transform could be resolved effectively by the time series forecasting method based on support vector regression machines which is superior to that based on neural networks.

The Hilbert–Huang transform is a new analysis method for non-stationary signals put forward by Huang [1], which contains empirical mode decomposition (EMD) and its corresponding Hilbert spectrum analysis method for signals. Firstly, EMD is applied to decompose the non-stationary signals into several intrinsic mode functions (IMFs). Secondly, Hilbert transform is carried out to each IMF component to get the instantaneous frequencies and instantaneous amplitudes. Finally, the instantaneous frequencies and amplitudes are reassembled to obtain the Hilbert spectrum. In signal analysis, time scale and the energy distribution along with the time scale are the two most important parameters to signals. The EMD method, which is based on the local characteristic time scale, can be used to decompose the complex signals into a number of IMF components. Since the decomposition is carried out according to the signals itself, the number of resulting IMF components are usually limited and each IMF component can reflect the intrinsic and real physical information of the signals, as a result of which the resulting Hilbert spectrum can indicate exactly the signal energy distributions in the space (or time) with diversified scales. Therefore, the Hilbert–Huang transform has been widely used in many fields, such as the analysis of the non-stationary sea wave data [2], earthquake signal and structure analysis, bridge and constructions state monitoring [3], and the fault diagnosis of machines, etc. [4] and [5]. Although Hilbert–Huang transform is quite suitable for non-stationary signal analysis, the end effects occur in the transform, which is represented in two aspects [1], [6], [7], [8], [9], [10] and [11]. First of all, the two ends of time series will disperse while the signal is decomposed by EMD method and this disperse would “empoison” in by the whole time series gradually which makes the results to get distortion [1], [6], [7], [8], [9], [10] and [11]. Additionally, the end effects also arise when the Hilbert transform is applied to the IMFs because of the inevitable window effects of the transform [7]. If the two end effects cannot be restrained effectively, the real characteristics of the original signals could not be reflected accurately by Hilbert spectrum. Huang et al. [1] forecast the time series by adding two characteristic waves at the ends of data, but fail to mention how to establish suitable characteristic waves. Although this method is effective some problems still exist [6]. Recently, many methods have been put forward to restrain the end effects, such as the time series forecasting based on neural networks [7], the forecasting based on AR model [8], the forecasting based on waveform matching method [9], the forecasting based on adding extreme [10], and the method by appending two parallels at the ends according to “balance place” of the series variety [11]. All the methods work well to a certain extent in the restraining of end effects. In EMD method, the original signals are forecast to insure that envelops are established completely by data within the ends and the distortion of envelops is limited to the least extent by discarding the end data in terms of the extremum situation during the decomposition. Also, the IMF components can be forecast before Hilbert transform and the end effects can be released out of the original signals by abandoning the end data. Hence the time series forecasting methods based on neural networks and AR model are both effective to deal with the end effects. However, the method based on AR is only suitable for stationary and simple non-stationary time series [12]. As far as the neural network is concerned, the local minimum point, over learning and the excessive dependence on experience about the choice of structures and types are its inevitable limitation [13], while support vector regression machines (SVRMs) get rid of these limitation [14] and has been applied to time series forecasting successfully [15], [16] and [17]. Therefore, it is an effective method to restrain the end effects of Hilbert–Huang transform in which SVRMs are adopted to establish models and the time series are forecast. In this paper, to target the end effects, the time series forecasting method based on support vector regression machines is put forward and confirmed to be effective by analysis results from the simulation and experimental signals. It also has been found by comparison study that the method based on support vector regression machines is superior to that based on neural networks in dealing with the end effects of Hilbert–Huang transform for the forecasting results of the latter depends excessively on the choice of structure and types while the former has satisfied results for different signals based on the same parameters with small error and less time.

For the end effects in Hilbert–Huang transform the original time series can be forecast by some method and during the process of EMD, the end data affected by end effects are abandoned constantly to achieve correct decomposition and the exact IMF components are obtained. And then Hilbert transform can be applied to each IMF component after it has been forecast again, respectively, and the two end data are abandoned to release the end effects out of the original signals and the exact Hilbert spectrum can be obtained. Hence, the key point is to find a suitable and effective forecasting method for the time series. The method based on AR is only available for stationary and simple non-stationary time series and the neural network has the inevitable limitation of over learning and the excessive dependence on experience about the choice of structures and types. However, SVRM can get rid of these limitations and can be used to forecast the time series effectively. By analysis of the three typical signals which include finality signals, stationary random signals and non-stationary random signals in practice it can be found that different types of the neural network have different forecasting effects for the same signal and the neural network with the same types and structures have different forecasting effect for the different signals either. So the selections of structures and types have great influence on the forecasting effect of the neural network and meanwhile these selections depend excessively on experience, which would limit the application of the neural network. However, satisfying forecasting results for different signals can be obtained by SVRM with the same parameters, which indicate that SVRM does not depend on the selection of parameters excessively as the neural network, and this brings convenience to the application of SVRM. Of course, the better forecasting results could be obtained by a certain type neural networks after its parameter and structure is optimised. But, the procedure of the optimisation needs much time. Furthermore, the optimised neural network cannot always be applicable to all problems. Therefore, in this paper, the forecasting results between the SVRM and two special types of neural networks are mainly compared in order to show the limitations of the neural network and the problem of the neural network optimisation was not mentioned. Nevertheless, it is just the application of SRVM in the processing of end effects of Hilbert–Huang transform that is involved primarily in this paper and the application of SRVM in time series forecasting has not been researched deeply any more, such as how to select different parameters (like the loss function, kernel function and its parameters, precision parameters and disciplinal parameters, etc.) and the influence of different parameters on forecasting effects and so forth. All these need a further study.