## 1 Introduction  
Gear and bearing are very important parts in mechanical systems, because their working conditions can directly affect safety and stable operations. Therefore, it is of great significance to detect and diagnose their operating states [1; 2; 3; 4]. The main denoising algorithms aim at one dimensional time domain vibration signals in the field of current mechanical signal noise reduction methods [5]. The vibration signals of rotary machinery have high computational complexity and large storage space, and many nonlinear or chaotic time series have an inherent broadband characteristic [6], which makes noise reduction so difficult. The nonlinear time series analysis method has been used in the fault diagnosis of rotary machinery. The analysis of vibration signal is vital for monitoring the operating condition of rolling bearings, and the accurate extraction of fault characteristic frequencies from vibration signals is particularly important [7]. In the process of monitoring rotating machinery, using signal processing methods to analyze and extract fault characteristic frequencies is becoming a common technique. Several signal processingmethods, such as wavelet transform (WT) [8; 9], independent component analysis (ICA) [10; 11], empirical mode decomposition (EMD) [12; 13], phase space reconstruction (PSR) [14], and sparse decomposition (SD) [15; 16], are widely used in this field. In the phase space reconstruction, one dimensional time series is reconstructed to high dimensional phase space to explore its underlying dynamical characteristics, hidden in one dimensional space [17]. [PERSON] and [PERSON] et al. [18] explained the ways of calculating fractal dimension and entropy in the nonlinear series to measure the complexity and regularity of time series. The proper orthogonal decomposition (POD) [19; 20; 21], which is also known as Karhunen-Loeve decomposition, has been widely applied to analyze nonlinear series fields as an effective tool in capturing the dominant modes and components. It is a statistical analysis technique in finding the coherent structures [22] in nonlinear data, which defines the optimum basis according to energy. The POD method has been advantageously applied in different fields, such as the stochastic finite element methods [23], the simulation of random fields [24], the damage identification [25], and the linear and nonlinear systems modal analysis [26]. In the nonlinear dynamics field, the POD is used to extract the dimensional information from an embedded attractor. In mechanical engineering and signal processing, the algorithm is conducted to identify subspaces [27] to obtain constrained clean trajectories in order to denoise the time series containing noises. Further studies lead to a modified decomposition called smooth orthogonal decomposition (SOD) [28; 29; 30], and it can be regarded as a projection of an ensemble of spatially distributed data. In smooth local decomposition, the one dimensional time series is reconstructed to high dimensional phase space, then noises in the data can be reduced by being projected to the tangent subspace [31]. The vector directions of the projection can make sure the variance is as maximum as possible, and the motions obtaining along these vector directions are the smoothest in terms of time. The vector directions can be defined as the eigenvectors of the eigenproblem [32] from the correlation matrices of the random field and the relevant time derivative. The basic idea of SOD derived from the proposed optimal tracking approach [33], and was formulated as a modal analysis tool. Then the smooth orthogonal approach [34] was evolved in terms of time continuous stationary random vector processes, and it was extended to time continuous and non-stationary random vector processes.