Shock and Vibration

Volume 2015, Article ID 469165, 8 pages

http://dx.doi.org/10.1155/2015/469165

## An Adaptive Support Vector Regression Machine for the State Prognosis of Mechanical Systems

^{1}School of Mechanical Engineering, Xi’an Jiaotong University, Xi’an 710049, China^{2}Key Laboratory of Education Ministry for Modern Design and Rotor-Bearing System, Xi’an Jiaotong University, Xi’an 710049, China^{3}State Key Laboratory for Manufacturing Systems Engineering, Xi’an Jiaotong University, Xi’an 710049, China

Received 12 December 2014; Accepted 20 January 2015

Academic Editor: Yanxue Wang

Copyright © 2015 Qing Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Due to the unsteady state evolution of mechanical systems, the time series of state indicators exhibits volatile behavior and staged characteristics. To model hidden trends and predict deterioration failure utilizing volatile state indicators, an adaptive support vector regression (ASVR) machine is proposed. In ASVR, the width of an error-insensitive tube, which is a constant in the traditional support vector regression, is set as a variable determined by the transient distribution boundary of local regions in the training time series. Thus, the localized regions are obtained using a sliding time window, and their boundaries are defined by a robust measure known as the truncated range. Utilizing an adaptive error-insensitive tube, a stabilized tolerance level for noise is achieved, whether the time series occurs in low-volatility regions or in high-volatility regions. The proposed method is evaluated by vibrational data measured on descaling pumps. The results show that ASVR is capable of capturing the local trends of the volatile time series of state indicators and is superior to the standard support vector regression for state prediction.

#### 1. Introduction

The state prognosis of mechanical systems is of critical importance in modern industry to prevent unexpected breakdowns, to improve machine availability, and to reduce maintenance costs. Generally, the working state of mechanical systems is represented by certain indicators, which are either acquired from monitoring devices or calculated from raw monitoring signals. The primary task of state prognosis is to estimate the actual development of the state by modeling the trend of state indicators. Then, the trend model can be extrapolated to predict the upcoming failure or estimate the remaining useful life. After an accurate prognosis is achieved, timely maintenance actions can be planned to avoid catastrophic failure.

Due to the unstable operating conditions and accidental disturbances, the time series of state indicators always exhibits random fluctuations, whether the monitored system works normally or not. Therefore, many intelligent methods have been proposed to extract hidden trends from the observed state indicators. Artificial neural network (ANN) is one of the widely used methods in the prognostics literatures [1]. Gebraeel et al. [2] developed a set of feed-forward backpropagation networks to model the degradation process of rolling element bearings and to estimate the failure time of partially degraded bearings. Tse and Atherton [3] used a recurrent neural network to determine the trend in the monitoring values and to predict the value at the next time step. Because the trend is learned and memorized by neurons and network weights, ANN provides a nontransparent solution to state prognosis, or rather, the way in which forecast results are inferred by a trained network cannot be observed. Random coefficient models are another category of prognosis method for mechanical systems. In these models, the trend in the state indicators is predefined as a linear, polynomial, exponential, or any other functional form [4, 5]. Then, the coefficients, which include the deterministic functional coefficients and the stochastic noise coefficients, are jointly estimated with historical state indicators. Due to the requirements for system-specific trend knowledge, the applications of random coefficient models are greatly restricted. Nonparametric regression models, in which the trend needs not to take a predetermined form, overcome the barriers of prior knowledge and are also commonly used for state prognosis [6]. Among this category of models, support vector regression (SVR) [7], which has good generalization ability even if training samples are not abundant, is the most widely accepted method. By training an SVR machine, the trend of the state indicators is represented as an explicit regression function and is easily extrapolated to obtain future values. The extrapolated values are used to prognose the state evolution. Generally, while the extrapolated values reach a failure threshold predefined by theoretical or experimental analysis, a prospective failure is deduced and the failure time is estimated [8]. Therefore, SVR has been extensively studied to tackle the prognosis problem of mechanical systems or components, such as bearings, gears, and pumps [9–12].

The life of mechanical systems can be divided into a normal working stage and a deterioration stage [13]. In the first stage, the state indicators are generally shown as a stationary time series. As initial defects emerge, the system steps into the deterioration stage, including unsteady evolution and abrupt changes. These nonstationary and transient phenomena are reflected in the time series of the state indicators. To trace the evolution of states, the regression model, which is capable of adapting to staged development and volatile state indicators, is required. However, the standard SVR seeks a globally optimized regression, in which the tolerance level for noise is fixed during the entire training time series. Therefore, it lacks the flexibility to capture the local trend of a data series with time-varying variance or staged characteristics. To improve the adaptability of volatile time series, several modified SVR machines, such as localized support vector regression (LSVR) [14] and piecewise support vector regression [15], have been developed and applied in the field of financial analysis. In this paper, a novel SVR machine, called adaptive support vector regression (ASVR), is proposed to model the trend of state indicators measured from mechanical systems. We will show that, by utilizing an adaptable error-insensitive tube, ASVR can provide satisfactory performance for regression and prediction while the system is in a deterioration stage.

The rest of the paper is organized as follows. We briefly introduce related studies of standard SVR and LSVR in Section 2. The methodologies of ASVR are described in Section 3. In Section 4, standard SVR, LSVR, and ASVR are applied to address the time series of vibration acquired from actual pumps. The regression and prediction results are compared and analyzed. Finally, conclusions and future work are discussed in Section 5.

#### 2. Related Studies

##### 2.1. Standard Support Vector Regression

Given a time series , where is the time tag, is the corresponding value, and is the number of data points, the goal of SVR is to find a function that has at most deviation from the actually obtained values for all time tags while being as flat as possible [7]. By mapping the time series into a high-dimensionality feature space, the regression function has the linear form aswhere denotes a weight coefficient, is the mapping function, and is the bias. To obtain the optimal function , a convex optimization problem is constructed as follows:In the constrained minimization problem, it is assumed that a function exists for which all data points in time series lie in a tube determined by . defines the width of the error-insensitive tube, or in other words, the precision of regression is . However, in many applications, it is preferred to accept a number of errors, which are caused by the data points outside the error-insensitive tube, to improve the generalization ability. Therefore, the concept of a soft margin is introduced, and the original optimization problem (2) is reformed aswhere is a positive constant, known as the regularization parameter, which determines the trade-off between the flatness of and the amount up to which deviations larger than are tolerated. and are called slack variables and measure the deviation of from the boundaries of the error-insensitive tube. Figure 1 provides a depiction of SVR with a soft margin.