Trend analysis of the power law process using Expectation–Maximization algorithm for data censored by inspection intervals

Abstract

Trend analysis is a common statistical method used to investigate the operation and changes of a repairable system over time. This method takes historical failure data of a system or a group of similar systems and determines whether the recurrent failures exhibit an increasing or decreasing trend. Most trend analysis methods proposed in the literature assume that the failure times are known, so the failure data is statistically complete; however, in many situations, such as hidden failures, failure times are subject to censoring. In this paper we assume that the failure process of a group of similar independent repairable units follows a non-homogenous Poisson process with a power law intensity function. Moreover, the failure data are subject to left, interval and right censoring. The paper proposes using the likelihood ratio test to check for trends in the failure data. It uses the Expectation–Maximization (EM) algorithm to find the parameters, which maximize the data likelihood in the case of null and alternative hypotheses. A recursive procedure is used to solve the main technical problem of calculating the expected values in the Expectation step. The proposed method is applied to a hospital's maintenance data for trend analysis of the components of a general infusion pump.

Introduction

Trend analysis is a common statistical method used to investigate the operation and changes in a repairable system over time. This method takes historical failure data of a system or a group of similar systems and determines whether the recurrent failures exhibit an increasing or decreasing trend. Both graphical methods and trend tests are used for trend analysis. The latter are statistical tests for the null hypothesis that the failure process follows a homogenous Poisson process (HPP) [1].

Crow/AMSAA test [2], [3] assumes that the failure process of a repairable system has a Weibull intensity function and finds the maximum likelihood estimates of the parameters. The shape parameter is then used as an indicator for a growth or deterioration in the system reliability. The Laplace and the Military Handbook tests [4] are used to test whether data follow a HPP. Kvaloy and Lindqvist [5] propose the Anderson–Darling trend test for a NHPP with a bathtub shaped intensity function based on the Anderson–Darling statistic test [6]. The Lewis–Robinson [7] is a modification of the Laplace test and is used as a general test to detect trend departures from a general renewal process [8]. The Mann test [9] corresponds to a renewal process null hypothesis and the monotonic trend as the alternative. Kvaloy et al. [10] declare that the Mann test is a more powerful test against decreasing trend. The MIL-HDBK-189 [11], [12] is used for testing a NHPP with a power law intensity function. Caroni [13] modifies the trend tests introduced by Kvaloy et al. [10] for data that end with the last of a random number of failures within a predetermined observation period. Louit et al. [14] review several tests available to assess the existence of trends, and proposes a practical procedure to discriminate between the use of statistical distributions to represent the time to failure. Regattieri et al. [15] introduce a framework defining a general approach for failure process modeling, and demonstrate the application of the proposed framework in a light commercial vehicle manufacturing system.

Most trend tests [4], [16] assume that failure times are known, so the failure data are complete. Currently in the literature, except for right censoring, there is no available method for estimating the parameters of a non-homogeneous Poisson process (NHPP), which incorporates left and interval censored failure data if repairs are not instantaneous or not performed immediately. However, it is expected that the data received from industry include missing and incomplete information. Sometimes a particular type of data of interest is not measured at all, or if measured may be incomplete or unreliable. For example, hidden failures which make up to 40% of all failure modes of a complex industrial system [17] are not evident to operators and remain dormant until they are rectified at scheduled inspections. In this case, the times of hidden failures are either left or interval censored, and the censoring interval is the interval between two consecutive inspections.

Weibull or power law [2] and log linear [18] intensity functions are two common models used to describe recurrent even data underlying NHPP. The maximum likelihood estimates of the parameters are obtained for two intensity parameters [12], [19]. In this paper, we assume that the failure process follows an NHPP with a power law intensity function. We use the Expectation–Maximization (EM) algorithm to estimate parameters of the power law process and propose a recursive method to calculate the expected values in the EM Expectation step. In the Maximization step, we use the Newton–Raphson method. We then apply the general likelihood ratio test to test HPP against NHPP alternative [12]. In some practical situations, the mid-points of the censoring intervals may be considered as the failure times and used to estimate the parameters of the failure process and to perform trend test analysis. If the inspection intervals are not short, this approximation of actual failure times can be inaccurate.

A detailed problem description is given in Section 2. Section 3 describes the EM algorithm and proposed trend test method, while Section 4 includes numerical examples using adapted data from the case study [20]. Section 5 presents our conclusions.