denoted by This deviation between and is denoted , and the distribution is f(x). f(x) estimated with the kernel density estimation approach [29]. For each cycle of current waveform signals, one KLD value is calculated and evaluated to determine if abnormality exists. The detection rule in Fig. 92 can be represented as follows. If the following equation is satisfied, an abnormality exists. 23 is the abnormality detection threshold. Dth Further details regarding the estimation of residual pdf by using the kernel density estimation and the calculation of KLD are provided in Appendix C and Appendix D, respectively. Selection of the Abnormality Detection Threshold The performance of the current method is highly dependent on a proper selection of the abnormality detection threshold. According to the signal detection theory [27], a proper threshold selection must account for an acceptable probability of false alarm. A false alarm occurs when normal data is classified as abnormal. For an arbitrary threshold value the false alarm probability in a certain period of Dth, time can be calculated as: 24 is the probability density function obtained by using KLD values calculated p(s) from waveform measurements with no abnormality, that were collected during a given period of time. The false alarm probability is a monotonic non-increasing function PFA of Dth. If the acceptable false alarm probability is the selected threshold must be the Dth α, smallest value that satisfies Mathematically, it can be expressed as PFA(Dth) ≤ α. follows: 25 A systematic threshold selection shall follow these steps: Step 1: Collect sets of the residual data of previous waveform cycles, NKLD where each set contains samples; N Step 2: For each set, estimate the probability density function of the samples by N employing the kernel density estimation. Then, calculate the divergence level (i.e., calculate the value of the KLD) between the kernel estimation and the theoretical Gaussian noise pdf. Each KLD value is a sample of function; p(s)
denoted by f(x). This deviation between and f(x) is denoted , and the distribution is estimated with the kernel density estimation approach [29]. For each cycle of current waveform signals, one KLD value is calculated and evaluated to determine if abnormality exists. The detection rule in Fig. 92 can be represented as follows. If the following equation is satisfied, an abnormality exists. 23 Dth is the abnormality detection threshold. Further details regarding the estimation of residual pdf by using the kernel density estimation and the calculation of KLD are provided in Appendix C and Appendix D, respectively. Selection of the Abnormality Detection Threshold The performance of the current method is highly dependent on a proper selection of the abnormality detection threshold. According to the signal detection theory [27], a proper threshold selection must account for an acceptable probability of false alarm. A false alarm occurs when normal data is classified as abnormal. For an arbitrary threshold value Dth, the false alarm probability in a certain period of time can be calculated as: 24 p(s) is the probability density function obtained by using KLD values calculated from waveform measurements with no abnormality, that were collected during a given period of time. The false alarm probability PFA is a monotonic non-increasing function of Dth. If the acceptable false alarm probability is α, the selected threshold Dth must be the smallest value that satisfies PFA(Dth) ≤ α. Mathematically, it can be expressed as follows: 25 A systematic threshold selection shall follow these steps: Step 1: Collect NKLD sets of the residual data of previous waveform cycles, where each set contains N samples; Step 2: For each set, estimate the probability density function of the N samples by employing the kernel density estimation. Then, calculate the divergence level (i.e., calculate the value of the KLD) between the kernel estimation and the theoretical Gaussian noise pdf. Each KLD value is a sample of p(s) function;
Signatures_Equipment_Failures_May2018.docx