Prognosis

Prognosis is the art of predicting the remaining useful life (RUL) of a system. Prognosis is related to diagnosis and depends on it. Diagnosis is the method of identifying and quantifying damage that has occurred (Sikorska, et al., 2011). Prognosis requires information from the diagnosis to forecast the future. However, in general, the degradation process cannot be directly observed or measured. It can only be investigated indirectly through the time series of features extracted from available process measurements (Yan, et al., 2010). This creates two major challenges for prognosis (Yan, et al., 2010):

1. How to design an appropriate degradation indicator (see Detection/Diagnosis) 2. How to establish a prediction model to estimate failure times

RUL Prediction methods can be classified according to the approach they use (Figure 11).

Figure 11: Main model categories for prediction of remaining useful life (Sikorska, et al., 2011)

Signal Analysis

A signal is an entity whose value can be measured and which conveys information (Owen, 2007).

Signals can represent sound, vibrations, colour values, temperature etc. There are two types of signals: analogue and digital. An analogue signal is continuous, and a digital signal has a finite

number of values. The process of transforming an analogue signal into a digital signal is called sampling. Sampling represents an analogue signal as several regularly spaced measurements or samples (Owen, 2007). Figure 12 shows the sampling of an analogue signal.

Figure 12: Signal sampling (Owen, 2007)

The number of regular spaced samples per second is the sampling rate measured in Hz. A signal has an amplitude and a phase. The amplitude is the sampling value, and the phase is the time delay between this motion and another motion of the same speed (Owen, 2007). Signals represented as in Figure 12 are in the time domain. It is possible to transform time signals so that they are represented in the frequency domain. In the frequency domain, the signal is represented by cosine and sine functions with different frequencies (Owen, 2007). The process converting the signal is called Fourier transform for analogue signals and discrete Fourier transform for digital signals.

Equation (1 shows the discrete Fourier transform.

𝑍(𝑓) = ∑ 𝑧(𝑘)𝑒^{−2𝜋𝑗𝑓𝑘𝑁}

𝑁−1

𝑘=0

(1)

where Z(f) is the Fourier coefficient at frequency f (Owen, 2007), N is the total number of samples and k is the current sample. z(k) is x(k) + jy(k), where x and y are the amplitude and the phase of the signal, respectively. It is possible to reverse the transform using Equation (2.

𝑧(𝑘) = 1

𝑁∑ 𝑍(𝑓)𝑒^{2𝜋𝑗𝑓𝑘𝑁}

𝑁−1

𝑗=0

(2)

It is also possible to treat the complex values as real values if the phase is unknown or zero. In the case of a real value signal, only ¹

𝑁 coefficients are independent because Z(N-f) and Z(f) are the same if only the real part is considered. In practice, this means 2N samples are needed to get N Fourier coefficients. Figure 13 shows a real value signal transformed into the frequency domain.

Figure 13: Time domain to frequency domain

An algorithm to compute the discrete Fourier transform on a computer is called the Fast Fourier Transform (FFT). The FFT requires that N is a power of two (Owen, 2007).

A filter is a process which changes the shape of a signal (Owen, 2007), often in the frequency domain. Usual types of filters are low-pass, high-pass or band-pass filters. Low-pass filters keep low frequency components of the signal and block high frequency ones. A high-pass filter blocks low frequencies and keeps high ones. A band-pass filter blocks all but a given range of frequencies (Owen, 2007). One way to apply a filter is to transform the time domain signal into the frequency domain, apply the filter and transform the signal back into the time domain.

Band-pass filters can be used to extract frequency components from a signal into a new signal. If multiple band-pass filters are applied to a signal to extract different frequencies, the filter is called a filter bank. The individual band-pass filters can have the same or different sizes (Rabiner &

Juang, 1993). Figure 14 shows a filter bank with equal sized band-pass filters, and Figure 15 shows a filter bank with band-pass filters of a different size.

Figure 14: Equal sized filter bank (Rabiner & Juang, 1993)

Figure 15: Variable sized filter bank (Rabiner & Juang, 1993) Feature Extraction

Feature extraction is the process of reducing the dimension of the initial input data to a feature set of a lower dimension that contains most of the significant information of the original data (Fonollosa, et al., 2013). This is done to extract features from noisy sensor data (Lin & Qu, 2000);

(Fu, 2011)and to avoid the problems caused by having too many input features (especially for vibration data) in the classifier learning phase (Yen & Lin, 2000). For these reasons, feature extraction is often a first and essential step for any classification (Yen & Lin, 2000).

Methods for feature extraction include extracting features from the time domain and the frequency domain (Fourier transformation, wavelet transformation (Fu, 2011)) and clustering them, if necessary. Basic features can be maximum, mean, minimum, peak, peak-to-peak interval

etc. (Jardine, et al., 2006). Complex feature extraction methods include principal component analysis (PCA), independent component analysis (ICA) and kernel principal component analysis (KPCA) (Widodo & Yang, 2007). Other feature extraction methods are: t-test, correlation matrix, stepwise regression and factor analysis (FA) (Tsai, 2009). A comparison of feature extraction methods is found in Arauzo-Azofra et al. (Arauzo-Azofra, et al., 2011).

Clustering is needed if the data samples from which the features are extracted have no information about what the data represent (Li & Elbestawi, 1996). In such cases, clustering methods can be applied to group the data into classes.

Selecting relevant features for classifiers is important for a variety of reasons, including generalization of performance, computational efficiency and feature interpretability (Nguyen &

De la Torre, 2010). Using all available features can result in over fitting and bad predictions, but it is not possible to look at each feature alone because many features are intercorrelated (Meiri &

Zahavi, 2006). Noise, irrelevant features or redundant features complicate the selection of features even more. Thus, features are often selected using methods taken from pattern recognition or heuristic optimization or a combination. Sugumaran et al. (2007) show how different technologies can be combined for a single goal; they use a decision tree for feature selection and a proximal support vector machine for classification. Widodo and Yang (2007) combine ICA/PCA plus SVM for feature extraction and classification. A combination of particle swarm optimization (PSO) and SVM is used for feature extraction and process parameter optimization by Huang and Dun (2008). Many algorithms combine genetic algorithms (GAs) with a pattern recognition method like decision trees (DTs), SVM or artificial neural networks (ANNs).

In these combinations, the GA is used to optimize the process parameters (Samanta, et al., 2003) (Huang & Wang, 2006) or to perform feature extraction and pattern recognition for classification (Samanta, 2004) (Saxena & Saad, 2007) (Jack & Nandi, 2002) (Samanta, 2004) Another popular approach is simulated annealing (SA) plus pattern recognition (Lin, et al., 2008) (Lin, et al., 2008).