with the final precision items of the machining process. Note that, with the four relays adopted in Figure 2.13 and Table 2.2, 15 M‐codes plus a reset code are allowed in this case to specify 15 pre‐defined M codes.
2.3.2 Cleaning
The second step, data cleaning, emerges after acquiring the segmented signals. Data cleaning effectively handles raw sensing signals with noises. Basically, a process observed and recorded via signals might contain a series of stationary and non‐stationary components, especially in a machinery manufacturing environment. If the properties of the process that generate the events do not change in time, then the process is defined as stationary.
The nonstationary process (such as edge, peak, break, and noise embedded in sensing signals) includes anything that does not satisfy the conditions for stationary. These result in low signal‐to‐noise (S/N) ratios of the raw data collected from sensors, and thereby affect the diagnosis or prediction accuracy.
Data cleaning attempts to cancel the noise in signal and improve the S/N ratio to prepare for post‐processing. Generally, data cleaning can be done by using various time‐domain or frequency‐domain techniques. Trend removal and wavelet thresholding methods are respectively introduced below.
2.3.2.1 Trend Removal
A signal with a trend is called nonstationary. A trend is a long‐term continuous increase or decrease embedded in signals over time, which is not a good situation for designing a stable process and achieving the high product quality. To identify the pattern of a trend helps to determine whether the changes resulted from equipment or environment are normal or not. By removing or correcting the unmeaning trend information from signals (de‐trending), problems can be simplified and model efficiency can be improved.
For example, Figure 2.14a demonstrates a direct current (DC) signal with 10 Hz in time‐domain, and Figure 2.14b shows its frequency spectrum, where the DC component, or the so‐called mean value, hardly contains any useful information to reflect mechanical faults due to an unchanged constant. In addition, the amplitude of the frequency spectrum is much larger than real characteristic frequency so that the critical information might be out of focus.
Figure 2.14 An DC signal: (a) in time‐domain; (b) in frequency‐domain; (c) in time‐domain after removing DC trend; and (d) in frequency‐domain after removing DC trend.
Therefore, this DC component must be removed in advance. Figure 2.14c removes the mean value from the original DC signals and Figure 2.14d indicates that after deleting the DC component, the real characteristic frequency of 10 Hz can be emphasized in the spectrum.
The other example is given in Figure 2.15. In Figure 2.15a, the waveshape of the collected signal is the composition of two cosine waves with the frequency of 0.5 and 1 Hz, respectively, and one linear function increases as time passes. Figure 2.15b indicates that the frequency spectrum for recognizing the characteristic frequencies at 0, 0.5, and 1 Hz. As the example shown in Figure 2.14b, the amplitude of the DC component is too large so that real characteristic frequencies are not obvious, which may result in miss detection.
Figure 2.15 A random signal: (a) in time‐domain; (b) in frequency‐domain; (c) in time‐domain after removing the linearly increasing trend; and (d) in frequency‐domain after removing the linearly increasing trend.
Therefore, this linear trend is unexpected and should be removed from the signal. The trend is obtained by computing the least‐squares fit of a straight line (or composite line for piecewise linear trends) to the signal, and then the trend is subtracted from the original signal as in Figure 2.15c. Figure 2.15d shows that the spectrum is more readable than Figure 2.15b since the DC component is removed.
2.3.2.2 Wavelet Thresholding
Wavelet thresholding, or the so‐called wavelet de‐noising, mainly adopts the discrete wavelet transform (DWT) technique [8–10] to filter noises in signals. DWT provides a multi‐resolution representation using wavelets, which can discretely capture rich information both in time and frequency domains.
The discrete wavelet coefficient capability of spare distribution and auto‐zooming in the time and frequency domains provided by DWT can be applied to deal with the non‐stationary signals for enhancing the S/N ratios. Thus, critical information behind signals with noises can be accurately obtained. More details can be referred to Sections 2.3.3.2 and 2.3.3.3.
Suppose that M sets of machining signals related to machining precision are collected and each set has data length N, denoted as s[i], i = 1, …, N. Let r and c represent the raw and cleaned data, respectively. Based on DWT, data cleaning adopts the wavelet de‐noising algorithm to purify the discrete raw sensor data of precision item p, denoted as
(2.1)
where
The general wavelet de‐noising process consists of three steps: decomposition, thresholding, and reconstruction. They are described as follows.
Decomposition
The wavelet coefficients are calculated by passing
