*2.4. Parameters Selection*

In this section, the regularization parameters *λ*i are set as [26]:

$$\begin{aligned} \lambda\_0 &= \beta\_0 \sigma \\ \lambda\_1 &= \beta\_1 \sigma = \gamma (1 - \beta\_0) \sigma \\ \lambda\_2 &= \beta\_0 \sigma \end{aligned} \tag{40}$$

where *β*0 and *β*1 are the constants so as to maximize the signal-to-noise-rate (SNR), *β*0 and *γ* are typically set up to be constants, i.e., *β*0 = [0.5, 1], *γ* = [7.5, 8], and *σ* is the standard deviation (SD) of the external noise. In practical engineering applications, the SD of the external noise in Equation (40) could be computed using the healthy data and fault data under the same operating environment. Moreover, when the healthy data is not available or unknown, the SD of the external noise can still be estimated by the following formula,

$$
\stackrel{\triangle}{\sigma} = MAD(y)/0.6745\tag{41}
$$

where the Equation (41) is a noise level estimator that used for traditional wavelet denoising in ref. [41], and *MAD*(*y*) represents the median absolute deviation (*MAD*) of observation signal *y*, i.e.,

$$MAD(y) = \text{median}(|y\_i - \text{median}(y)|), i = 1, 2, \dots, N \tag{42}$$

To better understand the asymmetric penalty sparse decomposition algorithm, a chromatogram signal is analyzed by the proposed APSD algorithm [38]. For illustration, Figure 3a exhibits the raw chromatogram signal. Figure 3b,c respectively shows the LFC and HFC that generated by the proposed APSD. Note that the main evolutionary trend, i.e., low frequency component, is well estimated as illustrated in Figure 3b, meanwhile, the estimated peaks, i.e., high frequency component, illustrated in Figure 3c, are well delineated. Therefore, the LFC and HFC components could be captured by the asymmetric penalty sparse decomposition algorithm.

**Figure 3.** The decomposition results of noisy chromatogram data using proposed APSD algorithm [38]. (**a**) The raw data with additive noise; (**b**) The estimated LFC signal; (**c**) The estimated HFC signal.
