Gray-Scale (GS) Encoding
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The Grey-scale encoding has been revived by [@wen2018] for the purpose of fault diagnosis in manufacturing systems using convolutional neural networks. The transformation consists of first, the reshaping of a 1D time series into a collection of $K$ sub-series of size $K_1$, and second, the rescaling of the values as color encoding values (8-bit integer). In this work, for consistency with other encodings, we aim at $K=K_1$ and compute the stride $s$ between the start of each sub-series accordingly. The GS encoding corresponds, therefore, to the following operation: $$GS_{i,j} = \mathrm{round}\left(P \cdot \frac{(x_{(i-1) \cdot s + j} - \mathrm{LB})}{\mathrm{UB} - \mathrm{LB}}\right),\ \ i,j\in \llbracket1,K\rrbracket,
\label{eq:gs}$$ where the operator $\mathrm{round}$ rounds the value to the nearest integer, $P$ is a scaling factor, $\mathrm{UB}$ and $\mathrm{LB}$ are respectively an upper and lower bound to be used for the scaling of $X$.

**Proposed modification:** In the original formulation, the authors suggested to use $\mathrm{UB}=\mathrm{max}(X)$, $\mathrm{LB}=\mathrm{min}(X)$ and $P=255$ such that the resulting values can be interpreted as 8-bit encoded grey scale images. In this work, we propose first, to define $\mathrm{UB}$ and $\mathrm{LB}$ over the entire training dataset, choosing these bounds in accordance with the GAF encoding. Furthermore, since most 