## 3 Obtaining the True Degradation Distribution via Deconvolution
Both limits of the regular grid are defined considering the domain of the deconvoluted random variable, such that the minimum value of the deconvoluted observation is 0, and the maximum value corresponds to a \(L_{0}\) as defined. Thus, the approximation of (9) can be viewed as the Riemann sum approximation [50] of the continuous IFT, as follows,  
\[\simeq\frac{2L_{0}}{P}\sum_{j=0}^{P-1}\varphi_{S}\bigg{(}\frac{2L_{0}}{P}(j-1 )-L_{0}\bigg{)}exp\bigg{\{}-i\zeta s\bigg{(}\frac{2L_{0}}{P}(j-1)\bigg{)}-L_{ 0}\bigg{\}}, \tag{10}\]  
where, the width of the \(p\) equally spaced sub-intervals is defined as \(2L_{0}/P\). The number of sub-intervals is considered to be a large enough integer number to obtain a good approximation of \(f\big{(}S_{i}\big{(}t_{j}\big{)}\big{)}\). The \"NormalGamma\" package [51] from R is used to implement the DFT in (10). As this package is defined for convolution operations, the original code was modified to implement the deconvolution operation. Fortunately, the FFT algorithm can be implemented in R to solve the proposed DFT. Specifically, the function is defined as follows for the sampled vector \(k=0,\ldots,P-1\)[26],  
\[W[k]=\frac{L_{0}}{P\pi}exp\{i\pi(k-1)\}\sum_{j=0}^{P-1}\varphi_{S}\bigg{(} \frac{2L_{0}}{P}(j-1)-L_{0}\bigg{)}exp\bigg{\{}\frac{2i\pi}{P}(j-1)(k-1)\bigg{\}}. \tag{11}\]  
In this paper, \(W[k]\) is considered to be an approximation of the true degradation, such that \(W[k]\) is governed by a certain stochastic process. The gamma process, inverse Gaussian (IG) process, geometric Brownian motion (GBM) process and the Wiener process may be considered and the best fitting model may be selected by assessing their respective goodness of fit. It should be noted that both the gamma and the IG processes are monotone processes, while the Wiener process is known to be non-monotone. If the observed degradation paths are monotone, then it is expected that the true degradation paths remain monotone. Which, can only be true when \(\sigma\) is small enough to sustain that \(\Delta Z_{i}\big{(}t_{j}\big{)}>\epsilon_{ij}\). If \(\sigma\) is large enough such that \(\Delta Z_{i}(t_{j})<e_{ij}\), then the degradation paths may become non-monotone, which in some case studies may not be reasonable (such as in crack propagation data). Given that in this paper it is considered that the observed degradation paths are governed by a gamma process, it is expected that the true degradation remains governed by a monotone stochastic process. In Figure 1, a comparison of observed degradation paths and deconvoluted true degradation paths is presented when \(\Delta Z_{i}(t_{j})>e_{ij}\) and \(\Delta Z_{i}(t_{j})<e_{ij}\). The paths with black lines were simulated from a gamma process. From Figure 1a, it can be noted that if \(\sigma\) is large enough the true deconvoluted degradation paths become non-monotone. While in Figure 1b, it can be noted that if \(\sigma\) is small enough, the deconvoluted paths remain monotone. The construction of the deconvoluted paths (red dashed lines in Figure 1) is performed considering that the deconvolution operation is performed at every \(t_{j}\) for every degradation measurement \(Z_{i}(t_{j})\). Then, random true measurements of \(S_{i}(t_{j})\) are generated at every \(t_{j}\) to construct the different paths, which represents cumulative sums of the generated random variables. Once the best fitting stochastic process of \(W[k]\) is defined, the true first-passage time distribution can be obtained. The lifetime of the true degradation is defined as \(T_{s}=inf\big{\{}t_{s}:S_{i}(t_{j})\geq\omega\big{\}}\). The first-passage time distribution will depend on the best fitting stochastic process.