# Performance Monitoring of Hybrid All-Optical Fiber/FSO Communication Systems
## 2 Simulation Setup Description
### Machine Learning Model  
In this work, the three parameters under study introduce amplitude variation. Therefore, in order to have a power ML model that can predict each parameter accurately, the model should be trained using features that preserve the amplitude information of the received signals. The amplitude histogram is used in this work as a feature for training the proposed model since it preserves the optical signal's amplitude information. To reduce the acquisition system cost, we utilize an AAH. The AAH samples the signal asynchronously as depicted in the inset in Figure 1, which reduces the sampling speed and hence allows using a low-speed sampler. Each realization is sampled at 500 Msample/s. The AAH feature not only helps build a cost-effective ML model but also eliminates the need for timing recovery at the receiver, thereby avoiding the need for additional hardware. Figure 2 shows examples of the AAH features extracted from different acquired signals using 100 bins. The signals are subjected to 500 ps/nm CD impairment and different OSNR impairment values. It is clear from the figures that the AAH signals have discrepancies, which can be exploited as features for the different types of impairments. The degree of correlation among distinct AAH features is a determining factor for the precision of the ML model in predicting the various parameters. In this work, we consider the GPR model for parameter prediction. GPR is a robust method with the ability to effectively handle complex problems, including nonlinearity, high dimensions, and small sample sizes. It has strong generalization capabilities and is more easily implemented compared to neural network approaches due to its fewer parameters. As a result, GPR is commonly utilized in a wide range of fields [23]. For a training dataset of size \(N\), each training data pair \((x_{i},y_{i})\) is assumed to have unknown distribution. A linear regression model in vector form is given by  
\[\mathbf{y}=\mathbf{x}^{T}\boldsymbol{\beta}+\boldsymbol{\epsilon}, \tag{1}\]  
where \(\mathbf{x}\) is the input vector to the model and \(\mathbf{y}\) is the predicted vector. \(\boldsymbol{\epsilon}\) is an error (noise) vector that has a normal distribution with zero mean and \(\sigma^{2}\) variance, and \(\boldsymbol{\beta}\) is a vector of weights. The \(\boldsymbol{\beta}\) and \(\boldsymbol{\epsilon}\) vectors are estimated from the training data. The GPR regression model performs regression operation by introducing latent variables \(f(x_{i})\) from explicit basis function \(h\) and a Gaussian process (GP). The smoothness of the response is reflected in a covariance function, \(k(\mathbf{x},\mathbf{x}^{\prime})\) of the latent variables, while the basis functions \(h\) maps the input vector \(\mathbf{x}\) into a feature space vector \(h(\mathbf{x})\). The covariance function is parameterized by a set of kernel parameters. The GPR converts (1) as follows [24]  
Figure 2: AAH features for an acquired signal under 500 ps/nm CD and (**a**) OSNR = 12 dB, (**b**) OSNR = 16 dB, and (**c**) OSNR = 20 dB. \[\mathbf{y}=h(\mathbf{x}^{T})\boldsymbol{\beta}+f(\mathbf{x}), \tag{2}\]  
where \(h(\mathbf{x})\) is a GP with zero mean and \(k\) covariance function. In this work, the GPR model uses a squared exponential covariance (kernel) function and a constant basis function. The initial values of \(\beta\) and \(\sigma\) are \(-17.19\) and \(0.55\), respectively. It is worth mentioning that although the GPR model is a powerful regression technique, it has limitations, including scalability and sensitivity to kernel function selection. The model can become computationally expensive when dealing with large datasets, making it impractical. Moreover, selecting the right kernel function can be challenging as the model's performance depends heavily on this choice. However, by using machine learning techniques such as quantum and graph neural networks, it may be possible to improve the performance of the GPR method in real-time scenarios.