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This research employs Gaussian Process Regression (GPR) with an ensemble kernel, integrating Exponential Squared, Revised Mat\'ern, and Rational Quadratic kernels to analyze pharmaceutical sales data. Bayesian optimization was used to identify optimal kernel weights: 0.76 for Exponential Squared, 0.21 for Revised Mat\'ern, and 0.13 for Rational Quadratic. The ensemble kernel demonstrated superior performance in predictive accuracy, achieving an \( R^2 \) score near 1.0, and significantly lower values in Mean Squared Error (MSE), Mean Absolute Error (MAE), and Root Mean Squared Error (RMSE). These findings highlight the efficacy of ensemble kernels in GPR for predictive analytics in complex pharmaceutical sales datasets., Comment: 6 pages, 5 figures

This study introduces an innovative Gaussian Process (GP) model utilizing an ensemble kernel that integrates Radial Basis Function (RBF), Rational Quadratic, and Mat\'ern kernels for product sales forecasting. By applying Bayesian optimization, we efficiently find the optimal weights for each kernel, enhancing the model's ability to handle complex sales data patterns. Our approach significantly outperforms traditional GP models, achieving a notable 98\% accuracy and superior performance across key metrics including Mean Squared Error (MSE), Mean Absolute Error (MAE), Root Mean Squared Error (RMSE), and Coefficient of Determination ($R^2$). This advancement underscores the effectiveness of ensemble kernels and Bayesian optimization in improving predictive accuracy, offering profound implications for machine learning applications in sales forecasting., Comment: 11 pages, 3 figures

The paper presents a Gaussian/kernel process regression method for real-time state estimation and forecasting of phase angle and angular speed in systems with a high penetration of solar generation units, operating under a sparse measurements regime on both sunny and cloudy days. The method treats unknown terms in the swing equations, such as solar power, as random processes, thereby transforming these equations into stochastic differential equations. The proposed method accurately forecasts and estimates both observed and unobserved operating states, delivering forecasts comparable to those of the standard data-driven Gaussian/kernel process for observed system states. Additionally, the method demonstrates improved accuracy with increased observation frequency and reduced measurement errors in the IEEE 14-bus test system.