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Enhancing Predictive Accuracy in Pharmaceutical Sales Through An Ensemble Kernel Gaussian Process Regression Approach

Shahin Mirshekari, Mohammadreza Moradi, Hossein Jafari, Mehdi Jafari, Mohammad Ensaf

TL;DR

The paper addresses forecasting pharmaceutical sales time series under pattern heterogeneity and uncertainty. It proposes Gaussian Process Regression with an ensemble kernel that linearly combines Exponential Squared, Revised Matérn, and Rational Quadratic kernels, with kernel weights learned through Bayesian optimization. The ensemble approach achieves near-perfect predictive accuracy (e.g., $R^2 \approx 1.0$) and substantially reduced error metrics compared to individual kernels, demonstrating the value of combining kernels for complex real-world data. The work highlights practical impact for healthcare analytics by enabling robust, uncertainty-aware forecasts in large, heterogeneous sales datasets. The dataset comprises hundreds of thousands of transactions across multiple drug categories, underscoring the method's scalability and relevance to time-series forecasting in the pharmaceutical domain.

Abstract

This research employs Gaussian Process Regression (GPR) with an ensemble kernel, integrating Exponential Squared, Revised Matérn, 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 Matérn, 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.

Enhancing Predictive Accuracy in Pharmaceutical Sales Through An Ensemble Kernel Gaussian Process Regression Approach

TL;DR

The paper addresses forecasting pharmaceutical sales time series under pattern heterogeneity and uncertainty. It proposes Gaussian Process Regression with an ensemble kernel that linearly combines Exponential Squared, Revised Matérn, and Rational Quadratic kernels, with kernel weights learned through Bayesian optimization. The ensemble approach achieves near-perfect predictive accuracy (e.g., ) and substantially reduced error metrics compared to individual kernels, demonstrating the value of combining kernels for complex real-world data. The work highlights practical impact for healthcare analytics by enabling robust, uncertainty-aware forecasts in large, heterogeneous sales datasets. The dataset comprises hundreds of thousands of transactions across multiple drug categories, underscoring the method's scalability and relevance to time-series forecasting in the pharmaceutical domain.

Abstract

This research employs Gaussian Process Regression (GPR) with an ensemble kernel, integrating Exponential Squared, Revised Matérn, 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 Matérn, and 0.13 for Rational Quadratic. The ensemble kernel demonstrated superior performance in predictive accuracy, achieving an 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.
Paper Structure (16 sections, 7 equations, 5 figures, 1 algorithm)

This paper contains 16 sections, 7 equations, 5 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Neural Networks vs. Gaussian Processes in Time Series Analysis
  3. Detailed Exposition of Gaussian Processes
  4. Insight into Gaussian Process Kernels
  5. Exponential Squared (ES) Kernel
  6. Revised Matérn Kernel
  7. Rational Quadratic (RQ) Kernel
  8. Ensemble Kernel Method in Gaussian Processes
  9. Bayesian Optimization with Gaussian Process Priors
  10. Gaussian Process as a Surrogate Model
  11. Bayesian Update and Acquisition Function
  12. Advantages and Applications
  13. Evaluation Metrics
  14. Simulation & Results
  15. Categorization and Time Series Analysis of Pharmaceutical Sales Data
  16. ...and 1 more sections

Figures (5)

  • Figure 1: Sales of Pharmaceutical Products Across Various States in the United States11
  • Figure 2: Time series forcastion with an ensemble kernel Gaussian Process Regression approach
  • Figure 3: Time Series Analysis of Pharmaceutical Sales by ATC Categories
  • Figure 4: Gaussian Process Regression Analysis Using RBF, Revised Matern, Rational Quadratic Kernels, and Their Ensemble
  • Figure 5: Comparative Analysis of GPR Kernels Using MSE, MAE, RMSE, and R2 Metrics