Forecasting Oncology Demand Trends with Boosting-Based Bayesian Conjugate Models

Technical Summary: Forecasting Oncology Demand Trends with Boosting-Based Bayesian Conjugate Models

Problem Statement
Accurate forecasting of healthcare demand, particularly in oncology, is critical for resource allocation and service planning. However, oncology units face significant challenges due to high variability in patient volumes driven by delayed diagnoses, individualized treatment protocols, and irregular follow-up visits. Traditional time series models, such as Autoregressive Integrated Moving Average (ARIMA) and Exponential Smoothing (ETS), often struggle to capture structural shifts, non-stationary trend dynamics, or to incorporate prior knowledge effectively. Furthermore, while machine learning models like Long Short-Term Memory (LSTM) networks and XGBoost offer flexibility, they may lack the analytical tractability and uncertainty quantification required for dynamic healthcare environments. The specific challenge addressed is the need for a model that can adapt to non-stationary patterns, track both short- and long-term trend shifts, and maintain closed-form updating capabilities.

Methodology
The authors propose a Bayesian inference framework that combines conjugate distributions with a residual-based boosting mechanism. The methodology operates in three sequential stages:

1. Base Model (Gamma–Poisson Conjugation):
   The core demand forecasting utilizes a Gamma–Poisson conjugate pair. Weekly appointment counts ($x_t$) are modeled as a Poisson process with an unknown rate parameter $\lambda_t$. A Gamma distribution serves as the conjugate prior for $\lambda$. This allows for sequential Bayesian updating where the posterior distribution remains in the Gamma family, providing a closed-form solution for the posterior mean forecast. This component captures the long-term memory of the demand process.

2. Residual Modeling (Gamma–Log-Normal Conjugation):
   To address systematic deviations between observed and predicted values, the model analyzes the residuals. The multiplicative residuals are transformed using a logarithmic function ($\log_p1(x) = \log(x+1)$) to approximate a Normal distribution. A Gamma–Log-Normal conjugate structure is then applied to these transformed residuals. Specifically, the precision of the error is modeled with a Gamma prior, and the mean error is modeled with a Normal distribution conditional on the precision. This yields a posterior mean for the error term that represents a weighted average of virtual and empirical evidence.

3. Boosting and Correction:
   The final forecast ($\hat{x}_{t+1}$) is generated by summing the base Poisson forecast ($\hat{\lambda}_{t+1}$) and the back-transformed error forecast. The error term is exponentiated and adjusted ($\exp(\hat{\mu}_{t+1}) - 1$) to return to the original scale, with a max(0, ...) function applied to prevent numerical instability. This additive correction allows the model to dynamically adapt to changes in the data-generating process while preserving the analytical tractability of the conjugate updates.

The algorithm is designed for online learning, updating posterior parameters ($\alpha, \beta$ for the base model; statistics for the error model) at each time step as new data arrives.

Key Contributions

• Hybrid Bayesian Framework: The integration of a Gamma–Poisson base model with a Gamma–Log-Normal residual boosting mechanism, enabling the tracking of both demand counts and systematic trend shifts.
• Analytical Tractability: The use of conjugate priors ensures closed-form, sequential updating, avoiding the computational costs of simulation-based methods like MCMC, which is advantageous for real-time or resource-constrained settings.
• Explicit Trend Direction Emphasis: The methodology explicitly prioritizes the direction of trends using the Percentage of Correct Direction (POCID) metric, rather than solely focusing on magnitude errors.
• Real-World Validation: The model was evaluated on real-world weekly oncology service data from São Vicente de Paula Hospital in Cariri, Ceará, Brazil, covering ten distinct cancer types (e.g., Prostate, Breast, Rectal) from January 2023 to December 2024.

Results
The proposed model was compared against five baselines: Naïve forecasting, Linear Regression, ARIMA, LSTM, and XGBoost.

• Trend Direction (POCID): The proposed model achieved the highest POCID values across most series, outperforming the second-best approach by up to 38.25% in specific cases. It demonstrated superior stability and accuracy in predicting the direction of demand changes compared to ARIMA, Linear Regression, and Naïve methods.
• Scale-Sensitive Metrics: The model showed significant reductions in error metrics compared to parametric and tree-based baselines. Median reductions included 74.5% in MAPE versus ARIMA, 77.6% versus Linear Regression, 75.7% versus LSTM, and 74.7% versus XGBoost.
• Statistical Significance: Using the Diebold–Mariano (DM) test based on Mean Squared Error (MSE), the proposed model statistically outperformed the second-best model in 7 out of 10 series. In the remaining cases, results were inconclusive, but the authors applied a fallback criterion favoring the proposed model.

Significance and Claims
The paper claims that the integration of prior knowledge with residual-based boosting provides a principled framework for handling non-stationary healthcare time series. The authors argue that this approach offers a distinct advantage in identifying demand trend directions where traditional models fail to adapt to structural variability.

The authors remain modest regarding the model's universality, noting that:

• Performance gains over the Naïve benchmark are series-dependent.
• The model's performance is sensitive to the initialization of prior distributions, particularly when data is scarce or highly variable.
• Certain evaluation metrics showed sensitivity to outliers, influencing final assessments.

Future work is proposed to explore systematic approaches for prior elicitation (potentially using empirical Bayes) and to integrate short- and long-term memory mechanisms inspired by LSTM networks to further enhance adaptability to evolving data structures.