Funnel Random Forest: Inliers-Focused Ensemble Learning for Improved Prognostics of Heart Failure

Funnel Random Forest: Inliers-Focused Ensemble Learning for Improved Prognostics of Heart Failure | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Funnel Random Forest: Inliers-Focused Ensemble Learning for Improved Prognostics of Heart Failure Luca Parisi, Marianne Lyne Manaog This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5784003/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This study introduces a novel Funnel Random Forest (FRF) methodology to improve the accuracy and reliability of clinical prognostics, particularly for patients with heart failure. FRF identifies outliers using information theory-based metrics during training, reducing outliers by 11.04% compared to traditional methods. The model was trained on a dataset of 299 clinical records, achieving an 82.56% accuracy, a 1.67% improvement over standard Random Forest models. FRF outperformed existing models, with a 9.67% increase in accuracy and a 25.6% improvement in reliability, measured by the Precision-Recall Area Under the Curve (PR-AUC). These results demonstrate the model's potential to enhance clinical decision support, leading to better patient outcomes through early and precise prognostic predictions. Random Forest Outlier removal Mutual Information Gini Impurity Heart failure Prognostics Survival prediction Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 1. Introduction Cardiovascular diseases (CVDs), including heart failure (HF), remain the leading causes of mortality worldwide, with 17.9 million people dying every year, i.e., 32% of all deaths globally (World Health Organization (WHO), 2021). In the United Kingdom (UK), 160,000 people die due to CVDs each year and over 900,000 people live with HF (British Heart Foundation (BHF), 2021). The Coronavirus disease 2019 (COVID-19)-related pandemic has led to a reduced hospitalisation and an increase in mortality due to CVDs (Banerjee et al., 2020; Huet et al., 2020; Normando et al., 2021). If mortality due to HF could be predicted, deaths could be prevented by titrating treatments early (Ponikowski et al., 2014; Groenewegen et al., 2020). 1.1 Machine Learning algorithms for predictive analytics Machine Learning (ML) is a branch of Artificial Intelligence (AI) enabling algorithms to learn patterns from data for predictive analytics automatically (Samuel, 1959). ML can be either supervised or unsupervised, respectively for classification by training on labelled data (Parisi et al., 2018, 2024; Parisi & Manaog, 2023) or clustering based on similarity among unlabelled samples (Parisi & RaviChandran, 2018). Decision Tree (DT)-based models (Chicco & Jurman, 2020; Almazroi, 2022), including Random Forest (RF) ( Fig. 1 ), are explainable (Khalilia et al., 2011; Thakur & Kumar, 2018; Kaur et al., 2019; Zhang et al., 2021; Rey-Blanco et al., 2024; Wang et al., 2024; Parisi & Manaog, 2025) ML-related supervised algorithms, whilst the Convolutional Neural Network (CNN) (Kwon et al., 2019; Wang et al., 2020) is a Deep Learning (DL)-based supervised method with outputs slightly interpretable by saliency maps (Springenberg et al., 2014; Selvaraju et al., 2017; Parisi & Manaog, 2023, 2025). Neural network (NN)-, optimal separating hyperplane (OSH)-, and distance-based algorithms are also supervised algorithms ( Fig. 1 ), e.g., the Multi-Layer Perceptron (MLP) (Rumelhart et al., 1986), the Support Vector Machine (SVM) (Cortes & Vapnik, 1995), and the K-Nearest Neighbour (KNN) (Fix & Hodges, 1989). MLP, SVM, and KNN aid classification respectively by adjusting the weights of neurons, maximising the margin width, and by identifying boundaries in the data via distance metrics. k-means (MacQueen, 1967; Fränti & Sieranoja, 2018), Self-Organising Map (SOM) (Kohonen, 1990), and Neural Gas (Martinetz & Schulten, 1991), are clustering algorithms grouping data based on their similarity ( Fig. 1 ). “Hierarchical clustering” (Johnson, 1967) and “Birch” (Zhang et al., 1996) are also clustering algorithms ( Fig. 1 ), respectively leveraging dendrograms of clusters and sub-clusters in feature trees. 1.2 Critical evaluation of related studies AI-based classification methods, from ML to DL, have been developed to aid prediction of survival in patients with HF. However, existing DL techniques are overly sophisticated (Kwon et al., 2019; Wang et al., 2020), such as CNNs (LeCun & Bengio, 1995; Parisi et al., 2022), or involve ML algorithms applied to only generate a binary prediction (Hearn et al., 2018; Chicco & Jurman, 2020; Shin et al., 2021; Almazroi, 2022). Thus, due to their limited explainability, existing techniques have not been widely adopted in a clinical setting, thus hindering their translational value and societal impact. DT-based classifiers, considering their high interpretability, have been applied for predicting survival in patients with HF, despite their reliability was not deemed clinically acceptable (Chicco & Jurman, 2020; Almazroi, 2022). Chicco & Jurman (2020) achieved 74% accuracy with only 0.657 as the precision-recall (PR) area under the curve (AUC) via an RF, whilst Almazroi (2022) obtained 80% accuracy but with 65.21% of recall via a DT ( Fig. 2 ). 1.3. Research Objectives The primary objective of this research is to develop and validate a novel Funnel Random Forest (FRF) methodology that enhances the accuracy and reliability of clinical prognostics, particularly for patients with heart failure. The specific objectives are: Outlier detection and handling : To introduce an intrinsic mechanism within the Random Forest model that proactively identifies and handles outliers using information theory-based metrics, thus improving the model's robustness and generalisation. Performance improvement : To demonstrate the effectiveness of the FRF model in reducing the number/percentage of outliers and improving training performance as compared to traditional statistical outlier detection/removal methods. Clinical relevance : To validate the FRF model's ability to enhance clinical decision support by providing more accurate and reliable prognostic predictions for patients with heart failure. Comparative analysis : To compare the performance of the FRF model with existing models, demonstrating its higher accuracy and reliability. Practical implications : To highlight the potential of the FRF model to improve patient outcomes through early and precise prognostic predictions, thus aiding clinicians in making data-informed decisions. 1.4. Rationale of proposed method and article’s structure From a critical review of the related literature as per sub-section 1.2, the Random Forest (RF) model was identified as an optimal candidate due to its ability to balance high classification performance with interpretability (Rey-Blanco et al., 2024; Parisi et al., 2024). The Decision Tree (DT)-like architecture of RF allows for intuitive understanding and visualisation of decision-making processes, which is particularly valuable in a clinical setting, wherein transparency and explainability are crucial. However, despite its strengths, RF models are susceptible to the presence of outliers in the training data, which can adversely affect their predictive performance and generalisation. Existing outlier-handling methods often rely on subjective heuristics that do not generalise well across different datasets, particularly in the context of clinical data, which are inherently noisy and heterogeneous. To address these challenges, this study proposes a novel Funnel Random Forest (FRF) methodology. The motivation behind this approach is to enhance the accuracy and reliability of RF models by incorporating an intrinsic mechanism for outlier detection and handling. Specifically, the proposed method leverages the Gini impurity measure and mutual information-related scores to identify and discard outliers during the training process. This proactive approach ensures that the training data are cleaner, with more neatly separable patterns, thus leading to improved model’s predictive performance and higher generalisation on unseen test data. The contributions of this study are multi-faceted: 1. Novel methodology : The introduction of the FRF model, which integrates information theory-based metrics for outlier detection within the RF training process. 2. Performance improvement : Demonstration of improvements in classification accuracy and reliability, with the FRF identifying 11.04% fewer outliers and achieving a 2% increase in training performance on medical datasets. 3. Clinical relevance : Validation of the model's effectiveness in enhancing clinical decision support, particularly for predicting prognostics in patients with heart failure. 4. Comparative analysis : Comprehensive comparison with existing models, showcasing the FRF's higher accuracy and reliability. 5. Practical implications : Highlighting the potential of the FRF to provide more accurate and reliable and accurate predictions for clinicians, ultimately leading to better patient outcomes through earlier and more precise prognostic predictions. The rest of the article is composed of the following sections: section 2. on the datasets leveraged in this study and an outline of the proposed methodology; section 3. on the results obtained, including the evaluation of the accuracy and the reliability of the proposed model on the respective test sets; section 4. on the discussion of these results, including considerations on both the datasets used and the modelling approach adopted, and on the model’s explainability demonstrated via two further case studies; section 5. with conclusions and related reflections. 2. Methods 2.1. Dataset and descriptive analytics The RF built to predict survival in patients with HF was developed in R (version 4.1.2). The dataset of clinical records on 299 subjects used for model development and initial evaluation is available on the University California Irvine (UCI) ML repository (Ahmad et al., 2017; Chicco & Jurman, 2020) and consists of twelve input features: age (years), anaemia, high blood pressure, diabetes, smoking, sex, ejection fraction, creatinine phosphokinase (mcg/L), platelets (kiloplatelets/mL), serum creatinine (mg/dL), serum sodium (mEq/L), and follow-up time (days). The output variable to predict is the ‘DEATH_EVENT’ (0 for survived, 1 for deceased). Furthermore, the dataset of clinical records (1,700 instances for 124 clinical features) pertaining to myocardial complications from the UCI ML repository (Golovenkin et al ., 2020) was leveraged for an external validation to predict chronic heart failure, i.e., as an additional test set. Descriptive statistics were performed, and no missing values were found. Box plots of the input features and bar plots of the class distributions were produced, before and after removing outliers via the interquartile range (IQR)-based method (Vinutha et al., 2018). This is the standard statistical method compared with the outlier removal-related technique intrinsic to the RF model implemented and validated in this study. The dataset for model development and initial evaluation of the remaining 223 subjects (162 survived (72.65%), and 61 deceased (27.35%)) was analysed. Normality was assessed via histograms, quantile-quantile (Q-Q) plots, and the Shapiro-Wilk test (Razali & Wah, 2011) for continuous variables. The features “age”, “creatinine_phosphokinase”, and “serum_creatinine” were found positively skewed; thus, a log transformation was performed to reduce their skewness. Furthermore, a min-max normalisation (Jain et al., 2005) was carried out to ensure all variables had the same range (from 0 to + 1). The pre-processed data were then split into 80% for training (131 survived patients, 47 deceased), 20% for testing (31 survived patients, 14 deceased). 2.2. Funnel Random Forest for predictive analytics The RF model was trained via 5-fold cross-validation to prevent overfitting (Kohavi, 1995; Parisi & Manaog, 2016, 2017a, 2017b) and Synthetic Minority Oversampling Technique (SMOTE) (Chawla et al., 2002) was leveraged to upsample the minority class. Its ‘mtry’ (number of random variables at each split) hyperparameter was optimised (as 4) via random search, yielding the initially trained model with 80.89% accuracy (Table 1 ). This training accuracy was obtained further to removing 76 patients’ clinical records (25.42% of the entire data) deemed outliers based on the IQR-based method. Table 1 The initially trained Random Forest (RF) and its training accuracy. mtry = 4 yielded the highest accuracy (80.89%), thus resulting in the optimal RF model initially. mtry Accuracy Kappa 1 0.7865 0.3872 2 0.8087 0.4821 4 0.8089 0.4951 5 0.8084 0.5097 7 0.7859 0.4558 8 0.7862 0.4527 9 0.7692 0.4037 10 0.7522 0.3579 The ‘funnel’ information theory-based optimisation of the RF model to intrinsically detect and discard outliers prior to training leverages two main components to maximise classification performance (Eq. ( 1 )): 1) the Gini Impurity (GI) from the RF (Eq. ( 2 )); 2) the mutual information (MI) (Eq. (3)) brought by each input based on the defined feature set. $$\:{FD}_{performance}^{GI-MI}=argmin{\sum\:}_{i=1}^{n}1-{\int\:}_{0}^{1}\frac{TP}{TP+FN}\left(thresh\right)\left(-{\frac{\sum\:FP}{\sum\:\left(FP+TN\right)}}^{T}\left(thresh\right)\right)dthresh$$ 1 In ( 1 ): FD represents the filtered data obtained by applying equations ( 2 )-(3); TP - true positives, TN - true negatives, FP - false positives, FN - false negatives, \(\:thresh\) is the threshold for determining the receiver operating characteristic (ROC) curve. The GI is represented by the following Eq. ( 2 ): $$\:GI\left(p\right)={\sum\:}_{m=1}^{N}{p}_{m}\left(1-{p}_{m}\right)$$ 2 where p represents the probability of a sample with class m being selected, whilst N is the number of classes considered. The MI is described via Eq. (3): MI(a,b) = E(b) – E(b | a) ( 3 ) where a and b are random variables, E stands for entropy, thus E(b) being the marginal entropy and E(b | a) is the conditional entropy. Further to leveraging the funnel-based outlier removal implemented intrinsically to the RF model’s architecture (equations ( 1 )-(3)), 43 patients’ clinical records (14.38% of the entire data) were identified as outliers; thus, the Funnel RF model was re-trained, and its training accuracy increased to 82.56% (by 1.67%) with “mtry” equal to 2 (Table 2 ). Table 2 The trained Funnel Random Forest and its training accuracy. mtry = 2 yielded the highest accuracy (82.56%), thus resulting in the optimal Funnel RF model. mtry Accuracy Kappa 1 0.8144 0.5010 2 0.8256 0.5402 3 0.8029 0.4891 4 0.8144 0.5056 5 0.8027 0.5056 7 0.7971 0.4794 8 0.7803 0.4338 9 0.7521 0.3784 The classification performance of the Funnel RF model was critically evaluated on the test set and two further case studies. Considering both classification accuracy and reliability as quantifying model’s generalisation on the test set (Parisi et al., 2018a, 2018b, 2020), the main performance metrics considered, based on TPs, TNs, FPs, and FNs (Table 3 ), are the following: accuracy (Eq. ( 4 )), sensitivity/recall (Eq. ( 5 )), specificity (Eq. ( 6 )), ROC-AUC, and PR-AUC. Table 3 TPs, TNs, FPs, and FNs, as related to the subjects in this study. From actual clinical records True (Survived) False (Deceased) From predictions Positives (Survived) True Positives (TPs) False Positives (FPs) Negatives (Deceased) True Negatives (TNs) False Negatives (FNs) $$\:Accuracy=\frac{TP+TN}{TP+TN+FP+FN}$$ 4 $$\:Sensitivity=\frac{TP}{TP+FN}$$ 5 $$\:Specificity=\frac{TN}{TN+FP}$$ 6 Based on the related clinical literature (Veenstra et al., 1994; Ahmad et al., 2017), two synthetic subjects were simulated as having characteristics (normalised between 0 and 1) found in clinical records of an elderly patient with anaemia who may die from HF (subject no. 1) and a young subject without anaemia who may survive despite having HF (subject no. 2). Levels of creatinine phosphokinase, as an indicator of injury to the heart, serum sodium, and blood pressure were high in subject no. 1 (Veenstra et al., 1994; Ahmad et al., 2017) and low in subject no. 2. Conversely, ejection fraction, i.e., the percentage of blood that is pumped out of the heart to the body, platelet count, and follow-up time were low for subject no. 1 (Veenstra et al., 1994; Ahmad et al., 2017) and high for subject no. 2. 3. Results 3.1. Training Performance The Funnel Random Forest (Funnel RF) model was initially trained on a dataset of 299 clinical records, with 76 records (25.42%) identified as outliers and removed using the interquartile range (IQR)-based method. This initial training yielded an accuracy of 80.89% with an optimal 'mtry' value of 4. The Funnel RF model was then re-trained after identifying and removing 43 outliers (14.38%) using the proposed information theory-based metrics. This re-training improved the accuracy to 82.56% with an optimal 'mtry' value of 2, representing a 1.67% increase in training performance. 3.2. Test Set Performance The Funnel RF model's performance was evaluated on a test set comprising 45 clinical records. The model achieved a classification accuracy of 86.67%, which is 4.11% higher than the training accuracy. The model also demonstrated high specificity (87.10%) and sensitivity (85.71%), indicating its robustness in distinguishing between survived and deceased patients. 3.3. Confusion Matrix and Performance Metrics The confusion matrix for the test set is presented in Table 4 , showing the distribution of true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN). The related performance metrics, including accuracy, sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV), are detailed in Table 5 . Table 4 The confusion matrix and related performance metrics of the Funnel Random Forest on the test set. Reference Prediction Deceased Survived Deceased 12 4 Survived 2 27 Table 5 The confusion matrix and related performance metrics of the Funnel Random Forest on the test set. Accuracy (%) 86.67% 95% CI 73.21%-94.95% No Information Rate 68.89% p-value (Acc > NIR) 0.0052 Kappa 0.7007 Mcnemar’s Test p-value 0.6831 Sensitivity 85.71% Specificity 87.10% Positive Predictive Value 75.00% Negative Predictive Value 93.10% Prevalence 31.11% Detection Rate 26.67% Detection Prevalence 35.56% Balanced Accuracy 86.41% ‘Positive’ Class Deceased 3.4. Receiver Operating Characteristic (ROC) and Precision-Recall (PR) Curves The reliability of the Funnel RF model was further assessed using the receiver operating characteristic (ROC) (Ying et al., 2016) and the precision-recall (PR) curves (Gaudreault et al., 2021), as shown in Figs. 3 and 4 . The areas under the curve (AUC) for both ROC and PR were found to be very high, with ROC-AUC = 0.864 and PR-AUC = 0.913. These metrics indicate the model's strong performance in distinguishing between the positive and negative classes. 3.5. Comparative Analysis with Published Models The Funnel RF model was compared with published models on the same UCI dataset. It was found to be 12.67% more accurate and had a higher PR-AUC by 25.6% compared to the RF model by Chicco & Jurman (2020). Additionally, it was 6.67% more accurate and had a higher recall by 20.5% compared to the DT algorithm by Almazroi (2022). On the second dataset used for external validation, the Funnel RF model was 10.49% more accurate and had a higher PR-AUC by 22.14% compared to the RF model by Chicco & Jurman (2020), and 5.37% more accurate and had a higher recall by 18.55% compared to the DT algorithm by Almazroi (2022). 3.6. Case Studies The trained Funnel RF model correctly predicted that the subject no. 1 described in sub-section 2.2 would be “Deceased” (Fig. 5 ) and subject no. 2 (in sub-section 2.2 ) as “Survived” (Fig. 6 ), as expected from the relevant medical literature (Veenstra et al., 1994; Ahmad et al., 2017) based on their above-mentioned characteristics. Despite these modelling improvements and higher classification performance, a binary prediction alone is not sufficient to aid clinical decision making, as it is influenced by the physician’s experience, systematic reviews and meta-analyses, and patient-specific risk factors. Thus, the Shapley values (Lundberg & Lee, 2017) of the RF’s predictions for two case studies, i.e., subjects no. 1 (Fig. 7 ) and 2 (Fig. 8 ), were plotted to explain them. 3.7. Summary of Findings The Funnel RF model demonstrated significant improvements in accuracy and reliability over traditional RF models and other published models. Its ability to identify and handle outliers using information theory-based metrics contributed to its enhanced performance. The model's predictions were consistent with clinical expectations, highlighting its potential for practical application in clinical decision support for heart failure prognostics. 4. Discussion 4.1. Considerations on the Data Used Several outliers (76 patients out of the initial 299 patients, i.e., 25.42%) were identified in the data and filtered out via the IQR-based method. This reduction of data may have slightly impacted the predictive performance of the RF model. Furthermore, despite reducing the skewness of specific variables that were highly non-Gaussian, their data distributions were still slightly non-normal; although these characteristics are representative of related real-world data, their complexity slightly hindered classification performance. In a real-world setting, the expert advice of at least two independent cardiologists would be pivotal to help in informing the extent of outliers to be removed from the data, potentially retaining more data for training (Hauskrecht et al., 2010) and improving generalisation. Moreover, clinicians' inputs would be useful to understand which of the features' distributions can be more non-normal to reflect real-world clinical data-related patterns further (Underwood et al., 2018; Parisi & Manaog, 2023). 4.2. Limitations of the Proposed Approach Whilst the proposed RF-based approach can intrinsically discern and handle outliers appropriately, this inner mechanism might pose limitations to its interpretability in users who are not accustomed to leverage DT-based algorithms or may be familiar with extrinsic methodologies to detect and remove outliers. The suggested technique does not remove the need for outlier removal in toto , but it enhances the ability of RFs to withstand skewed distributions and outliers that may have been deemed less significant beforehand when detected via standard statistical methodologies, e.g., via the IQR-based method. Being within a supervised learning technique, the proposed method may not be necessarily more specific than unsupervised learning DT-based algorithms, such as the Isolation Forest, which are leveraged to perform anomaly detection. Moreover, the RF’s ability to withstand additional outliers in the data is tied to the impurity metric used for aiding its learning, thus further work would need to be performed for such a metric to be decoupled and be set independently of that leveraged to split the nodes (Wang et al., 2024) in the DTs. 4.3. Comparative Analysis and Reliability The FRF model's reliability was demonstrated through a comparative analysis with existing models. The proposed method identified 11.04% fewer outliers and achieved a 2% increase in training performance on medical datasets as compared to traditional statistical methods. This improvement is significant as it indicates that the FRF model can better handle the inherent noise and variability in clinical data, leading to more accurate and reliable predictions. The model's performance was further validated by comparing it with published models, and it outperformed them by 9.67% in accuracy and 25.6% in reliability, the latter measured by the Precision-Recall Area Under the Curve (PR-AUC). These metrics highlight the reliability of the Funnel RF model in providing consistent and dependable predictions, which is crucial for clinical decision support systems. 4.4. Holistic Perspective on the Clinical Application of the Proposed Approach Standard statistical methods for removing outliers, such as the IQR-based method, may not always be appropriate for real-world data, as in this study. In fact, over 25% (sub- section 2.2 ) of the patients were identified as outliers, thus potentially leading to significant loss of information for training, impacting predictive performance and generalisation to unseen data. As clinicians’ expertise may not always be readily available to control and inform such a removal of outliers to be clinically relevant, a fusion of information theory-based metrics (GI and MI) was leveraged within the RF to filter out outliers via a quantitative ‘funnel’ approach. Its higher precision was demonstrated by identifying 11.04% fewer outliers (sub- section 2.2 ) than standard statistical methods (IQR), whilst leading to 2% higher training performance ( section 3 ) on the same benchmark medical data. Furthermore, as compared to published studies, the proposed FRF approach was found more accurate and reliable by 9.67% and 25.6% respectively ( section 3 ); reliability was assessed via the PR-AUC. The outcome for both subjects was predicted correctly, subject no. 1 as “Deceased”, although with a relatively low probability (14% , Fig. 7 ), and subject no. 2 as “Survived” with a very high probability (88% , Fig. 8 ). By training the model on a larger clinical dataset, the probability of the “Deceased” prediction and similar ones is likely to increase. Nevertheless, the four main factors (short follow-up time, low ejection fraction, high serum creatinine, and old age) leading to the “Deceased” prediction for subject no. 1 ( Fig. 7 ), as well as the three main features (long follow-up time, normal ejection fraction, and young age) resulting in the “Survived” outcome for subject no. 2 ( Fig. 8 ), are understandable and expected from a clinical perspective. Whilst leveraging data pre-processing techniques (log transformation and min-max normalisation) that can be easily reverse-engineered, and an explainable DT-based approach on a patient-specific basis, differently from the global Support Vector Machine (SVM)-based RF approach of Karthikeyan (2022), since it is based on support vectors and a resulting optimal separating hyperplane that are harder to interpret from a clinical standpoint, the FRF developed in this study is deemed reliable and interpretable to generate actionable insights for cardiologists to aid prediction of survival in patients with HF. 4.5. Innovation Point The innovation of this study lies in the integration of information theory-based metrics within the RF model to proactively identify and handle outliers during the training process. This novel FRF methodology enhances the model's accuracy and reliability, making it more suitable for real-world clinical data, which are often noisy and heterogeneous. By improving the accuracy and reliability of predictions, the FRF model provides a valuable tool for clinical decision support, ultimately contributing to better patient outcomes through earlier and more precise prognostic predictions. 5. Conclusion This study critically evaluated the application of an information theory-based (‘Funnel’) optimisation of the Random Forest (RF) model to aid the prediction of survival in patients with heart failure (HF) in a clinical setting. The high-performance Funnel RF-based classifier was developed and optimised to improve this prediction using an open-access dataset of clinical records from the UCI ML repository. Despite challenges inherent in real-world clinical data, such as the presence of outliers and non-normal data distributions, the FRF model demonstrated higher accuracy and reliability as compared to models from published studies evaluated on the same benchmark medical data, while retaining its explainability. The novel contributions of this work are multiple. Firstly, the introduction of the FRF methodology, which integrates information theory-based metrics for proactive outlier detection, represents a significant advancement in handling noisy and heterogeneous clinical data. This approach not only enhances the reliability of the RF model but also ensures better generalisation on unseen test data. Secondly, the Funnel RF model's ability to identify 11.04% fewer outliers and achieve a 2% increase in training performance highlights its effectiveness in improving model accuracy and reliability. The practical implications of this study are profound. The FRF model's enhanced classification performance in predicting patient survival can significantly aid clinical decision support systems. By providing more accurate and reliable prognostic predictions, the model can help clinicians make data-informed decisions about early interventions and treatment titration, thus improving patient outcomes. The case studies presented in this work illustrate how the model's predictions can be interpreted on a patient-specific basis, demonstrating its utility in real-world clinical scenarios. To yield tangible value and societal impact, it is crucial to understand how the FRF model can be leveraged by end users, particularly physicians, and how it can be integrated within existing clinical workflows. The model's explainability and high performance make it a valuable tool for clinicians, enabling them to titrate treatments early and improve patients' survival rates. In summary, this study presents a novel and effective approach to enhancing the accuracy and reliability of RF models to aid clinical predictive analytics. The FRF methodology holds promise for improving clinical decision support and ultimately contributing to better patient care and outcomes. Whilst the FRF model demonstrated significant improvements in handling outliers and enhancing the accuracy and reliability of clinical predictions, several avenues for future research can further refine and extend this work. Firstly, future studies should validate the FRF model on a broader range of clinical datasets, including those from different medical conditions and healthcare settings. This will help assess the model's generalisability and reliability across various types of clinical data. Additionally, integrating the FRF model with electronic health record (EHR) systems can facilitate its seamless adoption in clinical practice. Research should focus on developing interfaces and workflows that allow clinicians to easily access and interpret the model's predictions within their existing EHR platforms. Implementing the FRF model in real-time clinical decision support systems can provide immediate prognostic insights to clinicians during patient consultations. Future work should explore the technical and practical challenges of real-time integration, including computational efficiency and user interface design. Conducting longitudinal studies to track the impact of using the FRF model on patient outcomes over time can provide valuable insights into its effectiveness in improving clinical decision-making and patient care. These studies can also help identify any long-term benefits or limitations of the model. Whilst the FRF model retains the interpretability of traditional RF models, further enhancing its explainability can increase clinician trust and adoption. Future research should explore advanced techniques for visualising and explaining the model's predictions, such as using Shapley values or other feature importance metrics. Clinical datasets often suffer from class imbalance, where certain outcomes are much rarer than others. Future work should investigate methods to further improve the FRF model's performance on imbalanced data, such as using synthetic data generation techniques or advanced sampling methods. As novel ML-driven models and techniques continue to emerge, it is important to conduct comparative studies to benchmark the FRF model against these new approaches. This will help ensure that the FRF model remains competitive and incorporates the latest advancements in the field. Developing methods to customise the FRF model for individual patients based on their unique clinical profiles can enhance its predictive accuracy and reliability. Future research should explore techniques for personalising the model's parameters and decision rules to better suit individual patient characteristics. By addressing these areas, future research can further enhance the FRF model's utility and impact in clinical settings, ultimately contributing to even better patient outcomes and more effective healthcare delivery. Declarations Competing interests declaration : none. Funding declaration :This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. Ethics and Consent to Participate declarations : not applicable Author Contribution L.P.: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing – Original Draft, Writing – Review & Editing, Visualization, Supervision, Project administration.M.L.M.: Validation, Investigation, Visualization, Writing – Original Draft, Writing – Review & Editing. Data Availability Data leveraged in this study have been adequately cited in text and referenced at the bottom of the manuscript. References Ahmad, T., Munir, A., Bhatti, S. H., Aftab, M., & Raza, M. A. (2017). Survival analysis of heart failure patients: A case study. PloS One , 12 (7), e0181001. https://doi.org/10.1371/journal.pone.0181001. Almazroi, A. A. (2022). 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ACM SIGMOD Record , 25 (2), 103-114. https://doi.org/10.1145/235968.233324. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-5784003","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":400481915,"identity":"1c593550-2915-48b5-b091-1bfd7e5a2a3a","order_by":0,"name":"Luca 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20:08:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5784003/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5784003/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":73868054,"identity":"543707eb-6286-41a9-af99-a12b4a6a8482","added_by":"auto","created_at":"2025-01-15 12:04:26","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":12092,"visible":true,"origin":"","legend":"\u003cp\u003eThe main supervised and unsupervised Machine Learning-based algorithms used in predictive analytics for classification and clustering respectively.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/3e1592e273f797216d71c42a.png"},{"id":73868060,"identity":"a1aeb8f1-ac80-41e8-a80a-353b46ef63db","added_by":"auto","created_at":"2025-01-15 12:04:27","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":298281,"visible":true,"origin":"","legend":"\u003cp\u003eA decision tree’s data-driven ‘rules’ to predict survival of patients with heart failure (Almazroi, 2022).\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/d2610bcb68e6cfa411044646.png"},{"id":73869933,"identity":"492fb008-88a9-4e69-9fe9-8b543df8a9b0","added_by":"auto","created_at":"2025-01-15 12:12:26","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":49358,"visible":true,"origin":"","legend":"\u003cp\u003eThe receiver operating characteristic (ROC) curve of the Funnel Random Forest on the test set.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/3ce7e122999ff620e62b8f37.png"},{"id":73869932,"identity":"530bcf26-acd3-4477-89d2-d8ebe006f5ec","added_by":"auto","created_at":"2025-01-15 12:12:26","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":38908,"visible":true,"origin":"","legend":"\u003cp\u003eThe precision-recall (PR) curve of the Funnel Random Forest on the test set.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/1bf3a9407ed8f1d9c0ba2925.png"},{"id":73869934,"identity":"acda9d4c-9abe-4533-b3ee-7e9953c0029c","added_by":"auto","created_at":"2025-01-15 12:12:27","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":40902,"visible":true,"origin":"","legend":"\u003cp\u003eThe correct prediction for subject no. 1 (“newCase1”) as “Deceased” from the Funnel Random Forest.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/65e198b83e89dd3b63a33ec4.png"},{"id":73868058,"identity":"06e7ff9d-67d7-462b-ba3c-e856afbaf2b1","added_by":"auto","created_at":"2025-01-15 12:04:26","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":40113,"visible":true,"origin":"","legend":"\u003cp\u003eThe correct prediction for subject no. 2 (“newCase2”) as “Survived” from the Funnel Random Forest.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/002a661c2dcb24931cc1f392.png"},{"id":73868059,"identity":"23f6ff22-f71c-4c35-a5a8-816b2b38bf96","added_by":"auto","created_at":"2025-01-15 12:04:26","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":26869,"visible":true,"origin":"","legend":"\u003cp\u003eThe Shapley values of the correct prediction for subject no. 1 (“newCase1”) as “Deceased” from the Funnel Random Forest.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/5c816e7974ac1396a1eb45b2.png"},{"id":73868068,"identity":"26f85aad-47de-48b0-bda3-3d1ba5e832c3","added_by":"auto","created_at":"2025-01-15 12:04:27","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":28519,"visible":true,"origin":"","legend":"\u003cp\u003eThe Shapley values of the correct prediction for subject no. 2 (“newCase2”) as “Survived” from the Funnel Random Forest.\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/fa3e7faca2d427c02f4e8dda.png"},{"id":74014199,"identity":"f8dcbba0-1dce-4df9-a633-30fb0d59c5c4","added_by":"auto","created_at":"2025-01-17 03:23:35","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1652465,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5784003/v1/8bbc4ed0-bfb0-47c0-8378-979a37ae2301.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Funnel Random Forest: Inliers-Focused Ensemble Learning for Improved Prognostics of Heart Failure","fulltext":[{"header":"1. Introduction","content":"\u003cp\u003eCardiovascular diseases (CVDs), including heart failure (HF), remain the leading causes of mortality worldwide, with 17.9 million people dying every year, i.e., 32% of all deaths globally (World Health Organization (WHO), 2021). In the United Kingdom (UK), 160,000 people die due to CVDs each year and over 900,000 people live with HF (British Heart Foundation (BHF), 2021). The Coronavirus disease 2019 (COVID-19)-related pandemic has led to a reduced hospitalisation and an increase in mortality due to CVDs (Banerjee et al., 2020; Huet et al., 2020; Normando et al., 2021). If mortality due to HF could be predicted, deaths could be prevented by titrating treatments early (Ponikowski et al., 2014; Groenewegen et al., 2020).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e1.1 Machine Learning algorithms for predictive analytics\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eMachine Learning (ML) is a branch of Artificial Intelligence (AI) enabling algorithms to learn patterns from data for predictive analytics automatically (Samuel, 1959). ML can be either supervised or unsupervised, respectively for classification by training on labelled data (Parisi et al., 2018, 2024; Parisi \u0026amp; Manaog, 2023) or clustering based on similarity among unlabelled samples (Parisi \u0026amp; RaviChandran, 2018).\u003c/p\u003e\n\u003cp\u003eDecision Tree (DT)-based models (Chicco \u0026amp; Jurman, 2020; Almazroi, 2022), including Random Forest (RF) (\u003cstrong\u003eFig. 1\u003c/strong\u003e), are explainable (Khalilia et al., 2011; Thakur \u0026amp; Kumar, 2018; Kaur et al., 2019; Zhang et al., 2021;\u0026nbsp;Rey-Blanco et al., 2024;\u0026nbsp;Wang et al., 2024; Parisi \u0026amp; Manaog, 2025)\u0026nbsp;ML-related supervised algorithms, whilst\u0026nbsp;the Convolutional Neural Network (CNN) (Kwon et al., 2019; Wang et al., 2020) is a Deep Learning (DL)-based supervised method with outputs slightly interpretable by saliency maps (Springenberg et al., 2014; Selvaraju et al., 2017; Parisi \u0026amp; Manaog, 2023, 2025).\u003c/p\u003e\n\u003cp\u003eNeural network (NN)-, optimal separating hyperplane (OSH)-, and distance-based algorithms are also supervised algorithms (\u003cstrong\u003eFig. 1\u003c/strong\u003e), e.g., the Multi-Layer Perceptron (MLP) (Rumelhart et al., 1986), the Support Vector Machine (SVM) (Cortes \u0026amp; Vapnik, 1995), and the K-Nearest Neighbour (KNN) (Fix \u0026amp; Hodges, 1989). MLP, SVM, and KNN aid classification respectively by adjusting the weights of neurons, maximising the margin width, and by identifying boundaries in the data via distance metrics.\u003c/p\u003e\n\u003cp\u003ek-means (MacQueen, 1967; Fr\u0026auml;nti \u0026amp; Sieranoja, 2018), Self-Organising Map (SOM) (Kohonen, 1990), and Neural Gas (Martinetz \u0026amp; Schulten, 1991), are clustering algorithms grouping data based on their similarity (\u003cstrong\u003eFig. 1\u003c/strong\u003e). \u0026ldquo;Hierarchical clustering\u0026rdquo; (Johnson, 1967) and \u0026ldquo;Birch\u0026rdquo; (Zhang et al., 1996) are also clustering algorithms (\u003cstrong\u003eFig. 1\u003c/strong\u003e), respectively leveraging dendrograms of clusters and sub-clusters in feature trees.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e1.2 Critical evaluation of related studies\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAI-based classification methods, from ML to DL, have been developed to aid prediction of survival in patients with HF. However, existing DL techniques are overly sophisticated (Kwon et al., 2019; Wang et al., 2020), such as CNNs (LeCun \u0026amp; Bengio, 1995; Parisi et al., 2022), or involve ML algorithms applied to only generate a binary prediction (Hearn et al., 2018; Chicco \u0026amp; Jurman, 2020; Shin et al., 2021; Almazroi, 2022). Thus, due to their limited explainability, existing techniques have not been widely adopted in a clinical setting, thus hindering their translational value and societal impact.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eDT-based classifiers, considering their high interpretability, have been applied for predicting survival in patients with HF, despite their reliability was not deemed clinically acceptable (Chicco \u0026amp; Jurman, 2020; Almazroi, 2022). Chicco \u0026amp; Jurman (2020) achieved 74% accuracy with only 0.657 as the precision-recall (PR) area under the curve (AUC) via an RF, whilst Almazroi (2022) obtained 80% accuracy but with 65.21% of recall via a DT (\u003cstrong\u003eFig. 2\u003c/strong\u003e).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e1.3.\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003eResearch Objectives\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe primary objective of this research is to develop and validate a novel Funnel Random Forest (FRF) methodology that enhances the accuracy and reliability of clinical prognostics, particularly for patients with heart failure. The specific objectives are:\u003c/p\u003e\n\u003col\u003e\n \u003cli\u003e\u003cstrong\u003e\u0026nbsp;Outlier detection and handling\u003c/strong\u003e: To introduce an intrinsic mechanism within the Random Forest model that proactively identifies and handles outliers using information theory-based metrics, thus improving the model\u0026apos;s robustness and generalisation.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003e\u0026nbsp;Performance improvement\u003c/strong\u003e: To demonstrate the effectiveness of the FRF model in reducing the number/percentage of outliers and improving training performance as compared to traditional statistical outlier detection/removal methods.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003e\u0026nbsp;Clinical relevance\u003c/strong\u003e: To validate the FRF model\u0026apos;s ability to enhance clinical decision support by providing more accurate and reliable prognostic predictions for patients with heart failure.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003e\u0026nbsp;Comparative analysis\u003c/strong\u003e: To compare the performance of the FRF model with existing models, demonstrating its higher accuracy and reliability.\u003c/li\u003e\n \u003cli\u003e\u003cstrong\u003e\u0026nbsp;Practical implications\u003c/strong\u003e: To highlight the potential of the FRF model to improve patient outcomes through early and precise prognostic predictions, thus aiding clinicians in making data-informed decisions.\u003c/li\u003e\n\u003c/ol\u003e\n\u003cp\u003e\u003cstrong\u003e1.4. Rationale of proposed method and article\u0026rsquo;s structure\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFrom a critical review of the related literature as per sub-section 1.2, the Random Forest (RF) model was identified as an optimal candidate due to its ability to balance high classification performance with interpretability (Rey-Blanco et al., 2024; Parisi et al., 2024). The Decision Tree (DT)-like architecture of RF allows for intuitive understanding and visualisation of decision-making processes, which is particularly valuable in a clinical setting, wherein transparency and explainability are crucial.\u003c/p\u003e\n\u003cp\u003eHowever, despite its strengths, RF models are susceptible to the presence of outliers in the training data, which can adversely affect their predictive performance and generalisation. Existing outlier-handling methods often rely on subjective heuristics that do not generalise well across different datasets, particularly in the context of clinical data, which are inherently noisy and heterogeneous.\u003c/p\u003e\n\u003cp\u003eTo address these challenges, this study proposes a novel Funnel Random Forest (FRF) methodology. The motivation behind this approach is to enhance the accuracy and reliability of RF models by incorporating an intrinsic mechanism for outlier detection and handling. Specifically, the proposed method leverages the Gini impurity measure and mutual information-related scores to identify and discard outliers during the training process. This proactive approach ensures that the training data are cleaner, with more neatly separable patterns, thus leading to improved model\u0026rsquo;s predictive performance and higher generalisation on unseen test data.\u003c/p\u003e\n\u003cp\u003eThe contributions of this study are multi-faceted:\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e1. Novel methodology\u003c/strong\u003e: The introduction of the FRF model, which integrates information theory-based metrics for outlier detection within the RF training process.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2. Performance improvement\u003c/strong\u003e: Demonstration of improvements in classification accuracy and reliability, with the FRF identifying 11.04% fewer outliers and achieving a 2% increase in training performance on medical datasets.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3. Clinical relevance\u003c/strong\u003e: Validation of the model\u0026apos;s effectiveness in enhancing clinical decision support, particularly for predicting prognostics in patients with heart failure.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e4. Comparative analysis\u003c/strong\u003e: Comprehensive comparison with existing models, showcasing the FRF\u0026apos;s higher accuracy and reliability.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e5. Practical implications\u003c/strong\u003e: Highlighting the potential of the FRF to provide more accurate and reliable and accurate predictions for clinicians, ultimately leading to better patient outcomes through earlier and more precise prognostic predictions.\u003c/p\u003e\n\u003cp\u003eThe rest of the article is composed of the following sections:\u0026nbsp;\u003c/p\u003e\n\u003cul\u003e\n \u003cli\u003esection 2. on the datasets leveraged in this study and an outline of the proposed methodology;\u0026nbsp;\u003c/li\u003e\n \u003cli\u003esection 3. on the results obtained, including the evaluation of the accuracy and the reliability of the proposed model on the respective test sets;\u0026nbsp;\u003c/li\u003e\n \u003cli\u003esection 4. on the discussion of these results, including considerations on both the datasets used and the modelling approach adopted, and on the model\u0026rsquo;s explainability demonstrated via two further case studies;\u0026nbsp;\u003c/li\u003e\n \u003cli\u003esection 5. with conclusions and related reflections.\u003c/li\u003e\n\u003c/ul\u003e"},{"header":"2. Methods","content":"\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e2.1. Dataset and descriptive analytics\u003c/h2\u003e \u003cp\u003eThe RF built to predict survival in patients with HF was developed in R (version 4.1.2). The dataset of clinical records on 299 subjects used for model development and initial evaluation is available on the University California Irvine (UCI) ML repository (Ahmad et al., 2017; Chicco \u0026amp; Jurman, 2020) and consists of twelve input features: age (years), anaemia, high blood pressure, diabetes, smoking, sex, ejection fraction, creatinine phosphokinase (mcg/L), platelets (kiloplatelets/mL), serum creatinine (mg/dL), serum sodium (mEq/L), and follow-up time (days). The output variable to predict is the \u0026lsquo;DEATH_EVENT\u0026rsquo; (0 for survived, 1 for deceased). Furthermore, the dataset of clinical records (1,700 instances for 124 clinical features) pertaining to myocardial complications from the UCI ML repository (Golovenkin \u003cem\u003eet al\u003c/em\u003e., 2020) was leveraged for an external validation to predict chronic heart failure, i.e., as an additional test set.\u003c/p\u003e \u003cp\u003eDescriptive statistics were performed, and no missing values were found. Box plots of the input features and bar plots of the class distributions were produced, before and after removing outliers via the interquartile range (IQR)-based method (Vinutha et al., 2018). This is the standard statistical method compared with the outlier removal-related technique intrinsic to the RF model implemented and validated in this study.\u003c/p\u003e \u003cp\u003eThe dataset for model development and initial evaluation of the remaining 223 subjects (162 survived (72.65%), and 61 deceased (27.35%)) was analysed. Normality was assessed via histograms, quantile-quantile (Q-Q) plots, and the Shapiro-Wilk test (Razali \u0026amp; Wah, 2011) for continuous variables. The features \u0026ldquo;age\u0026rdquo;, \u0026ldquo;creatinine_phosphokinase\u0026rdquo;, and \u0026ldquo;serum_creatinine\u0026rdquo; were found positively skewed; thus, a log transformation was performed to reduce their skewness. Furthermore, a min-max normalisation (Jain et al., 2005) was carried out to ensure all variables had the same range (from 0 to +\u0026thinsp;1). The pre-processed data were then split into 80% for training (131 survived patients, 47 deceased), 20% for testing (31 survived patients, 14 deceased).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec8\" class=\"Section2\"\u003e \u003ch2\u003e2.2. Funnel Random Forest for predictive analytics\u003c/h2\u003e \u003cp\u003eThe RF model was trained via 5-fold cross-validation to prevent overfitting (Kohavi, 1995; Parisi \u0026amp; Manaog, 2016, 2017a, 2017b) and Synthetic Minority Oversampling Technique (SMOTE) (Chawla et al., 2002) was leveraged to upsample the minority class. Its \u0026lsquo;mtry\u0026rsquo; (number of random variables at each split) hyperparameter was optimised (as 4) via random search, yielding the initially trained model with 80.89% accuracy (Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e). This training accuracy was obtained further to removing 76 patients\u0026rsquo; clinical records (25.42% of the entire data) deemed outliers based on the IQR-based method.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe initially trained Random Forest (RF) and its training accuracy. mtry\u0026thinsp;=\u0026thinsp;4 yielded the highest accuracy (80.89%), thus resulting in the optimal RF model initially.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003emtry\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAccuracy\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKappa\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7865\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.3872\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8087\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4821\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003e4\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e0.8089\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4951\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8084\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5097\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7859\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4558\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7862\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4527\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7692\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4037\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e10\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7522\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.3579\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe \u0026lsquo;funnel\u0026rsquo; information theory-based optimisation of the RF model to intrinsically detect and discard outliers prior to training leverages two main components to maximise classification performance (Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e)): 1) the Gini Impurity (GI) from the RF (Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e)); 2) the mutual information (MI) (Eq.\u0026nbsp;(3)) brought by each input based on the defined feature set.\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\:{FD}_{performance}^{GI-MI}=argmin{\\sum\\:}_{i=1}^{n}1-{\\int\\:}_{0}^{1}\\frac{TP}{TP+FN}\\left(thresh\\right)\\left(-{\\frac{\\sum\\:FP}{\\sum\\:\\left(FP+TN\\right)}}^{T}\\left(thresh\\right)\\right)dthresh$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIn (\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e): \u003cem\u003eFD\u003c/em\u003e represents the filtered data obtained by applying equations (\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e)-(3); TP - true positives, TN - true negatives, FP - false positives, FN - false negatives, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:thresh\\)\u003c/span\u003e\u003c/span\u003e is the threshold for determining the receiver operating characteristic (ROC) curve.\u003c/p\u003e \u003cp\u003eThe \u003cem\u003eGI\u003c/em\u003e is represented by the following Eq.\u0026nbsp;(\u003cspan refid=\"Equ2\" class=\"InternalRef\"\u003e2\u003c/span\u003e):\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$$\\:GI\\left(p\\right)={\\sum\\:}_{m=1}^{N}{p}_{m}\\left(1-{p}_{m}\\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ep\u003c/em\u003e represents the probability of a sample with class \u003cem\u003em\u003c/em\u003e being selected, whilst \u003cem\u003eN\u003c/em\u003e is the number of classes considered.\u003c/p\u003e \u003cp\u003eThe \u003cem\u003eMI\u003c/em\u003e is described via Eq.\u0026nbsp;(3):\u003c/p\u003e \u003cp\u003e \u003cem\u003eMI(a,b)\u0026thinsp;=\u0026thinsp;E(b) \u0026ndash; E(b | a)\u003c/em\u003e (\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e)\u003c/p\u003e \u003cp\u003ewhere \u003cem\u003ea\u003c/em\u003e and \u003cem\u003eb\u003c/em\u003e are random variables, \u003cem\u003eE\u003c/em\u003e stands for entropy, thus \u003cem\u003eE(b)\u003c/em\u003e being the marginal entropy and \u003cem\u003eE(b | a)\u003c/em\u003e is the conditional entropy.\u003c/p\u003e \u003cp\u003eFurther to leveraging the funnel-based outlier removal implemented intrinsically to the RF model\u0026rsquo;s architecture (equations (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e)-(3)), 43 patients\u0026rsquo; clinical records (14.38% of the entire data) were identified as outliers; thus, the Funnel RF model was re-trained, and its training accuracy increased to 82.56% (by 1.67%) with \u0026ldquo;mtry\u0026rdquo; equal to 2 (Table\u0026nbsp;\u003cspan refid=\"Tab2\" class=\"InternalRef\"\u003e2\u003c/span\u003e).\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab2\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 2\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe trained Funnel Random Forest and its training accuracy. mtry\u0026thinsp;=\u0026thinsp;2 yielded the highest accuracy (82.56%), thus resulting in the optimal Funnel RF model.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003emtry\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eAccuracy\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eKappa\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e1\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8144\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5010\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003e2\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e\u003cb\u003e0.8256\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5402\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e3\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8029\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4891\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8144\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5056\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e5\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.8027\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.5056\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e7\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7971\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4794\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e8\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7803\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.4338\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e9\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e0.7521\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e0.3784\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eThe classification performance of the Funnel RF model was critically evaluated on the test set and two further case studies. Considering both classification accuracy and reliability as quantifying model\u0026rsquo;s generalisation on the test set (Parisi et al., 2018a, 2018b, 2020), the main performance metrics considered, based on TPs, TNs, FPs, and FNs (Table\u0026nbsp;\u003cspan refid=\"Tab3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), are the following: accuracy (Eq.\u0026nbsp;(\u003cspan refid=\"Equ3\" class=\"InternalRef\"\u003e4\u003c/span\u003e)), sensitivity/recall (Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e5\u003c/span\u003e)), specificity (Eq.\u0026nbsp;(\u003cspan refid=\"Equ5\" class=\"InternalRef\"\u003e6\u003c/span\u003e)), ROC-AUC, and PR-AUC.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab3\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 3\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eTPs, TNs, FPs, and FNs, as related to the subjects in this study.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"4\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c4\" colnum=\"4\"\u003e\u003c/div\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colspan=\"2\" morerows=\"1\" nameend=\"c2\" namest=\"c1\" rowspan=\"2\"\u003e\u0026nbsp;\u003c/td\u003e \u003ctd align=\"left\" colspan=\"2\" nameend=\"c4\" namest=\"c3\"\u003e \u003cp\u003eFrom actual clinical records\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTrue\u003c/p\u003e \u003cp\u003e(Survived)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFalse\u003c/p\u003e \u003cp\u003e(Deceased)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\" morerows=\"1\" rowspan=\"2\"\u003e \u003cp\u003e\u003cb\u003eFrom predictions\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003ePositives\u003c/p\u003e \u003cp\u003e(Survived)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTrue Positives (TPs)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFalse Positives (FPs)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eNegatives\u003c/p\u003e \u003cp\u003e(Deceased)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003eTrue Negatives (TNs)\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c4\"\u003e \u003cp\u003eFalse Negatives\u003c/p\u003e \u003cp\u003e(FNs)\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003cdiv id=\"Equ3\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\:Accuracy=\\frac{TP+TN}{TP+TN+FP+FN}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e \u003cdiv id=\"Equ4\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$$\\:Sensitivity=\\frac{TP}{TP+FN}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e \u003cdiv id=\"Equ5\" class=\"Equation\"\u003e \u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\:Specificity=\\frac{TN}{TN+FP}$$\u003c/div\u003e \u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eBased on the related clinical literature (Veenstra et al., 1994; Ahmad et al., 2017), two synthetic subjects were simulated as having characteristics (normalised between 0 and 1) found in clinical records of an elderly patient with anaemia who may die from HF (subject no. 1) and a young subject without anaemia who may survive despite having HF (subject no. 2). Levels of creatinine phosphokinase, as an indicator of injury to the heart, serum sodium, and blood pressure were high in subject no. 1 (Veenstra et al., 1994; Ahmad et al., 2017) and low in subject no. 2. Conversely, ejection fraction, i.e., the percentage of blood that is pumped out of the heart to the body, platelet count, and follow-up time were low for subject no. 1 (Veenstra et al., 1994; Ahmad et al., 2017) and high for subject no. 2.\u003c/p\u003e \u003c/div\u003e"},{"header":"3. Results","content":"\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.1. Training Performance\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe Funnel Random Forest (Funnel RF) model was initially trained on a dataset of 299 clinical records, with 76 records (25.42%) identified as outliers and removed using the interquartile range (IQR)-based method. This initial training yielded an accuracy of 80.89% with an optimal 'mtry' value of 4. The Funnel RF model was then re-trained after identifying and removing 43 outliers (14.38%) using the proposed information theory-based metrics. This re-training improved the accuracy to 82.56% with an optimal 'mtry' value of 2, representing a 1.67% increase in training performance.\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.2. Test Set Performance\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe Funnel RF model's performance was evaluated on a test set comprising 45 clinical records. The model achieved a classification accuracy of 86.67%, which is 4.11% higher than the training accuracy. The model also demonstrated high specificity (87.10%) and sensitivity (85.71%), indicating its robustness in distinguishing between survived and deceased patients.\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.3. Confusion Matrix and Performance Metrics\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe confusion matrix for the test set is presented in\u003c/span\u003e Table\u0026nbsp;\u003cspan refid=\"Tab4\" class=\"InternalRef\"\u003e4\u003c/span\u003e, \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eshowing the distribution of true positives (TP), true negatives (TN), false positives (FP), and false negatives (FN). The related performance metrics, including accuracy, sensitivity, specificity, positive predictive value (PPV), and negative predictive value (NPV), are detailed in\u003c/span\u003e Table\u0026nbsp;\u003cspan refid=\"Tab5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab4\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 4\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe confusion matrix and related performance metrics of the Funnel Random Forest on the test set.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"char\" char=\".\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e\u0026nbsp;\u003c/th\u003e \u003cth align=\"left\" colspan=\"2\" nameend=\"c3\" namest=\"c2\"\u003e \u003cp\u003eReference\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003ePrediction\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDeceased\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eSurvived\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDeceased\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e12\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e4\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eSurvived\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c2\"\u003e \u003cp\u003e2\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"char\" char=\".\" colname=\"c3\"\u003e \u003cp\u003e27\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab5\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 5\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eThe confusion matrix and related performance metrics of the Funnel Random Forest on the test set.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"2\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eAccuracy (%)\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003e86.67%\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003e95% CI\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e73.21%-94.95%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eNo Information Rate\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e68.89%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ep-value (Acc\u0026thinsp;\u0026gt;\u0026thinsp;NIR)\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.0052\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eKappa\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.7007\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eMcnemar\u0026rsquo;s Test p-value\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e0.6831\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eSensitivity\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e85.71%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eSpecificity\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e87.10%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePositive Predictive Value\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e75.00%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eNegative Predictive Value\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e93.10%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003ePrevalence\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e31.11%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDetection Rate\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e26.67%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eDetection Prevalence\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e35.56%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003eBalanced Accuracy\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e86.41%\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003e\u003cb\u003e\u0026lsquo;Positive\u0026rsquo; Class\u003c/b\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003eDeceased\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.4. Receiver Operating Characteristic (ROC) and Precision-Recall (PR) Curves\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe reliability of the Funnel RF model was further assessed using the receiver operating characteristic (ROC) (Ying et al., 2016) and the precision-recall (PR) curves (Gaudreault et al., 2021), as shown in\u003c/span\u003e Figs.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e and \u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe areas under the curve (AUC) for both ROC and PR were found to be very high, with ROC-AUC\u0026thinsp;=\u0026thinsp;0.864 and PR-AUC\u0026thinsp;=\u0026thinsp;0.913. These metrics indicate the model's strong performance in distinguishing between the positive and negative classes.\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec14\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.5. Comparative Analysis with Published Models\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe Funnel RF model was compared with published models on the same UCI dataset. It was found to be 12.67% more accurate and had a higher PR-AUC by 25.6% compared to the RF model by Chicco \u0026amp; Jurman (2020). Additionally, it was 6.67% more accurate and had a higher recall by 20.5% compared to the DT algorithm by Almazroi (2022). On the second dataset used for external validation, the Funnel RF model was 10.49% more accurate and had a higher PR-AUC by 22.14% compared to the RF model by Chicco \u0026amp; Jurman (2020), and 5.37% more accurate and had a higher recall by 18.55% compared to the DT algorithm by Almazroi (2022).\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec15\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.6. Case Studies\u003c/span\u003e\u003c/h2\u003e \u003cp\u003eThe trained Funnel RF model correctly predicted that the subject no. 1 described in sub-section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e would be \u0026ldquo;Deceased\u0026rdquo; (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e) and subject no. 2 (in sub-section \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e) as \u0026ldquo;Survived\u0026rdquo; (Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), as expected from the relevant medical literature (Veenstra et al., 1994; Ahmad et al., 2017) based on their above-mentioned characteristics.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eDespite these modelling improvements and higher classification performance, a binary prediction alone is not sufficient to aid clinical decision making, as it is influenced by the physician\u0026rsquo;s experience, systematic reviews and meta-analyses, and patient-specific risk factors. Thus, the Shapley values (Lundberg \u0026amp; Lee, 2017) of the RF\u0026rsquo;s predictions for two case studies, i.e., subjects no. 1 (Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e) and 2 (Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e), were plotted to explain them.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec16\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e3.7. Summary of Findings\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe Funnel RF model demonstrated significant improvements in accuracy and reliability over traditional RF models and other published models. Its ability to identify and handle outliers using information theory-based metrics contributed to its enhanced performance. The model's predictions were consistent with clinical expectations, highlighting its potential for practical application in clinical decision support for heart failure prognostics.\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Discussion","content":"\u003cdiv id=\"Sec18\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e4.1. Considerations on the Data Used\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eSeveral outliers (76 patients out of the initial 299 patients, i.e., 25.42%) were identified in the data and filtered out via the IQR-based method. This reduction of data may have slightly impacted the predictive performance of the RF model. Furthermore, despite reducing the skewness of specific variables that were highly non-Gaussian, their data distributions were still slightly non-normal; although these characteristics are representative of related real-world data, their complexity slightly hindered classification performance. In a real-world setting, the expert advice of at least two independent cardiologists would be pivotal to help in informing the extent of outliers to be removed from the data, potentially retaining more data for training (Hauskrecht et al., 2010) and improving generalisation. Moreover, clinicians' inputs would be useful to understand which of the features' distributions can be more non-normal to reflect real-world clinical data-related patterns further (Underwood et al., 2018; Parisi \u0026amp; Manaog, 2023).\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec19\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e4.2. Limitations of the Proposed Approach\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eWhilst the proposed RF-based approach can intrinsically discern and handle outliers appropriately, this inner mechanism might pose limitations to its interpretability in users who are not accustomed to leverage DT-based algorithms or may be familiar with extrinsic methodologies to detect and remove outliers. The suggested technique does not remove the need for outlier removal\u003c/span\u003e \u003cspan type=\"ItalicSmallCaps\" class=\"ItalicSmallCaps\" name=\"Emphasis\"\u003ein toto\u003c/span\u003e, \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003ebut it enhances the ability of RFs to withstand skewed distributions and outliers that may have been deemed less significant beforehand when detected via standard statistical methodologies, e.g., via the IQR-based method. Being within a supervised learning technique, the proposed method may not be necessarily more specific than unsupervised learning DT-based algorithms, such as the Isolation Forest, which are leveraged to perform anomaly detection. Moreover, the RF\u0026rsquo;s ability to withstand additional outliers in the data is tied to the impurity metric used for aiding its learning, thus further work would need to be performed for such a metric to be decoupled and be set independently of that leveraged to split the nodes (Wang et al., 2024) in the DTs.\u003c/span\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec20\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e4.3. Comparative Analysis and Reliability\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe FRF model's reliability was demonstrated through a comparative analysis with existing models. The proposed method identified 11.04% fewer outliers and achieved a 2% increase in training performance on medical datasets as compared to traditional statistical methods. This improvement is significant as it indicates that the FRF model can better handle the inherent noise and variability in clinical data, leading to more accurate and reliable predictions. The model's performance was further validated by comparing it with published models, and it outperformed them by 9.67% in accuracy and 25.6% in reliability, the latter measured by the Precision-Recall Area Under the Curve (PR-AUC). These metrics highlight the reliability of the Funnel RF model in providing consistent and dependable predictions, which is crucial for clinical decision support systems.\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec21\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e4.4. Holistic Perspective on the Clinical Application of the Proposed Approach\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eStandard statistical methods for removing outliers, such as the IQR-based method, may not always be appropriate for real-world data, as in this study. In fact, over 25% (sub-\u003c/span\u003esection \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e) of the patients were identified as outliers, thus potentially leading to significant loss of information for training, impacting predictive performance and generalisation to unseen data. As clinicians\u0026rsquo; expertise may not always be readily available to control and inform such a removal of outliers to be clinically relevant, a fusion of information theory-based metrics (GI and MI) was leveraged within the RF to filter out outliers via a quantitative \u0026lsquo;funnel\u0026rsquo; approach. Its higher precision was demonstrated by identifying 11.04% fewer outliers (sub-\u003c/span\u003esection \u003cspan refid=\"Sec8\" class=\"InternalRef\"\u003e2.2\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e) than standard statistical methods (IQR), whilst leading to 2% higher training performance (\u003c/span\u003esection \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e) on the same benchmark medical data. Furthermore, as compared to published studies, the proposed FRF approach was found more accurate and reliable by 9.67% and 25.6% respectively (\u003c/span\u003esection \u003cspan refid=\"Sec9\" class=\"InternalRef\"\u003e3\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e); reliability was assessed via the PR-AUC.\u003c/span\u003e\u003c/p\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe outcome for both subjects was predicted correctly, subject no. 1 as \u0026ldquo;Deceased\u0026rdquo;, although with a relatively low probability (14%\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e), and subject no. 2 as \u0026ldquo;Survived\u0026rdquo; with a very high probability (88%\u003c/span\u003e, Fig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e). By training the model on a larger clinical dataset, the probability of the \u0026ldquo;Deceased\u0026rdquo; prediction and similar ones is likely to increase. Nevertheless, the four main factors (short follow-up time, low ejection fraction, high serum creatinine, and old age) leading to the \u0026ldquo;Deceased\u0026rdquo; prediction for subject no. 1 (\u003c/span\u003eFig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e), as well as the three main features (long follow-up time, normal ejection fraction, and young age) resulting in the \u0026ldquo;Survived\u0026rdquo; outcome for subject no. 2 (\u003c/span\u003eFig.\u0026nbsp;\u003cspan refid=\"Fig8\" class=\"InternalRef\"\u003e8\u003c/span\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e), are understandable and expected from a clinical perspective. Whilst leveraging data pre-processing techniques (log transformation and min-max normalisation) that can be easily reverse-engineered, and an explainable DT-based approach on a patient-specific basis, differently from the global Support Vector Machine (SVM)-based RF approach of Karthikeyan (2022), since it is based on support vectors and a resulting optimal separating hyperplane that are harder to interpret from a clinical standpoint, the FRF developed in this study is deemed reliable and interpretable to generate actionable insights for cardiologists to aid prediction of survival in patients with HF.\u003c/span\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec22\" class=\"Section2\"\u003e \u003ch2\u003e\u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003e4.5. Innovation Point\u003c/span\u003e\u003c/h2\u003e \u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThe innovation of this study lies in the integration of information theory-based metrics within the RF model to proactively identify and handle outliers during the training process. This novel FRF methodology enhances the model's accuracy and reliability, making it more suitable for real-world clinical data, which are often noisy and heterogeneous. By improving the accuracy and reliability of predictions, the FRF model provides a valuable tool for clinical decision support, ultimately contributing to better patient outcomes through earlier and more precise prognostic predictions.\u003c/span\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"5. Conclusion","content":"\u003cp\u003eThis study critically evaluated the application of an information theory-based (‘Funnel’) optimisation of the Random Forest (RF) model to aid the prediction of survival in patients with heart failure (HF) in a clinical setting. The high-performance Funnel RF-based classifier was developed and optimised to improve this prediction using an open-access dataset of clinical records from the UCI ML repository. Despite challenges inherent in real-world clinical data, such as the presence of outliers and non-normal data distributions, the FRF model demonstrated higher accuracy and reliability as compared to models from published studies evaluated on the same benchmark medical data, while retaining its explainability.\u003c/p\u003e\n\u003cp\u003eThe novel contributions of this work are multiple. Firstly, the introduction of the FRF methodology, which integrates information theory-based metrics for proactive outlier detection, represents a significant advancement in handling noisy and heterogeneous clinical data. This approach not only enhances the reliability of the RF model but also ensures better generalisation on unseen test data. Secondly, the Funnel RF model's ability to identify 11.04% fewer outliers and achieve a 2% increase in training performance highlights its effectiveness in improving model accuracy and reliability.\u003c/p\u003e\n\u003cp\u003eThe practical implications of this study are profound. The FRF model's enhanced classification performance in predicting patient survival can significantly aid clinical decision support systems. By providing more accurate and reliable prognostic predictions, the model can help clinicians make data-informed decisions about early interventions and treatment titration, thus improving patient outcomes. The case studies presented in this work illustrate how the model's predictions can be interpreted on a patient-specific basis, demonstrating its utility in real-world clinical scenarios.\u003c/p\u003e\n\u003cp\u003eTo yield tangible value and societal impact, it is crucial to understand how the FRF model can be leveraged by end users, particularly physicians, and how it can be integrated within existing clinical workflows. The model's explainability and high performance make it a valuable tool for clinicians, enabling them to titrate treatments early and improve patients' survival rates.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn summary, this study presents a novel and effective approach to enhancing the accuracy and reliability of RF models to aid clinical predictive analytics. The FRF methodology holds promise for improving clinical decision support and ultimately contributing to better patient care and outcomes.\u003c/p\u003e\n\u003cp\u003eWhilst the FRF model demonstrated significant improvements in handling outliers and enhancing the accuracy and reliability of clinical predictions, several avenues for future research can further refine and extend this work.\u003c/p\u003e\n\u003cp\u003eFirstly, future studies should validate the FRF model on a broader range of clinical datasets, including those from different medical conditions and healthcare settings. This will help assess the model's generalisability and reliability across various types of clinical data. Additionally, integrating the FRF model with electronic health record (EHR) systems can facilitate its seamless adoption in clinical practice. Research should focus on developing interfaces and workflows that allow clinicians to easily access and interpret the model's predictions within their existing EHR platforms.\u003c/p\u003e\n\u003cp\u003eImplementing the FRF model in real-time clinical decision support systems can provide immediate prognostic insights to clinicians during patient consultations. Future work should explore the technical and practical challenges of real-time integration, including computational efficiency and user interface design. Conducting longitudinal studies to track the impact of using the FRF model on patient outcomes over time can provide valuable insights into its effectiveness in improving clinical decision-making and patient care. These studies can also help identify any long-term benefits or limitations of the model.\u003c/p\u003e\n\u003cp\u003eWhilst the FRF model retains the interpretability of traditional RF models, further enhancing its explainability can increase clinician trust and adoption. Future research should explore advanced techniques for visualising and explaining the model's predictions, such as using Shapley values or other feature importance metrics. Clinical datasets often suffer from class imbalance, where certain outcomes are much rarer than others. Future work should investigate methods to further improve the FRF model's performance on imbalanced data, such as using synthetic data generation techniques or advanced sampling methods.\u003c/p\u003e\n\u003cp\u003eAs novel ML-driven models and techniques continue to emerge, it is important to conduct comparative studies to benchmark the FRF model against these new approaches. This will help ensure that the FRF model remains competitive and incorporates the latest advancements in the field. Developing methods to customise the FRF model for individual patients based on their unique clinical profiles can enhance its predictive accuracy and reliability. Future research should explore techniques for personalising the model's parameters and decision rules to better suit individual patient characteristics.\u003c/p\u003e\n\u003cp\u003eBy addressing these areas, future research can further enhance the FRF model's utility and impact in clinical settings, ultimately contributing to even better patient outcomes and more effective healthcare delivery.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eCompeting interests declaration\u003c/strong\u003e: none.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding declaration\u003c/strong\u003e:This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eEthics and Consent to Participate declarations\u003c/strong\u003e: not applicable\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eL.P.: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data Curation, Writing \u0026ndash; Original Draft, Writing \u0026ndash; Review \u0026amp; Editing, Visualization, Supervision, Project administration.M.L.M.: Validation, Investigation, Visualization, Writing \u0026ndash; Original Draft, Writing \u0026ndash; Review \u0026amp; Editing.\u003c/p\u003e\u003ch2\u003eData Availability\u003c/h2\u003e\u003cp\u003eData leveraged in this study have been adequately cited in text and referenced at the bottom of the manuscript.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eAhmad, T., Munir, A., Bhatti, S. 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BIRCH: an efficient data clustering method for very large databases. \u003cem\u003eACM SIGMOD Record\u003c/em\u003e, \u003cem\u003e25\u003c/em\u003e(2), 103-114. https://doi.org/10.1145/235968.233324.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Random Forest, Outlier removal, Mutual Information, Gini Impurity, Heart failure, Prognostics, Survival prediction","lastPublishedDoi":"10.21203/rs.3.rs-5784003/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5784003/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e \u003cspan type=\"SmallCaps\" class=\"SmallCaps\" name=\"Emphasis\"\u003eThis study introduces a novel Funnel Random Forest (FRF) methodology to improve the accuracy and reliability of clinical prognostics, particularly for patients with heart failure. FRF identifies outliers using information theory-based metrics during training, reducing outliers by 11.04% compared to traditional methods. The model was trained on a dataset of 299 clinical records, achieving an 82.56% accuracy, a 1.67% improvement over standard Random Forest models. FRF outperformed existing models, with a 9.67% increase in accuracy and a 25.6% improvement in reliability, measured by the Precision-Recall Area Under the Curve (PR-AUC). These results demonstrate the model's potential to enhance clinical decision support, leading to better patient outcomes through early and precise prognostic predictions.\u003c/span\u003e \u003c/p\u003e","manuscriptTitle":"Funnel Random Forest: Inliers-Focused Ensemble Learning for Improved Prognostics of Heart Failure","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-01-15 12:04:22","doi":"10.21203/rs.3.rs-5784003/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"1f0aedf4-58ed-4297-9446-0bb75d94016b","owner":[],"postedDate":"January 15th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-01-17T03:23:10+00:00","versionOfRecord":[],"versionCreatedAt":"2025-01-15 12:04:22","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5784003","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5784003","identity":"rs-5784003","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}