The DBCV index is more informative than DCSI, CDbw, and VIASCKDE indices for unsupervised clustering internal assessment of concave-shaped and density-based clusters

The DBCV index

The DBCV is an index designed to internally evaluate the quality of density-based clustering algorithms (for example, DBSCAN (Ester et al., 1996) and HDBSCAN (McInnes, Healy & Astels, 2017)). It extends the concept of the Silhouette coefficient (Rousseeuw, 1987) by replacing the usual distance (that is, Euclidean) with density-based distances (or connectivity measures). Let $X=\{x_1,x_2,\dots ,x_n\}$ be a dataset of $n$ objects in ${\mathbb{R}}^d$, and let $C=\{C_1,\dots ,C_k\}$ be a clustering result with $l$ clusters.

The all-points-core-distance for an object $x$ in cluster $C_i$ is defined as:

$$\mathtt{A}\mathtt{P}\mathtt{C}\mathtt{D}(x)={\left(\frac{1}{|C_i|-1}\sum _{j=2}^{|C_i|}{\left(\frac{1}{\mathtt{K}\mathtt{N}\mathtt{N}(x,j)}\right)}^d\right)}^{-\frac{1}{d}}$$where $\mathtt{K}\mathtt{N}\mathtt{N}(x,j)$ is the distance from $x$ to its $j$-th nearest neighbor in $C_i$.

The mutual reachability distance between two objects $x_i$ and $x_j$ is:

$$\mathtt{M}\mathtt{R}\mathtt{D}(x_i,x_j)=max\left\{\mathtt{A}\mathtt{P}\mathtt{C}\mathtt{D}(o_i),\mathtt{A}\mathtt{P}\mathtt{C}\mathtt{D}(x_j),d(x_i,x_j)\right\}$$where $d(x_i,x_j)$ is the Euclidean distance between $x_i$ and $x_j$.

The minimum spanning tree (MST) is built for each cluster using mutual reachability distances. The density sparseness of cluster $C_i$ can be defined as:

$$\mathrm{D}\mathrm{S}\mathrm{C}(C_i)=max\left\{\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{o}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{M}\mathrm{S}\mathrm{T}\mathrm{o}\mathrm{f}{C}_i\right\}$$

The density separation between clusters $C_i$ and $C_j$ ( $i\ne j$) is the following:

$$\mathrm{D}\mathrm{S}\mathrm{P}\mathrm{C}(C_i,C_j)=min\left\{\mathrm{M}\mathrm{R}\mathrm{D}(x_p,x_q)\mid x_p\in C_i,x_q\in C_j\right\}$$

And the resulting cluster validity score for $C_i$ is:

$$\mathrm{V}\mathrm{C}(C_i)=\frac{{min}_{j\ne i}\mathrm{D}\mathrm{S}\mathrm{P}\mathrm{C}(C_i,C_j)-\mathrm{D}\mathrm{S}\mathrm{C}(C_i)}{max\left({min}_{j\ne i}\mathrm{D}\mathrm{S}\mathrm{P}\mathrm{C}(C_i,C_j),\mathrm{D}\mathrm{S}\mathrm{C}(C_i)\right)}$$

Finally, the overall DBCV index for the full clustering solution can be summarized as:

(1) $$\mathrm{D}\mathrm{B}\mathrm{C}\mathrm{V}(C)=\sum _{i=1}^k\frac{|C_i|}{|X|}\cdot \mathrm{V}\mathrm{C}(C_i)$$

(minimum and worst value: −1; maximum and best value: +1).

Values near $+1$ indicate well-separated, cohesive clusters (in density terms), values around $0$ suggest ambiguous cluster structure, values near $-1$ indicate poor clustering structure or misassignments.

Software packages

The original implementation of DBCV was in MATLAB (Jaskowiak, 2023), that unfortunately is a proprietary software: only users who buy a license can actually use it.

Fortunately, several versions of the DBCV index function are openly available on Python and R. In R, a function implementing the DBCV index recently published in the DBCVindex CRAN package (Chicco, 2025) and in the dbscan CRAN package (Hahsler). In Python, several packages reimplementing the original MATLAB version are available: FelSiq/DBCV (Siqueira, 2023), permetrics (van Thieu, 2024), k-DBCV (Kaufman Lab Columbia, 2024), and christopherjenness/DBCV (Jenness, 2023).

We tested the DBCV functions of all these software packages and found out that DBCVindex in R and FelSiq/DBCV in Python generated the most similar results to the original MATLAB implementation in reasonable time. We therefore decided to employ FelSiq/DBCV for this study.

The authors of CDbw and VIASCKDE released Python packages of their indices (Lashkov, 2019; Senol, 2024). DCSI was released as an R package (Gauss, 2023), and we used it in our Python environment through the rpy2 Python library (Gautier, 2025).

Clustering methods

To evaluate the reliability of the DBCV index and other metrics, we applied DBSCAN (Schubert et al., 2017) to artificial datasets, as detailed in the following sections. DBSCAN is a clustering algorithm that groups together points located in dense regions while identifying those in sparse areas as noise. It operates based on two main parameters: epsilon ( $\mathrm{ϵ}$) and minimal points. The $\mathrm{ϵ}$ parameter defines the maximum distance within which points are considered part of the same cluster. A larger $\mathrm{ϵ}$ value tends to merge more points into clusters, whereas a smaller $\mathrm{ϵ}$ can result in more fragmented clusters or isolated noise points. The minimal points parameter sets the minimum number of neighboring points required to form a cluster. A point is categorized as a core point if it has at least minimal points within its $\mathrm{ϵ}$-radius. Border points are those that are close to a core point but do not meet the minimal points threshold themselves, while noise points do not belong to any cluster. By carefully selecting $\mathrm{ϵ}$ and minimal points, DBSCAN is capable of detecting clusters of various shapes and managing outliers effectively.

Regarding the real-world medical data, we also employed HDBSCAN (McInnes, Healy & Astels, 2017) and MeanShift (Cheng, 1995). HDBSCAN is an advanced extension of DBSCAN that enhances cluster detection in datasets with variable density. Unlike DBSCAN, HDBSCAN builds a hierarchy of clusters based on density and then extracts the most stable partition automatically, making it more flexible and suitable for complex data.

Two key parameters of HDBSCAN are minimum cluster size and epsilon. The minimum cluster size parameter defines the minimum number of points required for a group to be considered a valid cluster. Higher values reduce the number of detected clusters and increase robustness against noise, while lower values allow the identification of smaller clusters. The epsilon parameter sets a distance threshold for separating clusters. If specified, this value ensures that two clusters are not merged if they are separated by a distance greater than $\mathrm{ϵ}$, providing finer control over clustering granularity.

MeanShift is a density-based clustering algorithm that iteratively shifts data points toward regions of higher density until convergence is reached. Unlike HDBSCAN, it does not require specifying the number of clusters beforehand and can identify arbitrarily shaped structures, making it useful for exploratory data analysis.

A fundamental parameter of MeanShift is bandwidth, which defines the width of the search window used to estimate local density. A higher bandwidth value results in larger and more generalized clusters, whereas a lower value allows for the detection of more detailed clusters but may increase noise and over-segmentation. Choosing the optimal bandwidth is crucial for obtaining meaningful clustering results.

Validation on artificial data clustering trends

To validate the DBCV index on concave clusters, we generated four distinct artificial datasets, each designed to model specific geometries and clustering challenges. We intentionally designed these datasets as compositions of different clusters clearly observable by anyone, applied DBSCAN, and measured its results through seven internal clustering metrics: the DBCV index (Davoud et al., 2014), Silhouette coefficient (Rousseeuw, 1987), Dunn index (Dunn, 1974), Davies-Bouldin index (Davies & Bouldin, 1979), Calinski-Harabasz index (Calinski & Harabasz, 1974), and Shannon Entropy (Shannon, 1948). We then manipulated the data by inserting noise so that these clusters would become almost undistinguishable, reapplied DBSCAN, and reassessed its results through the same metrics. This noise data manipulation worsened the clustering results, and we expected to see the metrics to produce worsening results, too. However, not all the metrics confirmed this worsening.

Here we list the artificial data shapes that we employed in these tests.

Half Moons: The moon-shaped datasets were generated using a function, which utilizes the make_moons utility from the scikit-learn (Scikit Learn, 2007) Python library. This function creates datasets with two interlocking crescent shapes. We used 1,000 points with different noise levels (0.0, 0.056, 0.111, 0.167, 0.222, 0.278, 0.333, 0.389, 0.444, 0.5).

For each noise level, a separate dataset was created. Increasing the noise progressively blurs the crescent shapes, making the clustering task more challenging.

Shifting Circles: It is used a function to generate datasets consisting of an outer circle and a shifted inner circle. This design emphasizes the spatial relationship between the clusters. We utilized 10 datasets, having 500 points for the outer circle and 500 points for the inner circle. We set 0.5 as inner radius and 1.5 as outer radius. The initial shift was (0, 0) and the incremental shift was (0.2, 0).

The inner circle is progressively shifted along the x-axis with each dataset by the incremental shift value. This systematic displacement introduces varying degrees of overlap between the two circles, ranging from fully separate to partially overlapping clusters. These datasets serve as a valuable testbed for evaluating clustering algorithms’ capability to discern subtle structural changes while maintaining the integrity of distinct clusters. By incrementally adjusting the shift, we simulate scenarios where the boundary between clusters becomes increasingly ambiguous, pushing the limits of clustering methods.

Sparse Circles: Noisy concentric circle datasets were created using a function, which utilizes the make_circles utility from scikit-learn (Scikit Learn, 2007). The function generates two concentric circles with added Gaussian noise. Here we used ten datasets having 1,000 samples and by using a noise increment of 0.0555.

The noise level increases progressively across datasets, starting from a base value of 0.05. This variation tests clustering algorithms’ robustness to noise while maintaining the underlying concentric circle structure.

Tulip shapes: A special fourth dataset was generated to further assess the DBCV index. This dataset builds on existing crescent-shaped data, incorporating additional points and noise to create a more challenging test case. The original data was obtained from the moon example of the CVIK MATLAB package (José-García & Gómez-Flores, 2023; José-García). The process involved the following steps:

• 1.

  The Moon.mat file was loaded, and the $x$ and $y$ coordinates were extracted and combined into a single array of points.

• 2.

  The original points were normalized to lie within the [0, 1] range by scaling them based on their minimum and maximum values.

• 3.

  Augmentation: We generated 10 datasets, and each dataset included:

  • •

    400 points sampled from the normalized original points.

  • •

    Gaussian noise added to these new points to keep them close to the original distribution. Noise levels ranged from 0.0 to 0.5 in steps of 0.056.

• 4.

  The noisy points were wrapped within the [0, 1] range using the modulo operation.

• 5.

  The noisy new points were combined with the original normalized points to create the final augmented datasets.

This augmentation strategy effectively preserves the original distribution while introducing variability, thereby challenging clustering algorithms to identify clusters amidst increased noise and data density variations.

The rationale beyond this validation algorithm is simple: as the trends worsen, we expected a consistent deterioration of the internal clustering metrics’ results. If the results of a specific metric worsened, we considered that metric reliable. If the results stayed stable or improved, we considered that metric unreliable.

Validation on real-world medical data clustering trends

Clinical records data play a crucial role in medical research, offering valuable insights for patient care, disease progression analysis, and predictive modeling. These datasets typically consist of unstructured information, including demographic details, laboratory results, diagnostic codes, treatment histories, and physician notes.

In this study, we utilize five publicly available clinical datasets derived from health records spanning different diseases: Neuroblastoma (Ma et al., 2018), Diabetes type one (Takashi et al., 2019; Cerono & Chicco, 2024), Sepsis & SIRS (Gucyetmez & Atalan, 2016; Mollura et al., 2024), Heart Failure and Depression (Jani et al., 2016) and Cardiac Arrest (Requena-Morales et al., 2017).

The first dataset concerns neuroblastoma (Ma et al., 2018), a malignant tumor of the sympathetic nervous system. This disease is characterized by high genetic and clinical heterogeneity, making the analysis of multiple variables crucial for accurate diagnosis and risk stratification. The dataset includes 168 patients and 13 clinical features.

The second dataset relates to diabetes (Takashi et al., 2019; Cerono & Chicco, 2024), a chronic metabolic disorder characterized by elevated blood glucose levels due to insufficient insulin production or utilization. Since diabetes is associated with numerous complications, the dataset includes 67 patients and 20 clinical.

Another dataset focuses on sepsis and Systemic Inflammatory Response Syndrome (SIRS) (Gucyetmez & Atalan, 2016; Mollura et al., 2024). Sepsis is a systemic inflammatory response triggered by a severe infection, whereas SIRS can occur without an infectious agent. Differentiating between these conditions is crucial for timely medical intervention. This dataset contains data from 1,257 patients with 16 clinical features relevant for identifying and managing these syndromes.

We also analyzed a dataset on heart failure and depression (Jani et al., 2016). Heart failure is a chronic condition in which the heart cannot pump blood efficiently, leading to inadequate tissue oxygenation. Depressive symptoms are common in these patients and can negatively impact prognosis. The dataset includes 425 patients with 15 clinical features to explore this complex relationship.

Finally, the dataset on cardiac arrest (Requena-Morales et al., 2017) provides information on 420 patients and 10 clinical features. Cardiac arrest is a sudden and potentially fatal event characterized by the cessation of cardiac activity. Analyzing this dataset is essential for identifying risk factors and improving emergency interventions.

We used datasets form electronic health records because of their high level of complexity. These data, in fact, were collected for clinical reasons and not for scientific purposes, through standard hospital laboratory tests (such as blood test). Moreover, these data consists of different data types (numeric, ordinal, categorical, binary) making their clustering particularly challenging. Nevertheless, machine learning and computational statistics applied to these data can be extremely useful for predicting prognosis or disease trends of patients (Patton & Liu, 2023; Steele et al., 2018; Wong et al., 2018).

Since these datasets are real-world clinical records with complex and heterogeneous structures, assessing the reliability of a clustering validation metric like DBCV becomes challenging. Unlike artificial datasets with well-defined clusters, medical data often contain overlapping patient profiles, missing values, and noisy features, making it difficult to determine a “ground truth” for clustering quality. The effectiveness of DBCV in such scenarios depends on its ability to capture meaningful patient groupings despite these complexities. Therefore, while DBCV provides valuable insights, its performance should be interpreted alongside other validation indices and domain knowledge to ensure clinically relevant results.

To assess the reliability of the DBCV metric, the proposed approach involves evaluating its stability as clustering parameters deteriorate. For each dataset, we selected a set of random hyperparameters for DBSCAN, HDBSCAN, and MeanShift. Each metric’s trend was then analyzed by comparing its values when transitioning from these random parameters to default settings. If the variation in the DBCV score aligns with the trend observed in the ARI, the metric is considered consistent; otherwise, it is deemed inconsistent.

We calculate the ARI values of the pairs: clusters generated by DBSCAN and clusters generated by HDBSCAN; clusters generated by DBSCAN and clusters generated by MeanShift; clusters generated by MeanShift and clusters generated by DBSCAN. We then calculate the average value of these three scores. We represent this process in Fig. 1.

In summary, if the clustering trends measured through ARI were worsening and the values of a specific metric were worsening too, we considered that metric reliable. The same thing for improvements: improving ARI trend and improving metric’s results made that metric reliable. In all the other cases, we considered the metric unreliable (Fig. 1).

Artificial clouds of points

The analysis involves generating multiple artificial datasets, each subjected to clustering using the DBSCAN algorithm (Ester et al., 1996). This method is particularly effective in identifying arbitrarily shaped clusters and managing noisy data.

For each dataset, three visualizations are produced:

• 1.

  Original dataset (left): A graphical representation of the generated points without clustering.

• 2.

  DBSCAN clustering (centre): A colored visualization illustrating how DBSCAN partitions the data into different clusters.

• 3.

  Evaluation metrics (right): A textual summary displaying various clustering validation indices.

Half moons

As observed in Figs. 2 and in 3, clustering performance progressively deteriorates with increasing noise levels. Initially, DBSCAN effectively identifies well-defined clusters, as reflected by high DBCV, VIASCKDE and DCSI scores. However, as noise intensifies, the quality of clustering declines, with a growing number of misclassified points and fragmented clusters. The number of detected clusters rises unpredictably, highlighting the algorithm’s sensitivity to noise. Additionally, the DBCV score steadily decreases, indicating a weakening of the density-based cluster structures. Among the validation metrics, DBCV initially aligns closely with the DCSI index, confirming well-defined clusters in low-noise conditions. However, as noise increases, DBCV deteriorates more gradually than the VIASCKDE score, which rapidly decrease, suggesting that DBCV is more robust in assessing density-based clustering structures. Interestingly, both VIASCKDE and DCSI exhibit a slight increase at the initial rise of noise, indicating a lower robustness of these metrics in the early stages of degradation.

The trends observed in Fig. 4 provide key insights into the effect of noise on clustering performance. The DBCV score remains relatively high at low noise levels, indicating well-defined clusters, but progressively declines as noise increases. Despite this decline, DBCV remains above other metrics in its ability to evaluate density-based structures, showing a more gradual deterioration. The VIASCKDE index performs reliably in the initial stages, decreasing appropriately as noise levels rise during the first four evaluation steps. However, it unexpectedly begins to increase afterwards—despite the dataset becoming progressively noisier. This behavior indicates a potential weakness in the metric’s ability to distinguish degraded clustering quality in higher-noise scenarios.

The CDbw index behaves counterintuitively: it reports lower values when there is no noise, and increases almost proportionally with the noise level. This trend challenges the expectation that higher values always reflect better clustering, and suggests that CDbw may not effectively penalize the fragmentation caused by noise.

Similarly, the DCSI index initially shows a correct downward trend as noise increases, signaling reduced cluster separability and connectivity. However, it too enters a phase of growth at higher noise levels, deviating from what one would expect in a scenario of worsening cluster structure. This pattern, shared with VIASCKDE, points to a reduced robustness when facing significant data degradation.

Shifting circles

As observed in Figs. 5, 6 and 7, the clustering performance remains relatively stable across different datasets, despite the gradual displacement of the inner circle. DBSCAN consistently detects two clusters, as evidenced by the DBCV score, which remains around 0.5 throughout the datasets. This stability suggests that the algorithm effectively captures the density-based structure, even as the spatial arrangement of the clusters shifts.

VIASCKDE remains relatively stable across the initial observations but shows a notable increase in its values when the two concentric circles begin to intersect. While the score appears consistent at first, it fails to maintain coherence with the actual cluster quality and does not accurately reflect good separation in the early, less noisy stages. This behavior suggests that VIASCKDE may lack sensitivity to subtle structural differences at low noise levels, and its sharp increase under higher noise may indicate an overestimation of clustering quality.

The CDbw index exhibits counterintuitive behavior throughout the evaluation. It starts with a very low value, then peaks unexpectedly, gradually declines, and eventually rises again. This non-monotonic trend diverges from the expected pattern of continuous degradation as noise increases. The observed increases may be due to a misinterpretation of dispersed noise points as inter-cluster density, artificially boosting the score. These fluctuations undermine the reliability of CDbw in scenarios involving progressive noise, suggesting it may not be well-suited for evaluating clustering quality in such contexts.

The DCSI score remains flat at zero across all evaluations, indicating no detected core structure or cluster separability according to its internal criteria. This persistent null response may stem from a conservative threshold for core point identification or a limited responsiveness to mild structural changes. Although DCSI may become more informative as the dataset degrades further, its lack of reaction in the early stages limits its practical utility under low-noise conditions.

Sparse circles

The trends observed in Figs. 8, 9 and 10 highlight the effects of increasing noise on clustering performance in sparse circular datasets.

The DBCV, VIASCKDE, and DCSI indices all show an initial decrease as noise increases, suggesting that they are all sensitive—at least in the early stages—to the loss of clear cluster structure. This common trend highlights how even moderate levels of noise can begin to erode the cohesion and separation that define meaningful clustering.

After this initial decline, DBCV tends to stabilize, maintaining moderately low values with only minor fluctuations. While it no longer reflects a strong cluster structure, its relative steadiness suggests it still captures some residual density information, even as noise increases.

VIASCKDE, on the other hand, displays a more irregular pattern after its initial drop. Its values begin to oscillate noticeably, alternating between rises and falls across different noise levels. This behavior might reflect a heightened sensitivity to local density changes, but also hints at instability—perhaps overreacting to the introduction of scattered noise rather than robustly tracking meaningful structure.

The DCSI index follows a similar fluctuating trend but does so more gently. Its variations are less pronounced, indicating a more conservative response to noise. While it does react to changes in the data, its limited range of movement suggests that it might miss more subtle transitions in cluster connectivity or underrepresented mild structural degradation.

In contrast, the CDbw index behaves differently. Instead of decreasing or oscillating, it gradually increases as noise rises. This unexpected trend could be due to the metric misinterpreting dispersed noise points as contributing positively to inter-cluster density, which artificially inflates its value.

Tulip dataset

In this scenario, all the evaluated metrics exhibit a consistent trend, showing similar behavior as noise increases Figs. 11 and 12. Despite some local fluctuations, the overall trajectory for each metric is downward, indicating a progressive degradation in clustering quality as noise disrupts the underlying data structure (Fig. 13).

Interestingly, even the CDbw index—which in previous cases displayed an opposite trend by increasing with noise—aligns with the behavior of the other metrics in this setting, showing a general decline. While some oscillations are still present and occasionally interrupt the monotonic descent, they do not significantly alter the overall pattern. These irregularities likely reflect residual sensitivity to local noise effects or internal variations in cluster structure, but the dominant trend remains one of decreasing cluster cohesion and separability in response to increasing noise.

Recap of the results on the artificial data tests. To assess the robustness of clustering validation metrics under increasing noise levels, a consistency criterion was established. A metric is deemed consistent if its values predominantly decrease as the noise increases, allowing for minor fluctuations.

The method involves a systematic analysis of each metric’s behavior, by counting the number of consecutive value pairs that decrease.

Metrics are classified as inconsistent if their values predominantly increase with higher noise levels, or if they remain constant across all noise levels—since such invariance suggests a lack of sensitivity to structural changes in the data.

Table 2 provides a summary of the consistency of various clustering validation metrics across different datasets. Each row corresponds to a specific metric, while the columns represent the datasets: Half Moons, Shifting Circles, Sparse Circles, and Tulip. The values in the cells indicate the number of times a given metric was found to be consistent within each dataset. The final column (Total) reports the overall number of consistent outcomes observed across all datasets as noise increased.

From the table, it is evident that DBCV is the most stable metric, showing the highest number of consistent trends both overall and within each individual dataset. Its values remained relatively stable and consistently outperformed the other metrics across noise levels.

VIASCKDE appears to follow closely, ranking as the second most consistent metric. In contrast, CDBW and DCSI exhibit limited stability, indicating a weaker responsiveness to changes in data structure under increasing noise.

Clinical records data tests

For each of the five datasets, a table is provided displaying the values of the various metrics, indicating whether the trend of the metric is consistent with that of the ARI. Additionally, a second table summarizes the frequency with which each metric is classified as consistent or inconsistent across the datasets.

Neuroblastoma dataset. We applied our validation approach described in subsection ‘Validation on real-world medical data clustering trends’ to a dataset derived from health records of patients with neuroblastoma (Ma et al., 2018).

We measured the trend of the ARI when moving from tests made through random parameters to tests made through default parameters (Table 3). The results showed that DBCV index resulted being consistent with the ARI trends on all three tests, followed by DCSI index and VIASCKDE on 2 tests (Table 4).

Diabetes type one dataset. We applied our validation approach described in ‘Validation on real-world medical data clustering trends’ to a dataset derived from health records of patients with diabetes type (Takashi et al., 2019; Cerono & Chicco, 2024).

We measured the trend of the Adjusted Rand Index when moving from tests made through random parameters to tests made through default parameters (Table 5). The results showed that no index was consistent with the ARI trend for all the tests made: DCSI index resulted consistent on 2 tests out of 3, while DBCV index only once (Table 6).

Sepsis and SIRS dataset. We applied our validation approach described in ‘Validation on real-world medical data clustering trends’ to a dataset derived from health records of patients with sepsis or SIRS (Gucyetmez & Atalan, 2016; Mollura et al., 2024).

We measured the trend of the Adjusted Rand Index when moving from tests made through random parameters to tests made through default parameters (Table 7). The results showed that DBCV index resulted informative on two tests, while DCSI index on one tests out of three (Table 8).

Heart Failure and Depression dataset. We applied our validation approach described in ‘Validation on real-world medical data clustering trends’ to a dataset derived from health records of the heart failure and depression among patients (Jani et al., 2016).

We measured the trend of the ARI when moving from tests made through random parameters to tests made through default parameters (Table 9). The results showed no test obtained consistent trends with the ARI trends on all the three tests (Table 10).

Cardiac arrest dataset. We applied our validation approach described in ‘Validation on real-world medical data clustering trends’ to a dataset derived from health records of patients with cardiac arrest (Requena-Morales et al., 2017).

We measured the trend of the Adjusted Rand Index when moving from tests made through random parameters to tests made through default parameters (Table 11). The results showed no test obtained consistent trends with the ARI trends on all the three tests (Table 12).

Recap of the medical data tests. We counted the number of times each clustering internal metric had consistent trend compared with the ARI trend (Fig. 14). Since we employed three clustering techniques (DBSCAN, HDBSCAN, and MeanShift) on five medical datasets (Neuroblastoma, Sepsis and SIRS, Heart Failure and Depression, Diabetes type one, Cardiac Arrest), we eventually had 15 tests for each metric. Our results indicated that the DBCV index is the most informative metric among the seven scores studied, by having a consistent trend on nine cases our of 15 (around 60%). The DCSI index obtained a 46.7% success rate, by resulting informative on seven datasets out of 15. VIASCKDE were coherent with the ARI trend on four times, while CDbw is not applicable to any of the datasets. These results prove that the DBCV index is more informative than these other clustering metrics considered in this study.