# Insurance Analytics with Clustering Techniques

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## Abstract

**:**

## 1. Introduction

## 2. K-Means Clustering

#### 2.1. A Holistic Overview

- Assignment step: In the e-th iteration, each observation ${\mathit{x}}_{i}$ is assigned to the cluster ${S}_{u}\left(e\right)$ whose centroid ${\mathit{c}}_{u}\left(e\right)$ is the closest.
- Update step: The K centroids ${\left({\mathit{c}}_{u}\right)}_{u=1:K}$ are replaced with the K new cluster means ${\left({\mathit{c}}_{u}(e+1)\right)}_{u=1:K}$.

Algorithm 1: K-means clustering. |

Algorithm 2: K-means++ initialization of centroids. |

#### 2.2. Burt Distance for Categorical Data

**Example**

**1.**

**Example**

**2.**

#### 2.3. Burt Distance for Mixed Data Types

## 3. Burt-Adapted K-Means Variants

#### 3.1. Mini-Batch K-Means

Algorithm 3: Mini-batch K-means algorithm. |

#### 3.2. Fuzzy K-Means

Algorithm 4: Fuzzy clustering. |

## 4. Burt-Adapted Spectral Clustering

- a set of vertices $V={\left\{{v}_{i}\right\}}_{i=1,\dots ,n}$ where each vertex ${v}_{i}$ represents one of the n data points in the dataset;
- a set of edges $E=\{{e}_{i,j}:{v}_{i}\u27f7{v}_{j}\}$ whose entries are equal to 1 if two vertices ${v}_{i}$ and ${v}_{j}$ are connected by an undirected edge ${e}_{i,j}$, and 0 otherwise;
- a set of weights $W=\{{w}_{ij}:{w}_{ij}\ne 0\phantom{\rule{4.pt}{0ex}}\mathrm{if}\phantom{\rule{4.pt}{0ex}}{v}_{i}\u27f7{v}_{j}\}$ that contains the similarity between two vertices linked by an edge.

- The $\u03f5$-neighborhood graph: Points are connected if the pairwise distance between them is smaller than a threshold $\u03f5$. In practice, we keep all similarities smaller than $exp\left(-{\displaystyle \frac{\u03f5}{\alpha}}\right)$, while setting others to zero.
- The (mutual) k-nearest neighbor graph: The vertex ${v}_{i}$ is connected to vertex ${v}_{j}$ if ${v}_{j}$ is among the k-nearest neighbors of ${v}_{i}$. Since the neighborhood relationship is not symmetric, and the graph must be symmetric, we need to enforce the symmetry. The graph is made symmetric by ignoring the directions of the edges, i.e., by connecting ${v}_{i}$ and ${v}_{j}$ if either ${v}_{i}$ is among the k-nearest neighbors of ${v}_{j}$ or ${v}_{j}$ is among the k-nearest neighbors of ${v}_{i}$, resulting in the k-nearest neighbor graph. Alternatively, the connection is made if ${v}_{i}$ and ${v}_{j}$ are mutual k-nearest neighbors, resulting in the mutual k-nearest neighbor graph.
- The fully connected graph: All points in the dataset are connected.

Algorithm 5: Spectral clustering. |

Input: Dataset XInit: Represent the dataset X as a graph $G=(V,E,W)$(1) Calculation of the $n\times n$ Laplacian matrix $L=D-A$ (2) Extract the eigenvector matrix U and diagonal matrix of eigenvalues $\Sigma $ from $L=U\Sigma {U}^{\top}$ (3) Fix k and build the $n\times k$ matrix ${U}^{\left(k\right)}$ of eigenvectors with the k eigenvalues closest to zero (4) Run the K-means (Algorithm 1) with the dataset of ${U}_{i,.}^{\left(k\right)}$ for $i=1,\dots ,n.$ (5) The ${i}^{th}$ data point is associated to the cluster of ${U}_{i,.}^{\left(k\right)}$ |

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Proof.**

## References

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**Figure 2.**Illustration of the partitioning of the numeric data into 10 clusters with the K-means algorithm (using Euclidean distance).

**Figure 3.**Illustration of the partitioning of the discretized numeric data (projected in Burt space) into 10 clusters with the Burt distance-based K-means algorithm.

**Figure 4.**Clustering results obtained with the Burt distance-based K-means algorithm on a simulated dataset.

**Figure 5.**Burt distance-based K-means metrics. (

**Left**) plot: evolution of the total intra-class inertia. (

**Right**) plot: evolution of the deviance.

**Figure 6.**Partition quality (in solid lines, measured by the average deviance over 10 random seeds) and running time (dotted lines) with respect to the number of clusters.

**Figure 7.**Illustration of the partition of non-convex data with the K-means and spectral clustering algorithms. Each cluster is identified by a color (black or green). The centroids are represented with a diamond shape.

**Figure 12.**(

**Left**) Partition of a non-convex dataset with spectral clustering. (

**Right**) Pairs of eigenvectors’ coordinates ${\left({U}_{i,1},{U}_{i,2}\right)}_{i=1,\dots ,n}$.

**Figure 13.**(

**Left**) Spectral clustering partitioning of a non-convex dataset that has been preliminarily reduced with the K-means algorithm. (

**Right**) Pairs of eigenvectors’ coordinates ${\left({U}_{i,1},{U}_{i,2}\right)}_{i=1,\dots ,n}$.

Rating Factors | Class | Class Description |
---|---|---|

Gender | M | Male (ma) |

K | Female (kvinnor) | |

Geographic area | 1 | Central and semi-central parts of Sweden’s three largest cities |

2 | Suburbs plus middle-sized cities | |

3 | Lesser towns, except those in 5 or 7 | |

4 | Small towns and countryside | |

5 | Northern towns | |

6 | Northern countryside | |

7 | Gotland (Sweden’s largest island) | |

Vehicle class | 1 | EV ratio –5 |

2 | EV ratio 6–8 | |

3 | EV ratio 9–12 | |

4 | EV ratio 13–15 | |

5 | EV ratio 16–19 | |

6 | EV ratio 20–24 | |

7 | EV ratio 25– |

**Table 2.**Example of a disjunctive table with $q=2$ variables and ${m}_{1}=2$, ${m}_{2}=3$ modalities, respectively.

Gender | Degree | ||||
---|---|---|---|---|---|

Policy | M | F | H | C | U |

1 | 1 | 0 | 0 | 1 | 0 |

2 | 0 | 1 | 0 | 0 | 1 |

**Table 3.**Burt matrix for the disjunctive Table 2.

Gender | Degree | |||||
---|---|---|---|---|---|---|

M | F | H | C | U | ||

Gender | M | ${n}_{1,1}$ | 0 | ${n}_{1,3}$ | ${n}_{1,4}$ | ${n}_{1,5}$ |

F | 0 | ${n}_{2,2}$ | ${n}_{2,3}$ | ${n}_{2,4}$ | ${n}_{2,5}$ | |

Degree | H | ${n}_{3,1}$ | ${n}_{3,2}$ | ${n}_{3,3}$ | 0 | 0 |

C | ${n}_{4,1}$ | ${n}_{4,2}$ | 0 | ${n}_{4,4}$ | 0 | |

U | ${n}_{5,1}$ | ${n}_{5,2}$ | 0 | 0 | ${n}_{5,5}$ |

**Table 4.**Table summary of K-modes clustering results (policy allocation, dominant features, and average claim frequencies per cluster). K-modes (Hamming distance) applied on data characterized by three categorical variables.

Cluster | % of Policies | Gender | Zone | Class | Frequency (%) |
---|---|---|---|---|---|

6 | 2.82 | K | 4 | 5 | 0.6003 |

1 | 5.73 | K | 4 | 3 | 0.627 |

5 | 2.77 | K | 3 | [3, 4] | 0.7855 |

10 | 13.39 | M | [2, 4] | 4 | 0.8559 |

3 | 17.03 | M | 4 | 3 | 0.8919 |

2 | 20.61 | M | 3 | 3 | 0.9374 |

9 | 19.99 | M | [3, 4] | 5 | 1.0532 |

7 | 2.73 | K | 2 | [3, 4] | 1.2236 |

8 | 5.06 | M | 2 | 1 | 1.8256 |

4 | 9.88 | M | [2, 4] | 6 | 1.9443 |

**Table 5.**Statistics of goodness of fit obtained by partitioning the categorical data into 10 clusters with K-modes.

Goodness of Fit | |
---|---|

Deviance | 6576.31 |

AIC | 8244.88 |

BIC | 9691.58 |

**Table 6.**Table summary of Burt distance-based K-means clustering results (policy allocation, dominant features, and average claim frequencies per cluster). Burt distance-based K-means applied on data characterized by three categorical variables.

Cluster | % of Policies | Gender | Zone | Class | Frequency (%) |
---|---|---|---|---|---|

2 | 8.21 | M | 4 | 3 | 0.3811 |

9 | 12.27 | M | 4 | [4, 5] | 0.5051 |

1 | 8.95 | M | 6 | [3, 5] | 0.6808 |

4 | 6.43 | K | 4 | [3, 5] | 0.6813 |

7 | 4.58 | M | 2 | 3 | 0.8621 |

6 | 11.82 | M | 4 | [1, 6] | 0.9237 |

10 | 8.75 | K | [2, 3] | [3, 4] | 1.0144 |

5 | 16.81 | M | 3 | [3, 5] | 1.0825 |

3 | 10.94 | M | 2 | [4, 6] | 2.1482 |

8 | 11.23 | M | 1 | [3, 4] | 3.0878 |

**Table 7.**Statistics of goodness of fit obtained by partitioning the categorical data into 10 clusters with Burt distance-based K-means.

Goodness of Fit | |
---|---|

Deviance | 6350.04 |

AIC | 8018.61 |

BIC | 9465.31 |

**Table 8.**Statistics of goodness of fit obtained by partitioning the numeric data into 10 clusters with the K-means algorithm (using Euclidean distance).

Goodness of Fit | |
---|---|

Deviance | 6097.18 |

AIC | 7485.75 |

BIC | 7666.59 |

**Table 9.**Table summary of K-means clustering results (policy allocation, dominant features, and average claim frequencies per cluster). K-means applied on data characterized by two continuous variables.

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

5 | 2.00 | M | [3, 4] | 1 | [48, 62] | [38, 48] | 0 |

9 | 4.85 | M | [3, 4] | [3, 4] | [40, 48] | [21, 26] | 0.2035 |

2 | 15.16 | M | [3, 4] | [3, 5] | [48, 53] | [13, 17] | 0.3698 |

10 | 6.46 | M | [3, 4] | [3, 5] | [60, 66] | [12, 18] | 0.45 |

3 | 14.94 | M | [3, 4] | [3, 5] | [42, 47] | [13, 17] | 0.51 |

7 | 2.09 | M | [2, 4] | 1 | [28, 43] | [37, 48] | 0.5106 |

6 | 13.54 | M | [2, 4] | [3, 4] | [44, 49] | [1, 5] | 0.8667 |

8 | 11.65 | M | [2, 4] | [3, 4] | [50, 55] | [1, 5] | 0.9183 |

4 | 14.42 | M | [2, 4] | [3, 5] | [25, 32] | [12, 16] | 1.5317 |

1 | 14.89 | M | [2, 4] | [3, 6] | [23, 29] | [1, 7] | 4.1332 |

**Table 10.**Statistics of goodness of fit obtained by partitioning the discretized numeric data (projected in Burt space) into 10 clusters with the Burt distance-based K-means algorithm.

Goodness of Fit | |
---|---|

Deviance | 6097.51 |

AIC | 7586.08 |

BIC | 8219.01 |

**Table 11.**Table summary of Burt distance-based K-means clustering results (policy allocation, dominant features, and average claim frequencies per cluster). Burt distance-based K-means applied on data characterized by two discretized continuous variables.

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

9 | 2.66 | M | [3, 4] | 1 | [35, 46] | [29, 46] | 0.2207 |

10 | 2.76 | M | [2, 4] | 1 | [29, 52] | [29, 45] | 0.3083 |

3 | 15.31 | M | [3, 4] | [3, 5] | [47, 52] | [13, 17] | 0.3257 |

2 | 15.23 | M | [3, 4] | [3, 5] | [42, 46] | [12, 17] | 0.408 |

7 | 6.41 | M | [3, 4] | [3, 5] | [58, 63] | [13, 17] | 0.456 |

6 | 12.96 | M | [2, 4] | [3, 5] | [47, 52] | [1, 6] | 0.906 |

5 | 10.46 | M | [2, 4] | [3, 4] | [42, 46] | [1, 5] | 0.915 |

8 | 4.69 | M | [2, 4] | [3, 4] | [58, 62] | [1, 5] | 0.9444 |

4 | 14.32 | M | [2, 4] | [3, 5] | [25, 32] | [12, 16] | 1.5324 |

1 | 15.21 | M | [2, 4] | [3, 6] | [23, 29] | [1, 7] | 4.0664 |

**Table 12.**Statistics of goodness of fit obtained by partitioning the mixed-type dataset into 15 clusters with the K-prototypes algorithm.

Goodness of Fit | |
---|---|

Deviance | 6161.47 |

AIC | 8050.04 |

BIC | 10,491.35 |

**Table 13.**Table summary of K-prototypes clustering results (dominant features and average claim frequencies per cluster).

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

13 | 4.45 | M | 4 | [1, 4] | [39, 51] | [23, 30] | 0.1754 |

5 | 3.50 | M | [3, 4] | 1 | [30, 53] | [38, 46] | 0.228 |

15 | 7.94 | M | 3 | 3 | [43, 53] | [13, 19] | 0.301 |

11 | 9.28 | M | 4 | 3 | [51, 62] | [1, 16] | 0.3856 |

14 | 9.24 | M | 4 | 5 | [43, 55] | [12, 17] | 0.3888 |

4 | 5.07 | M | 4 | 2 | [45, 60] | [13, 19] | 0.5943 |

12 | 5.82 | K | 4 | [3, 4] | [33, 46] | [12, 18] | 0.5985 |

7 | 9.01 | M | [2, 3] | 4 | [43, 52] | [1, 17] | 0.6748 |

2 | 4.56 | K | [1, 2] | 3 | [24, 48] | [1, 16] | 1.2525 |

6 | 4.85 | M | [2, 6] | 6 | [45, 54] | [1, 8] | 1.3813 |

9 | 5.30 | M | 4 | [1, 4] | [18, 29] | [12, 17] | 1.5103 |

1 | 7.37 | M | 1 | 3 | [39, 50] | [0, 15] | 1.6583 |

3 | 8.41 | M | 4 | 3 | [21, 29] | [1, 6] | 2.266 |

8 | 8.57 | M | 2 | 5 | [24, 34] | [1, 17] | 2.5845 |

10 | 6.63 | M | 3 | 6 | [23, 29] | [6, 13] | 3.8295 |

**Table 14.**Statistics of goodness of fit obtained by partitioning the mixed-type dataset into 15 clusters with the K-medoids algorithm (using Gower distance).

Goodness of Fit | |
---|---|

Deviance | 6290.92 |

AIC | 8179.49 |

BIC | 10,620.80 |

**Table 15.**Table summary of Burt distance-based K-means on mixed data types (dominant features and average claim frequencies per cluster).

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

13 | 4.83 | M | [3, 4] | 1 | [35, 55] | [29, 46] | 0.2512 |

2 | 9.86 | M | 4 | [4, 5] | [40, 52] | [12, 17] | 0.3141 |

3 | 9.52 | M | [1, 3] | [3, 5] | [47, 52] | [13, 17] | 0.3225 |

10 | 7.60 | M | [2, 3] | [3, 5] | [42, 46] | [12, 17] | 0.4842 |

9 | 8.09 | K | [3, 4] | [3, 5] | [25, 51] | [13, 17] | 0.5647 |

5 | 5.77 | M | 4 | [3, 5] | [47, 55] | [1, 6] | 0.5936 |

7 | 9.19 | M | [3, 4] | [3, 5] | [58, 62] | [1, 17] | 0.6645 |

8 | 4.23 | M | 4 | [3, 5] | [25, 34] | [11, 16] | 0.8449 |

4 | 6.46 | M | [1, 3] | [3, 4] | [42, 46] | [1, 5] | 1.0119 |

12 | 6.83 | K | [3, 4] | [3, 4] | [24, 52] | [1, 6] | 1.2278 |

15 | 7.23 | M | [2, 3] | [3, 4] | [47, 52] | [1, 5] | 1.2402 |

11 | 5.80 | M | [2, 3] | [3, 6] | [24, 31] | [11, 15] | 2.0485 |

6 | 4.38 | M | 4 | [3, 6] | [22, 29] | [1, 7] | 2.3001 |

14 | 6.56 | M | [2, 3] | [3, 6] | [23, 29] | [1, 8] | 4.2849 |

1 | 3.66 | M | 1 | [3, 4] | [25, 31] | [1, 9] | 8.084 |

**Table 16.**Statistics of goodness of fit obtained by partitioning the mixed-type dataset into 15 clusters with the Burt distance-based K-means algorithm.

Goodness of Fit | |
---|---|

Deviance | 5978.96 |

AIC | 8017.53 |

BIC | 11,136.99 |

**Table 17.**Burt-adapted mini-batch clustering. Policy allocation, dominant features, and average claim frequencies per cluster.

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

6 | 3.71 | M | 4 | 1 | [35, 64] | [13, 48] | 0.2625 |

11 | 7.94 | M | 4 | [3, 4] | [47, 59] | [12, 17] | 0.2759 |

2 | 3.31 | M | [1, 3] | 1 | [29, 60] | [29, 45] | 0.4523 |

10 | 6.91 | M | 4 | [5, 6] | [22, 46] | [12, 16] | 0.5052 |

15 | 6.77 | M | [1, 4] | [3, 4] | [42, 46] | [12, 17] | 0.5295 |

8 | 8.06 | K | [3, 4] | [3, 5] | [25, 51] | [13, 17] | 0.5661 |

5 | 9.85 | M | 4 | 3 | [47, 55] | [1, 5] | 0.656 |

9 | 8.58 | M | [1, 2] | [3, 6] | [47, 54] | [13, 17] | 0.7156 |

3 | 5.44 | M | 3 | 3 | [23, 63] | [13, 17] | 0.7496 |

14 | 8.26 | M | [2, 4] | [3, 5] | [42, 46] | [1, 6] | 0.9482 |

12 | 5.58 | M | [2, 3] | 5 | [24, 52] | [13, 17] | 1.0196 |

7 | 6.82 | K | [3, 4] | [3, 4] | [24, 52] | [1, 6] | 1.2301 |

4 | 5.81 | M | [2, 3] | [4, 5] | [47, 53] | [1, 7] | 1.6504 |

1 | 4.89 | M | 4 | [4, 6] | [22, 28] | [5, 10] | 3.2768 |

13 | 8.07 | M | [1, 3] | [3, 5] | [23, 29] | [1, 6] | 5.9861 |

**Table 18.**Statistics of goodness of fit obtained by partitioning the dataset into 15 clusters with the Burt-adapted mini-batch K-means.

Goodness of Fit | |
---|---|

Deviance | 6049.68 |

AIC | 8088.25 |

BIC | 11,207.71 |

**Table 19.**Burt-adapted fuzzy clustering. Policy allocation, dominant features, and average claim frequencies per cluster.

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

2 | 5.16 | M | 4 | [3, 5] | [47, 52] | [12, 17] | 0.2604 |

7 | 4.72 | M | [3, 4] | 1 | [35, 60] | [29, 46] | 0.2992 |

14 | 8.21 | M | [2, 3] | [3, 5] | [47, 52] | [13, 17] | 0.3552 |

6 | 4.92 | M | 4 | [3, 5] | [41, 46] | [12, 18] | 0.3586 |

9 | 6.02 | M | [3, 4] | [3, 5] | [58, 63] | [13, 17] | 0.449 |

12 | 7.6 | M | [2, 3] | [3, 5] | [42, 46] | [12, 17] | 0.4842 |

4 | 3.78 | M | [2, 4] | 3 | [47, 52] | [1, 5] | 0.6654 |

10 | 4.4 | M | [2, 4] | [3, 4] | [58, 62] | [1, 5] | 0.8285 |

1 | 4.23 | M | 4 | [3, 5] | [25, 34] | [11, 16] | 0.8449 |

11 | 14.88 | K | [3, 4] | [3, 4] | [24, 51] | [1, 17] | 0.8598 |

3 | 8.47 | M | [2, 4] | [3, 5] | [42, 46] | [1, 6] | 0.934 |

5 | 7.29 | M | [2, 4] | [4, 5] | [47, 52] | [1, 7] | 1.0891 |

13 | 4.44 | M | 4 | [3, 6] | [22, 29] | [1, 7] | 2.2784 |

15 | 7.08 | M | [2, 3] | [3, 5] | [25, 32] | [11, 15] | 2.4298 |

8 | 8.8 | M | [2, 3] | [3, 6] | [23, 29] | [1, 7] | 5.7983 |

**Table 20.**Statistics of goodness of fit obtained by partitioning the dataset into 15 clusters with the Burt-adapted fuzzy K-means.

Goodness of Fit | |
---|---|

Deviance | 6026.25 |

AIC | 8064.82 |

BIC | 11,184.27 |

**Table 21.**Burt-adapted spectral clustering on the Wasa dataset preliminarily reduced with the Burt-adapted K-means algorithm. Dominant features and average claim frequencies per cluster.

Cluster | % of Policies | Gender | Zone | Class | Owner Age | Vehicle Age | Frequency (%) |
---|---|---|---|---|---|---|---|

5 | 40.18 | M | 4 | [3, 5] | [42, 53] | [12, 18] | 0.3785 |

4 | 2.28 | M | [3, 4] | 1 | [60, 66] | [1, 45] | 0.4424 |

7 | 4.04 | K | [2, 4] | [3, 5] | [23, 33] | [13, 17] | 0.6854 |

9 | 6.04 | K | [3, 4] | [3, 4] | [42, 50] | [1, 16] | 0.704 |

2 | 1.78 | K | [3, 4] | [3, 6] | [23, 54] | [6, 11] | 1.1766 |

14 | 10.34 | M | [1, 3] | [3, 5] | [44, 52] | [1, 5] | 1.2747 |

1 | 5.43 | M | [1, 4] | 3 | [24, 60] | [1, 5] | 1.4537 |

6 | 9.56 | M | [2, 3] | [4, 6] | [25, 55] | [12, 16] | 1.539 |

12 | 4.89 | M | 4 | [4, 5] | [21, 58] | [2, 11] | 1.5788 |

3 | 4.71 | M | [2, 4] | 6 | [28, 57] | [7, 11] | 1.7313 |

10 | 1.64 | K | [1, 4] | 3 | [24, 54] | [1, 5] | 2.1993 |

15 | 1.56 | M | [2, 3] | [4, 5] | [28, 58] | [2, 3] | 2.6608 |

13 | 1.66 | M | [2, 4] | [2, 4] | [23, 57] | 1 | 3.1749 |

11 | 1.49 | M | 1 | [3, 4] | [28, 33] | [1, 5] | 6.5044 |

8 | 4.4 | M | [2, 3] | [4, 6] | [24, 27] | [4, 11] | 6.6864 |

**Table 22.**Statistic of the goodness of fit obtained by partitioning the dataset into 15 clusters with the Burt-adapted spectral algorithm.

Goodness of Fit | |
---|---|

Deviance | 6069.01 |

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**MDPI and ACS Style**

Jamotton, C.; Hainaut, D.; Hames, T.
Insurance Analytics with Clustering Techniques. *Risks* **2024**, *12*, 141.
https://doi.org/10.3390/risks12090141

**AMA Style**

Jamotton C, Hainaut D, Hames T.
Insurance Analytics with Clustering Techniques. *Risks*. 2024; 12(9):141.
https://doi.org/10.3390/risks12090141

**Chicago/Turabian Style**

Jamotton, Charlotte, Donatien Hainaut, and Thomas Hames.
2024. "Insurance Analytics with Clustering Techniques" *Risks* 12, no. 9: 141.
https://doi.org/10.3390/risks12090141