# Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. General Prototype-Based Clustering and Its Convergence

Algorithm 1: Prototype-based partitional clustering algorithm. |

Input: Dataset and the number of clusters K.Output: Partition of dataset into K disjoint groups.Select K points as the initial prototypes; repeat$\phantom{(}$1. Assign individual observation to the closest prototype; 2. Recompute the prototype with the assigned observations; until the partition does not change;$\phantom{(}$ |

**Proposition**

**1.**

**Proof.**

Algorithm 2: General K-means++-type initialization. |

Input: Dataset ${\left\{{\mathbf{x}}_{i}\right\}}_{i=1}^{N}$ and the number of clusters K.Output: Initial prototypes ${\left\{{\mathbf{b}}_{k}\right\}}_{k=1}^{K}$.1. Select ${\mathbf{b}}_{1}={\mathbf{x}}_{i}$ uniformly randomly, $i=1,\dots ,N$; for $k=2$, $k=k+1$, $k\le K$ do2. Select ${\mathbf{b}}_{k}={\mathbf{x}}_{i}$ with probability $\frac{{min}_{j=1,\dots ,k-1}{\parallel {\mathbf{x}}_{i}-{\mathbf{b}}_{j}\parallel}_{p}^{q}}{\mathcal{J}\left({\left\{{\mathbf{b}}_{j}\right\}}_{j=1}^{k-1}\right)}$, $i=1,\dots ,N$; end$\phantom{(}$ |

#### 2.2. Cluster Validation Indices

#### 2.3. On Computational Complexity

#### 2.4. About Earlier Validation Index Comparisons

## 3. Experimental Setup

## 4. Results

#### 4.1. CVIs for Synthetic Datasets

#### 4.2. CVIs for Real Datasets

#### 4.3. Convergence

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

cb, se, ec | KCE | WB | CH | DB | PBM | RT | WG |
---|---|---|---|---|---|---|---|

Unbalance | 5, 20, 5 | 5, 20, 5 | 5, 20, 5 | 8, 8, 8 | 5, 25, 5 | 8, 2, 8 | 8, 8, 8 |

a1 | 2, 20, 2 | 2, 20, 2 | 2, 20, 2 | 20, 19, 16 | 6, 24, 6 | 2, 18, 17 | 20, 20, 20 |

Sim5D10 | 3, 5, 3 | 3, 5, 3 | 3, 3, 3 | 3, 3, 3 | 5, 10, 5 | 3, 3, 3 | 3, 3, 3 |

Sim5D2 | 3, 5, 3 | 3, 5, 3 | 3, 3, 3 | 3, 3, 3 | 5, 16, 5 | 3, 3, 3 | 3, 3, 3 |

S1 | 2, 15, 2 | 15, 15, 15 | 2, 15, 2 | 15, 15, 15 | 15, 15, 15 | 15, 15, 15 | 15, 15, 15 |

S2 | 2, 15, 2 | 2, 15, 3 | 2, 15, 2 | 15, 15, 15 | 15, 15, 15 | 15, 15, 15 | 15, 15, 15 |

S3 | 2,15, 2 | 2, 15, 2 | 2, 15, 2 | 7, 13, 15 | 4, 15, 4 | 4, 4, 15 | 15, 15, 15 |

S4 | 2, 15, 2 | 2, 15, 3 | 2, 15, 2 | 17, 17, 15 | 5, 23, 5 | 17, 13, 15 | 16, 15, 16 |

DIM032 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

DIM1024 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 | 16, 16, 16 |

b2-sub-2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 20, 2 | 2, 2, 2 | 2, 2, 2 |

b2-sub-3 | 3, 3, 3 | 3, 3, 3 | 3, 3, 3 | 3, 3, 3 | 3, 3, 3 | 3, 3, 3 | 3, 3, 3 |

b2-sub-4 | 4, 4, 4 | 4, 4, 4 | 4, 4, 4 | 4, 4, 4 | 4, 4, 4 | 4, 4, 4 | 4, 4, 4 |

b2-sub-5 | 5, 5, 5 | 5, 5, 5 | 5, 5, 2 | 5, 5, 5 | 5, 5, 5 | 5, 5, 5 | 5, 5, 5 |

b2-sub-6 | 6, 6, 6 | 6, 6, 6 | 2, 6 ,2 | 5, 6, 5 | 6, 6, 6 | 5, 6, 6 | 6, 6, 6 |

b2-sub-7 | 7, 7, 7 | 7, 7, 7 | 2, 7, 2 | 5, 6, 6 | 7, 14, 7 | 2, 2, 2 | 7, 7, 7 |

b2-sub-8 | 8, 8, 8 | 8, 8, 8 | 2, 8, 2 | 6, 6, 7 | 8 ,17, 8 | 2, 2, 2 | 8, 8, 8 |

b2-sub-9 | 9, 19, 9 | 9, 19, 9 | 2, 9, 2 | 6, 7, 8 | 9, 21, 9 | 2, 7, 7 | 9, 9, 9 |

b2-sub-10 | 10, 21, 10 | 10, 21, 10 | 2, 21, 2 | 8, 8, 9 | 10, 25, 10 | 8, 8, 8 | 10, 10, 10 |

b2-sub-11 | 11, 23, 11 | 11, 23, 11 | 2, 23, 3 | 9, 9, 10 | 11, 24, 11 | 9, 9, 8 | 11, 11, 11 |

b2-sub-12 | 12, 25, 12 | 12, 25, 12 | 2, 25, 3 | 10, 10, 11 | 12, 25, 12 | 10, 10, 9 | 12, 12, 12 |

b2-sub-13 | 13, 13, 13 | 13, 24, 13 | 2, 13, 2 | 11, 11, 12 | 13, 24, 13 | 11, 11, 11 | 13, 13, 13 |

b2-sub-14 | 14, 14, 14 | 14, 14, 14 | 2, 14, 2 | 12, 12, 13 | 14, 14, 14 | 12, 12, 12 | 14, 14, 14 |

b2-sub-15 | 15, 15, 15 | 15, 15, 15 | 2, 15, 2 | 13, 13, 14 | 15, 15, 15 | 13, 13, 13 | 15, 15, 15 |

b2-sub-16 | 16, 16, 16 | 16, 16, 16 | 2, 16, 2 | 14, 14, 15 | 16, 16, 16 | 14, 14, 14 | 16, 16, 16 |

b2-sub-17 | 17, 17, 17 | 17, 17, 17 | 2, 17, 2 | 15, 15, 16 | 17, 17, 17 | 15, 2, 2 | 17, 17, 17 |

b2-sub-18 | 18, 18, 18 | 18, 18, 18 | 2, 18, 2 | 15, 15, 16 | 18, 18, 18 | 2, 2, 2 | 18, 18, 18 |

b2-sub-19 | 19, 19, 19 | 19, 19, 19 | 2, 19, 2 | 16, 16, 17 | 19, 19, 19 | 2, 2, 2 | 19, 19, 19 |

b2-sub-20 | 20, 20, 20 | 20, 20, 20 | 2, 20, 2 | 2, 2, 2 | 20, 21, 20 | 2, 2, 2 | 20, 20, 2 |

G2-1-10 | 2, 25, 2 | 2, 25, 2 | 2, 25, 2 | 2, 2, 2 | 25, 25, 25 | 2, 2, 2 | 2, 2, 2 |

G2-1-100 | 2, 25, 2 | 2, 25, 2 | 2, 25, 2 | 22, 25, 22 | 21, 25, 21 | 3, 3, 3 | 2, 2, 2 |

G2-2-10 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 20, 2 | 2, 2, 2 | 2, 2, 2 |

G2-2-100 | 2, 2, 2 | 2, 22, 2 | 2, 2, 2 | 21, 19, 25 | 8, 23, 2 | 10, 7, 7 | 2, 2, 2 |

G2-4-10 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 |

G2-4-100 | 2, 2, 2 | 2, 3, 2 | 2, 2, 2 | 2, 17, 23 | 2, 6, 2 | 2, 16, 16 | 2, 2, 2 |

G2-8-10 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 |

G2-8-100 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 |

⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ | ⋮ |

G2-1024-10 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 |

G2-1024-100 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 |

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Name | Notation | Intra | Inter | Formula |
---|---|---|---|---|

KCE [30] | KCE | $K\times {\mathcal{J}}_{K}$ | $Intra$ | |

WB-index [22] | WB | $K\times {\mathcal{J}}_{K}$ | $\underset{k=1}{\overset{K}{\Sigma}}}{n}_{k}{\parallel {\mathbf{c}}_{k}-m\parallel}_{p}^{q$ | $\frac{Intra}{Inter}$ |

Calinski–Harabasz [24] | CH | $(K-1)\times {\mathcal{J}}_{K}$ | $(N-K)\times {\displaystyle \underset{k=1}{\overset{K}{\Sigma}}}{n}_{k}{\parallel {\mathbf{c}}_{k}-m\parallel}_{p}^{q}$ | $\frac{Intra}{Inter}$ |

Davies–Bouldin [23] | DB | $\frac{1}{{n}_{k}}}{\mathcal{J}}_{K}^{k}+{\displaystyle \frac{1}{{n}_{{k}^{\prime}}}}{\mathcal{J}}_{K}^{{k}^{\prime}$ | $\parallel {\mathbf{c}}_{k}-{\mathbf{c}}_{{k}^{\prime}}{\parallel}_{p}^{q}$ | $\frac{1}{K}}{\displaystyle \underset{k=1}{\overset{K}{\Sigma}}\underset{k\ne {k}^{\prime}}{max}}\frac{Intra(k,{k}^{\prime})}{Inter(k,{k}^{\prime})$ |

Pakhira, Bandyopadhyay, and Maulik [45] | PBM | $K\times {\mathcal{J}}_{K}$ | $\underset{k\ne {k}^{\prime}}{max}}(\parallel {\mathbf{c}}_{k}-{\mathbf{c}}_{{k}^{\prime}}{\parallel}_{p}^{q})\times {\mathcal{J}}_{1$ | ${\left({\displaystyle \frac{Intra}{Inter}}\right)}^{2}$ |

Ray–Turi [25] | RT | $\frac{1}{N}}\times {\mathcal{J}}_{K$ | $\underset{k\ne {k}^{\prime}}{min}}{\parallel {\mathbf{c}}_{k}-{\mathbf{c}}_{{k}^{\prime}}\parallel}_{p}^{q$ | $\frac{Intra}{Inter}$ |

Wemmert– Gançarski ([46]) | WG | $\parallel {\mathbf{x}}_{i}-{\mathbf{c}}_{k}{\parallel}_{p}^{q}$ | $\underset{k\ne {k}^{\prime}}{min}}{\parallel {\mathbf{x}}_{i}-{\mathbf{c}}_{{k}^{\prime}}\parallel}_{p}^{q$ | $\frac{1}{N}}{\displaystyle \underset{k=1}{\overset{K}{\Sigma}}}max\left(\right)open="("\; close=")">0,\phantom{\rule{4pt}{0ex}}{n}_{k}-{\displaystyle \underset{i\in {I}_{k}}{\Sigma}}{\displaystyle \frac{Intra\left(i\right)}{Inter\left(i\right)}$ |

Data | Size | Dimensions | Clusters | Description |

S | 5000 | 2 | 15 | Varying overlap |

G2 | 2048 | 1–1024 | 2 | Varying overlap and dimensionality |

DIM | 1024 | 32–1024 | 16 | Varying dimensionality |

A | 3000–7500 | 2 | 20–50 | Varying number of clusters |

Unbalance | 6500 | 2 | 8 | Both dense and sparse clusters |

Birch | 100,000 | 2 | 1–100 | Varying structure |

Sim5 | 2970 | 2–10 | 5 | Small subclusters close to bigger ones |

Data | Size | Dimensions | Classes | Description |

Iris | 150 | 4 | 3 | Three species of iris |

Arrhythmia | 452 | 279 | 13 | Different types of cardiac arrhythmia |

Steel Plates | 1941 | 27 | 7 | Steel plates faults |

Ionosphere | 351 | 34 | 2 | Radar returns from the ionosphere |

USPS | 9298 | 256 | 10 | Numeric data from scanned handwritten digits |

Satimage (Train) | 6435 | 36 | 6 | Satellite images |

**Table 3.**Right suggestions for the 56 synthetic datasets (number of right suggestions/number of datasets).

Index | City-Block | Squared Euclidean | Euclidean |
---|---|---|---|

KCE | 85.7% | 87.5% | 85.7% |

WB | 87.5% | 80.4% | 87.5% |

CH | 58.9% | 85.7% | 57.1% |

DB | 60.7% | 60.7% | 62.5% |

PBM | 87.5% | 64.3% | 89.3% |

RT | 60.7% | 58.9% | 64.3% |

WG | 94.6% | 96.4% | 92.9% |

cb, se, ec | KCE | WB | CH | DB | PBM | RT | WG |
---|---|---|---|---|---|---|---|

Iris | 2, 3, 2 | 2, 3, 2 | 2, 3, 2 | 2, 2, 2 | 3, 22, 3 | 2, 2, 2 | 2, 2, 2 |

Arrhythmia | 2, 2, 2 | 2, 5, 2 | 2, 2, 2 | 25, 24, 17 | 2, 14, 2 | 2, 25, 3 | 2, 25, 25 |

Steel | 2, 2, 2 | 3, 5, 2 | 2, 2, 2 | 7, 3, 7 | 3, 7, 2 | 2, 2, 3 | 3, 2, 3 |

Ionosphere | 2, 2, 2 | 2, 2, 2 | 2, 2, 2 | 11, 23, 2 | 3, 20, 3 | 4, 4, 4 | 4, 2, 2 |

USPS | 2, 2, 2 | 2, 4, 2 | 2, 2, 2 | 2, 18, 11 | 2, 4, 4 | 2, 12, 7 | 2, 2, 7 |

Satimage (Train) | 2, 3, 2 | 3, 6, 2 | 2, 3, 2 | 3, 3, 3 | 3, 6, 3 | 3, 3, 3 | 3, 3, 3 |

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**MDPI and ACS Style**

Hämäläinen, J.; Jauhiainen, S.; Kärkkäinen, T.
Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering. *Algorithms* **2017**, *10*, 105.
https://doi.org/10.3390/a10030105

**AMA Style**

Hämäläinen J, Jauhiainen S, Kärkkäinen T.
Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering. *Algorithms*. 2017; 10(3):105.
https://doi.org/10.3390/a10030105

**Chicago/Turabian Style**

Hämäläinen, Joonas, Susanne Jauhiainen, and Tommi Kärkkäinen.
2017. "Comparison of Internal Clustering Validation Indices for Prototype-Based Clustering" *Algorithms* 10, no. 3: 105.
https://doi.org/10.3390/a10030105