# Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series

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## Abstract

**:**

## 1. Introduction

## 2. The SEPD-Based Clustering Approach

## 3. Application to Financial Time Series

#### 3.1. FTSE100 Stocks

#### 3.2. S&P500 Stocks: Industrial Sector

## 4. Portfolio Analysis

#### 4.1. FTSE100

#### 4.2. S&P500 Industrials

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. List of Stocks

ID | Name | Symbol | Sector |
---|---|---|---|

1 | 3i | III | Financial Services |

2 | Admiral Group | ADM | Nonlife Insurance |

3 | Anglo American plc | AAL | Mining |

5 | Ashtead Group | AHT | Support Services |

7 | AstraZeneca | AZN | Pharmaceuticals and Biotechnology |

8 | Auto Trader Group | AUTO | Media |

12 | B&M | BME | Retailers |

13 | BAE Systems | BA. | Aerospace and Defence |

17 | BHP | BHP | Mining |

25 | Compass Group | CPG | Support Services |

26 | CRH plc | CRH | Construction and Materials |

29 | Diageo | DGE | Beverages |

31 | Evraz | EVR | Industrial Metals and Mining |

33 | Ferguson plc | FERG | Support Services |

36 | GlaxoSmithKline | GSK | Pharmaceuticals and Biotechnology |

42 | IHG Hotels & Resorts | IHG | Travel and Leisure |

46 | International Airlines Group | IAG | Travel and Leisure |

56 | M&G | MNG | Asset Managers |

57 | Melrose Industries | MRO | Automobiles and Parts |

60 | NatWest Group | NWG | Banks |

68 | Prudential plc | PRU | Life Insurance |

74 | Rio Tinto | RIO | Mining |

83 | Severn Trent | SVT | Gas, Water, and Multi-utilities |

94 | Tesco | TSCO | Food & Drug Retailers |

97 | Vodafone Group | VOD | Mobile Telecommunications |

100 | WPP plc | WPP | Media |

ID | Symbol | Name |
---|---|---|

1 | MMM | 3M Company |

2 | AOS | A.O. Smith Corp |

16 | ALK | Alaska Air Group |

21 | ALLE | Allegion |

30 | AAL | American Airlines Group |

38 | AME | Ametek |

70 | BA | Boeing Company |

79 | CHRW | C. H. Robinson Worldwide |

88 | CARR | Carrier Global |

90 | CAT | Caterpillar Inc. |

107 | CTAS | Cintas Corporation |

123 | CPRT | Copart Inc |

128 | CSX | CSX Corp. |

129 | CMI | Cummins Inc. |

135 | DE | Deere and Co. |

136 | DAL | Delta Air Lines Inc. |

150 | DOV | Dover Corporation |

158 | ETN | Eaton Corporation |

164 | EMR | Emerson Electric Company |

168 | EFX | Equifax Inc. |

179 | EXPD | Expeditors |

184 | FAST | Fastenal Co |

186 | FDX | FedEx Corporation |

197 | FTV | Fortive Corp |

198 | FBHS | Fortune Brands Home & Security |

206 | GNRC | Generac Holdings |

207 | GD | General Dynamics |

208 | GE | General Electric |

216 | GWW | Grainger (W.W.) Inc. |

230 | HON | Honeywell Int’l Inc. |

233 | HWM | Howmet Aerospace |

237 | HII | Huntington Ingalls Industries |

238 | IEX | IDEX Corporation |

240 | INFO | IHS Markit |

241 | ITW | Illinois Tool Works |

244 | IR | Ingersoll-Rand |

257 | JBHT | J. B. Hunt Transport Services |

259 | J | Jacobs Engineering Group |

262 | JCI | Johnson Controls International |

265 | KSU | Kansas City Southern |

276 | LHX | L3Harris Technologies |

282 | LDOS | Leidos Holdings |

289 | LMT | Lockheed Martin Corp. |

301 | MAS | Masco Corp. |

333 | NLSN | Nielsen Holdings |

336 | NSC | Norfolk Southern Corp. |

338 | NOC | Northrop Grumman |

349 | ODFL | Old Dominion Freight Line |

353 | OTIS | Otis Worldwide |

354 | PCAR | Paccar |

356 | PH | Parker-Hannifin |

361 | PNR | Pentair plc |

387 | PWR | Quanta Services Inc. |

391 | RTX | Raytheon Technologies |

396 | RSG | Republic Services Inc. |

398 | RHI | Robert Half International |

399 | ROK | Rockwell Automation Inc. |

400 | ROL | Rollins Inc. |

401 | ROP | Roper Technologies |

415 | SNA | Snap-on |

417 | LUV | Southwest Airlines |

418 | SWK | Stanley Black and Decker |

433 | TDY | Teledyne Technologies |

438 | TXT | Textron Inc. |

449 | TT | Trane Technologies plc |

450 | TDG | TransDigm Group |

461 | UNP | Union Pacific Corp. |

462 | UAL | United Airlines Holdings |

463 | UPS | United Parcel Service |

464 | URI | United Rentals Inc. |

471 | VRSK | Verisk Analytics |

483 | WM | Waste Management Inc. |

491 | WAB | Westinghouse Air Brake Technologies Corp. |

500 | XYL | Xylem Inc. |

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**Figure 3.**Sample of stock returns time series included in the dastaset under consideration (FTSE100 data).

**Figure 4.**Empirical densities of the stocks shown in Figure 3.

**Figure 5.**Silhouette width criterion for different number of clusters C (distribution-based clustering)—experiment with FTSE100 stocks.

**Figure 6.**Silhouette width criterion for different number of clusters (correlation based clustering)—experiment with the FTSE100 stocks.

**Figure 7.**Sample of stock returns time series included in the dataset under consideration (S&P500 data).

**Figure 8.**Empirical densities of the stocks shown in Figure 7.

**Figure 9.**Silhouette width criterion for different number of clusters C (distribution-based clustering)—S&P500 stocks.

**Table 1.**Descriptive statistics and normality test of Jarque-Bera [51] for the FTSE100 stocks.

Mean | St. Dev. | Skewness | Kurtosis | JB Test | Length | |
---|---|---|---|---|---|---|

AAL | 0.0002 | 0.0326 | 0.4403 | 13.4706 | 19,141.2841 ${}^{***}$ | 2516 |

ADM | 0.0003 | 0.0161 | −0.5162 | 5.4498 | 3233.0866 ${}^{***}$ | 2516 |

AHT | −0.0010 | 0.0508 | 6.8974 | 177.1057 | 3,313,541.0732 ${}^{***}$ | 2516 |

AUTO | −0.0002 | 0.0488 | −0.7128 | 20.1631 | 42,911.4151 ${}^{***}$ | 2516 |

AZN | 0.0005 | 0.0151 | −0.4973 | 13.6616 | 19,707.8758 ${}^{***}$ | 2516 |

BHP | 0.0000 | 0.0208 | −0.3470 | 5.9551 | 3777.1783 ${}^{***}$ | 2516 |

BME | 0.0006 | 0.0143 | 0.1503 | 10.5756 | 11,758.3256 ${}^{***}$ | 2516 |

CPG | −0.0011 | 0.0344 | −1.7316 | 41.2409 | 179,865.2429 ${}^{***}$ | 2516 |

CRH | 0.0004 | 0.0204 | −0.7531 | 9.0937 | 8925.6488 ${}^{***}$ | 2516 |

DGE | 0.0003 | 0.0117 | −0.7707 | 8.7590 | 8309.3523 ${}^{***}$ | 2516 |

EVR | 0.0005 | 0.0229 | −0.1605 | 10.2988 | 11,152.9051 ${}^{***}$ | 2516 |

FERG | 0.0006 | 0.0189 | 0.0795 | 64.6506 | 438,904.5890 ${}^{***}$ | 2516 |

GSK | 0.0002 | 0.0122 | −0.6519 | 8.6100 | 7966.7005 ${}^{***}$ | 2516 |

IAG | −0.0006 | 0.0383 | −0.0550 | 2.7651 | 805.4615 ${}^{***}$ | 2516 |

IHG | 0.0005 | 0.0192 | −0.6035 | 15.9591 | 26,903.8564 ${}^{***}$ | 2516 |

III | 0.0002 | 0.0314 | −0.9284 | 20.7329 | 45,506.6850 ${}^{***}$ | 2516 |

MNG | −0.0003 | 0.0195 | −0.3442 | 9.4997 | 9530.2226 ${}^{***}$ | 2516 |

MRO | −0.0004 | 0.0319 | −2.7604 | 64.0144 | 433,505.1309 ${}^{***}$ | 2516 |

NWG | −0.0003 | 0.0272 | −0.9661 | 11.1644 | 13,485.1878 ${}^{***}$ | 2516 |

PRU | 0.0002 | 0.0213 | −0.8713 | 16.1174 | 27,602.6225 ${}^{***}$ | 2516 |

RIO | 0.0002 | 0.0218 | 0.0577 | 3.8344 | 1547.1291 ${}^{***}$ | 2516 |

SVT | 0.0001 | 0.0261 | 0.7359 | 15.7558 | 26,301.1559 ${}^{***}$ | 2516 |

TSCO | 0.0007 | 0.0185 | −0.1470 | 11.6275 | 14,210.8902 ${}^{***}$ | 2516 |

VOD | 0.0000 | 0.0160 | −0.4344 | 11.4206 | 13,780.2067 ${}^{***}$ | 2516 |

WPP | 0.0001 | 0.0194 | −1.7115 | 16.9938 | 31,561.2513 ${}^{***}$ | 2516 |

**Table 2.**MLE parameters estimation from a SEPD and assigned clusters according to the Entropy Weighting K-means–FTSE100 data.

Location | Scale | Shape | Skewness | Cluster | |
---|---|---|---|---|---|

AHT | 0.000938 | 0.038008 | 0.634380 | 1.020410 | 1 |

FERG | 0.000396 | 0.003676 | 0.737628 | 1.087450 | 1 |

III | 0.000215 | 0.030700 | 0.802500 | 1.005290 | 1 |

SVT | 0.005273 | 0.024976 | 0.571017 | 1.206776 | 1 |

AAL | 0.000173 | 0.031175 | 0.970264 | 0.978047 | 2 |

ADM | 0.000326 | 0.015780 | 0.980135 | 0.923800 | 2 |

AUTO | −0.007066 | 0.043761 | 0.840452 | 0.885689 | 2 |

AZN | 0.000486 | 0.014322 | 1.005312 | 0.994350 | 2 |

BHP | 0.000069 | 0.020380 | 1.131097 | 0.928945 | 2 |

BME | 0.000587 | 0.013721 | 0.980685 | 1.002382 | 2 |

CPG | −0.001029 | 0.032234 | 0.879535 | 0.976905 | 2 |

CRH | 0.000370 | 0.019841 | 1.089496 | 0.981539 | 2 |

DGE | 0.000287 | 0.011343 | 1.076945 | 0.925662 | 2 |

EVR | 0.000544 | 0.022103 | 1.022613 | 0.964463 | 2 |

GSK | 0.000174 | 0.011886 | 1.067266 | 0.966571 | 2 |

IAG | −0.000660 | 0.038030 | 1.171752 | 1.034922 | 2 |

IHG | 0.000484 | 0.018176 | 0.883195 | 0.953825 | 2 |

MNG | −0.000304 | 0.018965 | 1.066270 | 0.969299 | 2 |

MRO | −0.000229 | 0.029521 | 0.937861 | 0.975776 | 2 |

NWG | −0.000333 | 0.026422 | 1.009514 | 0.946825 | 2 |

PRU | 0.000284 | 0.020122 | 0.912030 | 0.975315 | 2 |

RIO | 0.000231 | 0.021566 | 1.139757 | 0.987474 | 2 |

TSCO | 0.000734 | 0.017782 | 1.042405 | 0.981452 | 2 |

VOD | 0.000242 | 0.015358 | 0.999362 | 1.011196 | 2 |

WPP | 0.000128 | 0.018252 | 0.952139 | 0.951217 | 2 |

Location | Scale | Shape | Skewness | |
---|---|---|---|---|

Cluster 1 | 0.253624 | 0.253463 | 0.245611 | 0.247302 |

Cluster 2 | 0.261146 | 0.260772 | 0.222667 | 0.255415 |

**Table 4.**Differences in the classification between the entropy weighted distribution-based and the correlation-based clustering approaches—FTSE100 data.

SEPD-Based Clustering | Correlation-Based Clustering | |
---|---|---|

AAL | 2 | 1 |

ADM | 2 | 1 |

AHT | 1 | 1 |

AUTO | 2 | 2 |

AZN | 2 | 3 |

BHP | 2 | 4 |

BME | 2 | 3 |

CPG | 2 | 5 |

CRH | 2 | 1 |

DGE | 2 | 3 |

EVR | 2 | 1 |

FERG | 1 | 6 |

GSK | 2 | 3 |

IAG | 2 | 4 |

IHG | 2 | 1 |

III | 1 | 1 |

MNG | 2 | 5 |

MRO | 2 | 5 |

NWG | 2 | 1 |

PRU | 2 | 1 |

RIO | 2 | 4 |

SVT | 1 | 7 |

TSCO | 2 | 3 |

VOD | 2 | 3 |

WPP | 2 | 1 |

**Table 5.**Descriptive statistics and normality test of Jarque-Bera [51] for the S&P500 stocks.

Stock | Mean | St. Dev. | Skewness | Kurtosis | JB Test | Length |
---|---|---|---|---|---|---|

MMM | 0.0002 | 0.0326 | 0.4403 | 13.4706 | 19,141.2841 ${}^{***}$ | 2517 |

AOS | 0.0006 | 0.0242 | −0.8093 | 17.4330 | 32,194.0594 ${}^{***}$ | 2517 |

ALK | 0.0005 | 0.0164 | −0.5533 | 9.9428 | 7494.6041 ${}^{***}$ | 1793 |

ALLE | 0.0006 | 0.0163 | −0.7401 | 16.1544 | 27,639.5014 ${}^{***}$ | 2517 |

AAL | 0.0008 | 0.0169 | −0.0805 | 3.4418 | 1248.2562 ${}^{***}$ | 2517 |

AME | 0.0006 | 0.0227 | −0.6764 | 27.5441 | 79,867.2387 ${}^{***}$ | 2517 |

BA | 0.0058 | 0.0393 | 0.9665 | 7.8917 | 562.3908 ${}^{***}$ | 200 |

CHRW | 0.0004 | 0.0183 | −0.4986 | 5.0412 | 2775.3550 ${}^{***}$ | 2517 |

CARR | 0.0001 | 0.0154 | −1.3744 | 11.4410 | 14,543.0106 ${}^{***}$ | 2517 |

CAT | 0.0004 | 0.0186 | −0.0322 | 8.3402 | 7308.2827 ${}^{***}$ | 2517 |

CTAS | 0.0010 | 0.0159 | −0.3440 | 14.9755 | 23,605.0368 ${}^{***}$ | 2517 |

CPRT | 0.0006 | 0.0183 | 0.1817 | 14.9407 | 23,459.7916 ${}^{***}$ | 2517 |

CSX | 0.0011 | 0.0162 | −0.5401 | 19.4430 | 39,825.4771 ${}^{***}$ | 2517 |

CMI | 0.0005 | 0.0260 | −0.7426 | 14.9902 | 23,833.2396 ${}^{***}$ | 2517 |

DE | 0.0006 | 0.0172 | −0.1867 | 8.2728 | 7204.9810 ${}^{***}$ | 2517 |

DAL | 0.0005 | 0.0173 | −0.6308 | 10.5860 | 11,939.0318 ${}^{***}$ | 2517 |

DOV | 0.0007 | 0.0159 | −1.2111 | 15.0746 | 24,484.0007 ${}^{***}$ | 2517 |

ETN | 0.0003 | 0.0173 | −0.9108 | 17.9497 | 34,187.4551 ${}^{***}$ | 2517 |

EMR | 0.0004 | 0.0181 | 0.0802 | 12.4383 | 16,253.7474 ${}^{***}$ | 2517 |

EFX | 0.0003 | 0.0150 | −0.3772 | 6.7968 | 4913.7515 ${}^{***}$ | 2517 |

EXPD | 0.0006 | 0.0173 | 0.1549 | 7.6491 | 6157.3116 ${}^{***}$ | 2517 |

FAST | 0.0009 | 0.0203 | −0.4309 | 8.8030 | 7638.9534 ${}^{***}$ | 2339 |

FDX | 0.0004 | 0.0182 | −0.6436 | 10.6534 | 12,096.2927 ${}^{***}$ | 2517 |

FTV | 0.0005 | 0.0178 | −0.3642 | 17.6651 | 14,804.7770 ${}^{***}$ | 1133 |

FBHS | 0.0004 | 0.0144 | −0.4477 | 6.3062 | 4262.9559 ${}^{***}$ | 2517 |

GNRC | −0.0001 | 0.0202 | −0.0884 | 8.9715 | 8458.9914 ${}^{***}$ | 2517 |

GD | 0.0012 | 0.0241 | 0.5907 | 10.1009 | 10,864.4486 ${}^{***}$ | 2517 |

GE | 0.0005 | 0.0173 | 0.0860 | 13.3704 | 18,780.3863 ${}^{***}$ | 2517 |

GWW | 0.0007 | 0.0175 | −0.4450 | 6.4335 | 4337.4524 ${}^{***}$ | 2463 |

HON | 0.0006 | 0.0148 | −0.1974 | 11.3107 | 13,454.9980 ${}^{***}$ | 2517 |

HWM | −0.0001 | 0.0264 | −0.4662 | 11.5068 | 13,999.6075 ${}^{***}$ | 2517 |

HII | 0.0007 | 0.0150 | −0.5132 | 6.5607 | 4633.3047 ${}^{***}$ | 2517 |

IEX | 0.0007 | 0.0160 | 1.9067 | 46.0650 | 146,908.9322 ${}^{***}$ | 1647 |

INFO | 0.0008 | 0.0251 | −0.3388 | 6.9870 | 1892.4783 ${}^{***}$ | 917 |

ITW | 0.0006 | 0.0150 | −0.1942 | 11.5515 | 14,032.6104 ${}^{***}$ | 2517 |

IR | 0.0003 | 0.0180 | −0.2123 | 5.1921 | 2852.1589 ${}^{***}$ | 2517 |

JBHT | 0.0005 | 0.0158 | −0.2285 | 8.1095 | 6931.2264 ${}^{***}$ | 2517 |

J | 0.0004 | 0.0158 | −0.4979 | 7.3329 | 5753.7195 ${}^{***}$ | 2517 |

JCI | 0.0006 | 0.0196 | −0.5279 | 12.0699 | 15,419.7615 ${}^{***}$ | 2517 |

KSU | 0.0006 | 0.0179 | −1.2609 | 19.1160 | 39,046.5207 ${}^{***}$ | 2517 |

LHX | 0.0007 | 0.0161 | −0.3588 | 10.6848 | 12,046.8015 ${}^{***}$ | 2517 |

LDOS | 0.0008 | 0.0134 | −0.6141 | 14.4475 | 22,082.3240 ${}^{***}$ | 2517 |

LMT | 0.0005 | 0.0208 | −0.4192 | 7.3590 | 5763.8034 ${}^{***}$ | 2517 |

MAS | 0.0007 | 0.0207 | −0.2718 | 5.2009 | 2873.8198 ${}^{***}$ | 2517 |

NLSN | 0.0004 | 0.0137 | −0.8574 | 11.6144 | 14,478.3288 ${}^{***}$ | 2517 |

NSC | 0.0001 | 0.0198 | −1.6786 | 27.7429 | 81,459.1085 ${}^{***}$ | 2500 |

NOC | 0.0007 | 0.0143 | −0.1255 | 8.0477 | 6811.0182 ${}^{***}$ | 2517 |

ODFL | 0.0006 | 0.0175 | −0.1828 | 10.7789 | 12,218.9127 ${}^{***}$ | 2517 |

OTIS | 0.0010 | 0.0181 | −0.0697 | 4.9432 | 2570.1778 ${}^{***}$ | 2517 |

PCAR | 0.0021 | 0.0255 | 0.3171 | 5.3143 | 245.0319 ${}^{***}$ | 200 |

PH | 0.0003 | 0.0167 | −0.1098 | 5.1097 | 2749.0581 ${}^{***}$ | 2517 |

PNR | 0.0005 | 0.0195 | −0.5206 | 12.0102 | 15,265.5206 ${}^{***}$ | 2517 |

PWR | 0.0004 | 0.0181 | −0.5505 | 13.2218 | 18,489.7124 ${}^{***}$ | 2517 |

RTX | 0.0005 | 0.0207 | −1.9917 | 34.1081 | 123,835.8960 ${}^{***}$ | 2517 |

RSG | 0.0004 | 0.0196 | −0.1265 | 9.3393 | 9169.7468 ${}^{***}$ | 2517 |

RHI | 0.0006 | 0.0189 | −0.2785 | 11.0507 | 12,860.5887 ${}^{***}$ | 2517 |

ROK | 0.0008 | 0.0153 | −0.4364 | 8.5542 | 7767.5581 ${}^{***}$ | 2517 |

ROL | 0.0007 | 0.0148 | −0.5568 | 8.8381 | 8336.3830 ${}^{***}$ | 2517 |

ROP | 0.0006 | 0.0121 | −1.7053 | 22.4745 | 54,267.9783 ${}^{***}$ | 2517 |

SNA | 0.0002 | 0.0164 | −0.4683 | 16.5103 | 28,722.5064 ${}^{***}$ | 2517 |

LUV | 0.0005 | 0.0167 | −0.1462 | 7.3703 | 5716.4328 ${}^{***}$ | 2517 |

SWK | 0.0005 | 0.0198 | −0.8209 | 22.7150 | 54,471.1200 ${}^{***}$ | 2517 |

TDY | 0.0011 | 0.0205 | −0.6358 | 26.5648 | 74,280.1975 ${}^{***}$ | 2517 |

TXT | 0.0009 | 0.0173 | −1.4434 | 25.9249 | 71,458.1550 ${}^{***}$ | 2517 |

TT | 0.0007 | 0.0175 | −0.4868 | 6.2835 | 4248.3077 ${}^{***}$ | 2517 |

TDG | 0.0003 | 0.0223 | −0.3101 | 10.5591 | 11,752.7020 ${}^{***}$ | 2517 |

UNP | 0.0002 | 0.0307 | −0.7200 | 16.6715 | 29,409.5251 ${}^{***}$ | 2517 |

UAL | 0.0007 | 0.0161 | −0.4693 | 8.5580 | 7786.9958 ${}^{***}$ | 2517 |

UPS | 0.0005 | 0.0137 | 0.0987 | 12.3985 | 16,151.3421 ${}^{***}$ | 2517 |

URI | 0.0009 | 0.0296 | −0.5009 | 6.2084 | 4155.5368 ${}^{***}$ | 2517 |

VRSK | 0.0007 | 0.0137 | −0.1206 | 13.4321 | 18,957.1646 ${}^{***}$ | 2517 |

WM | 0.0004 | 0.0200 | −0.5989 | 9.3161 | 9268.1082 ${}^{***}$ | 2517 |

WAB | 0.0006 | 0.0120 | −0.6692 | 14.2371 | 21,478.1235 ${}^{***}$ | 2517 |

XYL | 0.0007 | 0.0166 | −0.1198 | 8.1649 | 6462.2884 ${}^{***}$ | 2320 |

**Table 6.**MLE estimates of a skewed exponential power distribution and the entropy weighting clustering results—S&P500 data.

Location $\mathit{\mu}$ | Scale $\mathit{\sigma}$ | Shape p | Skewness $\mathit{\lambda}$ | Cluster | |
---|---|---|---|---|---|

BA | 0.000529 | 0.020121 | 0.809235 | 0.961483 | 1 |

CARR | 0.005634 | 0.037575 | 0.869097 | 1.022203 | 1 |

CTAS | 0.001057 | 0.014841 | 0.828099 | 0.972663 | 1 |

EFX | 0.000743 | 0.015000 | 0.910557 | 0.976399 | 1 |

FTV | 0.000516 | 0.016727 | 0.880068 | 1.010577 | 1 |

GE | −0.000323 | 0.019545 | 0.819186 | 0.987702 | 1 |

GNRC | 0.001210 | 0.023105 | 0.909975 | 1.013188 | 1 |

HON | 0.000665 | 0.014166 | 0.898461 | 0.999329 | 1 |

INFO | 0.000744 | 0.014293 | 0.862185 | 1.000896 | 1 |

LDOS | 0.000719 | 0.016602 | 0.907270 | 0.955210 | 1 |

MMM | 0.000428 | 0.013126 | 0.894845 | 0.965327 | 1 |

NLSN | 0.000161 | 0.018310 | 0.886294 | 1.006444 | 1 |

OTIS | 0.003455 | 0.024927 | 0.768527 | 1.186048 | 1 |

RTX | 0.000246 | 0.015165 | 0.830205 | 0.979384 | 1 |

SWK | −0.000646 | 0.018981 | 0.798267 | 0.880760 | 1 |

TDG | 0.001156 | 0.018331 | 0.820059 | 0.992412 | 1 |

TXT | 0.000295 | 0.021221 | 0.865774 | 1.001103 | 1 |

UAL | 0.000242 | 0.028890 | 0.904769 | 1.006104 | 1 |

VRSK | 0.000705 | 0.012904 | 0.881891 | 0.973572 | 1 |

WM | 0.000407 | 0.011229 | 0.862004 | 0.943475 | 1 |

AAL | 0.000173 | 0.031175 | 0.970261 | 0.978034 | 2 |

ALK | 0.000561 | 0.022819 | 0.941247 | 0.978997 | 2 |

ALLE | 0.000522 | 0.015682 | 0.980738 | 0.931036 | 2 |

AME | 0.000644 | 0.015499 | 0.929582 | 0.973628 | 2 |

AOS | 0.000757 | 0.016787 | 1.040460 | 0.949007 | 2 |

CAT | 0.000356 | 0.018032 | 1.043622 | 0.999351 | 2 |

CHRW | 0.000153 | 0.014709 | 0.959506 | 0.930355 | 2 |

CMI | 0.000387 | 0.018176 | 0.977557 | 0.988523 | 2 |

CPRT | 0.001047 | 0.014979 | 0.927105 | 0.992372 | 2 |

CSX | 0.000646 | 0.017462 | 1.043010 | 0.955829 | 2 |

DAL | 0.000517 | 0.024814 | 0.952124 | 0.980727 | 2 |

DE | 0.000556 | 0.016703 | 0.951225 | 0.983644 | 2 |

DOV | 0.000546 | 0.016845 | 0.988254 | 0.968131 | 2 |

EMR | 0.000267 | 0.016463 | 0.930394 | 0.967078 | 2 |

ETN | 0.000450 | 0.017423 | 0.995600 | 0.974408 | 2 |

EXPD | 0.000267 | 0.014679 | 1.011972 | 0.957943 | 2 |

FAST | 0.000560 | 0.016874 | 1.017518 | 0.999670 | 2 |

FBHS | 0.000868 | 0.019631 | 0.965633 | 0.978979 | 2 |

FDX | 0.000456 | 0.017386 | 0.963798 | 0.981288 | 2 |

GD | 0.000396 | 0.014028 | 1.047922 | 0.940179 | 2 |

GWW | 0.000490 | 0.016469 | 0.960258 | 0.995514 | 2 |

HII | 0.000664 | 0.017031 | 1.015763 | 0.935556 | 2 |

HWM | −0.000064 | 0.025139 | 0.909516 | 0.939999 | 2 |

IEX | 0.000699 | 0.014599 | 1.031668 | 0.962203 | 2 |

IR | 0.000843 | 0.024407 | 1.015252 | 0.942748 | 2 |

ITW | 0.000639 | 0.014417 | 0.922543 | 0.969491 | 2 |

J | 0.000343 | 0.017644 | 1.045491 | 0.976824 | 2 |

JBHT | 0.000498 | 0.015420 | 1.081486 | 1.014217 | 2 |

JCI | 0.000431 | 0.015401 | 1.046638 | 0.968509 | 2 |

KSU | 0.000603 | 0.018785 | 1.023660 | 0.994975 | 2 |

LHX | 0.000620 | 0.015306 | 0.974821 | 0.932664 | 2 |

LMT | 0.000783 | 0.012718 | 0.962104 | 0.975880 | 2 |

LUV | 0.000532 | 0.020178 | 0.986733 | 0.964936 | 2 |

MAS | 0.000680 | 0.020248 | 1.019500 | 0.984060 | 2 |

NOC | 0.000740 | 0.013834 | 1.019146 | 0.947251 | 2 |

NSC | 0.000618 | 0.016801 | 1.014521 | 0.964525 | 2 |

ODFL | 0.001056 | 0.017740 | 1.116417 | 0.950527 | 2 |

PCAR | 0.000295 | 0.016396 | 1.074364 | 0.986350 | 2 |

PH | 0.000523 | 0.018614 | 0.968202 | 0.976833 | 2 |

PNR | 0.000388 | 0.017448 | 1.044595 | 0.924451 | 2 |

PWR | 0.000523 | 0.019236 | 0.945752 | 0.953866 | 2 |

RHI | 0.000365 | 0.018797 | 0.971600 | 0.962834 | 2 |

ROK | 0.000568 | 0.018245 | 0.959031 | 0.993951 | 2 |

ROL | 0.000815 | 0.014646 | 0.979196 | 0.991513 | 2 |

ROP | 0.000716 | 0.014215 | 0.935390 | 0.964640 | 2 |

RSG | 0.000593 | 0.011185 | 0.925639 | 0.977828 | 2 |

SNA | 0.000507 | 0.016137 | 0.944876 | 0.962180 | 2 |

TDY | 0.000861 | 0.016373 | 1.002064 | 0.966281 | 2 |

TT | 0.000704 | 0.017061 | 0.988580 | 0.965844 | 2 |

UNP | 0.000661 | 0.015663 | 1.073618 | 0.996020 | 2 |

UPS | 0.000453 | 0.012969 | 0.923857 | 0.967459 | 2 |

URI | 0.000914 | 0.028867 | 1.027572 | 0.951293 | 2 |

WAB | 0.000425 | 0.019372 | 0.976249 | 0.969278 | 2 |

XYL | 0.000673 | 0.016063 | 1.013724 | 0.981268 | 2 |

**Table 7.**Distribution-based Distribution-based Entropy Weighting K-means for S&P500 stocks: resulting weights.

Location (${\mathit{w}}_{1}$) | Scale (${\mathit{w}}_{2}$) | Shape (${\mathit{w}}_{3}$) | Skewness (${\mathit{w}}_{4}$) | |
---|---|---|---|---|

Cluster 1 | 0.255808 | 0.255629 | 0.247364 | 0.241199 |

Cluster 2 | 0.258694 | 0.258503 | 0.229691 | 0.253112 |

**Table 8.**Final group assignment of the two alternative clustering approaches. The first column shows the results of the distribution-based approach, while the second column shows those of the correlation-based clustering (FTSE100 data).

SEPD-Based Clustering | Correlation-Based Clustering | |
---|---|---|

AAL | 2 | 1 |

ADM | 2 | 1 |

AHT | 2 | 1 |

AUTO | 2 | 2 |

AZN | 2 | 1 |

BHP | 2 | 3 |

BME | 2 | 1 |

CPG | 2 | 3 |

CRH | 2 | 1 |

DGE | 1 | 1 |

EVR | 2 | 1 |

FERG | 2 | 4 |

GSK | 1 | 1 |

IAG | 2 | 3 |

IHG | 2 | 1 |

III | 2 | 5 |

MNG | 2 | 3 |

MRO | 2 | 3 |

NWG | 2 | 3 |

PRU | 2 | 1 |

RIO | 2 | 3 |

SVT | 2 | 6 |

TSCO | 2 | 1 |

VOD | 1 | 1 |

WPP | 1 | 1 |

Sharpe Ratio | VaR | ES | Turnover | |
---|---|---|---|---|

Naive (SEPD)—Cluster 1 | 0.262155 | −0.014396 | −0.018755 | |

GMV (SEPD)—Cluster 1 | 0.278905 | −0.012489 | −0.016318 | 0.022555 |

Naive (SEPD)—Cluster 2 | 0.129284 | −0.018028 | −0.023003 | |

GMV (SEPD)—Cluster 2 | 0.123989 | −0.009490 | −0.012100 | 0.013587 |

Naive (corr)—Cluster 1 | 0.216579 | −0.016056 | −0.020760 | |

GMV (corr)—Cluster 1 | 0.290720 | −0.010547 | −0.013808 | 0.018170 |

Naive (corr)—Cluster 3 | 0.020610 | −0.026068 | −0.032775 | |

GMV (corr)—Cluster 3 | 0.054591 | −0.016854 | −0.021284 | 0.026502 |

**Table 10.**Final group assignment of the two alternative clustering approaches. The first column shows the results of the distribution-based approach, while the second column shows those of the correlation-based clustering (S&P500 data).

SEPD-Based Clustering | Correlation-Based Clustering | |
---|---|---|

AAL | 2 | 1 |

ALK | 1 | 1 |

AME | 2 | 2 |

AOS | 2 | 2 |

BA | 1 | 2 |

CAT | 2 | 2 |

CHRW | 2 | 2 |

CMI | 2 | 2 |

CPRT | 2 | 2 |

CSX | 2 | 2 |

CTAS | 2 | 2 |

DAL | 1 | 1 |

DE | 2 | 2 |

DOV | 2 | 2 |

EFX | 2 | 2 |

EMR | 2 | 2 |

ETN | 2 | 2 |

EXPD | 2 | 2 |

FAST | 2 | 2 |

FDX | 2 | 2 |

GD | 1 | 2 |

GE | 2 | 2 |

GNRC | 2 | 2 |

GWW | 2 | 2 |

HON | 2 | 2 |

HWM | 2 | 2 |

IEX | 2 | 2 |

ITW | 2 | 2 |

J | 2 | 2 |

JBHT | 2 | 2 |

JCI | 2 | 2 |

KSU | 2 | 2 |

LDOS | 2 | 2 |

LHX | 2 | 2 |

LMT | 1 | 2 |

LUV | 1 | 1 |

MAS | 2 | 2 |

MMM | 2 | 2 |

NOC | 1 | 2 |

NSC | 2 | 2 |

ODFL | 2 | 2 |

PCAR | 2 | 2 |

PH | 2 | 2 |

PNR | 2 | 2 |

PWR | 2 | 2 |

RHI | 2 | 2 |

ROK | 2 | 2 |

ROL | 2 | 2 |

ROP | 2 | 2 |

RSG | 2 | 2 |

RTX | 2 | 2 |

SNA | 2 | 2 |

SWK | 2 | 2 |

TDG | 2 | 2 |

TDY | 2 | 2 |

TT | 2 | 2 |

TXT | 2 | 2 |

UAL | 1 | 1 |

UNP | 1 | 2 |

UPS | 2 | 2 |

URI | 2 | 2 |

VRSK | 2 | 2 |

WAB | 2 | 2 |

WM | 2 | 2 |

Sharpe Ratio | VaR | ES | Turnover | |
---|---|---|---|---|

Naive (SEPD)—Cluster 1 | 0.201330 | −0.018444 | −0.023791 | |

GMV (SEPD)—Cluster 1 | 0.300182 | −0.014314 | −0.018771 | 0.027739 |

Naive (SEPD)—Cluster 2 | 0.147101 | −0.019142 | −0.024487 | |

GMV (SEPD)—Cluster 2 | 0.098290 | −0.009808 | −0.012460 | 0.020254 |

Naive (corr)—Cluster 1 | 0.131469 | −0.029350 | −0.037461 | |

GMV (corr)—Cluster 1 | 0.196732 | −0.023842 | −0.030730 | 0.037102 |

Naive (corr)—Cluster 2 | 0.155000 | −0.018541 | −0.023746 | |

GMV (corr)—Cluster 2 | 0.097379 | −0.009428 | −0.011975 | 0.026502 |

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**MDPI and ACS Style**

Mattera, R.; Giacalone, M.; Gibert, K. Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series. *Symmetry* **2021**, *13*, 959.
https://doi.org/10.3390/sym13060959

**AMA Style**

Mattera R, Giacalone M, Gibert K. Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series. *Symmetry*. 2021; 13(6):959.
https://doi.org/10.3390/sym13060959

**Chicago/Turabian Style**

Mattera, Raffaele, Massimiliano Giacalone, and Karina Gibert. 2021. "Distribution-Based Entropy Weighting Clustering of Skewed and Heavy Tailed Time Series" *Symmetry* 13, no. 6: 959.
https://doi.org/10.3390/sym13060959