Clustering tasks usually considers distances between objects, but mere distances are not effective in capturing the common patterns or moves of objects and clustering by those patterns. This article is going to show that correlation between objects can be used in clustering to cluster objects by their common patterns. This article is also going to show that this method has its theorecial background – correlation between two objects is equal to the Euclidean distance between row-wise standardized data. By this theoretical background, we can expect that this method will be effective especially in time-series data.
Motivation
I once took “Multiraviate Data Analysis” class in my university. One of the big topics in the class was exploratory factor analysis (EFA). Basically, it is a statistical framework which tries to find (unobserved) factors from the observed correlation matrix. For example, psychologists have been tried to find the factors which decides a person’s characteristics. So they made a survey which asked about peoples’ characteristics, and using the correlation between the answers of the survey, they could be able to divide person’s characteristics into several factors. Learning this EFA framework, I thought of some different thing – if we can find (unobserved) factors from a correlation matrix, why we cannot find so-called clusters from the correlation matrix? But my professor told me that, you should not use the EFA framework in that way, as the purpose of EFA is not clustering.
Interestingly, the next topic I learnt in that class was clster analysis. And surprisingly, in the textbook, there was a question which asks to perform agglomerative hierarchical clustering, using the correlation as a similarity measure. In that point, I wondered – is it really make sense to use correlation as a similarity measure? So I tried to find the reason that it works, but there was few materials related to this. So I tried to find the reason by myself.
(Actually there was some other papers which deals with this, but I found after self-studying it.)
Illustrative Example
Suppose you have the following time series data.
The left is original scale, and the right is log scaled one.
This data could be traffic data by time of day at each junction, or power usage data for each region over a single day, or anything else you can imagine. You may want to cluster those lines by their pattern. That is, if two lines increase and decrease at the same time, then they must be in the same cluster.
The plot on the left side is the original data. Can you capture the common patterns of the lines? It may be hard; but looking at the log-scaled data at the right side, you may notice that there are common patterns for these lines. The pink and red lines increases drastically for the first half and increases moderately for the second half, whereas the blue and skyblue lines increases drastically for the second half. The green line equally increases and decreases periodically. So these lines can be clustered in the three different clusters.
But, how can we compute these clusters? Mere k-means clustering cannot deal with finding patterns. This is because, it computes distances only based on absolute quantities. In the plot above, even though pink and red lines have similar patterns, their quantity differs a lot, and the red line is closer to the blue line regarding the quantities. So the red line will be clustered with the blue line rather than the pink line.
\[d(x_1, x_2) = d(red, pink) = [(1-10)^2 + (2-18)^2 + \cdots + (2-18)^2]^{1/2} = 389.099\] \[d(x_1, x_3) = d(red, blue) = [(1-1)^2 + (2-4)^2 + \cdots + (2-9)^2]^{1/2}=52.583\]There must be many different remedies to compute the clusters properly. One can compute rate of change and then cluster data onto the rate of change. Or we can use cosine similarity as the metric to cluster data. But, here I want to show another way; using correlation as a distance metric.
So, the main idea is like this: if two data have a similar pattern, then they must be positively correlated. It does make sense, as being highly-correlated means that two data are close to the each other, and it is reasonable to group them together into a single cluster. Look at the scatterplots above. We can notice that $x_1$(red) is strongly positively correlated with $x_2$(pink), whereas not that strongly correlated to the others. In fact, it is that
\[\begin{aligned}Corr(x_1, x_2) &= 0.971 \\ Corr(x_1, x_3) &= -0.028 \\ Corr(x_1, x_4) &= -0.019 \\ Corr(x_1, x_5) &= 0.015. \end{aligned}\]So this kind of approach must be helpful in capturing patterns and then clustering the data. To use the correlation as a distance metric in clsutering procedure, let’s use $1 - Corr(x, y)$ as a distance between data $x$ and $y$. This is reasonable, as
- (positivity) $0 \leq 1 - Corr(x, y)$, and $1-Corr(x, y) = 0$ only when $x$ and $y$ are perfectly positively correlated.
- (symmetry) $1 - Corr(x, y) = 1 - Corr(y, x)$.
The plot below is the heatmap of $1-Corr$ for the given data.
After computing $1 - Corr(x, y)$, we can deem this as the distance between $x$ and $y$ and and proceed to appropriate clustering methods. A typical method is agglomerative clustering. The plot at the below is the resulted dendrogram.
Here, we can conclude that 2 and 3 (blue and skyblue) are in the same cluster, and 0 and 1 (red and pink) are in the same cluster. So, in conclusion, the data are clustered with their patterns! The absolute quantities did not affect the result of clustering that much.
This kind of technique (using correlations as distances) is shown in some books. For example, a textbook by Johnson and Wichern (2002) showed an illustrative example of using the sample correlations are used as similarity measures (pp. 747-748). However, I couldn’t find a theoretical background to justify this technique. Is using correlation as a distance appropriate for clustering task?
How This Works
From the previous example, we found that clustering using correlations as a distance metric is somewhat effective in clustering by patterns. But, does it make sense to use correlation as a distance metric? Can we justify our decision?
In fact, the $1-Corr$ itself is not a distance metric. This is because it does not meet the triangle inequality.
Example: Correlation does not meet the triangle inequality
Suppose that
\[\begin{aligned} x_1 &= [4, 9, 2] \\ x_2 &= [2,4,7] \\ x_3 &= [0,1,4] \end{aligned}\]Then, the population correlations are
\[\rho = \begin{bmatrix}1 & & \\ -0.3857 & 1 & \\ -0.5329 & 0.9862 & 1 \end{bmatrix}\]so that the distance matrix is
\[d = \begin{bmatrix}0 & & \\ 1.3857 & 1 & \\ 1.5329 & 0.0138 & 1 \end{bmatrix}\]Here, $d(x_1, x_3) = 1.5329 > d(x_1, x_2) + d(x_2, x_3) = 1.3995$. Thus the triangle inquality does not hold here.
For this fact, we cannot safely use correlation as a distance metric in every cases, as it may lead to the somewhat strange clustering result. However, we can show that correlation can be a distance metric in a different perspective. Actually, $1 - Corr(x, y)$ is a distance between standardized $x$ and standardized $y$.
(1 - Correlation) is a squared distance between row-wise standardized data
Theorem. Suppose we are given sequence of data $x = (x_1, \cdots, x_k)$ and $y = (y_1, \cdots, y_k)$. Suppose these data are standardized ones, so that
\[E(x) = \dfrac{1}{k}\sum_i x_i = E(y) = \dfrac{1}{k}\sum_i y_i = 0,\]and
\[Var(x) = \dfrac{1}{k}\sum_i [x_i - E(x)]^2 = Var(y) = \dfrac{1}{k}\sum_i [y_i - E(x)]^2 = 1.\](Note that the variance is population variance, not sample.) Then,
\[||x - y||^2_2 \propto 1 - Corr(x, y),\]where $Corr(x, y)$ is the Pearson correlation between $x$ and $y$.
Proof. As $E(x) = E(y) = 0$ and $Var(x) = Var(y) = 1$, it is also that $E(x^2) = E(y^2) = 1$. Furthermore,
\[\begin{aligned} Corr(x, y) &= \dfrac{Cov(x, y)}{\sqrt{Var(x) Var(y)}}\\ &= Cov(x, y) \\ &=\dfrac{1}{k} \sum_i (x_i - E(x))(y_i - E(y)) \\ &= \dfrac{1}{k}\sum_i x_i y_i \\ &= E(xy). \end{aligned}\]Then,
\[\begin{aligned} ||x - y||^2_2 &= \sum_i (x_i - y_i)^2 \\ &= \sum_i(x_i^2 + y_i^2 - 2x_iy_i) \\ &= kE(x^2) + kE(y^2) - 2kE(xy) \\ &= 2k[1-Corr(x, y)]. \end{aligned}\]Hence $||x - y||^2_2 \propto 1 - Corr(x, y)$.
This theorem implies that, using (1 - correlation) between two objects as a distance is equivalent to use the distance between row-wise standardized data. Hence clustering data using correlation as similarity measure may make sense.
However, we must focus on that it is equivalent to distance between row-wise standardized data; but standardization is usually done on column-wise manner. This is because, generally, there is no point in standardizing observations in different columns. Imagine that, there is data looks like this:
| Age | Income | Number of children |
|---|---|---|
| 48 | 100000 | 2 |
Is standardizing this data by row make sense? It does not, as each column delivers very different features of each object.
However, time-series data is special, as each column is entangled with others. We may be able to predict the outcome at $t_5$ using the numbers in column $t_1, t_2, t_3$, and $t_4$. Therefore it does make sense to use correlation as a similarity measure in the case of time-series data.
Case Study: Clustering S&P 500 Companies
Let’s apply this method to real data. Stock prices data is typical time-series data. I am going to cluster S&P 500 Companies by their stock prices in 2023. The data is from Kaggle.
The raw data
For stock prices data, we usually consider return rate instead of the price itself. I calculated rise & fall rate for every day, and then compute correlation between each stock. Then I calculated the distance matrix, by (1 - correlation). And then, I conducted agglomerative hierarchical clustering for that distance matrix. I used the complete linkage agglomeration method. The plot below is the resulting dendrogram.
How should I decide the number of clusters? The simple way is, cutting the dendrogram at the height = 1. Height = 1 means correlation = 0, and it implies that two data are not correlated. Hence if two clusters are fused after the height 1, then they are actually not correlated.
The dendrogram cut by height 1.
We have got 10 clusters. Let’s look at each cluster.
Cluster 1
Symbols
Symbol Longname Sector
8 BRK-B Berkshire Hathaway Inc. Financial Services
12 JPM JPMorgan Chase & Co. Financial Services
22 HD The Home Depot, Inc. Consumer Cyclical
23 BAC Bank of America Corporation Financial Services
37 TMO Thermo Fisher Scientific Inc. Healthcare
38 WFC Wells Fargo & Company Financial Services
42 DHR Danaher Corporation Healthcare
56 MS Morgan Stanley Financial Services
59 GS The Goldman Sachs Group, Inc. Financial Services
63 UNP Union Pacific Corporation Industrials
66 HON Honeywell International Inc. Industrials
69 LOW Lowe's Companies, Inc. Consumer Cyclical
78 C Citigroup Inc. Financial Services
79 UPS United Parcel Service, Inc. Industrials
87 SCHW The Charles Schwab Corporation Financial Services
90 NKE NIKE, Inc. Consumer Cyclical
99 ADP Automatic Data Processing, Inc. Industrials
103 FI Fiserv, Inc. Technology
112 ICE Intercontinental Exchange, Inc. Financial Services
115 SHW The Sherwin-Williams Company Basic Materials
120 APH Amphenol Corporation Technology
121 CTAS Cintas Corporation Industrials
124 FDX FedEx Corporation Industrials
130 CMG Chipotle Mexican Grill, Inc. Consumer Cyclical
131 ITW Illinois Tool Works Inc. Industrials
134 PH Parker-Hannifin Corporation Industrials
135 USB U.S. Bancorp Financial Services
136 PNC The PNC Financial Services Group, Inc. Financial Services
138 ECL Ecolab Inc. Basic Materials
142 CSX CSX Corporation Industrials
146 MSI Motorola Solutions, Inc. Technology
153 ROP Roper Technologies, Inc. Technology
161 MMM 3M Company Industrials
162 DHI D.R. Horton, Inc. Consumer Cyclical
163 TFC Truist Financial Corporation Financial Services
167 F Ford Motor Company Consumer Cyclical
168 GM General Motors Company Consumer Cyclical
172 MET MetLife, Inc. Financial Services
177 NSC Norfolk Southern Corporation Industrials
182 CPRT Copart, Inc. Industrials
198 LEN Lennar Corporation Consumer Cyclical
199 GWW W.W. Grainger, Inc. Industrials
202 BK The Bank of New York Mellon Corporation Financial Services
203 TEL TE Connectivity Ltd. Technology
212 PRU Prudential Financial, Inc. Financial Services
214 PAYX Paychex, Inc. Industrials
218 ODFL Old Dominion Freight Line, Inc. Industrials
220 IQV IQVIA Holdings Inc. Healthcare
221 AMP Ameriprise Financial, Inc. Financial Services
224 OTIS Otis Worldwide Corporation Industrials
228 FICO Fair Isaac Corporation Technology
229 MSCI MSCI Inc. Financial Services
230 AME AMETEK, Inc. Industrials
231 IR Ingersoll Rand Inc. Industrials
233 FAST Fastenal Company Industrials
236 A Agilent Technologies, Inc. Healthcare
238 GLW Corning Incorporated Technology
243 CTSH Cognizant Technology Solutions Corporation Technology
244 GEHC GE HealthCare Technologies Inc. Healthcare
247 HPQ HP Inc. Technology
252 NDAQ Nasdaq, Inc. Financial Services
255 IT Gartner, Inc. Technology
259 LULU Lululemon Athletica Inc. Consumer Cyclical
261 MLM Martin Marietta Materials, Inc. Basic Materials
262 VMC Vulcan Materials Company Basic Materials
263 XYL Xylem Inc. Industrials
276 ROK Rockwell Automation, Inc. Industrials
281 PPG PPG Industries, Inc. Basic Materials
282 CBRE CBRE Group, Inc. Real Estate
287 DAL Delta Air Lines, Inc. Industrials
288 WAB Westinghouse Air Brake Technologies Corporation Industrials
289 TSCO Tractor Supply Company Consumer Cyclical
292 MTD Mettler-Toledo International Inc. Healthcare
293 FITB Fifth Third Bancorp Financial Services
294 MTB M&T Bank Corporation Financial Services
301 FTV Fortive Corporation Technology
304 NVR NVR, Inc. Consumer Cyclical
307 PHM PulteGroup, Inc. Consumer Cyclical
312 STT State Street Corporation Financial Services
313 DOV Dover Corporation Industrials
315 IFF International Flavors & Fragrances Inc. Basic Materials
322 BR Broadridge Financial Solutions, Inc. Technology
333 RJF Raymond James Financial, Inc. Financial Services
337 DECK Deckers Outdoor Corporation Consumer Cyclical
340 WY Weyerhaeuser Company Real Estate
342 HBAN Huntington Bancshares Incorporated Financial Services
351 CPAY Corpay, Inc. Technology
352 GDDY GoDaddy Inc. Technology
354 RF Regions Financial Corporation Financial Services
359 PFG Principal Financial Group, Inc. Financial Services
365 BLDR Builders FirstSource, Inc. Industrials
366 GPC Genuine Parts Company Consumer Cyclical
367 BBY Best Buy Co., Inc. Consumer Cyclical
368 CINF Cincinnati Financial Corporation Financial Services
372 APTV Aptiv PLC Consumer Cyclical
373 CFG Citizens Financial Group, Inc. Financial Services
374 ULTA Ulta Beauty, Inc. Consumer Cyclical
383 BAX Baxter International Inc. Healthcare
385 WAT Waters Corporation Healthcare
389 VRSN VeriSign, Inc. Technology
393 NTRS Northern Trust Corporation Financial Services
397 EXPD Expeditors International of Washington, Inc. Industrials
401 JBHT J.B. Hunt Transport Services, Inc. Industrials
405 LUV Southwest Airlines Co. Industrials
407 FDS FactSet Research Systems Inc. Financial Services
409 MAS Masco Corporation Industrials
414 UAL United Airlines Holdings, Inc. Industrials
419 IEX IDEX Corporation Industrials
422 KEY KeyCorp Financial Services
424 AKAM Akamai Technologies, Inc. Technology
427 SNA Snap-on Incorporated Industrials
431 VTRS Viatris Inc. Healthcare
437 NDSN Nordson Corporation Industrials
438 SWK Stanley Black & Decker, Inc. Industrials
439 RVTY Revvity, Inc. Healthcare
441 PNR Pentair plc Industrials
443 AOS A. O. Smith Corporation Industrials
446 POOL Pool Corporation Industrials
449 KMX CarMax, Inc. Consumer Cyclical
455 JKHY Jack Henry & Associates, Inc. Technology
458 LKQ LKQ Corporation Consumer Cyclical
460 TECH Bio-Techne Corporation Healthcare
462 BXP BXP, Inc. Real Estate
465 CRL Charles River Laboratories International, Inc. Healthcare
473 CHRW C.H. Robinson Worldwide, Inc. Industrials
475 RL Ralph Lauren Corporation Consumer Cyclical
479 GNRC Generac Holdings Inc. Industrials
480 TPR Tapestry, Inc. Consumer Cyclical
487 HSIC Henry Schein, Inc. Healthcare
489 BIO Bio-Rad Laboratories, Inc. Healthcare
492 BBWI Bath & Body Works, Inc. Consumer Cyclical
494 MHK Mohawk Industries, Inc. Consumer Cyclical
499 BWA BorgWarner Inc. Consumer Cyclical
503 AAL American Airlines Group Inc. Industrials
Cluster 2
Symbols
Symbol Longname Sector
1 AAPL Apple Inc. Technology
2 MSFT Microsoft Corporation Technology
4 GOOG Alphabet Inc. Communication Services
5 GOOGL Alphabet Inc. Communication Services
6 AMZN Amazon.com, Inc. Consumer Cyclical
7 META Meta Platforms, Inc. Communication Services
10 TSLA Tesla, Inc. Consumer Cyclical
28 NFLX Netflix, Inc. Communication Services
30 ADBE Adobe Inc. Technology
31 CRM Salesforce, Inc. Technology
45 ABT Abbott Laboratories Healthcare
48 AXP American Express Company Financial Services
54 BX Blackstone Inc. Financial Services
57 ISRG Intuitive Surgical, Inc. Healthcare
60 NOW ServiceNow, Inc. Technology
61 SPGI S&P Global Inc. Financial Services
72 SYK Stryker Corporation Healthcare
80 BLK BlackRock, Inc. Financial Services
86 BSX Boston Scientific Corporation Healthcare
93 KKR KKR & Co. Inc. Financial Services
98 MDT Medtronic plc Healthcare
117 MCO Moody's Corporation Financial Services
165 COF Capital One Financial Corporation Financial Services
176 EW Edwards Lifesciences Corporation Healthcare
213 DXCM DexCom, Inc. Healthcare
216 RCL Royal Caribbean Cruises Ltd. Consumer Cyclical
257 DFS Discover Financial Services Financial Services
268 EFX Equifax Inc. Industrials
279 CSGP CoStar Group, Inc. Real Estate
296 ANSS ANSYS, Inc. Technology
298 EBAY eBay Inc. Consumer Cyclical
311 TROW T. Rowe Price Group, Inc. Financial Services
318 CCL Carnival Corporation & plc Consumer Cyclical
332 ZBH Zimmer Biomet Holdings, Inc. Healthcare
335 TYL Tyler Technologies, Inc. Technology
336 STE STERIS plc Healthcare
338 LYV Live Nation Entertainment, Inc. Communication Services
363 SYF Synchrony Financial Financial Services
381 HOLX Hologic, Inc. Healthcare
382 COO The Cooper Companies, Inc. Healthcare
384 EXPE Expedia Group, Inc. Consumer Cyclical
410 GEN Gen Digital Inc. Technology
436 PODD Insulet Corporation Healthcare
450 BEN Franklin Resources, Inc. Financial Services
472 TFX Teleflex Incorporated Healthcare
490 NCLH Norwegian Cruise Line Holdings Ltd. Consumer Cyclical
500 ETSY Etsy, Inc. Consumer Cyclical
501 IVZ Invesco Ltd. Financial Services
Cluster 3
Symbols
Symbol Longname Sector
13 WMT Walmart Inc. Consumer Defensive
21 COST Costco Wholesale Corporation Consumer Defensive
25 ABBV AbbVie Inc. Healthcare
55 PM Philip Morris International Inc. Consumer Defensive
111 MO Altria Group, Inc. Consumer Defensive
113 HCA HCA Healthcare, Inc. Healthcare
208 STZ Constellation Brands, Inc. Consumer Defensive
209 KDP Keurig Dr Pepper Inc. Consumer Defensive
227 KR The Kroger Co. Consumer Defensive
297 DG Dollar General Corporation Consumer Defensive
328 DLTR Dollar Tree, Inc. Consumer Defensive
350 BF-B Brown-Forman Corporation Consumer Defensive
386 LH Labcorp Holdings Inc. Healthcare
406 DGX Quest Diagnostics Incorporated Healthcare
430 DPZ Domino's Pizza, Inc. Consumer Cyclical
456 UHS Universal Health Services, Inc. Healthcare
457 DVA DaVita Inc. Healthcare
467 LW Lamb Weston Holdings, Inc. Consumer Defensive
468 TAP Molson Coors Beverage Company Consumer Defensive
We can notice that most of the companies in this cluster are in the comsumer defensive sector.
Cluster 4
Symbols
Symbol Longname Sector
14 V Visa Inc. Financial Services
17 MA Mastercard Incorporated Financial Services
47 DIS The Walt Disney Company Communication Services
49 GE GE Aerospace Industrials
58 CMCSA Comcast Corporation Communication Services
64 UBER Uber Technologies, Inc. Technology
70 BKNG Booking Holdings Inc. Consumer Cyclical
76 TJX The TJX Companies, Inc. Consumer Cyclical
102 ABNB Airbnb, Inc. Consumer Cyclical
105 SBUX Starbucks Corporation Consumer Cyclical
116 ZTS Zoetis Inc. Healthcare
137 MAR Marriott International, Inc. Consumer Cyclical
151 PYPL PayPal Holdings, Inc. Financial Services
169 HLT Hilton Worldwide Holdings Inc. Consumer Cyclical
190 ROST Ross Stores, Inc. Consumer Cyclical
204 CHTR Charter Communications, Inc. Communication Services
219 FIS Fidelity National Information Services, Inc. Technology
234 IDXX IDEXX Laboratories, Inc. Healthcare
256 EL The Estée Lauder Companies Inc. Consumer Defensive
277 LVS Las Vegas Sands Corp. Consumer Cyclical
310 GPN Global Payments Inc. Industrials
324 AXON Axon Enterprise, Inc. Industrials
325 WST West Pharmaceutical Services, Inc. Healthcare
326 FSLR First Solar, Inc. Technology
347 WBD Warner Bros. Discovery, Inc. Communication Services
370 ALGN Align Technology, Inc. Healthcare
387 OMC Omnicom Group Inc. Communication Services
402 FOXA Fox Corporation Communication Services
403 FOX Fox Corporation Communication Services
404 ZBRA Zebra Technologies Corporation Technology
415 NWSA News Corporation Communication Services
416 NWS News Corporation Communication Services
425 ENPH Enphase Energy, Inc. Technology
428 MGM MGM Resorts International Consumer Cyclical
434 TRMB Trimble Inc. Technology
444 HST Host Hotels & Resorts, Inc. Real Estate
448 JBL Jabil Inc. Technology
466 IPG The Interpublic Group of Companies, Inc. Communication Services
470 ALB Albemarle Corporation Basic Materials
482 WYNN Wynn Resorts, Limited Consumer Cyclical
484 MTCH Match Group, Inc. Communication Services
485 PAYC Paycom Software, Inc. Technology
495 HAS Hasbro, Inc. Consumer Cyclical
497 PARA Paramount Global Communication Services
498 CZR Caesars Entertainment, Inc. Consumer Cyclical
Cluster 5
Symbols
Symbol Longname Sector
18 PG The Procter & Gamble Company Consumer Defensive
27 KO The Coca-Cola Company Consumer Defensive
32 PEP PepsiCo, Inc. Consumer Defensive
40 MCD McDonald's Corporation Consumer Cyclical
68 RTX RTX Corporation Industrials
73 PGR The Progressive Corporation Financial Services
88 LMT Lockheed Martin Corporation Industrials
92 MMC Marsh & McLennan Companies, Inc. Financial Services
94 CB Chubb Limited Financial Services
106 WM Waste Management, Inc. Industrials
107 MDLZ Mondelez International, Inc. Consumer Defensive
118 CL Colgate-Palmolive Company Consumer Defensive
119 GD General Dynamics Corporation Industrials
132 CME CME Group Inc. Financial Services
147 NOC Northrop Grumman Corporation Industrials
148 AON Aon plc Financial Services
150 RSG Republic Services, Inc. Industrials
152 ORLY O'Reilly Automotive, Inc. Consumer Cyclical
156 AJG Arthur J. Gallagher & Co. Financial Services
174 AFL Aflac Incorporated Financial Services
179 AZO AutoZone, Inc. Consumer Cyclical
180 MNST Monster Beverage Corporation Consumer Defensive
183 AIG American International Group, Inc. Financial Services
188 KMB Kimberly-Clark Corporation Consumer Defensive
197 TRV The Travelers Companies, Inc. Financial Services
205 ALL The Allstate Corporation Financial Services
211 LHX L3Harris Technologies, Inc. Industrials
222 KHC The Kraft Heinz Company Consumer Defensive
225 VRSK Verisk Analytics, Inc. Industrials
235 HSY The Hershey Company Consumer Defensive
251 ACGL Arch Capital Group Ltd. Financial Services
253 GIS General Mills, Inc. Consumer Defensive
254 YUM Yum! Brands, Inc. Consumer Cyclical
284 HIG The Hartford Financial Services Group, Inc. Financial Services
299 WTW Willis Towers Watson Public Limited Company Financial Services
305 BRO Brown & Brown, Inc. Financial Services
314 CHD Church & Dwight Co., Inc. Consumer Defensive
321 ROL Rollins, Inc. Consumer Cyclical
353 LDOS Leidos Holdings, Inc. Technology
360 K Kellanova Consumer Defensive
361 MKC McCormick & Company, Incorporated Consumer Defensive
362 WRB W. R. Berkley Corporation Financial Services
364 CBOE Cboe Global Markets, Inc. Financial Services
390 HRL Hormel Foods Corporation Consumer Defensive
391 L Loews Corporation Financial Services
399 CLX The Clorox Company Consumer Defensive
400 EG Everest Group, Ltd. Financial Services
429 CAG Conagra Brands, Inc. Consumer Defensive
433 CPB Campbell Soup Company Consumer Defensive
447 SJM The J. M. Smucker Company Consumer Defensive
474 HII Huntington Ingalls Industries, Inc. Industrials
496 GL Globe Life Inc. Financial Services
Cluster 6
Symbols
Symbol Longname Sector
3 NVDA NVIDIA Corporation Technology
11 AVGO Broadcom Inc. Technology
19 ORCL Oracle Corporation Technology
29 AMD Advanced Micro Devices, Inc. Technology
35 QCOM QUALCOMM Incorporated Technology
36 ACN Accenture plc Technology
39 CSCO Cisco Systems, Inc. Technology
41 TXN Texas Instruments Incorporated Technology
43 INTU Intuit Inc. Technology
50 AMAT Applied Materials, Inc. Technology
65 INTC Intel Corporation Technology
75 MU Micron Technology, Inc. Technology
81 LRCX Lam Research Corporation Technology
84 ADI Analog Devices, Inc. Technology
91 PANW Palo Alto Networks, Inc. Technology
96 ANET Arista Networks, Inc. Technology
97 KLAC KLA Corporation Technology
110 SNPS Synopsys, Inc. Technology
122 CDNS Cadence Design Systems, Inc. Technology
129 CRWD CrowdStrike Holdings, Inc. Technology
141 NXPI NXP Semiconductors N.V. Technology
175 ADSK Autodesk, Inc. Technology
192 MCHP Microchip Technology Incorporated Technology
196 SMCI Super Micro Computer, Inc. Technology
210 FTNT Fortinet, Inc. Technology
226 MPWR Monolithic Power Systems, Inc. Technology
241 EA Electronic Arts Inc. Communication Services
273 ON ON Semiconductor Corporation Technology
275 CDW CDW Corporation Technology
285 RMD ResMed Inc. Healthcare
300 HPE Hewlett Packard Enterprise Company Technology
306 TTWO Take-Two Interactive Software, Inc. Communication Services
308 NTAP NetApp, Inc. Technology
329 TER Teradyne, Inc. Technology
330 WDC Western Digital Corporation Technology
344 STX Seagate Technology Holdings plc Technology
349 PTC PTC Inc. Technology
379 SWKS Skyworks Solutions, Inc. Technology
453 JNPR Juniper Networks, Inc. Technology
454 EPAM EPAM Systems, Inc. Technology
463 QRVO Qorvo, Inc. Technology
476 FFIV F5, Inc. Technology
491 DAY Dayforce Inc Technology
Tech companies are clustered in this cluster.
Cluster 7
Symbols
Symbol Longname Sector
15 XOM Exxon Mobil Corporation Energy
26 CVX Chevron Corporation Energy
34 LIN Linde plc Basic Materials
51 CAT Caterpillar Inc. Industrials
53 IBM International Business Machines Corporation Technology
71 COP ConocoPhillips Energy
77 ETN Eaton Corporation plc Industrials
89 BA The Boeing Company Industrials
95 DE Deere & Company Industrials
126 TT Trane Technologies plc Industrials
128 EOG EOG Resources, Inc. Energy
133 SLB Schlumberger Limited Energy
139 TDG TransDigm Group Incorporated Industrials
144 EMR Emerson Electric Co. Industrials
145 FCX Freeport-McMoRan Inc. Basic Materials
154 CARR Carrier Global Corporation Industrials
155 CEG Constellation Energy Corporation Utilities
157 PSX Phillips 66 Energy
158 APD Air Products and Chemicals, Inc. Basic Materials
159 MPC Marathon Petroleum Corporation Energy
164 PCAR PACCAR Inc Industrials
166 OXY Occidental Petroleum Corporation Energy
171 WMB The Williams Companies, Inc. Energy
186 VLO Valero Energy Corporation Energy
187 OKE ONEOK, Inc. Energy
189 URI United Rentals, Inc. Industrials
191 KMI Kinder Morgan, Inc. Energy
194 HES Hess Corporation Energy
200 JCI Johnson Controls International plc Industrials
223 CMI Cummins Inc. Industrials
232 NUE Nucor Corporation Basic Materials
237 CTVA Corteva, Inc. Basic Materials
239 DOW Dow Inc. Basic Materials
240 PWR Quanta Services, Inc. Industrials
246 FANG Diamondback Energy, Inc. Energy
250 BKR Baker Hughes Company Energy
264 DD DuPont de Nemours, Inc. Basic Materials
266 GRMN Garmin Ltd. Technology
269 HWM Howmet Aerospace Inc. Industrials
270 ADM Archer-Daniels-Midland Company Consumer Defensive
272 LYB LyondellBasell Industries N.V. Basic Materials
278 HAL Halliburton Company Energy
280 TRGP Targa Resources Corp. Energy
283 DVN Devon Energy Corporation Energy
303 VST Vistra Corp. Utilities
355 STLD Steel Dynamics, Inc. Basic Materials
357 CTRA Coterra Energy Inc. Energy
358 HUBB Hubbell Incorporated Industrials
375 TDY Teledyne Technologies Incorporated Technology
378 J Jacobs Solutions Inc. Industrials
388 AVY Avery Dennison Corporation Consumer Cyclical
394 TXT Textron Inc. Industrials
395 PKG Packaging Corporation of America Consumer Cyclical
408 MRO Marathon Oil Corporation Energy
411 IP International Paper Company Consumer Cyclical
412 BG Bunge Global SA Consumer Defensive
413 EQT EQT Corporation Energy
418 NRG NRG Energy, Inc. Utilities
420 CE Celanese Corporation Basic Materials
440 CF CF Industries Holdings, Inc. Basic Materials
461 APA APA Corporation Energy
464 EMN Eastman Chemical Company Basic Materials
469 ALLE Allegion plc Industrials
477 MOS The Mosaic Company Basic Materials
488 AIZ Assurant, Inc. Financial Services
502 FMC FMC Corporation Basic Materials
Cluster 8
Symbols
Symbol Longname Sector
62 NEE NextEra Energy, Inc. Utilities
85 PLD Prologis, Inc. Real Estate
100 AMT American Tower Corporation Real Estate
108 SO The Southern Company Utilities
114 DUK Duke Energy Corporation Utilities
127 EQIX Equinix, Inc. Real Estate
149 WELL Welltower Inc. Real Estate
160 SPG Simon Property Group, Inc. Real Estate
170 NEM Newmont Corporation Basic Materials
173 PSA Public Storage Real Estate
178 DLR Digital Realty Trust, Inc. Real Estate
181 O Realty Income Corporation Real Estate
184 AEP American Electric Power Company, Inc. Utilities
185 SRE Sempra Utilities
193 PCG PG&E Corporation Utilities
201 MRNA Moderna, Inc. Healthcare
206 CCI Crown Castle Inc. Real Estate
217 D Dominion Energy, Inc. Utilities
242 PEG Public Service Enterprise Group Incorporated Utilities
245 SYY Sysco Corporation Consumer Defensive
248 EXR Extra Space Storage Inc. Real Estate
249 EXC Exelon Corporation Utilities
267 ED Consolidated Edison, Inc. Utilities
271 VICI VICI Properties Inc. Real Estate
274 XEL Xcel Energy Inc. Utilities
286 AVB AvalonBay Communities, Inc. Real Estate
290 EIX Edison International Utilities
291 IRM Iron Mountain Incorporated Real Estate
295 AWK American Water Works Company, Inc. Utilities
302 EQR Equity Residential Real Estate
309 WEC WEC Energy Group, Inc. Utilities
319 DTE DTE Energy Company Utilities
320 KEYS Keysight Technologies, Inc. Technology
323 ETR Entergy Corporation Utilities
331 FE FirstEnergy Corp. Utilities
334 SBAC SBA Communications Corporation Real Estate
339 INVH Invitation Homes Inc. Real Estate
341 ARE Alexandria Real Estate Equities, Inc. Real Estate
343 VTR Ventas, Inc. Real Estate
345 ES Eversource Energy Utilities
346 TSN Tyson Foods, Inc. Consumer Defensive
348 PPL PPL Corporation Utilities
356 AEE Ameren Corporation Utilities
369 BALL Ball Corporation Consumer Cyclical
371 ESS Essex Property Trust, Inc. Real Estate
376 ATO Atmos Energy Corporation Utilities
377 CNP CenterPoint Energy, Inc. Utilities
380 CMS CMS Energy Corporation Utilities
392 MAA Mid-America Apartment Communities, Inc. Real Estate
396 DRI Darden Restaurants, Inc. Consumer Cyclical
417 UDR UDR, Inc. Real Estate
421 DOC Healthpeak Properties, Inc. Real Estate
423 AMCR Amcor plc Consumer Cyclical
426 KIM Kimco Realty Corporation Real Estate
432 LNT Alliant Energy Corporation Utilities
435 NI NiSource Inc. Utilities
445 EVRG Evergy, Inc. Utilities
451 REG Regency Centers Corporation Real Estate
452 AES The AES Corporation Utilities
459 CPT Camden Property Trust Real Estate
471 CTLT Catalent, Inc. Healthcare
483 PNW Pinnacle West Capital Corporation Utilities
486 FRT Federal Realty Investment Trust Real Estate
493 MKTX MarketAxess Holdings Inc. Financial Services
This cluster is composed mainly of companies in the utilities sector and the real estate sector.
Cluster 9
Symbols
Symbol Longname Sector
20 JNJ Johnson & Johnson Healthcare
33 TMUS T-Mobile US, Inc. Communication Services
44 AMGN Amgen Inc. Healthcare
46 VZ Verizon Communications Inc. Communication Services
52 PFE Pfizer Inc. Healthcare
67 T AT&T Inc. Communication Services
82 REGN Regeneron Pharmaceuticals, Inc. Healthcare
104 GILD Gilead Sciences, Inc. Healthcare
109 BMY Bristol-Myers Squibb Company Healthcare
140 TGT Target Corporation Consumer Defensive
143 BDX Becton, Dickinson and Company Healthcare
265 BIIB Biogen Inc. Healthcare
442 INCY Incyte Corporation Healthcare
478 WBA Walgreens Boots Alliance, Inc. Healthcare
This cluster is mainly composed of healthcare companies.
Cluster 10
Symbols
Symbol Longname Sector
9 LLY Eli Lilly and Company Healthcare
16 UNH UnitedHealth Group Incorporated Healthcare
24 MRK Merck & Co., Inc. Healthcare
74 VRTX Vertex Pharmaceuticals Incorporated Healthcare
83 ELV Elevance Health, Inc. Healthcare
101 CI The Cigna Group Healthcare
123 MCK McKesson Corporation Healthcare
125 CVS CVS Health Corporation Healthcare
195 HUM Humana Inc. Healthcare
215 COR Cencora, Inc. Healthcare
258 CNC Centene Corporation Healthcare
327 CAH Cardinal Health, Inc. Healthcare
398 MOH Molina Healthcare, Inc. Healthcare
Only healthcare companies are clustered in this group.
Although the resulted clusters are not very accurate, we can notice that these clusters are appealing, and reflects the sectors. This shows that the proposed method is useful and has its potential.