Interpreting PCA Loadings and Variance in Equity Data
Article Quant Q&A · Author: Pelumi
Summary
The document discusses how to interpret principal component analysis when many pharmaceutical-company characteristics are analyzed together. The example reports component variance shares and loadings, and retains five components because their cumulative share exceeds 90%. It asks how those components can identify the most relevant variables.
The response emphasizes that component signs are arbitrary, so positive versus negative loading alone does not establish importance. Large absolute loadings indicate which variables are strongly associated with a component; the example notes that a later component can matter most for a particular variable even when it explains less total variance. PCA ranks components by variance captured, with the first component largest by construction. The response also cautions that interpreting components depends on the research goal: variance reduction does not automatically identify the most useful valuation metrics or explain stock prices causally.
Key ideas
- PCA orders components by the variance they explain, with the first component capturing the most.
- The sign of a principal component can flip without changing its meaning.
- Absolute loading magnitudes show which variables are most associated with each component.
- A component with a smaller overall variance share may still be most relevant to a particular variable.
- PCA results need to be interpreted in light of a specific research objective.
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Full text
# INTERPRETING PCA ANALYSIS
# INTERPRETING PCA ANALYSIS
I am having little trouble figuring our which variables are the most important when I am using PCA . What I am trying to do is see which variables explain the most variance when it comes to stock prices . What I have done is I took data from some stocks in the pharmaceutical sector (like JNJ,MRK ect.)and I took their P/S P/B ROE AND other variables and I would like to know which variables explain the most variance so I know what metrics to look at when I am analyzing the pharmaceutical sector . This was the result.
```
Importance of components:
PC1 PC2 PC3 PC4 PC5 PC6 PC7 PC8 PC9 PC10 PC11
Standard deviation 2.7288 1.7861 1.5533 1.1306 0.82578 0.71937 0.47079 0.34490 0.3150 0.18236 1.814e-16
Proportion of Variance 0.4654 0.1994 0.1508 0.0799 0.04262 0.03234 0.01385 0.00743 0.0062 0.00208 0.000e+00
Cumulative Proportion 0.4654 0.6648 0.8156 0.8955 0.93809 0.97043 0.98429 0.99172 0.9979 1.00000 1.000e+00
```
so I decided to keep 5 principle components because they explain over 90% of the data
My problem is how do I determine which variables are most important from these 5 principle components ? Could I use something like a weighted average of each variable making the weight based off of the proportion of variance and take the variables with the highest weights average ? Or is there a better way of determining which of my variables are the most relevant from a PCA analysis
these are the eigenvectors
```
Rotation (n x k) = (16 x 11):
PC1 PC2 PC3 PC4 PC5 PC6 PC7
Beta:M-3 -0.29790642 0.06702808 0.043897343 -0.23444949 0.26922306 -0.52278907 0.49126902
Debt/Equity LF -0.29683436 -0.20602790 0.209892052 0.17525538 0.17021916 0.22824284 0.13967642
P/S -0.13584463 0.37660138 0.371760998 -0.13147372 0.13251944 0.04785622 -0.07498348
P/B -0.32237181 0.04415421 0.294703959 0.01222229 -0.01493067 0.06389646 0.14404329
PM LF 0.31231588 0.13244909 0.280519000 -0.04766506 0.01057485 0.13851709 0.10567161
OPM LF 0.31383977 0.06628785 0.295610357 -0.05090070 0.03949326 0.20637913 0.07153909
R&D Exp T12M 0.13290870 -0.22363789 0.356892802 0.17596955 -0.70388205 -0.08328041 0.23325195
ROA LF 0.33128578 0.11822215 0.123556305 -0.15352214 0.15827191 0.04626632 0.41437205
ROE LF 0.35240443 0.05558026 0.054402901 -0.00381858 0.09509125 -0.03775416 0.29445397
Rev - 1 Yr Gr:Q -0.33213363 0.13173387 0.179946152 -0.04917693 -0.04327167 0.12309850 0.02891068
Dil EPS Frm Cont Op 1Y Gr LF -0.02935805 0.38139243 0.081834610 0.62033384 0.03574667 -0.03928716 -0.16499320
Curr Ratio LF 0.02859583 0.48002784 -0.038108130 -0.37888227 -0.06938542 0.33198272 -0.18806338
P/E 0.04731051 0.46627994 -0.033990479 0.41432037 -0.02392947 -0.30628066 0.12819470
Shrt Int Ratio:D-1 -0.33298583 0.01426119 0.008234069 0.10984204 -0.10586662 0.48387359 0.33537995
RSI: Period=14 0.06437034 0.09275629 -0.564351964 0.19383663 0.02711028 0.34214075 0.42631622
Tot Analyst Rec:D-1 0.15449771 -0.31749070 0.238832459 0.29221399 0.57364300 0.15502219 -0.09793154
```
hopefully you guys on stack exchange will share a little bit of your wisdom and help me out here . Thank you in advance .
## Answer by jason m (score 2, accepted)
https://quant.stackexchange.com/a/49520
IIRC, the signs of the PC are meaningless. +/-'ive doesn't itself tell you anything.
Rather, the cross-sectional, absolute max of the PCs will tell you which one is most important per item (eg: PC6 looks most important for Beta: M-3).
I think 6.6a and 6.6b in Cochrane's asset pricing touch on this (https://www.youtube.com/playlist?list=PLAXSVuGaw0KxVUym8IRkObSbUPEFaSbPt).
This blog post seems resonable: https://thequantmba.wordpress.com/2017/01/24/principal-component-analysis-of-equity-returns-in-python/
Generally, PC1, scaled to 100% weight, creates a long-short "Portfolio PC1".
These portfolios then tend to "look" like other observable assets.
Skimming this paper seems like it has some sample code, and was reviewed by someone: https://web.wpi.edu/Pubs/ETD/Available/etd-080614-144242/unrestricted/Chen,_Huanting_PCA_2014-07-31_FINAL_VERSION.pdf
The classic example is the level, slope and curvature exercise: https://faculty.chicagobooth.edu/john.cochrane/teaching/coursera_documents/bond_notes_2.pdf
You can likely recreate this with data on st louis fed's fred.
The larger question is "What are you trying to do?". PCA, IMO, is pretty confusing if you don't have a goal. PC1 is always the most important (in terms of variance explained) by default. That's just what PCA gives you.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.