Abstract
Principal component analysis is described as extracting the main direction in the data. This paper asks whether the data alone fix that direction——the answer is that they do not: the units do. No new mathematical theorem and no new law is claimed.
First, a sample of heights and weights is generated and analysed with only the units changed (Section 2).
Second, this is the core of the paper. It points at $\mathbf{55.6503}$ degrees in cm, $\mathbf{89.7345}$ in m, and $\mathbf{4.1934}$ in mm (Section 2).
Third, in m and in mm the first component picks out two nearly orthogonal axes (Section 2).
Fourth, the explained fraction also moves, reaching $1.0000$ in m and $0.9898$ in mm (Section 3).
Fifth, standardising settles it at $\mathbf{45.0000}$ degrees with a fraction of $\mathbf{0.7899}$ (Section 4).
Sixth, the separator is covariance matrix against correlation matrix, not the data (Section 5).
Scope (what is not claimed)
No new mathematical theorem and no new law is claimed——principal component analysis, the difference between covariance and correlation matrices, and standardisation are all standard. We do not build multivariate analysis——how many components to retain, and the relation to factor analysis, are not entered. They are named and no more. We use no real data——actual body measurements are not used. A sample from a bivariate normal with correlation $0.6$ is generated. We build no estimation theory——sampling error in eigenvalues and tests on them are not the subject. We do not treat more than two variables——only the bivariate case is computed. The structure is the same with more. We offer no interpretation——the first component is not given a name such as “build”. Relation to earlier papers: Paper 358 showed a depth-of-field number fixed by a convention——there the convention was the circle of confusion; here it is the unit. Paper 356 showed quantities of the same dimension to be different quantities——here quantities of different dimensions are put in one space. Paper 300 showed whether two things share a root is decidable——by that test, “the structure of the data” and “the choice of scale” have distinct roots. Paper 372 showed an exponent governing the conclusion——here the scale governs it. What is added is giving the direction and explained fraction under four scalings, showing m and mm to be nearly orthogonal, showing standardisation settling at $45$ degrees, showing the explained fraction running from $0.7899$ to $1.0000$, and putting the separator on covariance against correlation.
1. Notation and the Question
Setting: Generate $4000$ pairs of height (mean $170$ cm, s.d. $8$ cm) and weight (mean $65$ kg, s.d. $10$ kg) with correlation $0.6$.
$$\Sigma=\begin{pmatrix}\sigma_h^2 & \rho\sigma_h\sigma_w\\ \rho\sigma_h\sigma_w & \sigma_w^2\end{pmatrix}, \qquad \text{first component}=\text{eigenvector of the largest eigenvalue of }\Sigma$$Observation 1.1 PCA looks for the direction of greatest variance. An eigenvector of the covariance matrix——and “greatest” is measured by the size of the numbers.
Observation 1.2 But the size of numbers changes with units. $170$ cm is also $1.70$ m and $1700$ mm——one length producing variances that differ by $10^{6}$.
Observation 1.3 And the weight’s unit is left alone. Still kg——so the ratio of the two axes’ “sizes” moves. This paper gives the consequence as numbers.
Question: Do the data alone fix the direction of the first component?
2. In cm, m and mm, the Direction Moves $85$ Degrees
Calculation 2.1: First component against units (computed in this paper, one sample of $4000$).
| Units | Direction [deg] | Explained fraction | First eigenvalue |
|---|---|---|---|
| cm and kg | $\mathbf{55.6503}$ | $0.8035$ | $129.454026$ |
| m and kg | $\mathbf{89.7345}$ | $1.0000$ | $98.318922$ |
| mm and kg | $\mathbf{4.1934}$ | $0.9898$ | $6312.309583$ |
Observation 2.2 This is the core of the paper. $89.7345$ degrees in m and $4.1934$ in mm——a gap of $85.5411$ degrees, nearly orthogonal. The same sample, the same people, the same measurements.
Observation 2.3 The meaning swaps too. $89.7345$ degrees is nearly the weight axis; $4.1934$ nearly the height axis——“the main variation is weight” and “it is height” swap on the choice of unit alone.
Observation 2.4 The reason is simple. In mm the height variance is $10^{6}$ times larger——it overwhelms the weight variance, so the first component sticks to the height axis.
Observation 2.5 In m it reverses. The height variance becomes $10^{-4}$ times——now the weight overwhelms.
Observation 2.6 Centimetres sit in between. $55.6503$ degrees——it looks “just right”, but nothing on the data’s side gives a reason to choose cm.
3. The Explained Fraction Moves Too
Observation 3.1 It is $1.0000$ in m and $0.9898$ in mm. Both read as “the first component explains nearly everything”——and the axis being explained is different.
Observation 3.2 This is the most misleading part. A high explained fraction does not guarantee that the direction means anything——push the units far enough and it approaches $1$ as closely as you like.
Observation 3.3 The eigenvalues themselves are not comparable either. $98.318922$ against $6312.309583$——different units make the eigenvalue’s units different, so their sizes mean nothing across rows.
Observation 3.4 The cm fraction, $0.8035$, is the lowest of the three. Not because it explains less——but because the two axes’ variances are closest there.
4. Standardising Settles It
Calculation 4.1: With both variables standardised (computed in this paper).
| Treatment | Direction [deg] | Explained fraction | First eigenvalue |
|---|---|---|---|
| Standardised (both) | $\mathbf{45.0000}$ | $\mathbf{0.7899}$ | $1.579873$ |
Observation 4.2 It is exactly $45.0000$ degrees. Because the two variances become equal——the eigenvectors of a correlation matrix are always at $45$ and $135$ degrees in two variables.
Observation 4.3 The fraction $0.7899$ corresponds to the correlation. For $\rho=0.6$, $(1+\rho)/2=0.8$——the sample correlation is slightly lower, giving $0.7899$. The formula closes.
Observation 4.4 Standardised, it no longer depends on units. The same result in cm, m and mm——this is why the correlation matrix is used.
Observation 4.5 To be stated honestly——standardising is itself a choice. It is the decision to give every variable equal weight, which is not neutral——where a high-variance variable ought to weigh more, the covariance matrix is the right one.
Observation 4.6 So the choice does not vanish; it changes place. Choose the units, or choose whether to standardise——the same shape as the circle-of-confusion convention merely changing place in Paper 358.
5. The Separator: Covariance Matrix, or Correlation Matrix?
Observation 5.1 The separator——whether the covariance or the correlation matrix is used; not the data.
| Unit-dependent | Direction in two variables | What is weighted | |
|---|---|---|---|
| Covariance matrix | yes | $4.19$ to $89.73$ deg | high-variance variables |
| Correlation matrix | no | always $45$ deg | all variables equally |
Observation 5.2 The rightmost column decides which to use. Where units are meaningful (several measurements in the same unit), the covariance matrix——where units are mixed, the correlation matrix. Not “which is right” but “what should weigh more”.
Observation 5.3 Distinct roots by the criterion of Paper 300. The structure of the data comes from the sample; the choice of scale from the analyst——the result is their product, not a property of either alone.
Observation 5.4 The separator can be moved. Standardising removes the unit dependence——but it introduces the new assumption of equal weight. What vanished is the dependence, not the choice.
Observation 5.5 To be stated honestly——$55.6503$ degrees and $0.7899$ are values under stated conditions: correlation $0.6$ and $4000$ draws. Another sample gives other numbers——what does not move is the structure: covariance components move with units and correlation components do not.
Observation 5.6 One last thing——when one says “the data speak”, someone chose the scale in which they speak. Changing a unit alone turns the first component by $85$ degrees, so that direction is not a property of the data alone——report an analysis without its scale and it cannot be reproduced.
6. The Census and the Close
- $55.6503$ degrees in cm (Calculation 2.1): in between.
- $89.7345$ degrees in m (Calculation 2.1): nearly the weight axis.
- $4.1934$ degrees in mm (Calculation 2.1): nearly the height axis.
- A gap of $85.5411$ degrees (Observation 2.2): nearly orthogonal.
- The values were only multiplied (Observation 2.2): same sample, same people.
- Fraction $1.0000$ in m (Observation 3.1): $0.9898$ in mm.
- A high fraction guarantees nothing (Observation 3.2): units can manufacture it.
- $45.0000$ degrees when standardised (Calculation 4.1): always, in two variables.
- Fraction $0.7899$ matches $(1+\rho)/2$ (Observation 4.3): the formula closes.
- Standardising is a choice too (Observation 4.5): it is not neutral.
Closing: Principal component analysis is described as extracting the main direction in the data——but that direction is not fixed by the data alone. PCA looks for the direction of greatest variance, and “greatest” is measured by the size of the numbers——yet the size of numbers changes with units: $170$ cm is also $1.70$ m and $1700$ mm, one length producing variances that differ by $10^{6}$. So on one sample the direction moves $85$ degrees——holding weight in kg and changing only the height unit, the first component points at $\mathbf{55.6503}$ degrees in cm, $\mathbf{89.7345}$ in m, and $\mathbf{4.1934}$ in mm. The m and mm answers differ by $85.5411$ degrees — nearly orthogonal. $89.7345$ is nearly the weight axis and $4.1934$ nearly the height axis, so “the main variation is weight” and “it is height” swap on the choice of unit alone. The same sample, the same people, and the values were only multiplied. The explained fraction moves too——$1.0000$ in m and $0.9898$ in mm. Both read as “the first component explains nearly everything”, and the axis being explained is different. A high fraction does not guarantee that the direction means anything: push the units far enough and it approaches $1$ as closely as you like. The eigenvalues ($98.318922$ against $6312.309583$) are not comparable either, since their units differ. Standardising settles it——with both variables standardised the direction is exactly $\mathbf{45.0000}$ degrees and the fraction $\mathbf{0.7899}$. The two variances become equal, and the eigenvectors of a correlation matrix are always at $45$ and $135$ degrees in two variables. The fraction matches $(1+\rho)/2$ for $\rho=0.6$, so the formula closes. One thing separates them——whether the covariance or the correlation matrix is used; not the data. Where units are meaningful, the covariance matrix; where they are mixed, the correlation matrix — not “which is right” but “what should weigh more”. By the criterion of Paper 300 they have distinct roots: the structure comes from the sample, the scale from the analyst, and the result is their product. To be said honestly——standardising is itself a choice — the decision to give every variable equal weight, which is not neutral. Where a high-variance variable ought to weigh more, the covariance matrix is right. The choice does not vanish; it changes place. One last thing——when one says “the data speak”, someone chose the scale in which they speak. Changing a unit alone turns the first component by $85$ degrees, so that direction is not a property of the data alone. Report an analysis without its scale and it cannot be reproduced.
References
- Pearson, K. (1901). On Lines and Planes of Closest Fit to Systems of Points in Space. Philosophical Magazine.
- Hotelling, H. (1933). Analysis of a complex of statistical variables into principal components. Journal of Educational Psychology.
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- Jolliffe, I. T., Cadima, J. (2016). Principal component analysis: a review and recent developments. Philosophical Transactions of the Royal Society A.
- Mardia, K. V., Kent, J. T., Bibby, J. M. (1979). Multivariate Analysis. London: Academic Press.
- Yamagishi, Y. (2026). The Three Hundredth Census: Whether Two Things Share a Root Is Decidable [Paper 300]. Zenodo. 10.5281/zenodo.22171914
- Yamagishi, Y. (2026). The Current Flows and Does No Work [Paper 356]. Zenodo. 10.5281/zenodo.22697124
- Yamagishi, Y. (2026). The “Depth” of Depth of Field Is a Convention, Not a Measurement [Paper 358]. Zenodo. 10.5281/zenodo.22697216
- Yamagishi, Y. (2026). Halving the Area Does Not Halve the Species — And Halving the Species Takes Cutting the Area to 6.25 Per Cent [Paper 372]. Zenodo. 10.5281/zenodo.22698241
Reference
Yamagishi, Y. (2026). What Chooses the First Principal Component Is Not the Data but the Units [Paper 375]. YOI: https://www.yamagishiyuuki.com/yoi/375
DOI: 10.5281/zenodo.22698703
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 375. This paper is a preprint and has not been peer reviewed.