# Can PC1 explain more than 90% of variance?

I've been running some analysis using morphological data. However, when I run a PCA, PC1 explains more than 90% of variance. I'm not sure if this is possible. If not, what should I need to do? • Have you standardized your variables before the PCA? Feb 11 at 18:52
• just because no one has said it explicitly: this is really good news for you, in terms of there being a simple way to examine your problem (assuming everything is correct; as Serdash mentioned, if you haven't subtracted your mean from each column that could be manifesting itself here). Feb 11 at 19:43
• This is more of a subject matter question than a statistics question? Whether this is plausible or not really depends on the field and the specific data. I could imagine height and weight going up in near lockstep in some data set hence PC1 explains almost all the variation. I could also imagine a height, hair color, BMI dataset where there's a lot of unrelated variation among all the features. Feb 11 at 20:11
• Sure it can. It is also possible to produce an example where it totally explains the variance in the sample. Feb 12 at 10:41
• Often this arises when examining data that have a common source of variation. For morphological data, that common source is overall size. In many such applications that's of little interest. What you might prefer to do, if your intent is to study variation in shape, is to perform PCA on standardized variables. But since you haven't explained why you are doing PCA, there's little we can add in terms of recommendations.
– whuber
Feb 12 at 16:56

Certainly that can happen. Below is a simulated example in R with just two original variables. As long as they are strongly enough correlated, they can very well be summarized by the projection to a single line, which means that the first principal component explains an arbitrarily large proportion of total variance:

library(MASS)
set.seed(1)
dataset <- mvrnorm(1000,c(0,0),cbind(c(1,0.99),c(0.99,1)))
summary(prcomp(dataset))
plot(dataset,las=1,pch=19,cex=0.6)


Output:

Importance of components:
PC1     PC2
Standard deviation     1.460 0.10400
Proportion of Variance 0.995 0.00505
Cumulative Proportion  0.995 1.00000 I very much recommend Making sense of principal component analysis, eigenvectors & eigenvalues.

In case you need a non-simulated example: I did help.search("morpholog", agrep = FALSE) to find morphological data sets in the set of R packages I happen to have installed on my system. A useful one is ade4::tortues, a data set on 48 painted turtles from Jolicoeur and Mosimann 1960.

long, larg, and haut are the length, width, and height of the turtles in mm; because they're all measured in the same units, it shouldn't be necessary to scale the variable before/while computing principal components.

library(ade4)
## column 4 is sex
pairs(tortues[,1:3], col = tortues[,4], gap = 0)
summary(prcomp(tortues[,1:3]))


PC1 has 98% of the variance. (Jolicoeur and Mosimann get 97.61%; I get 97.16% if I set scale= TRUE; haven't checked for other issues [transcription errors etc.]).

Importance of components:
PC1     PC2     PC3
Standard deviation     25.3100 2.40272 2.26449
Proportion of Variance  0.9833 0.00886 0.00787
Cumulative Proportion   0.9833 0.99213 1.00000


## Jolicoeur, P. and Mosimann, J. E. (1960) Size and shape variation in the painted turtle. A principal component analysis. Growth, 24, 339-354.

• +1. Maybe add that long is the turtle's length, larg its width, and haut its height, all in millimeters? Feb 13 at 9:12

Let $$S \in \mathbb{R}^{p \times p}$$ be the sample covariance matrix (which equals to $$X'X$$ for centered and intercept-free design matrix $$X$$), and $$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_p \geq 0$$ be all the non-negative eigenvalues of $$S$$, then the total variation percentage explained by the first $$k$$ PCs is given by \begin{align} p_k = \frac{\sum_{j = 1}^k \lambda_j}{\sum_{j = 1}^p \lambda_j}, \; k = 1, 2, \ldots, p. \end{align} In particular, the total variation percentage explained by the first PC is \begin{align} p_1 = \frac{\lambda_1}{\sum_{j = 1}^p\lambda_j}. \tag{1} \end{align} In view of $$(1)$$, theoretically there are no upper bounds that are strictly less than $$1$$ for $$p_1$$. In other words, there exists $$X$$ such that $$p_1$$ can be arbitrarily close to $$1$$. For example, any design matrix $$X$$ in the form of $$(2)$$ below satisfies that its first PC explains $$1 - \varepsilon$$ total variation: \begin{align} X = U\begin{bmatrix} \operatorname{diag}(\sqrt{1 - \varepsilon}, c, \ldots, c) \\ 0 \end{bmatrix}V', \tag{2} \end{align} where $$c = \frac{\sqrt{\varepsilon}}{\sqrt{p - 1}}$$, $$U$$ and $$V$$ are arbitrary order $$n$$ and order $$p$$ orthogonal matrices. Therefore, it is not surprising at all to observe cases such that $$p_1 > 90\%$$.

• This is certainly technically correct but based on the way the question was framed may be too technical for the OP (but may be useful to later readers ...) Feb 13 at 18:00