Why the first principal component is mostly negative while the second component is mostly positive?

Question

I am running PCA for a fleet management data frame $X$, where each column is a city, each row is a date, there are 50 cities and 500 dates.

I run PCA on $A=X^{T}X$.

Then the first component $v_{1}$ is :

$Av_{1}=\lambda_{1}v_{1}$

However, I discovered that most of the values in the first component $v_{1}$ are negative; while most of the values of the second (and there after till 50) component $v_{2}$ are positive.

I am curious to understand from where the negativity comes from. What I understand about PCA is that, the principal components define new axis which are orthogonal to each other. In that case, the sign shouldn't matter, because if I 'flip' the component, it should still be orthogonal to the other components.

If we imagine that the data is projected into a 2-dim space, in my case, the first component is pointing toward the quadrant III;

Or in other words, what does the negative first component say about my input data? Does this imply that input data is skewed (at least , asymmetric) ?

Answer 1

The actual signs say nothing - they are arbitrary. For example, if you were analyzing data about the size of an object and you had height, breadth, depth, weight and so on, then you could get a first PC with all positive loadings and it would mean something like "bigness" or you could get one with all negative loadings and it would mean something like "smallness".

It is often the case that there is a first PC that accounts for some relatively obvious feature of the variables.