Solved – How to obtain the angle of rotation produced by a PCA on a 2D dataset

pcar

I want to find the principal direction in a set of spatial coordinates (i.e. in a 2D table). To do so I'm using PCA implemented in prcomp (because it allows missing values). prcomp returns a 'rotation' matrix which is said to be

the matrix of variable loadings (i.e., a matrix whose columns contain the eigenvectors).

From this, I would like to calculate the angle of the rotation achieved by the PCA, expressed as the clockwise angle from the North direction. I know the information is there, but I cannot get the calculation right. Any help would be appreciated.

EDIT: It seems that deg(asin(abs(rotation[1,1]))) works. My understanding of what I read elsewhere led me to that solution, but I don't feel like I would be able to correctly spell out an explanation for the benefit of others.

Best Answer

In a 2D case, the rotation matrix is $2\times 2$ and contains two eigenvectors as its columns. The first eigenvector $(x,y)$ is given by the first column. Its angle to the horizontal axis (abscissa) is given by $$\alpha = \operatorname{arctan} (y/x).$$As all eigenvectors are scaled to have unit length, this is equal to $$\alpha = \operatorname{arccos} (x/1)=\operatorname{arccos} (x).$$ You might call it a "counterclockwise" angle "from the East direction". If you want a "clockwise" angle "from the North direction", then you need $$\beta = 90^\circ - \alpha=90^\circ - \operatorname{arccos} (x)= \operatorname{arcsin} (x).$$ In R, it should be:

beta = asin(pc$rotation[1,1])*180/pi

which is more or less your formula too.

Related Solutions

Solved – PCA and component scores based on a mix of continuous and binary variables

I think Insanodag is right. I quote Jollife's Principal Component Analysis:

When PCA is used as a descriptive technique, there is no reason for the variables in the analysis to be of any particular type. [...] the basic objective of PCA - to summarize most of the 'variation' that is present in the original set of $p$ variables using smaller number of derived varaibles - can be achieved regardless of the nature of the original variables.

Multiplying the data matrix with the loadings matrix will give the desired result. However, I've had some problems with princomp() function so I used prcomp() instead.

One of the return values of the function prcomp() is x, which is activated using retx=TRUE. This x is the multiplication of the data matrix by the loadings matrix as stated in the R Documentation:

    rotation: the matrix of variable
              loadings (i.e., a matrix whose columns
              contain the eigenvectors).  The function ‘princomp’ returns
              this in the element ‘loadings’.

           x: if ‘retx’ is true the value of the rotated data (the centred
              (and scaled if requested) data multiplied by the ‘rotation’
              matrix) is returned.  Hence, ‘cov(x)’ is the diagonal matrix
              ‘diag(sdev^2)’.  For the formula method, ‘napredict()’ is
              applied to handle the treatment of values omitted by the
              ‘na.action’.

Let me know if this was useful, or if it needs further corrections.

I.T. Jollife. Principal Component Analysis. Springer. Second Edition. 2002. pp 339-343.

Solved – What do the arrows in a PCA biplot mean

Well it appears Kevin Wright should be given most of the credit to try to help explain the confusion (from R-help mail list);

The arrows are not pointing in the most-varying direction of the data. The principal components are pointing in the most-varying direction of the data. But you are not plotting the data on the original scale, you are plotting the data on the rotated scale, and thus the horizontal axis is the most-varying direction of the data.

The arrows are pointing in the direction of the variables, as projected into the 2-d plane of the biplot.

There is no bug.

Kevin Wright

Michael Greenacre has a very excellent free online book about biplots, Biplots in Practice, and simply reading the first chapter should help motivate where the coordinates of the arrows are taken from. There are also several other questions on the site that are similar and you may be interested in, see Interpretation of biplots in principal components analysis in R and Interpretation of MDS factor plot for two examples. Also look through the questions with biplot in search on the site, as there are a few more of potential interest (it appears maybe even making a biplot tag would be useful at this point given the number of questions it has come up in).