Q:

Imputation of missing values for PCA

I used the prcomp() function to perform a PCA (principal component analysis) in R. However, there's a bug in that function such that the na.action parameter does not work. I asked for help on stackoverflow; two users there offered two different ways of dealing with NA values. However, the problem with both solutions is that when there is an NA value, that row is dropped and not considered in the PCA analysis. My real data set is a matrix of 100 x 100 and I do not want to lose a whole row just because it contains a single NA value.
The following example shows that the prcomp() function does not return any principal components for row 5 as it contains a NA value.
d       <- data.frame(V1 = sample(1:100, 10), V2 = sample(1:100, 10), 
                      V3 = sample(1:100, 10))
result  <- prcomp(d, center = TRUE, scale = TRUE, na.action = na.omit)
result$x                                # $
d$V1[5] <- NA                           # $
result  <- prcomp(~V1+V2, data=d, center = TRUE, scale = TRUE, na.action = na.omit)
result$x

I was wondering if I can set the NA values to a specific numerical value when center and scale are set to TRUE so that the prcomp() function works and does not remove rows containing NA's, but also does not influence the outcome of the PCA analysis.
I thought about replacing NA values with the median value across a single column, or with a value very close to 0. However, I am not sure how that influences the PCA analysis.
Can anybody think of a good way of solving that problem?

A:

There is in fact a well documented way to deal with gappy matrices - you can decompose a covariance matrix $\textbf{C}$ contructed from of your data $\textbf{X}$, which is scaled by the number of shared values $n$:
$$
\textbf{C}=\frac{1}{n} \textbf{X} ^ {\text{T}} \textbf{X},~~~~~~~~~~~~~~~~ C_{jl} = \overline{X_{.j}Y_{.l}}
$$
and then expand the principal coefficients via a least squares fit (as @user969113 mentions). Here's an example.  
However, there are several problems with this method relating to the fact that the covariance matrix is no longer semipositive definite and the eigen/singular values tend to be inflated. A nice review of these problems can be found in Beckers and Rixen (2003), where they also propose a method of optimally interpolating the missing gaps - DINEOF (Data Interpolating Empirical Orthogonal Functions). I have recently written a function that performs DINEOF, and it really seems to be a much better way to go. You could perform DINEOF on your your dataset $\textbf{X}$ directly, and then use the interpolated dataset as input into prcomp.
Update
Another option for conducting PCA on a gappy dataset is "Recursively Subtracted Empirical Orthogonal Functions" (Taylor et al. 2013). It also corrects for some of the problems in the least squares approach, and is computationally much faster than DINEOF. This post compares the all three approaches in terms of the accuracy of the data reconstruction using the PCs.
References
Beckers, Jean-Marie, and M. Rixen. "EOF Calculations and Data Filling from Incomplete Oceanographic Datasets." Journal of Atmospheric and Oceanic Technology 20.12 (2003): 1839-1856.
Taylor, M., Losch, M., Wenzel, M., & Schröter, J. (2013). On the sensitivity of field reconstruction and prediction using Empirical Orthogonal Functions derived from gappy data. Journal of Climate, 26(22), 9194-9205.

A:

My suggestion depends on how much data is missing and why it is missing. But this has nothing to do with PCA, really. If there is very little data missing, then it won't much matter what you do. Replacing with the median isn't ideal, but if there is not much missing, it won't be much different from a better solution. You could try doing PCA with both median replacement and listwise deletion and see if there are major differences in the results.
Next, if there is more data missing, you should consider whether it is missing completely at random, missing at random, or not missing at random. I would suggest multiple imputation in the first two cases and some of the time in the third case - unless the data is highly distorted by its NMAR status, I think multiple imputation will be better than listwise deletion (Joe Schafer of Penn State has done a lot of work on missing data - I recall some work of his showing that multiple imputation worked pretty well even in some NMAR cases). However, if the data are MCAR or MAR, the properties of multiple imputation can be proven.
If you do decide to go with MI, one note is to be careful because the signs of the components in PCA are arbitrary, and a small change in the data can flip a sign. Then when you do the PCA you will get nonsense. A long time ago I worked out a solution in SAS - it isn't hard, but it's something to be careful about. 

A:

A recent paper which reviews approaches for dealing with missing values in PCA analyses is "Principal component analysis with missing values: a comparative survey of methods" by Dray &  Josse (2015). Two of the best known methods of PCA methods that allow for missing values are the NIPALS algorithm, implemented in the nipals function of the ade4 package, and the iterative PCA (Ipca or EM-PCA), implemented in the imputePCA function of the missMDA package. The paper concluded that the Ipca method performed best under the widest range of conditions. 
For your example syntax is :
For NIPALS :
library(ade4)
nipals(d[,c(1,2)])

For Ipca :
library(missMDA)
imputePCA(d[,c(1,2)],method="EM",ncp=1)

