I am have some difficulty understanding some steps in a procedure. They take coordinate data, find the covariance matrix, apply PCA, then extract the standard deviation from the square root of each eigenvalue in short. I am trying to re-produce this process, but I am stuck on the steps.
The Steps Taken
The data set consists of one matrix, R, that contains coordiante paris, (x(i),y(i)) with i=1,...,N for N is the total number of instances recorded. We applied PCA to the covariance matrix of the R input data set, and the following variables were obtained:
a) the principal components of the new coordinate system, the eigenvectors u and v, and
b) the eigenvalues (λ1 and λ2) corresponding to the total variability explained by each principal component.
With these variables, a graphical representation was created for each item. Two orthogonal segments were centred on the mean of the coordinate data. The segments’ directions were driven by the eigenvectors of the PCA, and the length of each segment was defined as one standard deviation (σ1 and σ2) around the mean, which was calculated by extracting the square root of each eigenvalue, λ1 and λ2.
My Steps
#reproducable data
set.seed(1)
x<-rnorm(10,50,4)
y<-rnorm(10,50,7)
# Note my data is not perfectly distirbuted in this fashion
df<-data.frame(x,y) # this is my R matrix
covar.df<-cov(df,use="all.obs",method='pearson') # this is my covariance matrix
pca.results<-prcomp(covar.df) # this applies PCA to the covariance matrix
pca.results$sdev # these are the standard deviations of the principal components
# which is what I believe I am looking for.
This is where I am stuck because I am not sure if I am trying to get the sdev output form prcomp() or if I should scale my data first. They are all on the same scale, so I do not see the issue with it.
My second question is how do I extract the standard deviation in the x and y direciton?
You don't apply prcomp to the covariance matrix, you do it on the data itself.
result= prcomp(df)
If by scaling you mean normalize or standardize, that happens before you do prcomp(). For more information on the procedure see this link that is introductory to the procedure: pca on R. That can walk you through the basics. To get the sdev use the the summary on the result object
summary(result)
result$sdev
You don't apply prcomp to the covariance matrix. scale=T bases the PCA on the correlation matrix and F on the covariance matrix
df.cor = prcomp(df, scale=TRUE)
df.cov = prcomp(df, scale=FALSE)
Related
I am trying to calculate PCA loadings of a dataset. The more I read about it, the more I get confused because "loadings" is used differently at many places.
I am using sklearn.decomposition in python for PCA analysis as well as R (using factomineR and factoextra libraries) as it provides easy visualization techniques. The following is my understanding:
pca.components_ give us the eigen vectors. They give us the directions of maximum variation.
pca.explained_variance_ give us the eigen values associated with the eigen vectors.
eigenvectors * sqrt(eigen values) = loadings which tell us how principal components (pc's) load the variables.
Now, what I am confused by is:
Many forums say that eigen vectors are the loadings. Then, when we multiply the eigen vectors by the sqrt(eigen values) we just get the strength of association. Others say eigenvectors * sqrt(eigen values) = loadings.
Eigen vectors squared tells us the contribution of variable to pc? I believe this is equivalent to var$contrib in R.
loading squared (eigen vector or eigenvector*sqrt(eigenvalue) I don't know which one) shows how well a pc captures a variable (closer to 1 = variable better explained by a pc). Is this equivalent of var$cos2 in R? If not what is cos2 in R?
Basically I want to know how to understand how well a principal component captures a variable and what is the contribution of a variable to a pc. I think they both are different.
What is pca.singular_values_? It is not clear from the documentation.
These first and second links that I referred which contains R code with explanation and the statsexchange forum that confused me.
Okay, after much research and going through many papers I have the following,
pca.components_ = eigen vectors. Take a transpose so that pc's are columns and variables are rows.
1.a: eigenvector**2 = variable contribution in principal components. If it's close to 1 then a particular pc is well explained by that variable.
In python -> (pow(pca.components_.T),2) [Multiply with 100 if you want percentages and not proportions] [R equivalent -> var$contrib]
pca.variance_explained_ = eigen values
pca.singular_values_ = singular values obtained from SVD.
(singular values)**2/(n-1) = eigen values
eigen vectors * sqrt(eigen values) = loadings matrix
4.a: vertical sum of squared loading matrix = eigen values. (Given you have taken transpose as explained in step 1)
4.b: horizontal sum of squared loading matrix = observation's variance explained by all principal components -How much all pc's retain a variables variance after transformation. (Given you have taken transpose as explained in step 1)
In python-> loading matrix = pca.components_.T * sqrt(pca.explained_variance_).
For questions pertaining to r:
var$cos2 = var$cor (Both matrices are same). Given the coordinates of the variables on a factor map, how well it is represented by a particular principal component. Seems like variable and principal component's correlation.
var$contrib = Summarized by point 1. In r:(var.cos2 * 100) / (total cos2 of the component) PCA analysis in R link
Hope it helps others who are confused by PCA analysis.
Huge thanks to -- https://stats.stackexchange.com/questions/143905/loadings-vs-eigenvectors-in-pca-when-to-use-one-or-another
I need to do a principal component analysis (PCA) with EQUAMAX-rotation in R.
Unfortunately the function principal() I normally use for PCA does not offer this kind of rotation.
I could find out that it may be possible somehow with the package GPArotation but I could not yet figure out how to use this in the PCA.
Maybe someone can give an example on how to do an equamax-rotation PCA?
Or is there a function for PCA in another package that offers the use of equamax-rotation directly?
The package psych from i guess you are using principal() has the rotations varimax, quatimax, promax, oblimin, simplimax, and cluster but not equamax (psych p.232) which is a compromise between Varimax and Quartimax
excerpt from the STATA manual: mvrotate p.3
Rotation criteria
In the descriptions below, the matrix to be rotated is denoted as A, p denotes the number of rows of A, and f denotes the number of columns of A (factors or components). If A is a loading matrix from factor or pca, p is the number of variables, and f is the number of factors or components.
Criteria suitable only for orthogonal rotations
varimax and vgpf apply the orthogonal varimax rotation (Kaiser 1958). varimax maximizes the variance of the squared loadings within factors (columns of A). It is equivalent to cf(1/p) and to oblimin(1). varimax, the most popular rotation, is implemented with a dedicated fast algorithm and ignores all optimize options. Specify vgpf to switch to the general GPF algorithm used for the other criteria.
quartimax uses the quartimax criterion (Harman 1976). quartimax maximizes the variance of
the squared loadings within the variables (rows of A). For orthogonal rotations, quartimax is equivalent to cf(0) and to oblimax.
equamax specifies the orthogonal equamax rotation. equamax maximizes a weighted sum of the
varimax and quartimax criteria, reflecting a concern for simple structure within variables (rows of A) as well as within factors (columns of A). equamax is equivalent to oblimin(p/2) and cf(#), where # = f /(2p).
now the cf (Crawford-Ferguson) method is also available in GPArotation
cfT orthogonal Crawford-Ferguson family
cfT(L, Tmat=diag(ncol(L)), kappa=0, normalize=FALSE, eps=1e-5, maxit=1000)
The argument kappa parameterizes the family for the Crawford-Ferguson method. If m is the number of factors and p is the number of indicators then kappa values having special names are 0=Quartimax, 1/p=Varimax, m/(2*p)=Equamax, (m-1)/(p+m-2)=Parsimax, 1=Factor parsimony.
X <- matrix(rnorm(500), ncol=10)
C <- cor(X)
eig <- eigen(C)
# PCA by hand scaled by sqrt
eig$vectors * t(matrix(rep(sqrt(eig$values), 10), ncol=10))
require(psych)
PCA0 <- principal(C, rotate='none', nfactors=10) #PCA by psych
PCA0
# as the original loadings PCA0 are scaled by their squarroot eigenvalue
apply(PCA0$loadings^2, 2, sum) # SS loadings
## PCA with Equimax rotation
# now i think the Equamax rotation can be performed by cfT with m/(2*p)
# p number of variables (10)
# m (or f in STATA manual) number of components (10)
# gives m==p --> kappa=0.5
PCA.EQ <- cfT(PCA0$loadings, kappa=0.5)
PCA.EQ
I upgraded some of my PCA knowledge by your question, hope it helps, good luck
Walter's answer helped a great deal!
I'll add some sidenotes for what it's worth:
R's psych::principal says under option "rotate", that more rotations are available. Under the linked "fa", there's in fact an "equamax". Sadly, the results are neither replicable with STATA nor with SPSS, at least not with the standard syntax I tried:
# R:
PCA.5f=principal(data, nfactors=5, rotate="equamax", use="complete.obs")
Walter's solution replicates SPSS' equamax rotation (Kaiser-normalized by default) in the first 3 decimal places (i.e. loadings and rotating matrix fairly equivalent) using the following syntax with m=no of factors and p=no of indicators:
# R:
PCA.5f=principal(data, nfactors=5, rotate="none", use="complete.obs")
PCA.5f.eq = cfT(PCA.5f$loadings, kappa=m/(2*p), normalize=TRUE) # replace kappa factor formula with your actual numbers!
# SPSS:
FACTOR
/VARIABLES listofvariables
/MISSING LISTWISE
/ANALYSIS listofvariables
/PRINT ROTATION
/CRITERIA FACTORS(5) ITERATE(1000)
/EXTRACTION PC
/CRITERIA ITERATE(1000)
/ROTATION EQUAMAX
/METHOD=CORRELATION.
STATA's equamax - Kaiser-normalized and unnormalized - is replicable at least in the first 4 decimal places with Kappa .5 irrespective of your actual number of factors and indicators which seems to contradict their manual (c.f. Walter's citation).
# R:
PCA.5f=principal(data, nfactors=5, rotate="none", use="complete.obs")
PCA.5f.eq = cfT(PCA.5f$loadings, kappa=.5, normalize=TRUE)
# STATA:
factor listofvars, pcf factors(5)
rotate, equamax normalize # kick the "normalize" to replicate R's "normalize=FALSE"
mat list e(r_L)
I have 2 two dissimilarity matrices. One with observed data comparing among 111 sites and another generated using a null model.
I would like to use the adnois function in vegan to test whether the observed dissimilarities differ significantly from those expected by the null model. However the adonis function will only take one dissimilarity matrix on the left side of the formula.
Does anyone have any idea how to model this test?
Thanks
The answer to this problem was:
meanjac <- function(x) mean(vegdist(x, method='jaccard', diag=TRUE))
test <- oecosimu(x, nestfun=meanjac, method="r1", nsimul = 10^3, statistic='adonis')
which passes a function to get the mean of jaccard dissimilarity matrix to oecosimu, which then uses the 'r1' method to generate null community matrices by randomly shuffling the binary community matrix but assigning the probability of species occupancies based on their observed occupancy and comparing this to the observed dissimilarity matrix.
Thanks Jari for pointing me in the right direction...
I'd like to calculate multivariate distance from a set of points to the centroid of those points. Mahalanobis distance seems to be suited for this. However, I get an error (see below).
Can anyone tell me why I am getting this error, and if there is a way to work around it?
If you download the coordinate data and the associated environmental data, you can run the following code.
require(maptools)
occ <- readShapeSpatial('occurrences.shp')
load('envDat.Rdata')
#standardize the data to scale the variables
dat <- as.matrix(scale(dat))
centroid <- dat[1547,] #let's assume this is the centroid in this case
#Calculate multivariate distance from all points to centroid
mahalanobis(dat,center=centroid,cov=cov(dat))
Error in solve.default(cov, ...) :
system is computationally singular: reciprocal condition number = 9.50116e-19
The Mahalanobis distance requires you to calculate the inverse of the covariance matrix. The function mahalanobis internally uses solve which is a numerical way to calculate the inverse. Unfortunately, if some of the numbers used in the inverse calculation are very small, it assumes that they are zero, leading to the assumption that it is a singular matrix. This is why it specifies that they are computationally singular, because the matrix might not be singular given a different tolerance.
The solution is to set the tolerance for when it assumes that they are zero. Fortunately, mahalanobis allows you to pass this parameter (tol) to solve:
mahalanobis(dat,center=centroid,cov=cov(dat),tol=1e-20)
# [1] 24.215494 28.394913 6.984101 28.004975 11.095357 14.401967 ...
mahalanobis uses the covariance matrix, cov, (more precisely the inverse of it) to transform the coordinate system, then compute Euclidian distance in the new coordinates. A standard reference is Duda & Hart "Pattern Classification and Scene Recognition"
Looks like your cov matrix is singular. Perhaps there are linearly-dependent columns in "dat" that are unnecessary? Setting the tolerance to zero won't help if
the covariance matrix is truly singular. The first thing to do, instead, is look for columns that might be a rescaling of some other column, or might be just a sum of 2 or more other columns and remove them. Such columns are redundant for the mahalanobis distance.
BTW, since mahalanobis distance is effectively a rescaling and rotation, calling the scaling function looks superfluous - any reason why you want that?
Now i have a 7000*7000 correlation matrix and I have to do PCA on this in R.
I used the
CorPCA <- princomp(covmat=xCor)
, xCor is the correlation matrix
but it comes out
"covariance matrix is not non-negative definite"
it is because i have some negative correlation in that matrix.
I am wondering which inbuilt function in R that i can use to get the result of PCA
One method to do the PCA is to perform an eigenvalue decomposition of the covariance matrix, see wikipedia.
The advantage of the eigenvalue decomposition is that you see which directions (eigenvectors) are significant, i.e. have a noticeable variation expressed by the associated eigenvalues. Moreover, you can detect if the covariance matrix is positive definite (all eigenvalues greater than zero), not negative-definite (which is okay) if there are eigenvalues equal zero or if it is indefinite (which is not okay) by negative eigenvalues. Sometimes it also happens that due to numerical inaccuracies a non-negative-definite matrix becomes negative-definite. In that case you would observe negative eigenvalues which are almost zero. In that case you can set these eigenvalues to zero to retain the non-negative definiteness of the covariance matrix. Furthermore, you can still interpret the result: eigenvectors contributing the significant information are associated with the biggest eigenvalues. If the list of sorted eigenvalues declines quickly there are a lot of directions which do not contribute significantly and therefore can be dropped.
The built-in R function is eigen
If your covariance matrix is A then
eigen_res <- eigen(A)
# sorted list of eigenvalues
eigen_res$values
# slightly negative eigenvalues, set them to small positive value
eigen_res$values[eigen_res$values<0] <- 1e-10
# and produce regularized covariance matrix
Areg <- eigen_res$vectors %*% diag(eigen_res$values) %*% t(eigen_res$vectors)
not non-negative definite does not mean the covariance matrix has negative correlations. It's a linear algebra equivalent of trying to take square root of negative number! You can't tell by looking at a few values of the matrix, whether it's positive definite.
Try adjusting some default values like tolerance in princomp call. Check this thread for example: How to use princomp () function in R when covariance matrix has zero's?
An alternative is to write some code of your own to perform what is called a n NIPLAS analysis. Take a look at this thread on the R-mailing list: https://stat.ethz.ch/pipermail/r-help/2006-July/110035.html
I'd even go as far as asking where did you obtain the correlation matrix? Did you construct it yourself? Does it have NAs? If you constructed xCor from your own data, do you think you can sample the data and construct a smaller xCor matrix? (say 1000X1000). All these alternatives try to drive your PCA algorithm through the 'happy path' (i.e. all matrix operations can be internally carried out without difficulties in diagonalization etc..i.e., no more 'non-negative definite error msgs)