Example dataset - Hapmap Phase III

Description

The example dataset is from the Hapmap Phase III project (https://www.ncbi.nlm.nih.gov/variation/news/NCBI_retiring_HapMap/). Our training sample consisted of unrelated individuals from two different populations: a) Utah residents with Northern and Western European ancestry (CEU), and b) Toscans in Italy (TSI). We present the eigenvalues and PC scores obtained from performing PCA on the SNPs on chromosome 7.

Format

This example dataset is a list containing the following elements:

train.eval

Sample eigenvalues of the training sample.

trainscore

PC scores of the training sample. This has PC1 and PC2 scores for 198 observations.

testscore

We obtained the predicted scores by leaving one observation out at a time, applying PCA to the rest of the data and then predicting the PC score of the left out observation. This has PC1 and PC2 scores of 198 observations.

nSamp

Number of observations in the training set = 198.

nSNP

Number of SNPs on chromosome 7.

High-dimensional PCA estimation

Description

Estimates the population eigenvalues, angles between the sample and population eigenvectors, correlations between the sample and population PC scores, and the asymptotic shrinkage factors. Three different estimation methods can be used.

Usage

hdpc_est(samp.eval, p, n, method = c("d.gsp", "l.gsp", "osp"), 
n.spikes, n.spikes.max, n.spikes.out, nonspikes.out = FALSE, smooth = TRUE)

Arguments

Details

The different choices for method are:

• "d.gsp": d-estimation method based on the Generalized Spiked Population (GSP) model.

• "l.gsp": \lambda-estimation method based on the GSP model.

• "osp": Estimation method based on the Ordinary Spiked Population (OSP) model.

At least one of n.spikes and n.spikes.max must be provided. If n.spikes is provided then n.spikes.max is ignored, else n.spikes.max is used to find out the number of distant spikes using select.nspike.

The argument nonspikes.out is ignored if method="d.gsp".

The argument smooth is useful when the user assumes the population spectral distribution to be continuous.

Value

Author(s)

Rounak Dey, deyrnk@umich.edu

References

Dey, R. and Lee, S. (2019). Asymptotic properties of principal component analysis and shrinkage-bias adjustment under the generalized spiked population model. Journal of Multivariate Analysis, Vol 173, 145-164.

See Also

select.nspike,pc_adjust

Examples

data(hapmap)
#n = 198, p = 75435 for this data

####################################################
## Not run: 
train.eval<-hapmap$train.eval
n<-hapmap$nSamp
p<-hapmap$nSNP

m<-select.nspike(train.eval,p,n,n.spikes.max=10,evals.out=FALSE)$n.spikes
out<-hdpc_est(train.eval, p, n, method = "d.gsp", 
n.spikes=m, n.spikes.out=2, nonspikes.out = FALSE)	#Output 2 spikes, no non-spike

out<-hdpc_est(train.eval, p, n, method = "l.gsp", 
n.spikes=m, nonspikes.out = FALSE)	#Output m many spikes, no non-spike

out<-hdpc_est(train.eval, p, n, method = "l.gsp", 
n.spikes.max=10, nonspikes.out = TRUE)	#Output all eigenvalues

out<-hdpc_est(train.eval, p, n, method = "osp", 
n.spikes=m, n.spikes.out=2, nonspikes.out = TRUE)	#Output m many spikes, no non-spike

## End(Not run)

Adjusting shrinkage in PC scores

Description

Adjusts the shrinkage bias in the predicted PC scores based on the estimated shrinkage factors.

Usage

pc_adjust(train.eval, p, n, test.scores, method = c("d.gsp", "l.gsp", "osp"),
n.spikes, n.spikes.max, smooth = TRUE)

Arguments

Details

The different choices for method are:

• "d.gsp": d-estimation method based on the Generalized Spiked Population (GSP) model.

• "l.gsp": \lambda-estimation method based on the GSP model.

• "osp": Estimation method based on the Ordinary Spiked Population (OSP) model.

The (i,j)^{th} element of test.scores should denote the j^{th} predicted PC score for the i^{th} subject in the test sample.

At least one of n.spikes and n.spikes.max must be provided. If n.spikes is provided then n.spikes.max is ignored, else n.spikes.max is used to find out the number of distant spikes using select.nspike.

The argument nonspikes.out is ignored if method="d.gsp" or "osp".

The argument smooth is useful when the user assumes the population spectral distribution to be continuous.

Value

A matrix containing the bias-adjusted PC scores. The dimension of the matrix is the same as the dimension of test.scores.

A printed message shows the number of top PCs that were adjusted for shrinkage bias.

Author(s)

Rounak Dey, deyrnk@umich.edu

References

Dey, R. and Lee, S. (2019). Asymptotic properties of principal component analysis and shrinkage-bias adjustment under the generalized spiked population model. Journal of Multivariate Analysis, Vol 173, 145-164.

See Also

hdpc_est,select.nspike

Examples

data(hapmap)
#n = 198, p = 75435 for this data

####################################################
## Not run: 
#First estimate the number of spikes and then adjust test scores based on that
train.eval<-hapmap$train.eval
n<-hapmap$nSamp
p<-hapmap$nSNP
trainscore<-hapmap$trainscore
testscore<-hapmap$testscore

m<-select.nspike(train.eval,p,n,n.spikes.max=10,evals.out=FALSE)$n.spikes
score.adj.o1<-pc_adjust(train.eval,p,n,testscore,method="osp",n.spikes=m)
score.adj.d1<-pc_adjust(train.eval,p,n,testscore,method="d.gsp",n.spikes=m)
score.adj.l1<-pc_adjust(train.eval,p,n,testscore,method="l.gsp",n.spikes=m)

#Or you can provide an upper bound n.spikes.max
score.adj.o2<-pc_adjust(train.eval,p,n,testscore,method="osp",n.spikes.max=10)
score.adj.d2<-pc_adjust(train.eval,p,n,testscore,method="d.gsp",n.spikes.max=10)
score.adj.l2<-pc_adjust(train.eval,p,n,testscore,method="l.gsp",n.spikes.max=10)

#Plot the training score, test score, and adjusted scores
plot(trainscore,pch=19)
points(testscore,col='blue',pch=19)
points(score.adj.o1,col='red',pch=19)
points(score.adj.d2,col='green',pch=19)

## End(Not run)

Finding Distant Spikes

Description

Estimates the number of distant spikes in the population based on the Generalized Spiked Population model. A finite upper bound (n.spikes.max) of the number of distant spikes must be provided.

Usage

select.nspike(samp.eval, p, n, n.spikes.max, evals.out = FALSE, smooth = TRUE)

Arguments

Details

The function searches between 0 and n.spikes.max to find out the number of distant spikes in the population. It also estimates both non-spiked and spiked eigenvalues based on the \lambda-estimation method.

The argument smooth is useful when the user assumes the population spectral distribution to be continuous.

Value

Author(s)

Rounak Dey, deyrnk@umich.edu

References

Dey, R. and Lee, S. (2019). Asymptotic properties of principal component analysis and shrinkage-bias adjustment under the generalized spiked population model. Journal of Multivariate Analysis, Vol 173, 145-164.

See Also

hdpc_est,pc_adjust

Examples

data(hapmap)
#n = 198, p = 75435 for this data

####################################################
## Not run: 
#If you just want the estimated number of spikes
train.eval<-hapmap$train.eval
n<-hapmap$nSamp
p<-hapmap$nSNP

select.nspike(train.eval,p,n,n.spikes.max=10,evals.out=FALSE)

#If you want the estimated spikes and non-spikes
out<-select.nspike(train.eval,p,n,n.spikes.max=10,evals.out=TRUE)

## End(Not run)