Super-sparse principal component analyses for high-throughput genomic data
- Donghwan Lee^{1},
- Woojoo Lee^{2},
- Youngjo Lee^{1} and
- Yudi Pawitan^{2}Email author
DOI: 10.1186/1471-2105-11-296
© Lee et al; licensee BioMed Central Ltd. 2010
Received: 22 February 2010
Accepted: 2 June 2010
Published: 2 June 2010
Abstract
Background
Principal component analysis (PCA) has gained popularity as a method for the analysis of high-dimensional genomic data. However, it is often difficult to interpret the results because the principal components are linear combinations of all variables, and the coefficients (loadings) are typically nonzero. These nonzero values also reflect poor estimation of the true vector loadings; for example, for gene expression data, biologically we expect only a portion of the genes to be expressed in any tissue, and an even smaller fraction to be involved in a particular process. Sparse PCA methods have recently been introduced for reducing the number of nonzero coefficients, but these existing methods are not satisfactory for high-dimensional data applications because they still give too many nonzero coefficients.
Results
Here we propose a new PCA method that uses two innovations to produce an extremely sparse loading vector: (i) a random-effect model on the loadings that leads to an unbounded penalty at the origin and (ii) shrinkage of the singular values obtained from the singular value decomposition of the data matrix. We develop a stable computing algorithm by modifying nonlinear iterative partial least square (NIPALS) algorithm, and illustrate the method with an analysis of the NCI cancer dataset that contains 21,225 genes.
Conclusions
The new method has better performance than several existing methods, particularly in the estimation of the loading vectors.
Background
Principal component analysis (PCA) or its equivalent singular-value decomposition (SVD) is widely used for the analysis of high-dimensional data. For such gene expression data with an enormous number of variables, PCA is a useful technique for visualization, analyses and interpretation [1–4].
Lower dimensional views of data made possible, via the PCA, often give a global picture of gene regulation that would reveal more clearly, for example, a group of genes with similar or related molecular functions or cellular states, or samples of similar or connected phenotypes, etc. PCA results might be used for clustering, but bear in mind that PCA is not simply a clustering method, as it has distinct analytical properties and utilities from the clustering methods. Simple interpretation and subsequent usage of PCA results often depends on the ability to identify subsets with nonzero loadings, but this effort is hampered by the fact that the standard PCA yields nonzero loadings on all variables. If the low-dimensional projections are relatively simple, many loadings are not statistically significant, so the nonzero values reflect the high variance of the standard method. In this paper our focus on the PCA methodology is constrained to produce sparse loadings.
where D is n × p matrix with (i, i )th element d_{ i }; the columns of Z = UD = XV are the principal component scores, and the columns of the p × p matrix V are the corresponding loadings. The vector v_{ k }in (1) is the k-th column of V.
Each principal component in (2) is a linear combination of p variables, where the loadings are typically nonzero so that PCA results are often difficult to interpret. To get sparse loadings, [5] proposed to use L_{1}-penalty, which corresponds to the least-absolute shrinkage and selection operator (LASSO; [6]). [7] proposed to use the so-called elastic-net (EN) penalty. However, LASSO and EN may not be satisfactory either, because it can still gives too many nonzero coefficients. [8] proposed the smoothly-clipped absolute deviation (SCAD) penalty for oracle variable selection. Recently, in regression setting, [9] proposed a new random-effect model using a gamma scale mixture, which gives various types of penalty, including the normal-type (bell-shaped for ridge penalty), cusped-type (LASSO and SCAD-type), and a new (singular) unbounded penalty at the origin. [9] showed that the new unbounded penalty can yield very sparse estimates that are better than LASSO both in prediction and sparsity.
In this paper we use the random-effect model approach of [9] for sparse PCA (SPCA); the model gives unbounded gains for zero loadings at the origin, so it forces many estimated coefficients to zero. We improve the estimation further by shrinking the singular values from the SVD of the data; the resulting procedure is called super-sparse PCA (SSPCA). We provide some simulation studies that indicate that these SPCA methods perform better than existing ones, and illustrate their use using a cancer gene-expression dataset with 21,225 genes. We also show how to modify the ordinary NIPALS algorithm [10] to implement these methods computationally.
Results
Numerical studies
where , Σ_{22} = ϕI_{p-4}and J_{ k } is the k × k matrix of ones. Here we consider cases (n, p) = (80, 20) for n > p and (n, p) = (50, 200) for n <p. Based on 100 simulated data, we compare our new sparse PCA method using the h-likelihood (HL; See Methodology section) with the LASSO and EN penalties for both SPCA and SSPCA methods. We also tried the SCAD method but the results are very similar to LASSO, so we do not report results for SCAD.
When v_{1} = , dist (v_{1}, ) = 0.
Simulation results: estimation
SPCA | |||||||
---|---|---|---|---|---|---|---|
n | p | PCA | HL | LASSO | EN | ||
80 | 20 | 2.0 | 0.1 | 0.054 (0.010) | 0.023 (0.011) | 0.022 (0.010) | 0.025 (0.013) |
0.5 | 0.1 | 0.109 (0.021) | 0.045 (0.021) | 0.051 (0.022) | 0.055 (0.029) | ||
50 | 200 | 2.0 | 0.1 | 0.223 (0.022) | 0.029 (0.014) | 0.035 (0.015) | 0.056 (0.028) |
0.5 | 0.1 | 0.424 (0.041) | 0.062 (0.032) | 0.080 (0.033) | 0.122 (0.058) | ||
PCA* | HL | SSPCA LASSO | EN | ||||
80 | 20 | 2.0 | 0.1 | 0.055 (0.010) | 0.020 (0.009) | 0.021 (0.010) | 0.022 (0.010) |
0.5 | 0.1 | 0.113 (0.020) | 0.042 (0.020) | 0.050 (0.023) | 0.050 (0.026) | ||
50 | 200 | 2.0 | 0.1 | 0.218 (0.025) | 0.026 (0.013) | 0.032 (0.014) | 0.055 (0.030) |
0.5 | 0.1 | 0.993 (0.010) | 0.063 (0.030) | 0.083 (0.044) | 0.866 (0.000) |
Simulation results: model selection
SPCA | SSPCA | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
n | p | PCA | HL | LASSO | EN | PCA* | HL | LASSO | EN | ||
80 | 20 | 2.0 | 0.1 | 0 | 72 | 12 | 64 | 0 | 95 | 14 | 99 |
0/16 | 16/16 | 14/16 | 16/16 | 0/16 | 16/16 | 15/16 | 16/16 | ||||
0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | ||||
0.5 | 0.1 | 0 | 77 | 1 | 56 | 0 | 100 | 43 | 99 | ||
0/16 | 16/16 | 12/16 | 16/16 | 0/16 | 16/16 | 15/16 | 16/16 | ||||
0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | ||||
50 | 200 | 2.0 | 0.1 | 0 | 73 | 0 | 88 | 0 | 100 | 27 | 87 |
0/196 | 196/196 | 184.5/196 | 196/196 | 0/196 | 196/196 | 194/196 | 196/196 | ||||
0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | ||||
0.5 | 0.1 | 0 | 79 | 0 | 70 | 0 | 97 | 84 | 0 | ||
0/196 | 196/196 | 185.5/196 | 196/196 | 0/196 | 196/196 | 196/196 | 196/196 | ||||
0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 0/4 | 3/4 |
Simulation results: prediction
SSPCA | |||||||
---|---|---|---|---|---|---|---|
n | p | PCA | HL | LASSO | EN | ||
80 | 20 | 2.0 | 0.1 | 7.979 (0.831) | 7.998 (0.837) | 7.996 (0.842) | 7.970 (1.116) |
0.5 | 0.1 | 2.050 (0.213) | 2.057 (0.222) | 2.055 (0.225) | 2.088 (0.283) | ||
50 | 200 | 2.0 | 0.1 | 7.907 (1.633) | 8.242 (1.599) | 8.242 (1.601) | 8.149 (1.386) |
0.5 | 0.1 | 1.769 (0.362) | 2.143 (0.418) | 2.140 (0.414) | 2.071 (0.349) | ||
SSPCA | |||||||
PCA* | HL | LASSO | EN | ||||
80 | 20 | 2.0 | 0.1 | 7.954 (1.125) | 7.999 (0.849) | 7.997 (0.850) | 7.978 (1.115) |
0.5 | 0.1 | 2.062 (0.292) | 2.057 (0.226) | 2.057 (0.225) | 2.088 (0.280) | ||
50 | 200 | 2.0 | 0.1 | 7.564 (1.718) | 8.243 (1.593) | 8.243 (1.597) | 7.928 (1.755) |
0.5 | 0.1 | 0.242 (0.075) | 2.137 (0.452) | 2.149 (0.424) | 0.503 (0.316) |
Analysis of NCI data
In the analysis of microarray data it is often of interest to co-regulated genes, since they will point to some common involvement in molecular functions or biological processes or cellular states. PCA is a useful tool for such analyses [1–4]; since interpretation depends on comparing the relative sizes of the loading vectors, the sparse loadings in SPCA are much easier to interpret than ordinary PCA. Furthermore, the previous section also shows that SPCA has better estimation characteristics than the ordinary PCA. For illustrations we consider the so-called NCI-60 microarray data downloaded from the CellMiner program package, National Cancer Institute http://discover.nci.nih.gov/cellminer/. Only n = 59 of the 60 human cancer cell lines were used in the analysis, as one of the cell lines had missing microarray information. The cell lines consist of 9 different cancers and were used by the Developmental Therapeutics Program of the U.S. National Cancer Institute to screen > 100,000 compounds and natural products. The number of genes is p = 21,225.
Analyses of NCI data: number of zero loadings
PCA | SPCA | SSPCA | ||
---|---|---|---|---|
HL | LASSO | HL | LASSO | |
214/21225 | 7966/21225 | 650/21225 | 19965/21225 | 1144/21225 |
(1.01) | (37.53) | (3.06) | (94.06) | (5.39) |
To select the number of principal components, we use a permutation approach as follows. First, we randomly permute the expression values within each sample (row) of X to create permuted data X_{ perm }. Then PCA is performed on X_{ perm }to get the singular values . We perform P = 1000 permutations, from which we can compute the p-values of the observed d_{ k }'s. The number of principal components, k_{0}, is such that the p-value of d_{ k }'s is less than 0.001 when k ≤ k_{0}.
Analysis of NCI data: number of zero loadings
Principal component scores | Z _{1} | Z _{2} | Z _{3} | Z _{4} | Z _{5} | Z _{6} | Z _{7} | Z _{8} |
---|---|---|---|---|---|---|---|---|
PCA | ||||||||
Number of nonzero loadings | 21011 | 20385 | 19226 | 21099 | 20948 | 20817 | 20945 | 20997 |
Adjusted Variance (%) | 12.3 | 10.2 | 6.6 | 4.1 | 3.6 | 3.2 | 2.9 | 2.6 |
Cumulative adjusted Variance (%) | 12.3 | 22.5 | 29.1 | 33.2 | 36.8 | 40.0 | 42.9 | 45.5 |
SPCA - HL | ||||||||
Number of nonzero loadings | 13259 | 4086 | 15362 | 13547 | 13946 | 10445 | 9890 | 10958 |
Adjusted Variance (%) | 20.6 | 13.4 | 11.5 | 6.4 | 6.1 | 4.9 | 4.0 | 4.1 |
Cumulative adjusted Variance (%) | 20.6 | 34.0 | 45.5 | 51.9 | 58.0 | 62.9 | 66.9 | 71.0 |
SSPCA - HL | ||||||||
Number of nonzero loadings | 1260 | 681 | 375 | 290 | 47 | 58 | 33 | 3434 |
Adjusted Variance (%) | 22.3 | 8.7 | 6.1 | 6.5 | 1.3 | 0.4 | 0.0 | 1.6 |
Cumulative adjusted Variance (%) | 22.3 | 31.0 | 37.1 | 43.6 | 44.9 | 45.3 | 45.3 | 46.9 |
Gene Ontology analysis
Number | GO ID | GO Term | P-value(1) | P-value(2) | P-value(3) |
---|---|---|---|---|---|
1 | GO:0048856 | anatomical structure development | 1.6e-10 | 1.5e-09 | 4.5e-07 |
2 | GO:0009653 | anatomical structure morphogenesis | 2.9e-10 | 4.8e-06 | |
3 | GO:0008283 | cell proliferation | 1.3e-09 | ||
4 | GO:0050793 | regulation of developmental process | 1.7e-09 | 9.4e-06 | |
5 | GO:0032502 | developmental process | 3.8e-09 | 8.1e-08 | 4.9e-06 |
6 | GO:0042127 | regulation of cell proliferation | 5.8e-08 | 3.9e-06 | |
7 | GO:0048513 | organ development | 6.6e-08 | ||
8 | GO:0048869 | cellular developmental process | 1e-07 | ||
9 | GO:0048731 | system development | 1.1e-07 | 3.6e-07 | 5.3e-06 |
10 | GO:0007155 | cell adhesion | 1.3e-07 | 7.6e-07 | |
11 | GO:0022610 | biological adhesion | 1.3e-07 | 7.6e-07 | |
12 | GO:0051093 | negative regulation of developmental process | 1.9e-06 | ||
13 | GO:0048519 | negative regulation of biological process | 2.8e-06 | ||
14 | GO:0048523 | negative regulation of cellular process | 3.4e-06 | ||
15 | GO:0009605 | response to external stimulus | 2.8e-07 | ||
16 | GO:0043065 | positive regulation of apoptosis | 7.4e-06 | ||
17 | GO:0043068 | positive regulation of programmed cell death | 8.6e-06 | ||
18 | GO:0042981 | regulation of apoptosis | 9.6e-06 | ||
19 | GO:0032501 | multicellular organismal process | 1.3e-06 | ||
20 | GO:0007275 | multicellular organismal development | 4.3e-06 |
Comparative GO analyses from the ordinary PCA are given in the Additional file 1. We use the same number of top-ranking nonzero loadings as for the SSPCA, which are 1,260, 681 and 375 for the first 3 principal components, respectively. Out of these, the number of overlapping probes between the SSPCA and PCA are 462, 194 and 60. These overlaps are substantially more (up to 8 times more) than expected under random re-arrangement. However, there is sufficiently large number of distinct probes in the two methods, so the GO analyses could be different. The P-values from the SSPCA-based GO analyses are more significant than those from the ordinary PCA; this may be due to better estimation of the loadings, so that the SSPCA has better power than the ordinary PCA in revealing biologically-important grouping of genes.
Discussions and Conclusions
PCA is one of the most important tools in multivariate statistics, where it has been used, for example, in data reduction or visualization of high-dimensional data. The emergence of ultra-high dimensional data such as in genomics, involving 10,000s of variables but with only a few samples has brought new opportunities for PCA applications. However, there are new challenges also, particularly on the interpretation of results. If we treat PCA quantities such as the loading vectors as parameter estimates, the large-p-small-n applications typically produce very noisy estimates. This is obvious since the loading vectors are a statistic derived from the sample covariance matrix, and the latter is not well estimated.
It is well known that improved estimation can come by imposing constraints, and in this case sparsity constraint is natural. As PCA scores capture some underlying biological processes, we do not expect every gene in the genome to be involved. Out of possibly 30,000 genes we can expect only a small fraction, probably less than 1,000, to be involved in a cellular process. Hence sparsity constraint can help in reducing the number of loading parameters to estimate.
Imposing statistical constraints can be achieved by applying a penalty approach as used by the ridge regression or the LASSO methods [6]. In this paper we have investigated a random-effect model approach using a gamma scale mixture, which leads to a class of penalties that includes the ridge and LASSO penalties as special cases. One significant property is that it can produce unbounded penalties on the origin, which leads to stronger constraints and more sparse estimates. From our results it seems clear that the penalty approach alone is not able to yield sufficiently sparse PCA for high-dimensional genomic data. Additionally we also need the shrinkage on the singular values of the data matrix. In simulation studies we show that the proposed methods outperform existing methods both in estimation and model selection. Hence we believe that the new SPCA methods are promising tools for high-dimensional data analyses.
For future works, it will be of interest to apply super-sparse technique in this paper to locally-linear methods of dimensionality reduction (e.g. [13]]), partial-least squares (PLS) regression and classification methods (e.g. [14]), or other high-throughput data analysis method where dimensionality reduction is used (e.g. [15]).
Methodology
NIPALS algorithm for PCA
Standard algorithms for SVD (e.g. [16]) give the PCA loadings, but if p is large and we only want to obtain a few singular vectors, the computation to obtain the whole set of singular vectors may be impractical. Furthermore, with these algorithms it is not obvious how to impose sparsity on the loadings. [10] described a NIPALS algorithm that works like a power method ([17], p.523) for obtaining the largest eigenvalue of a matrix and its associated eigenvector. The NIPALS algorithm computes only a singular vector at a time, so it is efficient if we only want to extract a few singular vectors. Also the steps are recognizable in regression terms, so the algorithm is immediately amenable to random-effect modification as needed to obtain the various SPCA methods proposed in this paper.
- 1.
- 2.
- 3.
Find z _{1}: z _{1} ← X ^{ T } v _{1}
- 4.
Repeat steps 1 to 3 until convergence.
To obtain the second-largest singular value, first compute residual , then apply the NIPALS algorithm above by replacing X by X_{2}.
Sparse PCA via random-effect models
where p_{ λ }(·) is a penalty function. For example, p_{ λ }(|v_{1j}|) = λ|v_{1j}| gives LASSO, gives ridge, and gives EN, where λ, λ_{1} and λ_{2} are tuning parameters. For the prediction the ridge-type penalty is effective and for sparse estimation the LASSO-type penalty is recommended, so that EN [19] has been recommended as a compromise between the ridge and LASSO methods. [7] proposed to use EN for sparse (SPCA), but it gives less sparse estimates than LASSO.
using and λ = ϕ/θ. In random-effect model approach, the penalty function p_{ λ }(|v_{1j}|) stems from a probabilistic model . As noted previously the proposed penalty p_{ λ }(|v_{1j}|) is nonconvex. However, by expressing the model for p_{ λ }(|v_{1j}|) hierarchically as (i) v_{1j}|u_{ j }is normal and (ii) u_{ j }is gamma, both models can be fitted by convex GLM optimizations. Thus, the proposed IWLS algorithm overcomes the difficulties of a nonconvex optimization by solving two-interlinked convex optimizations [22].
Other methods for sparse principal component analysis
for deriving the first sparse loading vector v_{1}. Given θ, this optimization problem becomes a naive elastic net problem for v_{1}. Given v_{1} , θ can updated from SVD of X^{ T }X v_{1}. These two steps are repeated until v_{1} converges. Following [25], (11) is different from our objective function (5) even when we use the same penalty function. In fact, (5) is very close to the objective function of [26], but we put the normalization constraint of the loading inside iterated procedure so that it could make a different result. In this paper, we used the function spca() in the R-package elasticnet for the EN method in the simulation studies.
Condition-number constraint for SPCA
where Λ = diag(l_{1}, ..., l_{ p }and for i = 1, ..., p is the eigenvalues of S_{ X }in non-increasing order (l_{ 1 } ≥ … ≥ l_{ p }≥ 0). Let the p × 1 random vectors x_{1}, …, x_{ n }be rows of X that have zero mean vector and true covariance matrix Σ with the non-increasing eigenvalues, λ_{1} ≥ ... ≥ λ_{ p }. When our goal is to estimate Σ, the sample covariance matrix S_{ X }can be used. Many applications require a covariance estimate that is not only invertible but also well-conditioned. An immediate problem arises when n <p, where the estimate S_{ X }is singular. Even when n > p, the eigen-structure tends to be systematically distorted unless p/n is small [27], resulting in ill-conditioned estimator for Σ.
where the eigenvalues . To estimate the shrinkage parameter κ_{max}, they proposed to use the K-fold cross validation.
where D* is n × p matrix with (i, i)th diagonal element . Thus, for condition-number constrained PCA we use X* instead of the original data matrix X. As the procedure yields extremely sparse loading vectors, we call it SSPCA, for super-sparse PCA.
[29] considered the estimation of covariance matrix when p is not very large. However, for large p such as over 10,000 in gene expression data, it becomes computationally too intensive. Because the aim is to obtain a few singular vectors, not whole p singular vectors, when p > n in this paper we propose to apply the above algorithm to X^{ T }and the results are transformed back appropriately.
Modified NIPALS algorithm for SPCA and SSPCA
where is defined in (9). For SSPCA we also apply this modified step, but replace X by X* defined in (14).
Tuning parameter selection
where is the estimated loadings from the k th training sets (the whole data without the k th validation set) and S_{X[k]}is the sample variance based on the k th validation set. For the numerical studies in Section we use K = 5.
Declarations
Acknowledgements
This research is partially funded by a grant for the Swedish Science Foundation.
Authors’ Affiliations
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