Penalized weighted lowrank approximation for robust recovery of recurrent copy number variations
 Xiaoli Gao^{1}Email author
Received: 22 September 2015
Accepted: 23 November 2015
Published: 10 December 2015
Abstract
Background
Copy number variation (CNV) analysis has become one of the most important research areas for understanding complex disease. With increasing resolution of arraybased comparative genomic hybridization (aCGH) arrays, more and more raw copy number data are collected for multiple arrays. It is natural to realize the coexistence of both recurrent and individualspecific CNVs, together with the possible data contamination during the data generation process. Therefore, there is a great need for an efficient and robust statistical model for simultaneous recovery of both recurrent and individualspecific CNVs.
Result
We develop a penalized weighted lowrank approximation method (WPLA) for robust recovery of recurrent CNVs. In particular, we formulate multiple aCGH arrays into a realization of a hidden lowrank matrix with some random noises and let an additional weight matrix account for those individualspecific effects. Thus, we do not restrict the random noise to be normally distributed, or even homogeneous. We show its performance through three real datasets and twelve synthetic datasets from different types of recurrent CNV regions associated with either normal random errors or heavily contaminated errors.
Conclusion
Our numerical experiments have demonstrated that the WPLA can successfully recover the recurrent CNV patterns from raw data under different scenarios. Compared with two other recent methods, it performs the best regarding its ability to simultaneously detect both recurrent and individualspecific CNVs under normal random errors. More importantly, the WPLA is the only method which can effectively recover the recurrent CNVs region when the data is heavily contaminated.
Keywords
Background
Copynumber variations (CNVs) are changes in the number of copies of DNA in some genome regions. The size of those variations including both deletion and amplification can vary from size of 1kb to a complete chromosome arm. CNVs in genomic DNA have been considered as a major source of genomic variation [1, 2] and can be linked to the susceptibility or resistance to certain disease such as cancer, Alzheimer and Parkinson’s disease [3–5].
Three main types of technologies have been developed to detect CNVs: array comparative genomic hybridization (aCGH) arrays [6, 7], SNP genotyping arrays [8, 9] and genome resequencing [10–13]. Among these technologies, aCGHs have remained the most frequently used methods for CNVs identification and genotyping of personal CNVs [9, 14, 15], because of their accuracy and costeffectiveness [16]. A typical aCGH experiment includes DNA labeling, hybridization, and scanning. During the experiment, the fluorescence signal intensities from both the test and sample DNA at given probes are measured. After some appropriate preprocessing procedures including normalization, the raw DNA copy number data from an aCGH experiment is generally in the form of log2 ratios of those intensities between test and reference DNA samples. Thus, a probe with a positive (negative) log2 ratio indicates a possible occurring of DNA amplification (deletion), while a zero value means no copy number variation is involved at this probe, that is, the copy number in the target agrees with one in the control.
However, the observed log2 intensities are often noisy surrogates of the true copy number. The detection of CNVs is to recover the underlying true copy number from the random noise at those measured positions. Many methods have been developed to analyze singlesample aCGH data, including the changepoint models [17–20], smoothing methods [21–24], Haarbased wavelets [25], and Hidden Markov models [26]. Lai et al. [27] reviewed and compared the performance of some of those existing approaches.
The aforementioned methods analyze DNA CNVs data for each sample individually, which may be inefficient in the search for diseasecritical genes. Recently, simultaneous analysis of multiple aCGH arrays draws considerable attention since DNA copy number regions shared by multiple samples are more likely to harbor diseasecritical genes [7, 28–30]. In a multiple sample aCGH data analysis, our interest is to recover those recurrent CNV regions from the noisy data. Here a recurrent CNV region is defined as a genome region where CNVs exist for a group of samples [31].
Many of above listed individual sample CNV detection methods have been extended to recurrent CNV detection for multiple samples. Examples include hypothesis testing methods [28, 32, 33], Multiplechain HMM [34], joint segmentation [35], and matrix factorization methods [36, 37].
Most of previous methods for recurrent CNVs detection either assume the random noise to be Gaussian distributed or ignore the coexistence of individualspecific CNVs. A multiple aCGH data is often contaminated due to either a nonnormal random noise or the coexistence of outliers, i.e. heteroscedasticity, among some probes. In statistics, an outlier is an observation that is distant from other observations [38]. Some outliers may be due to intrinsic variability of the data (individualspecific effect), while others may indicate errors such as experimental error and data entry error. In a raw log2 ratio copy number data from multiple aCGH arrays, it is natural to consider the coexistence of two types of CNVs regions: recurrent CNVs regions among multiple samples and individualspecific CNVs belonging to different probes of different samples. Although the detection of recurrent CNVs is our main target, identifying some individualspecific CNVs can also help to improve our understanding of complex diseases. Moreover, the existence or mixture of individual CNVs may eventually corrupt the model fitting of recurrent CNVs, without being addressed.
Finding common or recurrent CNA regions from a noisy data remains a challenge both computationally and conceptually, not mentioning the noisy data is contaminated by both individualspecific and nonGaussian random errors. There are much less robust methods on CNV detection from individual samples or multiple samples [39, 40]. Very recently, researchers formulated the recurrent CNV detection into a matrix decomposition problem with hidden signals being lowrank [41, 42]. For example, to address the data contamination, [41] included a individualspecific effect matrix into the model and consider the observed CNV data matrix D as an addition of three matrices: lowrank true recurrent CNVs matrix X, individualspecific effect matrix E, and random noise matrix ε. Penalization optimization are adopted to recover X and E iteratively. In particular, a soft threshold operator [43] is adopted to update E. We denote this method as RPLA in this paper to differentiae it from ours. Mohammadi et al. [44] proposed another robust recurrent CNVs detection method using a Correntropy induced metric. We denote this method as CPLA in this paper. After using a HalfQuadratic technique [37, 45], CPLA is eventually reduced into a similar optimization problem as the one for RPLA. Instead of solving E using a soft threshold operator, CPLA updates E using a minimizer function δ during the iteration.
As explained in the last paragraph, both RPLA and CPLA introduce an individualspecific effect matrix E in the model needed to be estimated. She and Owen [46] has demonstrated by linear regression analysis that a Lasso penalty on the mean shift parameter cannot reduce both the masking (outliers are not detected) and swamping (normal observations are incorrectly identified as outliers) effects for the outlier detection. This justifies some limitations of RPLA, where E plays the same role as outliers in multiple regression in [46]. Additionally, the minimizer function ρ used in CPLA does not encourage the sparsity of the matrix E. Thus, CPLA itself does not have any ability of detecting individualspecific CNVs. This phenomenon will be further addressed in Section ‘Results and discussion’.
In this paper, we propose a novel method for robust recovery of the recurrent CNVs using a penalized weighted lowrank approximation (WPLA). Instead of using a mean shift parameter to represent each individual effect, we consider to assign a weight parameter to each probe of every sample. Thus, all the individual effects are related to a weight matrix W, where a weight value of 1 indicates a normal probe for a normal sample without individualspecific effect, and a weight value less than 1 indicates possible individualspecific effect occurring at this probe. We propose to shrink all individualspecific effects in the direction of the recurrent effects by penalizing the weight matrix W. As a result, a robust detection of recurrent CNVs is obtained by simultaneous identification of both individualspecific CNVs and recurrent ones.

It can perform both the recurrent CNV and individual effect detection simultaneously and efficiently;

It has strong robustness in terms of recurrent CNV detection. When the data is heavily contaminated, WPLA performs consistently better than the two aforementioned methods (CPLA and RPLA).
The rest of the paper is organized as follows. In Section ‘Methods’, we introduce our model formulation with some properties. We also provide its computation algorithm in this section. In Section ‘Results and discussion’, we demonstrate the performance of WPLA by both synthetic data analysis in multiple scenarios and two real data analysis. Finally, we conclude our paper with some discussions in Section ‘Conclusions’.
Methods
Formulation
where ε _{ ij } is assumed to have mean 0 and variance \(\sigma _{\textit {ij}}^{2}=\sigma ^{2}/w_{\textit {ij}}^{2}\) for all i and j. Here 0<w _{ ij }≤1 is a weight parameter at probe j of sample i. A major relaxation of model (1) from existing recurrent CNVs detection methods is that we do not restrict all random errors to be homogenously distributed with the same variance. In fact, the variance can go to infinity when w _{ ij } goes to 0.

(P1) For each sample, the hidden log2 intensities tend to be the same at nearby probes. This property is incorporated into the model by assuming each row in the hidden signal matrix, x _{ i }, to be piecewise constant with only a few breakpoints.

(P2) Most samples include recurrent CNV, and the number of unique recurrent CNV regions is small. This property is incorporated into the model by assuming the hidden signal matrix X to be lowrank.

(P3) Most probes are observed with homogeneous random errors with mean 0, some of them may be contaminated and include individualspecific effects. This feature is incorporated into the model by assuming most w _{ ij }s to be 1, except a few of them being smaller than 1.

The hidden signal total variation term \(\alpha _{2}\\mathbf {x}_{i}\_{\text {TV}}=\alpha _{2}\sum _{j=2}^{p}x_{i,j}x_{i,j1}\) is to enforce a piecewise constant estimation of all x _{ i } along the sequence for all 1≤i≤n, where α _{2}>0 is a tuning parameter controlling the the number of breakpoints among all n sequences. The larger α _{2} is, the less number of breakpoints are encouraged. This term is to realize the above feature P1.

The nuclear norm term \(\alpha _{1}\\mathbf {X}\_{*}=\alpha _{1}\sum _{i=1}^{r} \sigma _{i}\) is adopted to realize the above feature P2 and obtain a reduced rank estimation of X. Here σ _{ i } for 1≤i≤r are all r singular values of X and α _{1}>0 is the tuning parameter controlling the effective rank of matrix X. The larger α _{1} is, the lower rank of X with stronger recurrent properties is encouraged.

The last ℓ _{1} norm penalty \(\beta \\log (\mathbf {W})\_{1}=\beta \sum _{i=1}^{n}\) \(\sum _{j=1}^{p}\log (w_{\textit {ij}})\) is adopted to control the number of heterogeneous CNVs with individual effects. The larger β is, the less the individual effects is encouraged. Due to the fact that 1−w _{ ij }≈ log(w _{ ij }) when w _{ ij } is close to 1, this term can be also replaced by an alternative β∥1−W∥_{1}, where 1 is a n×p matrix with all elements being 1.
Robust property
This ρ _{ β }(t) function in (5) produces strong robust property since \(\frac {d \rho _{\beta }(\cdot)}{dt}\) is approaching 0 when t→∞. An additional pdf file shows more details (see Section 1 in Additional file 1).
Adaptive WPLA
where \(\beta _{\textit {ij}}=\beta /\sqrt {\log (w^{(0)}_{\textit {ij}})}\) and \(w^{(0)}_{\textit {ij}}\) is an initial value of w _{ ij }. In implementation, we can obtain \(w^{(0)}_{\textit {ij}}\)s from \(x^{(0)}_{\textit {ij}}\)s using (8) to be presented in Section ‘Algorithm’, where \(x^{(0)}_{\textit {ij}}\) is an initial estimate of x _{ ij }. For example, we can obtain \(x^{(0)}_{\textit {ij}}\)s from either RPLA or CPLA. If 0 occurs at the denominator, we replace it by 0.001 by convention. In the next section, we will present more details on choice of initial values following the computation algorithm for (6).
Algorithm
Once W is fixed, solving X is a convex optimization problem. Due to the coexistence of both nuclear norm and total variation, instead of solving the optimization problem directly, we adopt the Alternating Direction Method of Multipliers (ADMM, [43]) in our algorithm. ADMM was also used in both RPLA and CPLA. We now divide the optimization problem in (3) in separate steps. Some more details on mathematical computations can be found in Additional file 1 (Section 3).
where Y is the dual variable matrix, Z is an auxiliary variable matrix, and <Y,X−Z>=Tr(Y ^{′}(X−Z)) denotes the inner product between Y and X−Z. Here ρ is a tuning parameter controlling the convergence of the algorithm (ρ is adaptively tuned during the iteration; one can refer Eq. (10) in Section 3.4.1 of [43] for more details).
An additional pdf file shows more detailed computation (see Section 3 in Additional file 1).
where \(\mathbf {C}=(1+\rho)^{1}\sum _{i=1}^{n}\sum _{j=1}^{p} a_{\textit {ij}}b_{\textit {ij}}\mathbf { A}_{\textit {ij}}=(1+\rho)^{1}\mathbf {A}\cdot \mathbf { A}\cdot \mathbf { B}\) and α1′=α _{1}/(1+ρ).
where r is the rank of C and (a)_{+}= max{a,0}.
Thus for each 1≤i≤n, we can update z _{ i } using the fused lasso signal approximation algorithm in [49].
We summarize the above iterations in the following Algorithm 1.
The objective function in (3) is a biconvexity optimization problem. Two matrices W and X are alternatingly updated with the other held fixed until reaching convergence. However, the initial points X ^{(0)} and W ^{(0)} may affect the final solution. Rousseeuw and Driessen [50] suggests a multistart iterative strategy in general. In our applications, we found initial values generated from both RPLA and CPLA work very well.
Tuning parameter selection
Tuning parameter selection is always challenging in penalized optimization. It is computationally expensive to extensively search an optimal (α _{1},α _{2},β) in (7). Here we propose some strategies on selecting three types tuning parameters. This strategy works surprisingly well during the implementation of both synthetic data and two real data analysis.
1) Type 1 tuning parameters including α _{1} and α _{2} control the hidden recurrent copy number structure. Between them, α _{1} is the most important one. We follow the discussions in [41] and let \(\alpha _{1}=(\sqrt {n}+\sqrt {p})\widehat {\sigma }\), where \(\widehat {\sigma }=1.48MAD\), where M A D=median{D−median(D)}. After α _{1} is determined, we fix α _{2}=0.1α _{1}.
2) Type 2 tuning parameter includes β, controlling the ratio of individual CNVs (outliers). Refer to (Gao X, Fang Y: Penalized weighted least squares for outlier detection and robust regression, Under revision), we provide a Bayesian understanding of the weight parameter penalty term in a multivariate regression framework, where the robust estimation of the coefficients vector is a posterior mode if ν=1/w when ν has a Type I Pareto prior distribution π(ν)∝ν ^{1−β } I(ν≥1), where β _{0}=1 is the uniform prior and β _{0}=2 is the Jeffrey’s prior. This motivates us to select the tuning parameter \(\beta =\widehat {\sigma }^{2}\beta _{0}\) with β _{0}=1 or 2, where \(\widehat {\sigma }\) is a robust measurement of the noises’ variability. For example, we let \(\widehat {\sigma }=1.4826 MAD\) in real implementation [51]. An additional pdf file shows more details on this Bayesian understanding (see Section 2 in Additional file 1).
3) Type 3 tuning parameter ρ is the parameter controls the convergence of the algorithm. We let ρ=0.1σ _{ D }, where σ _{ D } is the maximum singular value of matrix D, and adaptively tuned during the iteration following [43]. On the one hand, if the primal residual (the maximal singular value of X−Z) is 10 times of the dual residual (the maximal singular value of \(\rho ({\mathbf Z}\widetilde {\mathbf Z})\)), then ρ is doubled. On the other hand, if the dual residual is 10 times of the primal residual, then ρ is halved. We keep updating ρ during the iteration steps in Algorithm 1 until converge.
Results and discussion
Results
Synthetic data sets
This means that the synthetic data is consistent with the mean shift model studied in RPLA, and our WPLA model is actually misspecified under (13). However, we will show that WPLA still performs the best among all these three methods in all numerical experiments.

(Recurrent CNVs: X) Six different types recurrent CNV regions are listed in Table 1. Here all gains (+) and losses () have the true signal value of 1 and 1, respectively.Table 1
12 Synthetic data settings: “+” means gains, “” means losses
Case 1 & 7
Case 2 & 8
Sample
[76, 85]
[86, 95]
[96, 105]
[106, 125]
[76, 85]
[86, 95]
[96, 105]
[106, 125]
110
+
+


1120
+
+


2130
+
+
+
+
3140
+
+
+
+
4150
+
+
+
+
Case 3 & 9
Case 4 & 10
Sample
[76, 85]
[86, 95]
[96, 105]
[106, 125]
[76, 85]
[86, 95]
[96, 105]
[106, 125]
110


+
+
+
1120


+
+
+
2130
+
+
+
+
3140
+
+


4150
+
+


Case 5 & 11
Case 6 & 12
Sample
[76, 85]
[86, 95]
[96, 105]
[106, 125]
[76, 85]
[86, 95]
[96, 105]
[106, 125]
110
+
+
+
+
+
+
1120



2130



3140
+
+
+
+
+
4150
+
+
+



(Individual CNVs: E) Each sample includes an individual CNV region with a length of 20 probes. This region is randomly located outside of recurrent regions, with intensities randomly selected from {2, 1, 1, 2}.

(Random error: ε) Two types of noises are considered for all probes: 1) Case 16 have Gaussian noises with σ=0.3; 2) Case 712 have contaminated noises from 0.5t(1), where t(1) indicates a t distribution with degrees of freedom of 1. We take t(1) distribution as a heavily contaminated example since tdistribution is often used for quantifying the thicker tails of genetic data as mentioned in [52, 53]. In addition, no finite variance for t(1). Thus Case 712 are corresponding heavily contaminated scenarios parallel to Case 16.
We have the following findings. 1) When the random noise is Gaussian, all three methods perform well in terms of the recurrent CNVs detection; there is still some advantages of WPLA and CPLA over RPLA by producing smaller false discovery rate. 2) Among all three methods, CPLA performs the worst in terms of individual CNVs detection. This is not surprising since CPLA is only designed for robust detection of recurrent CNVs, not for individual CNVs. 3) When the data is heavily contaminated in addition to the existence of individual CNVs, WPLA beats both two other methods considerablyby producing much higher ROC curves and much lower FDR curves. In this situation, CPLA loses control of detecting recurrent CNVs completely and treat the input data as noise most of the time.Therefore, some of CPLA curves disappear in Fig. 3 since CPLA does not produce any recurrent CNVs region when data is heavily contaminated. Overall, WPLA has strong robust properties when data is heavily contaminated; it is as efficient as existing robust methods when the random noise is normally distributed. In addition, WPLA has the ability of simultaneous detection of both recurrent and individual CNVs.
Real applications
We apply WPLA to three independent real data sets: chromosome 17 from Pollack data [54], chromosome 17 from Bekhouche data [55], and chromosome 11 from Wang data [16]. The first two data sets were also analyzed by [41]. The Pollack data consists of log2 intensities at 382 probes from 44 breast tumors samples, while the Bekhouche data is a much larger data set including 7727 probes from 173 breast tumors. Compared with the other two data sets, Wang data is the newest with the highest resolution, and has much smaller sample size with only 12 pig samples: one Asian wild boar population, six Chinese indigenous breeds, and two European commercial breeds. Due to the small sample size and high resolution, we only analyze one segment of chromosome 11 (between 62,001,001 and 71,997,810), where three CNVs regions were confirmed by the quantitative realtime PCR analysis. Thus, Wang data consists of 3766 probes from 12 samples.
For the Pollack data and Wang data, three methods produce more discrepant CNV regions, where results from WPLA are smoother than two other methods. For the Bekhouche data, all three methods produce more consistent recurrent CNV regions. Overall, among all detected recurrent CNV regions, WPLA turns to produce higher frequency ratio among all samples, which is reasonable for recurrent CNVs detection. For example, WPLA detects the recurrent CNVs region where gene ERBB2 and C17orf37 are located (at around probe 3460). This result is consistent with scientific discoveries since both of those two genes are claimed to be related breast cancer [41, 56]. It is worthwhile to point it out that WPLA detects this recurrent region at a much higher frequency ratio than both RPLA and CPLA.
Discussion
All numerical experiments have been done under fixed tuning parameters. There is still some potential improvement if all parameters are optimally tuned. However, the computation must be much more expensive. It is worthwhile to investigate some more effective ways of tuning parameter selection methods in future studies.
Conclusions
In this paper, we propose a novel robust method for recurrent copy number variation detection. This method is unique by assigning a weight parameter to each probe of every sample. Thus, all the individual effects are related to a weight matrix W, which is estimated data adaptively, together with the lowrank approximation. As a result, a robust detection of recurrent CNVs is obtained by shrinking all weights from some small values (because of individualspecific effects) to 1 (no individual effects). This proposed method has two important results: efficient detection of both recurrent CNVs and individualspecific CNVs, strong robustness in dealing with severe data contamination.
We have applied the proposed method to three real data sets and twelve synthetic data sets generated from six different types of recurrent CNVs associated with either normal random errors or heavily contaminated errors. The numerical experiment has demonstrated its superior performance of recovering recurrent CNV patterns from raw data under different scenarios.Compared with two other recent methods, it has the best ability of simultaneous detection of both recurrent and individualspecific CNVs under normal random errors. More importantly, the proposed method is the only one which can effectively recover the recurrent CNVs region when the data is heavily contaminated.
Declarations
Acknowledgements
The author gratefully acknowledges Simons Foundation (#359337) and UNC Greensboro (New Faculty Grant) for their support in this Project.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
Authors’ Affiliations
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