Gene expression prediction using lowrank matrix completion
 Arnav Kapur^{1}Email author,
 Kshitij Marwah^{1} and
 Gil Alterovitz^{1, 2}
https://doi.org/10.1186/s1285901611066
© Kapur et al. 2016
Received: 20 November 2015
Accepted: 28 May 2016
Published: 17 June 2016
Abstract
Background
An exponential growth of highthroughput biological information and data has occurred in the past decade, supported by technologies, such as microarrays and RNASeq. Most data generated using such methods are used to encode large amounts of rich information, and determine diagnostic and prognostic biomarkers. Although data storage costs have reduced, process of capturing data using aforementioned technologies is still expensive. Moreover, the time required for the assay, from sample preparation to raw value measurement is excessive (in the order of days). There is an opportunity to reduce both the cost and time for generating such expression datasets.
Results
We propose a framework in which complete gene expression values can be reliably predicted insilico from partial measurements. This is achieved by modelling expression data as a lowrank matrix and then applying recently discovered techniques of matrix completion by using nonlinear convex optimisation. We evaluated prediction of gene expression data based on 133 studies, sourced from a combined total of 10,921 samples. It is shown that such datasets can be constructed with a low relative error even at high missing value rates (>50 %), and that such predicted datasets can be reliably used as surrogates for further analysis.
Conclusion
This method has potentially farreaching applications including how biomedical data is sourced and generated, and transcriptomic prediction by optimisation. We show that gene expression data can be computationally constructed, thereby potentially reducing the costs of gene expression profiling. In conclusion, this method shows great promise of opening new avenues in research on lowrank matrix completion in biological sciences.
Keywords
Background
A tremendous growth in biomedical information and datasets has been observed in the last two decades [1]. This growth is supported by the development of new technologies that profile gene expressions in an automated manner. Such technologies have significantly evolved in the past 20 years, from initially monitoring less than 50 features per slide [2] to whole genome expression analysis with new generation microarrays having more than 10^{6} features, such as GeneChip oligonucleotide probe based arrays and high density bead arrays [3]. This evolution has persisted in the form of nextgeneration sequencing (NGS) methods being used to quantify RNA in a sample [4] and have proven to be advantageous in terms of performing discoverybased experiments and having a larger dynamic range.

The cost of commercial RNAseq and microarray services remain prohibitive and limits their wider adoption in research and clinical applications alike.

There is a challenge in data storage requirements and high analysis complexity that is associated with datasets sourced from nextgeneration sequencing (NGS) methods.

Despite microarray experiments being more economical in terms of cost and data volume, missing data is an inevitable phenomenon in such experiments, and adversely affects downstream analysis. The prevailing missing value imputation algorithms successfully recover expression levels albeit at low missing value rates (only up to 15 % of the expression values).
As of 2015, commercial microarray services cost approximately $450 per sample, and prices vary for different platforms [5–7]. Profiling is generally performed using multiple tests to increase the statistical power of the measurement [8], thus increasing the combined cost of the experiment. The MammaPrint test, a microarray based gene expression test used to predict the risk of recurrence in patients with breast cancer, costs approximately $4,200. Similarly, the Oncotype DX costs more than $3,000 [9]. RNASeq is even more expensive than conventional DNA microarray based tests used for gene expression measurements. The cost of RNA sequencing services directly increases with number of reads per sample [10]. There is an upward trend to increase the capacity of such platforms, with manufacturers pushing for higher number of reads and probes per sample, inadvertently increasing the cost per sample. We explore if there is merit to this surge in number of reads and probes to create high dimensional gene expression datasets. For gene expression profiling experiments, it is often the case that a new experiment is designed and performed to capture any novel aspect of interest. We explore a potential possibility of modelling already sourced datasets, and extrapolating these insilico to discover expression levels of interest.
In this paper, we propose a computational framework to estimate gene expression data using only a selected fraction of gene expression measurements. We demonstrate that the expression levels of certain genes selected from the collection of genes of interest can be used to accurately estimate the remaining expression levels. We show that conclusions regarding expression levels can be derived from partial measurements. We also show that further analysis can be performed using such predicted data, thus enabling the conduction of whole genome expression analysis, using such data. This framework allows for customisation because selected genes can be isolated for analysis. We believe that this method has applications in how biomedical data is sourced and in turn is relevant in the areas of differential gene analysis (class comparison), class prediction, cancer investigation, and noninvasive diagnosis.
Benefits and contributions

We demonstrate that gene expression data can be modelled as an approximate lowrank data matrix, in order to computationally predict expression values.

We show that sparse gene expression measurements (“known” expression levels) could be used to artificially construct the gene expression dataset using nonlinear convex optimisation, and report prediction results on diverse expression datasets sourced from multiple experiments. This is in contrast with current biochemical methods which directly measure all expression values.

We conduct differential gene analysis and Bayesian network analysis on predicted datasets, and compare our results with those obtained using original datasets, to show that the prediction capabilities of the reconstructed and the original datasets are not significantly different.

This can be used to computationally predict behaviour of genes subject to a condition, given a set of measurements. This also has potential applications in consolidating multiple datasets with common phenotypes to infer new transcriptomic behaviour, using lowrank prediction.

This framework allows for construction of expression datasets using a fraction of known values thereby reducing the number of measurements (in terms of number of probes and reads) required to capture such data.

We believe that these techniques can potentially reduce the cost of experiments, thus saving millions of dollars, and open a new avenue for research on data completion in other domains, where the observable data is scarce.

This has applications in high dimensional expression data compression and reconstruction, and can be used to impute missing gene expression data even at high missing value rates.
Related work
Biological data and machine learning Plenty of biological data has generated a need for computational methods to extract useful knowledge from such heterogeneous information. This has led to advancements in machine learning techniques in making predictions particularly applied to data involving proteomics, genomics, and microarrays [11]. Computational models have been successfully used in gene finding [12–14] and prediction of proteins with a secondary structure [15, 16]. More recently, Alipanahi et al. used advancements in deep learning to predict DNA and RNA binding proteins [17]. In the case of expression data, Bayesian networks are effective in modelling relationships between expression profiles for prognosis prediction [18] and inference [19]. Machine learning techniques have been extensively used in expression pattern identification [20, 21] classification [22, 23], and network analysis of expression data [24]. However, the process of measuring expression levels and generating profiles is primarily devoid of any considerable learning or the use of optimisation.
Lowrank matrix recovery The objective of recovering a lowrank matrix from a few data samples can be described as an optimisation problem. This is used in various practical scenarios and is a motivation for this study. The Netflix problem is a popular example of how such techniques are applied to recommendation systems [25]. The user–movie data matrix in this case consists of movie ratings (integral values of 1–5) provided by different users for various movies. Because users tend to rate very few movies, the entries in the matrix are sparsely filled. Predicting movie ratings based on such data is used to recommend other movies to the user by posing it as a collaborative filtering problem [26]. The user–movie matrix is assumed to be a lowrank matrix because each movie has a few linearly independent parameters on which the users generally rate the movie. Therefore, only a few samples can be used to predict all the values in the rating matrix.
Lowrank modelling has been applied to computer vision [27] to improve face recognition methods and has been used in novel camera architecture to create highresolution light fields from a single coded image [28]. In 2003, Basri and Jocobs assumed their highdimensional image data of convex Lambertian surfaces under different lighting illuminations to exist in a lowdimensional subspace [29]. The concept of low dimensionality has been used to improve background subtraction [30] and motion segmentation [31]. In addition, lowrank matrix recovery is applied for estimating the distance matrix in a triangulation problem when the data available is partial [32, 33].
Gene expression prediction In 2004, Nir Friedman proposed a model for predicting gene expression levels by using probabilistic graphical models [34]. Although the method is robust, the performance of accurate prediction is moderate. Approaches involving the information theory [35] have been proposed to identify transcriptional interactions between genes in microarray data, which are computationally inexpensive. However, these approaches do not accurately estimate the expression levels. Methods for estimating missing values in large dimensional expression data are available. For example, the least square imputation method, LL Simpute, involves the combination of similar genes and selects a gene of interest by using knearest neighbours [36]. Oba et al. used Bayesian principal component analysis, BPCA, to estimate the missing values in expression profiles [37]. The prevailing methods estimate the gene expression values at very high observabilities of data, that is, unknown values predicted using these methods are extremely few (only up to 10 % of the values). To the best of our knowledge, missing rates of 5 %–10 % are considered moderate and those more than 15 % affect prediction and interpretation [38, 39]. In this study, we attempt to predict highdimensional expression matrices with only sparse data, with as high as 90 % of the data unknown.
Methods
In this section, we introduce the principals involved in modeling lowrank matrix completion and artificial construction of the gene expression dataset from known sparse expression levels. We further analyse parameters to improve the prediction performance.
Model
A gene expression study yields measurements of mRNA levels that represent gene expression values under contrasting experimental conditions, and experiments on multiple samples are consolidated to form a gene expression data matrix. We propose approaching the problem of prediction as recovery from known values as distributed entries in this data matrix. The yet unknown values constitute the complete matrix. The expression data to be predicted can be represented as M _{ m×n }, where m and n describe the genes and sample instances respectively. The locations of the known values in the data matrix, also referred to as checkpoint expression values hereafter, are encoded in Ω, where (i,j)∈Ω if expression value is hitherto known.
where (i,>j)∈Ω the nuclear norm is the tightest convex relaxation of the rank function, and therefore its ideal replacement. The advantage of the nuclear norm is that it is convex, and its global optimum can be efficiently computed. Candès and Recht showed that solution obtained using convex heuristic is the same as that obtained using rank minimisation heuristic, and the replacement holds good under certain conditions [32]. If the predicted gene expression matrix is assumed to be of rank r, a lower bound is set on the number of measurements as \(\Omega  \geq Cm^{6/5}r \log m\) for a positive constant C and where m is the number of distinct genes in the dataset.
Why low rank?
It is universally known that in any biological process, genes do not act in a solitary manner and rather act in concert [41, 42]. Groups of genes interact in any biological setting, and consequently, the expression levels of genes are interdependent. The association between gene expressions has been studied and analysed in many forms, such as association network structures [24, 43] and pairwise correlations [44]. We believe interdependent factors contribute to the behaviours of transcription factors, thereby influencing the expression of genes and resulting in a highly correlated data matrix. We assume that the gene expression values lie on a lowdimensional linear subspace and the data matrix thus formed may be a lowrank matrix. We later show that this assumption can be considered true to approximately predict these values.
Expression prediction
where u _{ i } and v _{ i } are the left singular vectors and right singular vectors of X, respectively. The sequence of iterations converges to the desired expression matrix that would minimise (4).
Parameters
The parameters can be further optimised to enhance the prediction performance. To reduce the computation time and the time required for implementations on modest desktop computers, iterations with different values can be performed within a defined range on similar test datasets, pivoted on values determined using (8) and (9). Nevertheless, we demonstrate that the aforementioned relations can be used as is for high accuracy gene expression prediction.
The convergence criterion was empirically set. In our implementation, the tolerance in the error of expression levels was maintained at 10^{−8}. An upper limit of the number of iterations was contingent on the available computational power, which was set to 750 iterations.
Robustness to noise
Prediction results with additive noise
Ratio  Observability (%)  Relative error 

0.003  50  4.22 ×10^{−4} 
0.03  50  4.21 ×10^{−3} 
0.3  50  1.78 ×10^{−2} 
0.003  10  1.21 ×10^{−2} 
0.03  10  1.57 ×10^{−2} 
0.3  10  1.91 ×10^{−1} 
Data preprocessing
Data preprocessing can often lead to significant improvement in model performance, and is therefore an imperative step, with normalisation and transformation characteristic to gene expression analysis. The input gene expression data was logtransformed prior to prediction. The distribution of gene expression measurements is heavily skewed, and the values are better correlated after logtransformation, increasing accuracy of lowrank recovery. A variety of normalisation techniques exist for gene expression data analysis, with no clear consensus on a singular strategy. The performance of prediction is enhanced after normalisation; for example, the prediction accuracy with Robust Multiarray Average (RMA) on microarray expression datasets and transforming RNAseq raw reads into Reads Per Kilobase of transcript per Million mapped reads (RPKM) has a higher prediction accuracy, as compared to prediction performed using raw values. Although, the range of normalisation approaches would be qualified in the case of very low observability of the expression data, data preprocessing with normalisation and transformation is highly recommended for superior results.
Results and discussion
We present the results of the method in two major parts. First, we evaluated the prediction accuracy on real expression data by using lowrank recovery. Second, we verified whether this predicted dataset can be used as a surrogate of the original dataset for further analysis. We answered this by comparing the results of differential expression analysis obtained using predicted datasets with those obtained using original datasets. Finally, we used Bayesian network modelling for both groups of datasets and compared their results to further address the question.
Gene expression prediction
where M and X are the original and recovered expression matrices, respectively.
where P _{ Ω } is the orthogonal projection matrix.
For the same number of iterations in the prediction algorithm, the predicted datasets that converged with a low relative error had a corresponding low omega error, and vice versa for outlier datasets with a high relative error (Fig. 2). Therefore, the error in the convergence of checkpoint expression levels can be used as an indicator of the extent to which predicted expression levels coincide with real values (measured using highdensity arrays and RNAseq). Crossvalidation using holdout rows and columns on a single dataset, and sophisticated methods using weighted Nonnegative Matrix Factorisations would give further insight into prediction accuracy [55]. The datasets that did not converge and therefore were not constructed were detected using the omega error.
Differential expression analysis
We attempt to replicate gene expression profiling experiments using partial measurements, and predicted expression levels basis on these measurements. We identified differentially expressed unique genes by using datasets predicted through lowrank completion and compared the results with those obtained using the original dataset. We also append differential analysis results solely on observed measurements without any prediction or learning to highlight the advantage of such prediction methods.
Differential analysis on predicted expression datasets. Top unique differentially expressed genes upregulated in lesional skin compared with those in nonlesional skin when ranked according to log2foldchange in (a) original dataset, (b) predicted dataset with 60 % observability and (c) sparse knownvalue (checkpoint) dataset without prediction at 60 % observability
Original dataset  Recovered dataset (60 %)  Checkpoint dataset (60 %)  

Gene  Probe ID  Symbol FC  log  Adj.  Probe ID  Symbol  log FC  Adj.  Probe ID  Symbol  log FC  Adj. 
ranking  PValue ×10^{−10}  PValue ×10^{−10}  PValue  
1  205863_at  S100A12  9.79929  < 1  205863_at  S100A12  8.99648  < 1  211906_s_at  SERPINB4  6.21118  3.3×10^{−10} 
2  211906_s_at  SERPINB4  9.60376  < 1  211906_s_at  SERPINB4  8.67119  < 1  205863_at  S100A12  5.48282  3.3×10^{−9} 
3  205513_at  TCN1  8.65788  < 1  205513_at  TCN1  8.12271  < 1  205513_at  TCN1  5.07988  4.8×10^{−9} 
4  232220_at  S100A7A  8.21988  < 1  232220_at  S100A7A  7.92112  < 1  204385_at  KYNU  5.06729  3.3×10^{−10} 
5  205660_at  OASL  7.94647  < 1  205660_at  OASL  7.4045  < 1  1569555_at  GDA  4.75835  4.8×10^{−9} 
6  220664_at  SPRR2C  7.87929  < 1  220664_at  SPRR2C  7.3366  < 1  205844_at  VNN1  4.70129  3.3×10^{−10} 
7  207602_at  TMPRSS11D  7.64471  < 1  1569555_at  GDA  7.11896  < 1  209719_at  SERPINB3  4.67529  1.6×10^{−4} 
8  1569555_at  GDA  7.39506  < 1  207602_at  TMPRSS11D  7.10503  < 1  234699_at  RNASE7  4.57012  2.9×10^{−7} 
Probabilistic modelling and classification
The problem of the classification of samples into biological classes of tissues and diseases has been a crucial topic of research. We explored the possibility of using data that is artificially constructed to train the classifier instead of the original gene expression data. We used Bayesian networks for modelling the expression levels of genes and class prediction. Bayesian networks provide a means to model the stochastic nature of biological data and capture causal relationships between expression levels of genes for inference on new unseen data and for classifying owing to high prediction accuracies [58, 59].
The analysis comprised many steps. We first preprocessed both groups of datasets by discretising the gene expression levels into three states, underexpressed, baseline, and overexpressed [58]. We trained Bayesian network classifiers on reduced datasets of 100 gene variables from the predicted datasets and the corresponding original datasets to shrink the search space of dependent networks. Classification accuracy was determined in a multiple run 10 fold cross validation analysis. We include comparison of Bayesian network classification trained on microarray datasets of lung adenocarcinoma [60], myelodysplastic syndrome [61], pancreatic ductal adenocarcinoma [62], psoriasis [56], pulmonary fibrosis [63] with corresponding lowrank predicted datasets and datasets sampled from a uniform distribution. The performances of the classifiers obtained using lowrank recovered datasets matched with those of classifiers obtained using corresponding original datasets (Table 4). Furthermore, we compared the class predictions and probability distributions of individual test instances (see Additional file 1: Table S4). In this section, we presented the results at low observabilities to demonstrate lower bound cases.
Top unique differentially expressed genes upregulated in lesional skin compared with those in nonlesional skin when ranked according to log2foldchange in (a) original dataset, (b) predicted dataset with 30 % observability, and (c) sparse knownvalue (checkpoint) dataset without prediction at 30 % observability
Original dataset  Recovered dataset (30 %)  Checkpoint dataset (30 %)  

Gene  Probe ID  Symbol  log FC  Adj.  Probe ID  Symbol  log FC  Adj.  Probe ID  Symbol  log FC  Adj. 
ranking  PValue ×10^{−10}  PValue ×10^{−10}  PValue  
1  205863_at  S100A12  9.79929  < 1  205863_at  S100A12  8.48947  < 1  207367_at  ATP12A  3.17871  0.02 
2  211906_s_at  SERPINB4  9.60376  < 1  211906_s_at  SERPINB4  7.98211  < 1  201086_x_at  SON  3.12259  0.17 
3  205513_at  TCN1  8.65788  < 1  220664_at  SPRR2C  7.17109  < 1  213356_x_at  NA  3.06212  0.29 
4  232220_at  S100A7A  8.21988  < 1  232220_at  S100A7A  6.77508  < 1  209719_x_at  SERPINB3  2.98365  0.15 
5  205660_at  OASL  7.94647  < 1  204385_at  KYNU  6.4279  < 1  33322_i_at  SFN  2.89353  0.36 
6  220664_at  SPRR2C  7.87929  < 1  207602_at  TMPRSS11D  6.41765  < 1  213523_at  KIAA0368  2.88306  0.29 
7  207602_at  TMPRSS11D  7.64471  < 1  207367_at  ATP12A  6.40415  < 1  210413_x_at  CCNE1  2.83059  0.06 
8  1569555_at  GDA  7.39506  < 1  210413_x_at  NA  6.39934  < 1  217388_s_at  NA  2.82118  0.19 
Comparison of the results of classification obtained using Bayesian networks learnt on low observability predicted datasets with those in which networks were learnt on original datasets
Study  Dataset  True positive rate  False positive rate  Precision  Recall  Fmeasure  AUROC 

Lung adenocarcinoma  Original  0.944  0.057  0.944  0.944  0.944  0.988 
Lowrank prediction  0.944  0.057  0.944  0.944  0.944  0.996  
(O = 60 %)  
Sampled Uniform distribution  0.757  0.256  0.758  0.757  0.755  0.777  
(O = 60 %)  
Myelodysplastic syndrome  Original  0.865  0.866  0.844  0.865  0.854  0.673 
Lowrank prediction  0.865  0.92  0.833  0.865  0.849  0.675  
(O = 40 %)  
Sampled Uniform distribution  0.85  0.868  0.842  0.85  0.846  0.425  
(O = 40 %)  
Pulmonary hypertension  Original  0.638  0.121  0.633  0.638  0.635  0.854 
Lowrank prediction  0.681  0.118  0.645  0.681  0.659  0.897  
(O = 60 %)  
Sampled Uniform distribution  0.267  0.372  0.213  0.267  0.218  0.424  
(O = 60 %)  
Pancreatic ductal  Original  0.782  0.218  0.784  0.782  0.782  0.886 
adenocarcinoma  Lowrank prediction  0.821  0.179  0.821  0.821  0.82  0.905 
(O = 50 %)  
Sampled Uniform distribution  0.397  0.603  0.389  0.397  0.385  0.417  
(O = 50 %)  
Psoriasis  Original  0.912  0.088  0.913  0.912  0.912  0.96 
Lowrank prediction  0.912  0.088  0.912  0.912  0.912  0.956  
(O = 40 %)  
Sampled Uniform distribution  0.641  0.359  0.641  0.641  0.641  0.648  
(O = 40 %) 
The results indicated that Bayesian networks constructed using lowrank recovered datasets closely resemble those constructed using original datasets, irrespective of classifier accuracy. For instance, the area under the receiver operating characteristic curve (AUROC) of the network constructed using the original and predicted Myelodysplastic syndrome datasets were 0.673 and 0.675 (Table 3, Pvalue < 0.01), respectively, whereas the AUROC of the original and predicted pulmonary hypertension dataset were 0.854 and 0.897 (Table 3, Pvalue < 0.001), respectively.
Conclusions
In this article, we described the modelling of biological datasets as lowrank matrices subject to their inherent dependencies. These datasets can be recovered using the mathematics of lowrank matrix completion. We used random samples as checkpoints. However, quantitatively derived checkpoints can function more satisfactorily than random samples. This provides a foundation for future work in which prediction accuracy, particularly at low observabilities, could be further improved.
Moreover, we see a clear scenario in which such techniques can be applied to other datasets in biomedicine. This framework allows for prediction of biomedical quantities, in likeness to recommender systems, given a set of observable values. Such a framework also has applications in fields in which data collection is precious and prediction could be made using partial measurements. The method can be further developed to manage data volumes sourced from highthroughput sequencing methods. The method can be used as an imputation method, when there is partial data loss as is prevalent in using microarrays today. A major concern in current convex algorithms is the computational requirement. However, datasets with hundreds of millions of points can be accurately predicted using highly parallel processing using GPUs and the cloud.
We believe that this study will open new avenues in research on lowrank matrix completion in biological sciences. We show how much information is inherently present in the actual matrix for gene expression thereby telling us how many measurements we really need to make. We believe biomedical researchers will design actual experiments based on this information opening up new avenues in research on such techniques.
Abbreviations
AUROC, area under receiver operating curve; BPCA, Bayesian principal component analysis; LLSimpute, local least square imputation; NCBI, National Center for Biotechnology Information; NGS, next generation sequencing; NP, nondeterministic polynomial time; RKPM, reads per kilobase of transcript per million mapped reads; RMA, robust multiarray average
Declarations
Acknowledgements
We thank the anonymous reviewers for their detailed comments, and for their help in strengthening the analysis.
Availability of data and materials
The data used to evaluate prediction of expression levels was sourced from multiple publically available studies which are listed in Additional file 2.
Authors’ contributions
AK conceptualised and developed the framework, conducted analysis and wrote the manuscript. KM contributed to framework development and helped review the manuscript. GA provided overall guidance. All authors have read and approved this manuscript.
Competing interests
The authors declare that they have no competing interests.
Consent for publication
Not applicable.
Ethics approval and consent to participate
Not applicable.
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
References
 Marwah K, Zollanvari A, Alterovitz G. Hyperexperiments: Bayesian inference and annotation over geo. In: Proceeding of Annual Medical Informatics Joint Summits on Translational Science: 2012.Google Scholar
 Schena M, Shalon D, Davis RW, Brown PO. Quantitative monitoring of gene expression patterns with a complementary dna microarray. Science. 1995; 270(5235):467–70.View ArticlePubMedGoogle Scholar
 Miller MB, Tang YW. Basic concepts of microarrays and potential applications in clinical microbiology. Clin Microbiol Rev. 2009; 22(4):611–33.View ArticlePubMedPubMed CentralGoogle Scholar
 Wang Z, Gerstein M, Snyder M. Rnaseq: a revolutionary tool for transcriptomics. Nat Rev Genet. 2009; 10(1):57–63.View ArticlePubMedPubMed CentralGoogle Scholar
 Affymetrix GeneChips™ Pricing. http://www.bumc.bu.edu/microarray/pricing. Accessed 15 Nov 2015.
 Science Exchange. Affymetrix RNA Microarray. 2015. https://www.scienceexchange.com/services/affymetrixrnamicroarray. Accessed 15 Nov 2015.
 Science Exchange. Illumina RNA Microarray. 2015. https://www.scienceexchange.com/services/illuminarnamicroarray. Accessed 15 Nov 2015.
 Jung SH, Bang H, Young S. Sample size calculation for multiple testing in microarray data analysis. Biostatistics. 2005; 6(1):157–69.View ArticlePubMedGoogle Scholar
 Marchionni L. Impact of gene expression profiling tests on breast cancer outcome: DIANE Publishing; 2009.Google Scholar
 Applied Biological Materials, Inc. RNA Sequencing. https://www.abmgood.com/RNASequencingService.html. Accessed 15 Nov 2015.
 Larrañaga P, Calvo B, Santana R, Bielza C, Galdiano J, Inza I, et al.Machine learning in bioinformatics. Brief Bioinform. 2006; 7(1):86–112.View ArticlePubMedGoogle Scholar
 Salzberg S. Locating protein coding regions in human dna using a decision tree algorithm. J Comput Biol. 1995; 2(3):473–85.View ArticlePubMedGoogle Scholar
 Korf I. Gene finding in novel genomes. BMC Bioinformatics. 2004; 5(1):59.View ArticlePubMedPubMed CentralGoogle Scholar
 Hyatt D, Chen GL, LoCascio PF, Land ML, Larimer FW, Hauser LJ. Prodigal: prokaryotic gene recognition and translation initiation site identification. BMC Bioinformatics. 2010; 11(1):119.View ArticlePubMedPubMed CentralGoogle Scholar
 Jain P, Garibaldi JM, Hirst J. Supervised machine learning algorithms for protein structure classification. Comput Biol Chem. 2009; 33(3):216–23.View ArticlePubMedGoogle Scholar
 Chen C, Chen L, Zou X, Cai P. Prediction of protein secondary structure content by using the concept of chou’s pseudo amino acid composition and support vector machine. Protein Peptide Lett. 2009; 16(1):27–31.View ArticleGoogle Scholar
 Alipanahi B, Delong A, Weirauch MT, Frey BJ. Predicting the sequence specificities of dnaand rnabinding proteins by deep learning. Nat Biotechnol. 2015.Google Scholar
 Gevaert O, Smet FD, Timmerman D, Moreau Y, Moor BD. Predicting the sequence specificities of dnaand rnabinding proteins by deep learning. Bioinformatics. 2006; 22(14):184–90.View ArticleGoogle Scholar
 Bansal M, Belcastro V, AmbesiImpiombato A, di Bernardo D. How to infer gene networks from expression profiles. Mol Syst Biol. 2007; 3(1):78.PubMedPubMed CentralGoogle Scholar
 McNicholas PD, Murphy T. Modelbased clustering of microarray expression data via latent gaussian mixture models. Bioinformatics. 2010; 26(21):2705–12.View ArticlePubMedGoogle Scholar
 Valafar F. Pattern recognition techniques in microarray data analysis. Ann N Y Acad Sci. 2002; 980(1):41–64.View ArticlePubMedGoogle Scholar
 Nanni L, Brahnam S, Lumini A. Combining multiple approaches for gene microarray classification. Bioinformatics. 2012; 28(8):1151–7.View ArticlePubMedGoogle Scholar
 Duval B, Hao JK. Advances in metaheuristics for gene selection and classification of microarray data. Brief Bioinform. 2010; 11(1):127–41.View ArticlePubMedGoogle Scholar
 Gill R, Datta S, Datta S. A statistical framework for differential network analysis from microarray data. BMC Bioinformatics. 2010; 11(1):95.View ArticlePubMedPubMed CentralGoogle Scholar
 Bennett J, Lanning S. The netflix prize. In: Proceedings of KDD Cup and Workshop: 2007.Google Scholar
 Zhou Y, Wilkinson D, Schreiber R, Pan R. Largescale parallel collaborative filtering for the netflix prize. In: Algorithmic Aspects in Information and Management. Berlin Heidelberg: Springer: 2008. p. 337–48.Google Scholar
 Zhou X, Yang C, Zhao H, Yu W. Lowrank modeling and its applications in image analysis. ACM Comput Surv (CSUR). 2014; 47(2):36.View ArticleGoogle Scholar
 Marwah K, Wetzstein G, Bando Y, Raskar R. Compressive light field photography using overcomplete dictionaries and optimized projections. ACM Trans Graphics (TOG). 2013; 32(4):46.View ArticleGoogle Scholar
 Basri R, Jacobs DW. Lambertian reflectance and linear subspaces. Pattern Anal Mach Intell IEEE Trans. 2003; 25(2):218–33.View ArticleGoogle Scholar
 Cui X, Huang J, Zhang S, Metaxas DN. Background subtraction using low rank and group sparsity constraints. In: Computer Vision–ECCV 2012. Berlin Heidelberg: Springer: 2012. p. 612–25.Google Scholar
 Vidal R, Hartley R. Motion segmentation with missing data using powerfactorization and gpca. In: Computer Vision and Pattern Recognition, 2004. CVPR 2004. Proceedings of the 2004 IEEE Computer Society Conference on IEEE Vol. 2: 2004. p. 310.Google Scholar
 Candès EJ, Recht B. Exact matrix completion via convex optimization. Found Comput Math. 2009; 9(6):717–72.View ArticleGoogle Scholar
 So AMC, Ye Y. Theory of semidefinite programming for sensor network localization. Math Prog. 2007; 109(2–3):367–84.View ArticleGoogle Scholar
 Friedman N. Inferring cellular networks using probabilistic graphical models. Science. 5659; 303:799–805.View ArticleGoogle Scholar
 Margolin AA, Wang K, Lim WK, Kustagi M, Nemenman I, Califano A, et al.Reverse engineering cellular networks. Nat Protoc. 2006; 1(2):662–71.View ArticlePubMedGoogle Scholar
 Kim H, Golub GH, Park H. Missing value estimation for dna microarray gene expression data: local least squares imputation. Bioinformatics. 2005; 21(2):187–98.View ArticlePubMedGoogle Scholar
 Oba S, Sato MA, Takemasa I, Monden M, Matsubara K, Ishii S. A bayesian missing value estimation method for gene expression profile data. Bioinformatics. 2003; 19(16):2088–96.View ArticlePubMedGoogle Scholar
 Moorthy K, Mohamad M, Deris SB. A review on missing value imputation algorithms for microarray gene expression data. Curr Bioinformatics. 2014; 9(1):18–22.View ArticleGoogle Scholar
 Liew AC, Law B, Yan H. Missing value imputation for gene expression data: computational techniques to recover missing data from available information. Brief Bioinform. 2011; 12(5):498–513.View ArticlePubMedGoogle Scholar
 Gillis N, Glineur F. Lowrank matrix approximation with weights or missing data is nphard. SIAM J Matrix Anal Appl. 2011; 32(4):1149–65.View ArticleGoogle Scholar
 Staiger C, Cadot S, Györffy B, Wessels LF, Klau GW. Current compositefeature classification methods do not outperform simple singlegenes classifiers in breast cancer prognosis. Front Genet. 2013; 4:289.View ArticlePubMedPubMed CentralGoogle Scholar
 Silver M, Chen P, Li R, Cheng CY, Wong TY, Tai ES, et al.Pathwaysdriven sparse regression identifies pathways and genes associated with highdensity lipoprotein cholesterol in two asian cohorts. PLoS Genet. 2013; 9(11):1003939.View ArticleGoogle Scholar
 Xiong M, FeghaliBostwick CA, Arnett FC, Zhou X. A systems biology approach to genetic studies of complex diseases. FEBS Lett. 2005; 579(24):5325–32.View ArticlePubMedGoogle Scholar
 Weckwerth W, Loureiro ME, Wenzel K, Fiehn O. Differential metabolic networks unravel the effects of silent plant phenotypes. Proc Natl Acad Sci U S A. 2004; 01(20):7809–14.View ArticleGoogle Scholar
 Cai JF, Candès EJ, Shen Z. A singular value thresholding algorithm for matrix completion. SIAM J Optim. 2010; 20(4):1956–82.View ArticleGoogle Scholar
 Brennecke P, Anders S, Kim JK, Koodziejczyk AA, Zhang X, Proserpio V, Baying B, Benes V, Teichmann SA, Marioni JC, et al.Accounting for technical noise in singlecell rnaseq experiments. Nat Methods. 2013; 10(11):1093–5.View ArticlePubMedGoogle Scholar
 Kim JK, Kolodziejczyk AA, Illicic T, Teichmann SA, Marioni JC. Characterizing noise structure in singlecell rnaseq distinguishes genuine from technical stochastic allelic expression. Nat Commun. 2015; 6.Google Scholar
 Tang VT, Yan H. Noise reduction in microarray gene expression data based on spectral analysis. Int J Mach Learn Cybernet. 2012; 3(1):51–7.View ArticleGoogle Scholar
 He Z, Zhou J. Empirical evaluation of a new method for calculating signaltonoise ratio for microarray data analysis. Appl Environ Microbiol. 2008; 74(10):2957–66.View ArticlePubMedPubMed CentralGoogle Scholar
 Kitchen RR, Sabine VS, Simen AA, Dixon JM, Bartlett JM, Sims AH. Relative impact of key sources of systematic noise in affymetrix and illumina geneexpression microarray experiments. BMC Genomic. 2011; 12(1):589.View ArticleGoogle Scholar
 Klebanov L, Yakovlev A. How high is the level of technical noise in microarray data. Biol Direct. 2007; 2(9):1–9.Google Scholar
 MAQCConsortium. The microarray quality control (maqc) project shows inter and intraplatform reproducibility of gene expression measurements. Nat Biotechnol. 2006; 24(9):1151–61.View ArticlePubMed CentralGoogle Scholar
 Edgar R, Domrachev M, Lash AE. Expression omnibus: Ncbi gene expression and hybridization array data repository. Nucleic Acids Res. 2002; 30(1):207–10.View ArticlePubMedPubMed CentralGoogle Scholar
 Kolesnikov N, Hastings E, Keays M, Melnichuk O, Tang YA, Williams E, Dylag M, Kurbatova N, Brandizi M, Burdett T, Megy K, Pilicheva E, Rustici G, Tikhonov A, Parkinson H, Petryszak R, Sarkans U, Brazma A. Arrayexpress update—simplifying data submissions. Nucleic Acids Res. 2015; 43(D1):1113–6. doi:10.1093/nar/gku1057.View ArticleGoogle Scholar
 Kanagal B, Sindhwani V. Rank selection in lowrank matrix approximations: A study of crossvalidation for nmfs. Proc Conf Adv Neural Inf Process. 2010; 1:10–15.Google Scholar
 SuárezFariñas M, Li K, FuentesDuculan J, Hayden K, Brodmerkel C, Krueger JG. Expanding the psoriasis disease profile: interrogation of the skin and serum of patients with moderatetosevere psoriasis. J Investigative Dermatolog. 2012; 132(11):2552–64.View ArticleGoogle Scholar
 Boyle JO, Gümüş ZH, Kacker A, Choksi VL, Bocker JM, Zhou XK, et al.Effects of cigarette smoke on the human oral mucosal transcriptome. Cancer Prevent Res. 2010; 3(3):266–78.View ArticleGoogle Scholar
 Friedman N, Linial M, Nachman I, Pe’er D. Using bayesian networks to analyze expression data. J Comput Biol. 2000; 7(3–4):601–20.View ArticlePubMedGoogle Scholar
 Helman P, Veroff R, Atlas SR, Willman C. A bayesian network classification methodology for gene expression data. J Comput Biol. 2004; 11(4):581–615.View ArticlePubMedGoogle Scholar
 Landi MT, Dracheva T, Rotunno M, Figueroa JD, Liu H, Dasgupta A, et al.Gene expression signature of cigarette smoking and its role in lung adenocarcinoma development and survival. PloS ONE. 2008; 3(2):1651.View ArticleGoogle Scholar
 Pellagatti A, Cazzola M, Giagounidis A, Perry J, Malcovati L, Della Porta MG, et al.Deregulated gene expression pathways in myelodysplastic syndrome hematopoietic stem cells. Leukemia. 2010; 24(4):756–64.View ArticlePubMedGoogle Scholar
 Badea L, Herlea V, Dima SO, Dumitrascu T, Popescu I. Combined gene expression analysis of wholetissue and microdissected pancreatic ductal adenocarcinoma identifies genes specifically overexpressed in tumor epithelia. Hepatogastroenterology. 2008; 55(88):2016.PubMedGoogle Scholar
 Mura M, Anraku M, Yun Z, McRae K, Liu M, Waddell TK, et al.Gene expression profiling in the lungs of patients with pulmonary hypertension associated with pulmonary fibrosis. CHEST J. 2012; 141(3):661–73.View ArticleGoogle Scholar