In the validation set, there was excellent concordance between the RUO scores and their commercial counterparts (rc(RS) = 0.96, 95% CI 0.93–0.97 with level of agreement (LoA) of −7.69 to 8.12; rc(EP) = 0.97, 95% CI 0.96–0.98 with LoA of −0.64 to 1.26 and rc(ROR) = 0.97 (95% CI 0.94–0.98) with LoA of −8.65 to 10.54). There was also a strong agreement in risk stratification: . In conclusion, the calibrated algorithms recapitulate the commercial RS and EP scores on individual biopsies and ROR scores on samples based on subgroup-centreing method using NanoString expression data.Multi-gene prognostic signatures including the Oncotype® DX Recurrence Score (RS), EndoPredict® (EP) and Prosigna®  are widely used to predict the likelihood of distant recurrence in patients with oestrogen-receptor-positive (ER+), HER2-negative breast cancer. Here, we describe the development and validation of methods to recapitulate RS, EP and ROR scores from NanoString expression data. RNA was available from 107 tumours from postmenopausal women with early-stage, ER+, HER2− breast cancer from the translational Arimidex, Tamoxifen, Alone or in Combination study (TransATAC) where previously these signatures had been assessed with commercial methodology. Gene expression was measured using NanoString nCounter. For RS and EP, conversion factors to adjust for cross-platform variation were estimated using linear regression. For ROR, the steps to perform subgroup-specific normalisation of the gene expression data and calibration factors to calculate the 46-gene ROR score were assessed and verified. Training with bootstrapping ( However, a substantial risk remains for relapse.Over 80% of breast cancer patients in the developed western world have oestrogen receptor (ER)-positive disease4, EndoPredict (EP/EPclin)5 and Prosigna® Risk Of Recurrence score (ROR)6. Each of these have been endorsed for prognostic use in authoritative guidelines 8.Multi-parameter gene-expression-based prognostic signatures are often used to estimate the residual risk of recurrence after surgery to guide patient management. Amongst the most widely used prognostic signatures in ER+ breast cancer are the Oncotype DX Recurrence Score (RS)4. The HER2 and proliferation modules are thresholded such that quantitative read-outs from those modules are only differentiated in tumours with high expression for the respective modules. The 16 genes were selected from 250 candidates on tumours collected from a mixed cohort of 447 patients collected including the tamoxifen arm of NSABP B-20 trial. The cut-points for individual gene expression were based on the results from NSABP B-20 trial4. The final RS algorithm was validated on 668 tamoxifen-treated patients from the NSABP B-14 trial, an ER+, node-negative cohort that contained both HER2-positive and -negative patients. RS does not include clinico-pathological information beyond the molecular score. An RS-pathology-clinical model was developed that improves the prognostic performance of RS9, but this seems to be used infrequently in clinical practice. Cut-off points for RS were established to classify patients into low (RS < 18), intermediate (18 ≤ RS ≤ 31) and high risk (RS > 31). The RS has been validated in NSABP B-2010, TransATAC11 and SWOG 881412. More recently the TAILORx study reported findings for the reduced cut-points of 11 and 26, respectively13, showing women with hormone receptor-positive, HER2-negative, axillary node-negative breast cancer, and a high RS of 26 to 100 had better prognosis when treated with ET with adjuvant CT regimens than expected with ET alone; however, there was a lack of CT benefit in patients with RS < 2614.RS consists of 16 prognostic genes and five reference genes assessed by RT-PCR. Thirteen of the prognostic genes are grouped into four modules , which allow these features to be weighted differently in the signature algorithm with scale between 0 and 1005. The clinically applicable EPclin, combines the molecular EP score with tumour size and nodal status where the EPclin value of 3.3 splits patients into low- or high-risk categories5. EP and EPclin were trained in an ER+/HER2− population that included patients with both node-negative and -positive disease (n = 664). EP and EPclin were validated in ABCSG-6 and -8 trials5 and in TransATAC15.The molecular EP score consists of eight prognostic and four reference genes measured by RT-PCR, and ranges between 0 and 15 with a cut-off value of 5 categorising patients into low- and high-risk groups16. The PAM50 algorithm is based on measurement of 50 genes optimally selected to classify the five intrinsic subtypes and eight reference genes6. The clinically applied Prosigna® score uses 46 of the 50 intrinsic subtype genes, as measured by the NanoString nCounter, integrated with a proliferation score and tumour size information based on the ROR-PT formula18 and referred as ROR score here. The ROR score was validated in TransATAC19, ABCSG-817 and DBCG cohorts20 to predict risk of relapse for postmenopausal women with hormone receptor-positive, node-negative (Stage I or II) or node-positive (Stage II or IIIA) early-stage breast cancer to be treated with adjuvant ET. The final ROR score ranges from 0 to 100 and the risk is calculated differently depending on the nodal status; specifically, node-negative cancers are classified as low (0–40), intermediate (41–60) or high (61–100) risk, 1–3 node-positive cancers are classified as low (0–15), intermediate (16–40) or high (41–100) risk and 4+ node-positive cancers are classed as high risk. The PAM50 genes are measured by the NanoString nCounter which allows the detection of up to 800 RNA species in extracts from formalin-fixed tissues by utilising molecular ‘barcodes’, direct hybridisation, and single molecule imaging without amplification steps that might introduce bias21. NanoString is widely used for clinical and biomarker research studies of gene expression in FFPE22.Breast cancers can be classified into one of five intrinsic subtypes, namely Luminal A, Luminal B, Basal-like, HER2-enriched and Normal-like by gene expression profilingHere, we describe the derivation and validation of RS and EP and ROR scores within ER+/HER2− tumours using gene expression data simultaneously generated by the NanoString nCounter System. We also provide a step-by-step protocol as reference and a guide for computing these scores to allow other researchers to assess the prognostic and predictive value of these scores in research studies.n = 59) are listed in Tables n = 59) there were linear relationships between the RS gene expression levels measured by RT-PCR and NanoString for all genes with the exception of CTSL2 and GSTM1 due to the lack of expression detected in some samples by the NanoString assay   was used for the validation of the gene-wise conversion factors. The averaged correction coefficients  NanoString RS module scores were highly correlated; (ccc) rc was 0.91 0.93, 0.98, 0.88, 0.91 and 0.95 for the proliferation, thresholded proliferation, oestrogen, invasion, HER2 and thresholded HER2 modules, respectively  = 0.96 (95% CI 0.93–0.97) and rc(EP) = 0.97 (95% CI 0.96–0.98)  Fig. . The hig98) Fig. . For RS,4 and EP5 scores. For RS, comparing the commercial and RUO risk groups, of the 48 validation samples, there were 6 (12.5%) cases misclassified by the RUO RS when compared with the commercial RS classification results ; p < 0.0001. Since the report of TAILORx trial results, the cut-offs for RS to identify patients who may benefit to additional CT are 26 for postmenopausal women and 16 for premenopausal, we had repeated our comparisons based on these cut-offs (Supplementary Table κ = 1.00 (95% CI 1.00–1.00) for cut-off at 26 and κ = 0.83 (95% CI 0.68–0.99) for cut-off at 16, respectively.The validation set samples were classified into risk groups according to the pre-specified cut-points for the RSlts Fig. . The kapκ = 0.87 . The equivalent of the clinically applicable EPclin may be calculated using the EPclin algorithm5.For EP, there were three patients 3/48, 6.3%) at low risk according to the commercial EP score who were categorised into the high-risk group by the RUO EP , with ccc rc(ROR) = 0.95 (95% CI 0.92–0.96)  with the ccc rc(ROR) = 0.97 (95% CI 0.94–0.98)  Fig. . The dis98) Fig.  and the κ = 0.92  in the validation set . The Prosigna® test does not use Normal-like subtype for subtype assignment. When a sample is classified as Normal-like subtype, the subtype with the second highest correlation coefficient  is used for assignment. Supplementary Table Of the 59 TransATAC cases within the training set, there were 19 (32.2%) cases defined as Luminal B, 26 44.1%) as Luminal A, 8 (13.6%) as HER2-enriched and 6 (10.2%) as Normal-like subtypes. Of the 48 TransATAC cases within the validation set, there were 15 (31.3%) cases defined as Luminal B, 14 (29.2%) as Luminal A, 9 (18.8%) as HER2-Enriched, 3 (6.3%) as Basal-like and 7 (14.6%) as Normal-like subtypes. The intrinsic subtype assignments by commercial Prosigna® and RUO ROR were compared for all TransATAC samples (4.1% as Lc = 0.85 (95% CI 0.57–0.96)   ) within this cohort of mainly high-risk samples . Additionally, it would allow the head to head comparative analysis of newly discovered genomic signatures with these three clinically well-established molecular signatures. However, we stress that the presented RUO prognostic scores should not be used for patient management.There is great interest in computing accurate RS, EP and ROR scores for academic clinical research. Attempts have been made to recapitulate these and other commercial prognostic scores using expression data derived by gene expression microarrayOur study has strengths and weaknesses. Strengths include that the samples used in this study are from a well-characterised and documented set of samples. RNA samples have been assayed by the commercial tests’ developers using their commercial proprietary methods used in the clinic. For each patient the same aliquot of RNA was used for the measurements of the commercial assays  and for the NanoString assay to calculate RUO scores. This eliminated issues related to intratumour heterogeneity and allowed us to calculate more accurate RUO scores. For ROR calculations we used a 229 patient ER+/HER2− scaling cohort which enabled the robust derivation of RUO ROR scores relevant for this TransATAC cohort. The accuracy of conversion factors was confirmed by an independent validation set of 48 samples.29 than a random population of early ER+/HER2− breast cancer.Weaknesses include that the RUO adjustment parameters are optimised for NanoString-based assay, and if working with expression data obtained on different platforms , the normalisation and adjustment factors described here are not applicable. The commercial Prosigna® subtype correlation data and confidence of subtype calls are not available for this TransATAC dataset and it is therefore not possible to accurately determine the robustness of the RUO subtype classification, as samples may have a high confidence correlation with more than one subtype. Meanwhile, our development work had eliminated issues related to intratumour heterogeneity, and the possible variability generated by different RNA quality among a central and a peripheral lab may have been underestimated. In addition, it was a challenge to identify and gain access to external datasets that had been subjected to commercial RS, EP and ROR assays with residual RNA available to assess the accuracy of the conversion factors on data generated by custom gene expression panels in other laboratories. While we were able to assess the variability of ROR scores calculated based on the RUO ROR algorithm using external cohorts from Hospital Clinic of Barcelona, this cohort was biased with much higher riskc = 0.77 (95% CI 0.59–0.88); Supplementary Fig. A RUO EP and RS can be calculated for individual biopsies; however, for RUO ROR to be calculated for individual samples, improvement is needed for better precision as demonstrated on a testing set of 20 samples  trial evaluated efficacy and safety of anastrozole vs tamoxifen given for 5 years in postmenopausal women with localised primary breast cancern = 59) for development and validation set (n = 48) to determine the correlation coefficient between the RUO and their commercial scores. In order to detect a positive relationship (i.e. correlation >0.4) with 85% power and 0.05 significance level, we would need at least 42 cases. Furthermore, as a secondary analysis, 48 cases would also have approximately 86% power to detect 20% variability of the model at a 0.05 significance level. The Hospital Clinic of Barcelona cohorts are three sets of patient samples with ER+/HER− disease previously tested with Prosigna®. This study was approved by the South-East London Research Ethics Committee and all patients provided written informed consent for their tissue to be used in translational research.Our analytical approach is to compute conversion factors by modelling the gene expression data values measured by NanoString with those assessed by the commercial assays , a custom 60 genes panel (n = 29) and the NanoString Breast Cancer 360™ panel (n = 107).RNA for the TransATAC samples was extracted by Genomic Health Inc. (GHI)31.Data normalisation was performed using R version 3.6.3. Raw NanoString output data was normalised using the NanoStringNorm R packagen = 59), the RS and EP signature genes were normalised, respectively, with the following settings:CodeCount = ‘geo.mean’Background = ‘mean’SampleContent = ‘housekeeping.geo.mean’round.values = FALSEtake.log = TRUEverbose = TRUEFor the training set (The commands performed the following: (1) Normalisation of raw data with the geometric mean of the positive controls; (2) Background correction by subtracting the mean of the negative probes; (3) Normalisation with the geometric mean of the housekeeping genes of the respective signature; (4) Setting of <1 values to 1; and (5) Log2 transformation of normalised data.Additionally, from this set of 59 samples we calculated the average of geometric means of the positive controls before normalisation across the samples (TransATAC-pos) and the average of TransATAC housekeeping geometric means across the samples (TransATAC-hkgm). These parameters were used for the normalisation procedure of the validation set.CodeCount = ‘geo.mean’CodeCount.summary.target = TransATAC-pos*Background = ‘mean’SampleContent = ‘housekeeping.geo.mean’SampleContent.summary.target = TransATAC-hkgm*round.values = FALSEtake.log = TRUEverbose = TRUEFor the independent validation set of 48 TransATAC samples the RS and EP signature genes were normalised, respectively, with the following parameters:These commands performed the following: (1) Normalisation of raw data with the geometric mean of the positive controls setting the target to TransATAC-pos; (2) Background correction by subtracting the mean of the negative probes; (3) Normalisation with the geometric mean of the housekeeping genes of the respective signature setting the target to TransATAC-hkgm; (4) Setting of <1 values to 1; and (5) Log2 transformation of normalised data. *TransATAC-hkgm was 5627.777; the TransATAC-pos for RS was 2461.699 and for EP 8621.201. To utilise the conversion factors described in this study we recommend applying the same normalisation parameters as used for the validation set.ACTB, GUS, MRPL19, PSMC4, PUM1, RPLP0, SF3A1 and TFRC) were subtracted from all log2-transformed expression values.Raw data was normalised with the following parameters; expression counts plus 1 were log2 transformed and then the geometric mean of the log2 transformed eight housekeeping genes (n = 48) in this study. The publicly available intrinsic subtype centroids were derived from microarray data, we had scaled our NanoString dataset to a technical calibration factor   and (2) validation of the conversion factors on an independent validation set (n = 48)  estimation of conversion factors in the training set (I = 30) using a cross-validation approach. The TransATAC training dataset of 59 samples was split into training  and test  by random sampling . For each iteration, gene-wise conversion factors (intercept and slope) were obtained using linear regression models and were applied to adjust NanoString data using Eq. , β0,zi is the intercept of gene (i) and βzi is the linear coefficient of gene (i) in iteration z = 1–30.Computation of conversion factors were repeated 30 times  for the 20 test set samples, geneziRTPCR, is the RT-PCR mRNA level of gene (i) in iteration z = 1–30 for the 20 test set samples.The accuracy of conversion factors were evaluated by calculating the % error between the adjusted NanoString gene expression levels and the commercial RT-PCR gene expression level using Eq. :3\documeβ0,ci and βci are the average of β0,zi and βzi conversion factors giving error of <10% for gene (i) in iteration z = 1–30, n is the number of the coefficients averaged.For each gene the conversion factors (intercept and slope) giving error of <10% were averaged and equate to the final conversion factor using Eq. :4\documeβ0,ci and βci of each gene) on the independent validation set (n = 48) using Eq.  and tumour size is the pathological tumour size (coded as 0 if ≤2 cm or 1 if >2 cm).Following subtype assignment, the RUO ROR score was calculated using Eq. 6\documenUsing the 59 TransATAC cases, we also calculated adjustment factors based on the linear regression Eq. :7\documeThe correlation and agreement between the commercial and RUO prognostic scores were measured by the concordance correlation coefficient and Bland-Altman plot. The agreement between the risk groups defined by the commercial and RUO risk scores was also evaluated with the weighted kappa statistic. STATA and R were used for statistical calculations. Normalisation of NanoString data and image generation were performed with R version 3.6.1.Further information on research design is available in the Supplementary Figures and Tables.Supplementary Data 1.Reporting summary.


Multi-omics studies, which explore the interactions between multiple types of biological factors, have significant advantages over single-omics analysis for their ability to provide a more holistic view of biological processes, uncover the causal and functional mechanisms for complex diseases, and facilitate new discoveries in precision medicine. However, omics datasets often contain missing values, and in multi-omics study designs it is common for individuals to be represented for some omics layers but not all. Since most statistical analyses cannot be applied directly to the incomplete datasets, imputation is typically performed to infer the missing values. Integrative imputation techniques which make use of the correlations and shared information among multi-omics datasets are expected to outperform approaches that rely on single-omics information alone, resulting in more accurate results for the subsequent downstream analyses. In this review, we provide an overview of the currently available imputation methods for handling missing values in bioinformatics data with an emphasis on multi-omics imputation. In addition, we also provide a perspective on how deep learning methods might be developed for the integrative imputation of multi-omics datasets. Recent technological developments in high-throughput biology have generated large-scale multi-omics datasets in genomics, epigenomics, transcriptomics, proteomics, metabolomics, metagenomics, and phenomics. Several special collections of publicly available multi-omics datasets have been provided by projects such as Scientific Data  and The Multi-omics integration analyses can reveal the connections between different biological factors and provide researchers with a systematic view of the cell and human disease. A recent review presented meaningful integration strategies to elucidate the molecular mechanisms of osteoporosis . By provWhile many challenges and their solutions have been reviewed elsewhere , to our In this review, we present a comprehensive and systematic view of the currently available imputation methods for handling missing values in bioinformatics data. This review aims to deliver a summary of current imputation strategies and to serve as a starting point for applying deep learning methods to the imputation of multi-omics datasets. The remaining sections of the paper are organized as follows: (1) an overview of single-omics imputation methods for genotype, gene expression, epigenomic, and proteomic data; (2) an overview of integrative imputation approaches which make use of the inter-omics correlations embedded in the multi-omics datasets; and (3) a discussion that focuses on comparing and contrasting imputation techniques for both single-omics and multi-omics data, followed by an outlook of research trends with an emphasis on multi-model networks and autoencoders.In this section, we briefly introduce single-omics imputation methods and organize them by different data types, including genotype, gene expression, epigenomic, and proteomic data. First, we discuss genotype imputation methods of two broad categories: reference-based and reference-free methods. Second, we review three types of imputation methods for gene expression data: statistical methods, classic ML methods, and deep learning methods. Third, we organize methods for imputing epigenomic data into two categories: statistical methods and deep learning-based methods. Finally, we discuss proteomic data imputation and list examples of methods in three categories: single-digit replacement, local and global similarity-based approaches.As an essential tool in genome-wide association studies (GWAS), genotype imputation has facilitated developments in fine-mapping and identification of causal variants, meta-analysis for discovering trait-associated loci, and boosting the statistical power of association tests . MissingCurrent genotype imputation approaches can be categorized into two groups depending on their requirements for using reference panels . The metIn contrast, other genotype imputation techniques require a reference panel constructed from whole genome sequencing samples  and have the advantage of making full use of key genetic characteristics such as linkage patterns (or the ordering of genes on chromosomes), mutations, and recombination hotspots . The basRecently, genotype imputation has greatly benefited from the increased availability of publicly available genetic reference panels and is now a standard tool for human genome analysis. In order to make genotype imputation simpler and more accessible, the University of Michigan, Trans-Omics for Precision Medicine (TOPMed) project, and Wellcome Sanger Institute have provided users with three different web-based imputation servers. The Michigan and TOPMed imputation servers are based on Minimac3/Minimac4, while the Sanger imputation server is based on Positional Burrows Wheeler Transform (PBWT) . AlthougTranscriptomic profiles are typically acquired using bulk RNA sequencing (RNA-seq) or single-cell RNA sequencing (scRNA-seq), which measure gene expression with different resolutions. The gene expression data imputation methods that we discuss here can be divided into three categories: statistical methods, classic ML methods, and deep learning methods .Traditional bulk RNA-seq (or micro-array) technology analyzes the RNA of an entire cell population, i.e., the gene expression profile that represents the average expression values, weighted by the unknown proportions of different cell types, across the heterogeneous cell population. The popular statistical methods for imputing missing values in bulk RNA-seq datasets can be classified into five general strategies : (1) impDue to the limitations of bulk RNA-seq such as low resolution and inability to study the cellular heterogeneity of a tissue sample, current transcriptome analysis has made the leap from bulk population-based studies to studying gene expression on a single-cell level via scRNA-seq. However, there are particular challenges that arise in scRNA-seq analysis, including high dropout rate and the curse of dimensionality . For scRThere are a number of unique challenges for gene expression data imputation that are distinct from those for genotype imputation. First, in contrast to genotype imputation, there are seldom external reference panels that can be used to facilitate the imputation of gene expression datasets. Second, unlike genotype data where a given SNP is clearly either genotyped or not genotyped, for gene expression data it is typically impossible to completely distinguish between the true zeros and the dropout zeros. Third, methods for gene expression data imputation have been reported to under-correct or over-correct for data noise in some circumstances, potentially resulting in false positive signals . TherefoWhile the overwhelming majority of transcriptomic studies evaluate gene expression profiles at a single time point, studying repeated measures of gene expression on the same individuals at multiple time points can provide novel insights into the dynamics of complex biological processes. However, the imputation of time series expression profiles poses unique challenges because the observations from different time points are highly correlated. The autoregressive least squares imputation (ARLSimpute)  was deveGenome-wide maps are constructed by using epigenetic data that describe chromatin accessibility, histone modifications, and DNA methylation . DespiteOverall, current methods for epigenomic data imputation can be categorized into two classes as shown in In recent years, the field of mass spectrometry (MS)-based proteomics has quickly progressed. Using high-resolution MS techniques, it is possible for modern proteomics studies to detect and quantify vast amounts of proteins and peptides in a single run. These methods can be roughly partitioned into the two broad groups of label-based and label-free quantification . HoweverAs comprehensively reviewed previously , commonlIn this section, we first provide a general overview of integrative imputation. Then, we focus specifically on methods that perform imputation using information obtained by combining transcriptomic data with genomic, epigenomic, or proteomic data. Finally, we discuss matrix factor-based imputation and the strengths and limitations of a few methods which use these techniques.According to the central dogma of molecular biology, DNA encodes RNA (known as transcription) and RNA encodes proteins (known as translation) . This paSimilar to single omics data, multi-omics profiles may be collected at either the bulk tissue or single-cell level. Current multi-omics integration approaches for the bulk tissue level data have previously been discussed in detail . From a On the other hand, the recent maturation of single-cell multi-omics technologies has provided unique opportunities for integrative methods capable of learning from combinations of various data types. These approaches provide researchers with a state-of-the-art tool for profiling different sources of omics data such as DNA, RNA, and proteins on the single-cell level in parallel. Currently, there are a variety of single-cell multi-omics techniques, such as scNMT-seq  , CITE-seHowever, integrative analysis of multi-omics datasets at both the bulk tissue level and the single-cell level may be hindered by missing values due to technical errors and cost limitations. The underlying principle of multi-omics data imputation is to take advantage of the correlations between different types of biological features measured on the same subjects/cells  and complex human traits and diseases, the mechanisms governing these associations are still largely undetermined . Recent TWAS integrate large reference panels including paired genotype and gene expression datasets from the same individuals to uncover gene-trait associations . UsuallyTranscriptomic data imputation relies on correlations between genotype and gene expression data and involves the first two steps of TWAS : weightsYg represents the levels of gene expression , X represents the SNP matrix, w represents the corresponding weights (or cis-eQTL effect sizes) vector, and ε represents the error term. Next, the genetically regulated gene expression (GReX) is imputed bywhere xnew represents the new GWAS sample dataset. After imputing the GReX, TWAS conduct gene-based tests by testing associations between GReX and the trait of interest. From a gene-based test perspective, TWAS are equivalent to a weighted burden test  and Ridge (L2) penalties to perform variable selection  such as the identification of novel therapeutic treatments in GReX-based drug repositioning predictions.where ∥⋅∥The PrediXcan model is based on the following assumptions: (1) all loci have equal contributions in their roles as possible expression quantitative trait loci (eQTLs), even though they may regulate gene expression through different functional mechanisms, and (2) different alleles have an equal influence on gene expression . These aIn contrast to PrediXcan, which is based on the individual level genotype data, S-PrediXcan (or MetaXcan)  was devew as follows   is an imS2  . CoMM adS2 , 2 was d, UTMOST  and Mult, UTMOST , have beIn each of the above methods, the number of genes that can be accurately imputed largely depends on the sample size of the training reference panels and the quality of the training data. For example, PrediXcan only has prediction models available for genes with expression values in GTEx. Therefore, improving the sample sizes of the available references will lead to improved coverage of these imputation methods across the genome. Currently available reference panels include GEUVADIS , GTEx, DGN (Depression Genes and Networks), METSIM (Metabolic Syndrome in Men), YFS (Young Finns Study), ROSMAP (Religious Orders Study and Memory and Aging Project), NFBC1966 (Northern Finland Birth Cohort 1966) and the 1000 Genomes Project. It is also worth noting that validation of transcriptomic prediction accuracy based on independent datasets is critically important. However, multiple large expression panels are currently not yet available for tissues other than whole blood . Non-traEpigenetic factors such as DNA methylation, histone modification, and chromatin accessibility play an indispensable role in contributing to the discovery of distinct biological functions and complex human diseases. A growing number of results demonstrate that cis-regulatory elements (CREs) such as promoters and enhancers, which exhibit highly enriched levels of risk variants associated with disease, regulate gene expression .By extending single-omics imputation methods to take advantage of useful features from multi-omics data, Lin et al. developed a novel ensemble learning method (Lin) that uses correlations between various types of multi-omics datasets such as miRNA, mRNA, and DNA methylation for multi-omics imputation . This apEpiXcan  improvesP is the weight diagonal matrix whose entries are the penalty factors obtained from rescaled SNP priors. j corresponding to the penalty factor of the j-th SNP. m represents the number of cis-SNPs. Particularly, if P=I , the WENet model becomes a standard ENet model. From this point of view, WENet is the more general model and ENet is one of its special cases.Here, TOBMI (Trans-Omics Block Missing data Imputation)  is a KNNCurrent integrative imputation methods for epigenomic and transcriptomic data can be broadly categorized into two groups: ML-based regression models   is a traX denotes the normalized scRNA count matrix, Y denotes the normalized protein abundance matrix, F denotes the mapping function, and ∥⋅∥1 denotes L1 norm.where L2 normalization; (3) identify pairs of mutual nearest neighbors (MNN) in the shared space referred to as “anchors” to guide data integration; (4) calculate a score for each pair of anchors accounting for their mutual neighborhood structure for each pair of cells; (5) generate a weight matrix reflecting the strength of association by using the anchor score and the distance between the anchor and the query cell; and (6) transfer information from a reference to the query dataset. One of this method’s applications is to predict cell surface protein abundances based on the cellular transcriptomes in human bone marrow cells by using CITE-seq, which can simultaneously measure immunophenotypes and transcriptomes with single-cell resolution using DNA-barcoded antibodies.Seurat v3  is an anThe comparison results of imputation accuracy between Seurat v3 and cTP-net using the benchmark dataset of CITE-seq PBMC  show that the performance of cTP-net is comparable to that of Seurat v3, with cTP-net performing slightly better . The cTPSeveral multi-omics imputation methods are based on multi-view matrix factorization techniques. For instance, MI-MFA   involvesm represents each omics data matrix, Z represents the shared factor matrix, Wm represents the weight matrix for each omics data m, and εm represents the residual noise term.Where YMulti-view matrix factorization methods for multi-omics imputation can be applied to more than two omics datasets. For example, MOFA has been demonstrated to identify factors that elucidate variance throughout multi-omics datasets such as bulk genomic, RNA expression, and DNA methylation data obtained from individuals suffering from chronic lymphocytic leukemia . HoweverWithin the area of single-omics imputation, genotype imputation provides a relatively more mature and standard toolbox for GWAS studies, thanks to the availability of large, public genetic reference panels. In contrast, gene expression data imputation, especially scRNA-seq data imputation, is a more active and challenging field. While scRNA-seq has the advantages of a wide range of technologies for sensitive, combinatorically barcoded or highly multiplexed profiling , it faceCompared with single-omics imputation, integrative imputation of multi-omics datasets has great potential for helping researchers uncover many informative functional mechanistic pathways, from the original root cause of diseases to functional consequences or relevant interactions. Therefore, it could help to provide a more comprehensive and clearer view for the downstream analysis of multi-omics studies. However, integration of multi-omics data across different samples, experiments, and types of measurements is highly challenging due to the dynamic and complicated mechanisms of biological processes. According to the type of measurements (or omics), samples, experiments (or laboratories), and cells, There are a number of factors that affect the accuracy of multi-omics imputation: missing value mechanism , missing rate, sample size, degree of overlap of samples, and noise level. Since the principle of multi-omics imputation is to make use of the shared information across the biological datasets, it is critical to use multi-omics datasets measured on the same set of samples. For example, Another emerging challenge is the application of state-of-the-art deep learning methods to multi-omics imputation. Since different omics datasets, such as genomics, epigenomics, transcriptomics, proteomics, and phenomics have their own distinct biological roles and functional mechanisms, it is impossible to use one single, uniform deep learning framework to extract all specific features from these different omics datasets. On the other hand, deep learning methods for multi-omics imputation should have the ability to leverage the correlation or similarity among different omics datasets in order to better impute the missing values in multi-omics datasets. Deep learning methods, especially autoencoders, have already shown great improvements in single-omics imputation. For instance, Chen et al. proposed an autoencoder-based SCDA model for genotype imputation without a reference panel . AnotherAs presented in a recent review of ML methods for integrating multi-view data, the DNN-based multi-model structure is a powerful tool for integrating heterogeneous sets of features with multi-omics data and for capturing their high-level correlations, which can be used for imputation and prediction . EssentiMulti-model DNN has four attractive advantages for multi-omics data integration . First, For our future plans, we propose to generate a multi-view autoencoder model for multi-omics imputation by combining multi-model DNN and autoencoders. For example, this novel model may use genotype and gene expression data as two inputs, with at least one of those inputs having missing values, and then produce the two corresponding imputed outputs . The two input layers would be separately encoded in the first hidden layer to learn their own specific features before they are concatenated in the second hidden layer. Then, the combined common features could be processed through the bottleneck of the autoencoder in order to leverage the correlation between these two omics data. Finally, the missing values would be imputed after decoding each omics data according to the symmetry structure of the autoencoder. Specifically, we can use different sub-encoder layers, such as convolutional encoders or denoising encoders, and assign different loss functions for different types of omics data in order to better learn their distinct features.H-WD, CZ, and PG conceived and supervised this study. MS and JG carried out literature survey and drafted the manuscript. PG, JL, CZ, and H-WD revised the first draft manuscript. WZ, CW, and HS provided insightful inputs to the original and revised manuscript. All authors contributed to manuscript revisions in response to peer-review comments and approved the final version for publication.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


This is particularly important when mutation counts are low and sampling variance is high . (2) sigLASSO uses L1 regularization to parsimoniously assign signatures, leading to sparse and interpretable solutions. (3) It fine-tunes model complexity, informed by data scale and biological priors. (4) Consequently, sigLASSO can assess model uncertainty and abstain from making assignments in low-confidence contexts.Multiple mutational processes drive carcinogenesis, leaving characteristic signatures in tumor genomes. Determining the active signatures from a full repertoire of potential ones helps elucidate mechanisms of cancer development. This involves optimally decomposing the counts of cancer mutations, tabulated according to their trinucleotide context, into a linear combination of known signatures. Here, we develop sigLASSO (a software tool at  The next generation sequencing has provided the opportunity to look for signatures of carcinogenesis on a genome wide scale. Here, the authors develop the algorithm, sigLASSO, that provides confidence in assigning mutational signatures when the mutation count is low and the samples used are variable. Notably, these processes have characteristic mutational nucleotide context biases6. Sequencing cancer samples at presentation revealed all mutations accumulated over lifetime; these include somatic alterations generated by multiple mutational processes both before cancer initiation and during cancer development. In a generative model, multiple latent mutational processes generate mutations over time, drawing from their corresponding nucleotide context distributions (mutation signature)5. Here, a mutation signature is a multinomial probability distribution of mutations of a set of nucleotide contexts. In cancer samples, mutations from various mutational processes are mixed and observable by sequencing.Mutagenesis is a fundamental process underlying cancer development. Examples of mutational mechanisms include spontaneous deamination of cytosines, the formation of pyrimidine dimers by ultraviolet (UV) light, and the crosslinking of guanines by alkylating agents. Multiple endogenous and exogenous mutational processes drive cancer mutagenesis and leave distinct fingerprints7. Many signatures have been linked with mutational processes with known etiologies, such as aging, smoking, or ApoBEC activity. Investigating the fundamental processes underlying mutagenesis could help elucidate the initiation and development of cancer.By applying unsupervised methods such as non-negative matrix factorization (NMF) and clustering to large-scale cancer studies, researchers have decomposed the mutation mixture and identified at least 30 distinct mutational signatures8 decomposes the mutation profile into a signature mixture using binary search to iteratively test coefficients one-by-one and then hard pruning signatures with low estimated contribution to achieve sparsity. Other approaches use linear programming9 or iterate all combinations by brute force10. None of these approaches explicitly formulates sampling uncertainty into the model or uses efficient regression techniques. Moreover, no off-the-shelf implementation of these methods, besides deconstructSigs, is available.A major task in cancer research is to leverage signature studies on large-scale cancer cohorts and efficiently select and attribute active signatures to new cancer samples. A popular previously published method, deconstructSigs,Although we do not fully know the latent mutational processes in cancer samples, we can make reasonable and logical assumptions that facilitate our method design. Here, we aimed to design a computational framework, sigLASSO, which could meet these criteria. First, we assumed that the set of estimated mutational mechanisms should be small, as de novo studies indicate that not all signatures can be active in a single sample or even a given cancer type. In most cancer samples, only a few signatures are identified in the original de novo discovery studies. An ideal tool should generate solutions that follow this sparse distribution. Moreover, the number of detectable mutation signatures is limited by the amount of data support. Too many signatures lead to overfitting and unstable solutions. We aimed for a sparser solution as it explains observations in a simpler fashion. Second, the estimated mutational mechanisms should be biologically interpretable and reflect some cancer type specificity. For example, we should not observe UV-associated signatures in tissues that are not exposed to UV. Likewise, we only expect to observe activation-induced cytidine deaminase mutational processes, which are biologically involved in antibody diversification, in B-cell lymphomas. Finally, we felt the solution should be robust and the data should control the model complexity.8. Low mutation count results in high sampling variance, leading to an unreliable estimation of the mutation context probability distribution, which is the target for signature fitting. A desirable signature identification tool should model the sampling process and take sampling variance into consideration.In particular, reliably recovering the signature composition is challenging when the mutation number is low11) and L1-based regularization (SparseSignatures12). The most straightforward way to do this would be to use the L0 norm , but this approach cannot be effectively optimized. Conversely, using the L2 norm flattened out at small values leads to many tiny, non-zero coefficients, which do not resemble the sparse signature distribution in the original de novo studies and are hard to interpret biologically. sigLASSO uses the L1 norm, which promotes sparsity. The L1 norm is convex, and thus allows efficient optimization14. In addition, this approach is able to harmoniously integrate prior biological knowledge into the solution by fine-tuning penalties on the coefficients. Compared with the approach of subsetting signatures before fitting, our soft thresholding method is more flexible to noise and unidentified signatures. Finally, sigLASSO is aware of data complexity such as mutational number and patterns in the observation. It is able to abstain  and defer to the human researcher to decide. Our method is automatically parameterized empirically on performance, allowing data complexity to inform model complexity. In this way, our approach also promotes result reproducibility and fair comparison of data sets.In this work, we formulated the task as a joint optimization problem with L1 regularization. First, by jointly fitting signatures and the parameters of a multinomial sampling process, sigLASSO takes into account the sampling uncertainty. Cooperatively fitting a linear mixture and maximizing the sampling likelihood enables knowledge transfer and improves performance. Specifically, signature fitting imposes constraints on the previously unconstrained multinomial sampling probability distribution. Conversely, a better estimation of the multinomial sampling probability improves signature fitting. This property is especially critical in high sampling variance settings, for example, when we only observe low mutation counts in whole-exome sequencing (WES). Existing methods use continuous relaxation, which makes the model invariant to different mutation counts. Second, sigLASSO penalizes the model complexity and achieves variable selection by regularization. Regularization is essential for this fitting problem with a large amount of signatures to achieve proper variable selection and avoid overfitting. Using regularization to promote sparsity and prevent overfitting is also a standard practice in de novo signature discovery. A few recent examples are rule-based constraints (SigProfiler), a Bayesian variant of NMF that penalizes on model complexity .Mutational processes leave mutations in the genome within distinct nucleotide contexts. We denoted the total number of contexts as K latent mutational processes. Large-scale pan-cancer analyses identified K = 30 COSMIC signatures by NMF  and clustering3. Here, our objective was to leverage the pan-cancer analysis and decompose mutations from new samples into a linear combination of signatures. Mathematically, we formulated the following non-negative regression problem, maintaining the original Frobenius norm:n trinucleotide contexts. ith category. n × K signature matrix, containing the mutation probability in 96 trinucleotide contexts of the 30 signatures. Then there are 8 neglecting the discrete nature of mutation counts. This approach essentially transforms observed mutations into a multinomial probability distribution, making model estimation insensitive to the total mutation count. However, the total mutation count has a critical role in inference. Assuming mutations are drawn from a latent probability distribution, which is the mixture of several mutational signatures, the mutations follow a multinomial distribution. The total mutation count is the sample size of the distribution, thus greatly affecting the variance of the inferred distribution.In practice, this problem is optimized using continuous relaxation for efficiency and simplicity,For instance, 20 mutations within the 96 categories give us very little confidence in inferring the underlying mutation distribution. By contrast, if we observed 2000 mutations, we would have much higher confidence. Methods using continuous relaxation treat these two conditions indifferently. Here, we aimed to use a likelihood-based approach to acknowledge the sampling variance and design a tool sensitive to the total mutation count.mi(i = 1…n) to denote the mutation count of the ith category. The vector pj, denoting the probability of the jth category. The total number of mutations is N.We divided the data generation process into two parts. First, multiple mutational signatures mix together to form an underlying latent mutation distribution. Second, we observed a set of categorical data (mutations), which is a realization of the underlying mutation distribution. We used λ. LASSO is mathematically justified and can be computationally solved efficiently14. Adding an L1 norm regularizer is equivalent to placing a Laplacian prior on 19. K penalty weights , each indicating the strength to penalize the coefficient of a certain signature. This vector should be tuned to reflect the level of confidence in the prior knowledge. For example, a smaller penalty weight represents a stronger prior, reflecting higher confidence and vice versa.To achieve variable selection and promote sparsity and interpretability of the solution, sigLASSO adds an L1 norm regularizer on the weights α = 1/σ2. We can infer α from the residual errors from linear regression (see parameter tuning). Meanwhile, because of its continuous nature, 20 by initializing βOLS, where βOLS are the coefficients from non-negative ordinary least square. Our aim was to obtain a less-biased estimator by applying smaller penalties on variables with larger coefficients.Overall, from the generative model, we can determine the likelihood function for a single sample.By jointly optimizing both the sampling process and signature fitting, sigLASSO is aware of the sampling variance and infers an underlying mutational context distribution mentclass2pt{minimmentclass2pt{minimWe illustrated how the mutation count affects the estimation of Because linear fitting and sampling likelihood optimization mutually inform each other, concurrently learning an auxiliary sampling likelihood improves performance. We compared the accuracy of the estimation of mentclass2pt{minimWe first evaluated sigLASSO on simulated data sets. Both sigLASSO (with and without priors) and deconstructSigs performed better with higher mutation number and lower noise Fig. . A decreAlthough the recall was slightly lower with sigLASSO than deconstructSigs in noisy settings, in very low and no noise situations sigLASSO achieved, a higher recall owing to its ability to adapt to different noise levels. By contrast, deconstructSigs, which assumes a fixed noise level, overly pruned the signatures in the post-fitting step when the noise was low. At last, adding correct priors helped boost both precision and recall significantly.We next explored how different priors affect sigLASSO’s performance. Our experiments showed that using known signatures as priors to tune the weights boosts performance. Priors improved performance even when we included only a small fraction of true signatures or blended in a large number of wrong signatures Fig. . As the We next moved from synthetic data sets to real cancer mutational profiles, which are likely noisier than simulations and exhibit a highly non-random distribution of signatures.21. The median mutation count was 4528 (range: 912–9257). We found that without prior knowledge, both sigLASSO and deconstructSigs showed high contributions from signatures 3 and 5 . Adding prior from COSMIC (kidney cancer combined) helps sigLASSO to put more weight on Signature 5, and increase the number of signatures assigned to 4.06.We benchmarked the two methods using 35 WGS papillary kidney cancer samplesHowever, if we naively subset the signatures and took the ones that were found to be active in previous studies, the signature profile was completely dominated by signature 5, to which only ~10% mutations on average were assigned with other signatures. Moreover, the model assigned highly similar signature profiles to all samples. This finding suggests an overly simple, underfitted model.To show that sigLASSO is sensitive to fitting uncertainty, able to abstain, and is robust, we performed subsampling on 30 pRCC samples  to 0.97 (20 mutations). The model complexity, as reflected by the mean number of signatures assigned also increased from 4.43 to 4.70. The agreement between subsamples dropped significantly as the subsampling size decreased, indicating the solution is unstable and potentially overfitted.We next aimed to evaluate the two methods on 182 WES esophageal carcinoma (ESCA) samples with >20 mutations. The median mutation count was 175.5 (range: 28–2146), which is considerably lower than WGS but typical for WES.In sigLASSO, the L1 penalty strength was tuned based on model performance. In a low mutation count setting, ESCA WES data set, the model variance was high, which pushed up the penalty. As expected by its ability to abstain, sigLASSO assigned a lower fraction of mutations with signatures than deconstructSigs : 0.56–0.75 and 0.86–0.94, respectively, Supplementary Figure 8. We demonstrated that sigLASSO generates a sparser but comparable result with wider signature 1, and 16 gaps between the subtypes  tumors , we expect the signature assignment to be very sparse as the uncertainty is high, the model should from making assignments. sigLASSO indeed assigned low number of signatures to the samples . In comparison, the solutions of deconstructSigs were less sparse . In all, 18.5% samples were fitted by deconstuctSigs with six signatures or more, compared with 0.4% by sigLASSO. We provide a kidney cancer example and detailed paired analysis in the Supplement  samples, smoking samples carry significantly higher signature 4 (“smoking signatures”) fractions than non-smoking ones , we also found a weaker but statistically significant trend of higher signature 4 in smokers  preventing over- and underfitting on only one sample, often with very few mutations , (2) achieving proper variable selection from a large amount of recognized signatures, and (3) promoting interpretability.First, with less than a few hundred mutations, sampling variance becomes a significant factor in reliably identifying signatures. Therefore, the fitting scheme should be aware of the sampling variance, which is especially pronounced in low mutation count scenarios (WES or cancer types with low mutation burden). Ideally, the tool should be able to attribute the signatures by flexibly inferring the underlying true mutation distribution given the sampling variance and the signature-fitting performance. Second, the large number of available signatures necessitates a proper variable selection, especially with a limited amount of observed data. Signature studies on large-scale cancer data sets have revealed that mutational signatures are not all active in one sample or cancer type; in most tumor cases, only a few signatures prevail. A recent signature summary suggested that 2–13 known signatures are observed in a given cancer type (based on 30 COSMIC signatures), which might include hundreds and even thousands of samples. Therefore, sparse solutions are in line with previous de novo studies, and are biologically sound and interpretable. In addition, sparse solutions can give better predictions owing to lower estimator variance. Third, a desirable method should be parameterized according to the data complexity to achieve optimum fitting. Finally, mutational signatures are not orthogonal owing to their biological nature. Collinearity of the signatures will lead to unstable fittings that change erratically with even a slight perturbation of the observation.23.deconstructSigs was the first tool that could identify signatures even in a single tumor. Instead of regression, this tool uses binary search to iteratively tune coefficients. To achieve sparsity, deconstructSigs performs post hoc pruning with a preset 6% cutoff value. The mutation spectrum is normalized before fitting, thus making mutation counts invariant to the model. Moreover, the binary search operates on unbounded coefficients and uses a hardcoded upper and lower bounds, which provides no guarantee of finding the optimal solution. Finally, the greedy nature of stepwise coefficient tuning is prone to eliminating valuable predictors in later steps that are correlated with previously selected onesHere, we describe sigLASSO, which simultaneously optimizes both the sampling process and an L1 regularized signature fitting. By explicitly formulating a multinomial sampling likelihood into the optimization, we designed sigLASSO to take into account the sampling variance. Meanwhile, sigLASSO uses the L1 norm to penalize the coefficients, thus achieving effective variable selection and promoting sparsity. By fine-tuning the penalizing terms using prior biological knowledge, sigLASSO is able to further exploit previous signature studies from large cohorts and promote signatures that are believed to be active in a soft thresholding manner.Jointly optimizing a mutation sampling process enables sigLASSO to be aware of the sampling variance. By additionally modeling an auxiliary multinomial sampling process and corresponding distribution, we demonstrated that sigLASSO achieves better and more stable signature attribution, especially in cases with low mutation counts. In cancer research, WES data are abundant, but it also suffers from undersampling in signature attribution. In these cases, sigLASSO generates more reliable and robust solutions. We showcase the successful application of sigLASSO on 8893 TCGA WES samples. Overall, sigLASSO achieves better sparsity than deconstructSigs and promote structures in the previous signature studies by injecting priors into the optimization process. The final signature distribution of neither sigLASSO nor deconstructSigs is identical to the original signature studies, which is probably owing to biases in the original signature discovery, sequencing/variants calling, and unknown hypermutational processes. We noticed deconstructSigs assigned signatures to 100% of the mutations in most samples, while in sigLASSO assignment fraction correlated with cancer types. This abstain pattern indicates the presence of potentially unknown signatures and cancer subtypes.24. The vast amount of cancer genomics data will give scientists larger power to discern unknown or rare signatures. The growing number of signatures will eventually make the signature matrix underdetermined . A traditional simple solver method would give infinitude (noiseless) or unstable (noisy) solutions in this underdetermined linear system. However, by assuming the solution is sparse, we were able to apply regularization to achieve a simpler, sparser, and more stable solution. Furthermore, sigLASSO is very efficient and thus able to handle a larger number of samples and signatures, as well as hyperparameter optimization.As the cost of WGS drops rapidly, we expect an even greater number of cancer samples to be sequenced25 In both subsampling and simulation experiments, sigLASSO is able to abstain under high uncertainty, providing solutions with consistently high precision and robustness under various scenarios.Moreover, sigLASSO does not specify a noise level explicitly beforehand, but instead empirically tunes parameters based on model performance. By contrast, deconstructSigs specifies a noise level of 0.05 to derive a cutoff of 0.06 to achieve sparsity. In general, sigLASSO lets the data itself control the model complexity and leave any post hoc filtering to users. Abstention is a desired trait in machine learning models. In addition to the objectives, the model learns what it knows and what it does not (KWIK: Knows What It Knows),12 also uses L1 regularization but does not penalize on background signatures. sigLASSO could use this piece of information in its priors to better assign signatures when working together with SparseSignatures. We showcased the performance of sigLASSO on real cancer data sets. Although we lack the ground truth of the operative mutational signatures in tumors, we have several reasonable beliefs about the signature solution. sigLASSO produced signature solutions that are biologically interpretable, properly align with our current knowledge about mutational signatures, and well distinguish cancer types and histological subtypes.Next, owing to the collinearity nature of signatures, pure mathematical optimization might lead algorithms to select wrong signatures that are highly correlated with truly active ones. To overcome this problem, sigLASSO allows researchers to incorporate domain knowledge to guide signature identification. This input could be cancer type-specific signatures or patient clinical information . Furthermore, we transformed the binary classification (active or inactive) to a continuous space, with weights indicating the strength of the prior. These weights could be effectively learned using patient information or other assays . Moreover, sigLASSO can adapt to sparsity promoting schemes in de novo discovery tools through priors. For example, SparseSignatures26. We found that elastic net did not improve the performance  in our simulations using the current 30 COSMIC signatures, likely because the correlations between them are not too high. However, with more cancer samples sequenced every day, researchers will gain power to discern highly correlated mutational processes and grow the size of the signature set significantly. Therefore, elastic net might be beneficial in the near future.We also implemented elastic net with hyperparameter optimization in sigLASSO. Elastic net blends L2 with L1 regularization and in principle demonstrates better performance and stability than LASSO on strongly correlated features5. By doing so, sigLASSO is unbiased in detecting signatures in cancer samples. Nonetheless, the initial discovery suffers from several limitations. First, the actual number of mutational processes is likely higher than 30, the current number of COSMIC signatures. This is because the power of the de novo discovery is bounded by the amount of available data. Second, the nature of mutagenesis leads to some mutational processes being more prevalent than others. Some of the signatures are cancer type specific. The original signature discovery did not use a balanced data set with equal numbers of cancer types. Third, the original NMF objective function employs quadratic loss, which might lead to bias towards flat signatures over spiky ones. In addition to the original NMF approach, researchers have proposed other decomposition methods for signature discovery. For example, SomaticSignatures27 uses PCA. Because the loss function is also quadratic, we expect sigLASSO to work seamlessly with SomaticSignatures.Finally, sigLASSO uses quadratic loss, which follows the previous de novo studiesOur simulation reveals that sigLASSO does not have a discernable preference to assign to either spiky or smooth signatures  and the sampling likelihood. In biological experiments, low-count observations on discrete variables are common. For example, in single cell RNA sequencing (scRNAseq), the measured discrete mRNA counts are often very low or even zero (owing to undersampling). Our joint optimization approach could have further implications in these scenarios. We, indeed, find some similar work in scRNAseq31.The negative log-likelihood is convex in respect to both sigLASSO algorithmλ is parameterized empirically.Specifically, to begin the iteration, we initialized pi is satisfied by only retaining the non-negative root of the solution.Next, we solved the mi), the goodness of the signature fit (as reflected in α), and the overall shapes of Intuitively, in the In the Λ to satisfy the linear constraint of i = 1, 2. . . n) and Λ. This resulted in n + 1 equations.We added the Lagrangian multiplier n quadratic equations are given byThe roots of the first α = 1/σ2 is strictly positive and mi is non-negative. Therefore, if mi = 0, there exists only one zero root and mi = 0. If mi > 0, there is exactly one negative and one positive root. Because we required  ∀Λ.We plugged all the roots into the last equation  and used the R function uniroot to solve λ by repeatedly splitting the nucleotide contexts into training and testing sets and testing the performance. Because mutations of the same single-nucleotide substitution context are often correlated, we split the data set into eight subsets. Each subset contained two of each single-nucleotide substitution. We then held one subset as the testing data set and only fit the signatures on the remaining ones. After circling all eight subsets and repeating the process 20 times, we used the largest λ (which leads to a sparser solution) that gave an MSE 0.5 or 1 SD from the minimum MSE. λ was tuned whenever λ when the deviation was small to avoid the inherited randomness in subsetting affecting convergence.We tuned α = 1/σ2, σ2 is estimated using k is the number of non-zero coefficients in the LASSO estimator and SSE is the sum of squared errors32. sigLASSO updates α after every LASSO linear fitting step. To avoid grossly overestimating σ2 (thus underestimating α) in the initial steps when α value. In addition, because in practice factors such as unknown signatures violate the assumptions of linear signature regression, α tends to get overestimated. So we multiplied it by a confidence factor that represents how confident we are about the goodness of the signature fit. By default, we set min α = 400 and the confidence factor to 0.1. Users can further tune these values based on the strength of prior belief of noise level and confidence level of the signature model.26. Therefore, we also implemented elastic net in sigLASSO. The objective function then became:Adding an L2 regularizer  might improve and stabilize the performance when variables are highly correlatedγ by grid search together with λ. We always picked the largest γ and the largest λ (for better sparsity) that gave an MSE that was 0.5 SD from the minimal MSE. In simulations, we did not find that introducing L2 regularization added additional benefits. The model gave almost identical solutions to using L1 only. Nonetheless, we kept elastic net as an option in highly correlated signature scenarios to the user in our implementation.We tuned the hyperparameter http://cancer.sanger.ac.uk/cosmic/signatures). We created a simulated data set by randomly and uniformly drawing two to eight signatures and corresponding weights (minimum: 0.02). The reason for picking up to eight signatures is because (1) empirically, in cancer signature studies, most samples have only a few signatures. (2) We further confirmed the sparsity of signatures by running deconstructSigs on large-scale TCGA data and found 99.96% samples got assigned with eight or less signatures. (3) Moreover, biologically, we believe that it is unreasonable for more than eight of the processes to act simultaneously in one sample. Many of these processes describe biological disjoint conditions, e.g., smoke, UV radiation and tissue-specific cellular processes. We simulated additive Gaussian noise at various levels with a positive normal distribution of up to 25  randomly selected trinucleotide contexts. Then, we summed all the signatures and noise to form a mutation distribution. We sample mutations from this distribution with different mutation counts.We downloaded 30 previously identified COSMIC signatures v2 .We ran deconstructSigs according to the original publicationhttps://portal.gdc.cancer.gov/). A detailed list of files used in this study can be found in Supplementary Table To assess the performance of our method on real-world cancer data sets, we used somatic mutations from 33 cancer types from TCGA. We downloaded MAF files from the Genomic Data Commons Data Portal (http://cancer.sanger.ac.uk/cosmic/signatures).We extracted prior knowledge on active signatures in various cancer types from a previous pan-cancer signature analysis  and their weights. “siglasso” uses 30 COSMIC signatures by default. Users are also given the option to supply customized signature files. It is computationally efficient; using default settings, the program can successfully decompose a whole genome sequencing (WGS) cancer sample in less than a few seconds on a regular laptop. For time profiling purposes, we ran siglasso and deconstructSigs on an Intel Xeon E5-2660 (2.60 GHz) CPU. We employed the R package “microbenchmark” to profile the function call siglasso and whichSignatures. For each setup, we generated 100 noiseless simulated data sets and repeated the process 100 times for each evaluation.http://github.com/gersteinlab/siglasso.We made siglasso source code available at One of the limitations of cancer signature research is that the ground truth of real samples typically cannot be obtained. Previous large-scale signature studies have relied largely on mutagen exposure association from patient records and biochemical knowledge on mutagenesis. Here, besides using simulation, we illustrated the outputs of different models and compared the results on real data set with existing signature knowledge and distributions. Although no gold standard exists to evaluate the performance, we do have a few reasonable expectations about the solution:Sparsity: one or more signature should be active in a given cancer sample and type. However, not all signatures should be active. A sparse distributed signature set among cancer samples and types is defined in previous de novo discovery studies. Any signature-fitting tool should follow and produce a similar signature distribution. Moreover, mutational processes are discrete in nature and tied with certain endogenous and environmental factors. An obvious example is that the UV signature should not exist in unexposed tissues. Existing signature-identifying methods aim to implicitly achieve sparse solutions by dropping signatures with small coefficients or pre-selecting a small signature subset for fitting.The ability to abstain : signature assignment issues are often undefined owing to collinearity of the signatures and a larger number of possible signatures. Overfitting is a serious concern, especially when the mutation counts are low  or the fitting is poor . A good solution should refuse to fully assign signatures to every mutation when it does not have enough confidence to do so and, instead, defer to the human researchers to decide.Robustness: solutions should be robust and reproducible. Signatures are not orthogonal, thus simple regression might lead to solutions that change erratically when a small perturbation is made in the observation. Moreover, the solution should reflect the level of ascertainment. Especially in WES, low mutation count is often a severe obstacle for assigning signatures owing to undersampling. In particular, under low mutation count, not all of the operative signatures would be reliably discovered. It is better to abstain under high uncertainty.Biological interpretability: the solution should be biological interpretable. Because of the biological nature of collinearity in the signatures, simple mathematical optimization might pick the wrong signature. Researchers now tackle this problem by simply removing the majority of predictors they believe to be inactive. sigLASSO allows users to supply domain knowledge to guide the variable selection in a soft thresholding manner, leaving space for noise and rare or unknown signatures. For instance, we expected to find divergent signature distributions in different cancer types, which is reported in previous de novo discovery studies. Various tissues have divergent endogenous biological features, are exposed to diverse mutagens, and undergo mutagenesis in dissimilar fashions. Signature patterns should be able to distinguish between cancer types. This can be achieved by using different cancer type-specific priors.These expectations are not quantitative, but they help direct us to recognize the most plausible solution as well as the less-favorable ones.Supplementary InformationPeer Review File


PhenMap) to identify unsupervised clinically-relevant  subtypes and biomarkers by simultaneously integrating clinical phenotypes and single omics data, mainly, transcriptome. This integrative analysis provides the opportunity to simultaneously identify robust and context-specific subtypes and associated phenotypes  without statistically losing information. We demonstrate its utility using breast cancer cell lines and patient samples to identify functional subtypes associated with specific drug responses (including CD4/6 inhibitor) and prognosis. These subtypes may potentially predict clinical outcomes with further validation. This tool can be applied to other omics data such as methylomics, genomics and radiomics along with any phenotypic data.Tumours are heterogeneous that reflect variable patient prognosis and treatment responses (phenotypes). Since these variable phenotypes are outcomes of genomics, it is essential to integrate genome and phenome jointly. In this study, we report the development and application of a new machine learning tool , which jointly integrates single omics data with phenotypic information using three published breast cancer datasets (n = 2045). The PhenMap framework uses a modified factor analysis method that is governed by a key assumption that, features from different omics data types are correlated due to specific “hidden/mapping” variables (context-specific mapping variables (CMV)). These variables can be simultaneously modeled with phenotypic data as covariates to yield functional subtypes and their associated features  and phenotypes. In one example, we demonstrate the identification and validation of six novel “functional” (discrete) subtypes with differential responses to a cyclin-dependent kinase (CDK)4/6 inhibitor and etoposide by jointly integrating transcriptome profiles with four different drug response data from 37 breast cancer cell lines. These robust subtypes are also present in patient breast tumors with different prognosis. In another example, we modeled patient gene expression profiles and clinical covariates together to identify continuous subtypes with clinical/biological implications. Overall, this genome-phenome machine-learning integration tool, PhenMap identifies functional and phenotype-integrated discrete or continuous subtypes with clinical translational potential.One of the major challenges in defining clinically-relevant and less heterogeneous tumor subtypes is assigning biological and/or clinical interpretations to etiological (intrinsic) subtypes. Conventional clustering/subtyping approaches often fail to define such subtypes, as they involve several discrete steps. Here we demonstrate a unique machine-learning method, phenotype mapping ( Cancers represent a heterogeneous collection of diseases with different molecular features, prognoses, and responses to treatment ,2,3,4,5.PhenMap. This framework is an unsupervised machine-learning approach to simultaneously integrate cancer omics and phenome data to reveal clinically relevant and functional  subtypes with defined clinical indications. Here, we demonstrate PhenMap using two different examples and three different breast cancer datasets (n = 2045). Using PhenMap on one of the breast cancer datasets, we demonstrate the identification of subtypes that may predict response to a cyclin-dependent kinase (CDK)4/6 inhibitor. These biomarkers with further extensive validation may serve as biomarkers for the CDK4/6 inhibitors that are currently available for breast cancer patients with hormone receptor-positive metastatic/advanced breast cancer , ξi~MVNp , and εi~MVNq , respectively, where Σ = diag(σ12… p2σ) and Φ = diag(ϕ12… p2ϕ) are residual variances for both the observed data and CMVs, respectively. Bayesian methodology was employed to fit this model as it is known to perform better than deterministic algorithms when the data is of small sample size, a common scenario in biology.Gene expression data for sample PhenMap is that the observed correlations between features are due to CMVs such that conditional on these CMVs the features are independent of each other. Hence, the estimated loadings and regression coefficients  can be used to identify features and phenotypes, respectively, associated with CMVs.The key assumption behind the model within W and β, respectively. For features, the automatic relevance determination (ARD) prior , where j = 1…p, k = 1…q and λjk is a precision hyper-parameter which controls the contribution (loading) of feature j on the kth CMV, wjk. The λjk has a gamma prior distribution p = G. Large values of λjk result in shrinkage of wjk towards zero, inducing sparsity in W . The other variance parameters in Σ and Φ are also allowed to have independent gamma priors.In an effort to allow for automatic feature and phenotype selection, we introduced sparsity through priors on the elements of D) prior , i.e., iβ, g-priors [β and, also, parameter estimation using g-priors is invariant to changes in the scales of phenotypes [l + 1)-dimensional multivariate Gaussian g-prior distribution centered at zero, with a parameter g to control the prior covariance, p(βk|g) = MVNl+1[βk|0, g(XTX)−1β. The role of g is to shrink the effect of non-informative phenotypes towards zero.For regression coefficients g-priors  were adoenotypes . An , where ll is the maximum log likelihood, C and n represent the number of parameters and samples, respectively. The highest BIC value indicates the best model. A regularized version of BIC was used in PhenMap models to select the optimal number of CMVs. This regularized BIC method avoids the disadvantage of using standard BIC, which performs poorly in high-dimensional data settings or when the model has multiple parameters. Additionally, this modification in BIC evaluates the likelihood of the maximum aposteriori (MAP) estimate instead of the maximum likelihood estimate [PhenMap . Trace plots (PhenMap model). PhenMap were mixing very well , and hence, were sampling from the target distribution (model convergence).The Markov chain Monte Carlo (MCMC) algorithm allows us to sample from the target distribution  breast ribution  between PhenMap is complex and MCMC sampling techniques are required to produce samples from this complex distribution. The full conditional distributions of all the parameters exist in standard form. Hence, a Gibbs sampler [s = 1…S, the number of iterations or cycles;The posterior distribution of the model in  sampler  was cons(1)U(s), from a Gaussian full conditional distribution,Derive the CMVs, (2)Λ(s) from a Gamma full conditional distribution,Derive the precision hyper-parameters (3)W(s) from a Gaussian full conditional distribution,Derive the loadings matrix (4)Σ(s) from an inverse-Gamma full conditional distribution,Derive the error covariance (5)β(s) from a Gaussian full conditional distribution, andDerive the regression coefficients (6)Φ(s) from an inverse-Gamma full conditional distribution.Derive the CMV covariance parameters 50 values (drug response information) were unit scaled for them to be comparable. ConsensusClusterPlus [R package was used to repeatedly (1000 times) cluster the data to eliminate the issue of dependence on initial conditions associated with K-means. The number of subtypes were varied from two to seven and both cophenetic coefficient [The CMVs and drug response information of the associated drugs (etoposide and fascaplysin) were jointly clustered using K-means consensus clustering. Firstly, the CMVs and the (growth inhibitory concentration) GIsterPlus  R packagfficient  and silhfficient  measuresfficient . Silhouefficient . High vaPhenMap, respectively. PAM centroid represents scaled average expression of the signature in each subtype [PhenMap) was generated for both classifiers. The expression pattern of a new sample was correlated to the PAM centroid and assigned to the subtype with highest Pearson correlation coefficients.In order to assign new samples into UF-subtypes and CMV-2-subtypes, we developed classifiers based on prediction analysis of microarray (PAM ) centroi subtype . Using fPublished gene expression data of breast cancer cell lines and drug response data from our original publication  were usehttp://www.cbioportal.org/datasets; access date: 2017/05/10) gene expression data sets with their corresponding clinical information were used. Hence, in total, 2043 breast cancer samples were used in this work .PhenMap resource: The PhenMap package is available as an R package on github . Overall, PhenMap has the significant advantages over conventional clustering approaches of being able to simultaneously derive discrete or continuous functional subtypes, associate subtypes with phenotypes, provide biological/clinical implications, define subtype-specific biomarkers/signatures, and provide context-specific information.Our tool will be applicable clinically for stratifying cancers or other diseases provided large-scale data generated by The Cancer Genome Atlas (TCGA), International Cancer Genome Consortium (ICGC), and other similar consortia. Although this model can be applied to other individual- or multi-omics data types , here, we used gene expression data that is widely available and clinically applicable to illustrate its potential. Currently, the model is limited to a single quantitative omics data type (containing continuous data) at a time. However, there can be multiple sample-matched phenotypic data of any types. Like any other factor-analytic model,


As a proof-of-concept, we also tested chromatin accessibility data from acute myeloid leukemia (AML) and successfully obtained unique cell types associated with AML progression. Furthermore, we showed that chromatin accessibility represents more essential characteristics in the identification of cell types than gene expression. Taken together, DeconPeaker as a powerful tool has the potential to combine different datasets  and define different cell types in mixtures. The Python package of DeconPeaker is now available at https://github.com/lihuamei/DeconPeaker.While our understanding of cellular and molecular processes has grown exponentially, issues related to the cell microenvironment and cellular heterogeneity have sparked a new debate concerning the cell identity. Cell composition (chromatin and nuclear architecture) poses a strong risk for dynamic changes in the diseased condition. Since chromatin accessibility patterns play a major role in human diseases, it is therefore anticipated that a deconvolution tool based on open chromatin data will provide better performance in identifying cell composition. Herein, we have designed the deconvolution tool “DeconPeaker,” which can precisely define the uniqueness among subpopulations of cells using open chromatin datasets. Using this tool, we simultaneously evaluated chromatin accessibility and gene expression datasets to estimate cell types and their respective proportions in a mixture of samples. In comparison to other known deconvolution methods, we observed the lowest average root-mean-square error (RMSE = 0.042) and the highest average correlation coefficient ( Human diseases are multifactorial and complex processes in which genetic–epigenetic components are significantly involved. To date, several key biological pathways regulating cellular functions have been defined; however, the knowledge about the behavior of individual cells is still very limited. Furthermore, the diversity among intracellular and intercellular interactions creates a significant challenge toward understanding of this multicellular network. To mention, the lack of defined gene signature and biological characteristics of bulk tissues from the histological district subtypes lead to the suboptimal–mediocre results in human diseases .Several disease association studies have suggested the cell type composition as a confounding factor . For insTo gain better statistical insight into the composition of the cell types in a sample mixture, many methods  have previously been developed. Most of these approaches use gene expression data by focusing on estimating the proportions and/or pure expression states, which can be divided into two subclasses, “partial” and “complete” . The forConsidering these factors, we generated DeconPeaker, a partial deconvolution method that resolves relative proportions of different cell types in the peak intensity profiles (chromatin accessibility) from the measurement of mixture samples. Compared to other known deconvolution methods, DeconPeaker is reliable and applicable to both chromatin accessibility as well as gene expression data from mixtures.m = B × f, where m is the measurement for the mixtures, B represents the signature matrix, and f denotes a vector of unknown proportions that needs to be estimated  to align pair-end reads to the hg19 genome and remove duplicate reads; (2) to call narrow peaks with MACS2  to avoid false positives as previously described . The peaescribed , a tool escribed  was usedTo avoid batch effects, we performed quantile normalization on the count matrix across all reference samples and excluded peaks below the median value of globally normalized intensities.A statistically reliable and significant CTSP represents an open chromatin region that tends to be more pronounced in one cell type than others. Recently, Zamanighomi et al. used a Poisson regression model combined with hypothesis testing to call CTSPs in single-cell ATAC-seq (scATAC-Seq) data , which wB, which represents the signature matrix, as mentioned above. In a typical human ATCA-Seq, thousands of CTSPs can introduce noise, which can be avoided by shrinking the total number of CTSPs simultaneously, while considering the stability of the linear system to avoid extreme sensitivity to small fluctuations  and mixtures (m)], the accuracy and reliability of the deconvolution are difficult to guarantee. Therefore, we employed an automatic transformation method to enhance the adaptability of DeconPeaker, primarily based on the optimal parameter λ, estimated by the Box–Cox algorithm  and the measurement of the mixture (m), which was conducted using the function “mvr” in the R packages “pls” (f), negative regression coefficients were set to 0, and the remaining coefficients were normalized to sum to 1.In DeconPeaker, SIMPLS  was emples “pls” . To obtafp) and the known composition of cell type (ft) were calculated to evaluate the model, which were commonly used to measure differences between the ground truth and estimate. In addition, we employed an asymptotic test for the Wasserstein distance between observer m and prediction P-value of the deconvolution using Monte Carlo sampling. Wasserstein metric reflects the degree of similarity between two distributions, in which smaller differences and smaller assigned P-values indicated greater significance of the model. The details of estimating P-value are described in Root-mean-square error (RMSE) and Pearson correlation coefficient (PCC) between the prediction (We simulated 195 synthetic mixture samples with cell type-known ATAC-Seq data by sampling the cell type fractions using Dirichlet distribution from primary blood cells (GSE74912) with SAMTOOLS . Cell tyTo further validate the performance of DeconPeaker using chromatin accessibility data, we retrieved a dataset of transposase-accessible chromatin profiles for 695 individual mouse cardiac progenitor cells (PRJEB23303) covering E7.5 to E9.5 of five cell types . From thGSE74912 , as demonstrated in GSE74912  and fromgenitors . The foritors   , n−1 refAdditionally, the signature peak intensities showed a strong positive correlation between the pairs HSC and multipotent progenitor (MPP), granulocyte–monocyte progenitor (GMP), and lymphoid–primed multipotent progenitor (LMPP) , which lThe performance of DeconPeaker was also validated on single-cell ATAC-Seq datasets (PRJEB23303). This dataset contained ATAC-Seq data for 695 mouse single cardiac progenitor cells covering E7.5 to E9.5 of five cell types  see “Ma. For theSince lack of ATAC-Seq data of cell type proportion-known mixture samples, the evaluation of our tool is based on eight known gene expression benchmarking datasets (two RNA-Seq and six microarray data), which have been widely used to test deconvolution algorithms . AlthougWe used DeconPeaker and CIBERSORT to derive signature matrices with the reference samples of each benchmarking dataset to predict the cell type proportions in each mixed sample. In our analysis for RNA-Seq, for Liu’s data, both DeconPeaker and CIBERSORT showed good performance , while iNotably, from the RMSE perspective, the predictions on datasets of Parsons’, Shi’s, and Newman’s showed higher deviations from true proportions than on others . SeveralWe compared DeconPeaker against nine other deconvolution models, four of which were accessed through the CellMix R package , includiFor the first scenario mentioned above, the acquisition of signatures corresponding to different methods includes the following aspects. For DeconPeaker, CIBERSORT, and dtangle, the signature matrices were derived by their own specific strategy, while in the case of other models, the signature peaks (or genes) or signature matrices were provided by CIBERSORT. Since, some algorithms are preferred by certain platforms, such as CIBERSORT (to microarray), DeconRNASeq (to RNA-Seq), and dtangle (to both). If the deconvolution algorithms are applied to cross-platform data, their prior assumptions in the models may be destroyed. In this comparison, we have considered different configurations of the algorithms to reduce the impacts of cross-platforms on the models, and the details are provided in Shi’s = 0.054, RMSEParson’s = 0.091)  and found that DeconPeaker showed the lowest average RMSEs on the Shen-Orr’s and Newman’s data. Although DeconPeaker performed slightly low on Shi’s and Parsons’ data, it still performed quite well (RMSE= 0.091) , indicat= 0.091)  and the = 0.091) . For the= 0.091) . All of To evaluate the effect of signature matrices derived by DeconPeaker, the signature matrix containing 1,768 peaks from GSE74912  was usedt-test by minimizing condition number to derive signature matrices , an automatic data transformation method was introduced, and the deconvolution performance was evaluated. Here, we only applied “None” (without transform), “Auto” (Automatic transform), “Log2” (log2-transform), and “Sqrt” (square-root transform) transformations on the signature matrix and the mixtures of each benchmarking data. The results indicated that transformation “Auto” has the lowest average RMSEs, the highest average PCCs, and the most compact distribution , indicatWe used a dataset of HSCs from AML . This daFor the ATAC-Seq data (GSE74912), we built the signature matrix  based on 77 normal samples, covering 13 primary blood cell types (workflow shown in For RNA-Seq data (GSE74246), we built the signature matrix  with 49 normal transcriptomes to deconvolute the mixed transcriptomes of AML . The resTo further evaluate the capacity of chromatin accessibility data and gene expression data to identify cell types, we visualized the distribution of individual samples of different cell types with PCA, using either the signature peaks (from GSE74912)  or the sGene expression deconvolution methods are ideal to define unique cell types in transcriptomes of samples with mixed cell types. Likewise, DNA methylation data is also highly cell type specific and can reveal hidden components of tissue mixtures. Studies have shown that models using DNA methylation consistently outperformed the gene expression-based methods . NotablyTo validate DeconPeaker, we analyzed marker cell types of AML at different stages based on the chromatin accessibility  and mRNA expression data (GSE74246). The conclusions derived from chromatin accessibility were consistent with previous reports , suggestIn addition, we must also point out the possible biases when using a constant signature matrix to deconvolute samples under different disease states or sequencing platforms. Therefore, we propose a signature matrix based on purified data from multiple platforms and from both healthy and disease samples to reduce the biological and technique bias .Taken together, “DeconPeaker” is amenable to chromatin accessibility data measured with ATAC-Seq and gene expression datasets. Primarily, due to its flexible statistical approach, it will enable researchers to measure bulk biospecimens, in particular, the samples with a mixture of cell types. Notably, to enhance the performance, additional test using DeconPeaker, particularly on more ATAC-Seq datasets containing both reference and cell type proportion-known mixture samples should be conducted in the future.Publicly available datasets were analyzed in this study. This data can be found at NCBI: GSE74912, GSE74246, GSE5350, GSE29832, GSE19830, GSE11058, GSE19380, GSE65133, and GSE64098. This data can be available from the ENA repository: PRJEB8231 and PRJEB23303.HDL and XS designed the study. HML coded the algorithms. HML, HDL, and AS wrote and revised the manuscript. HML conducted the data analysis. XS, KL, and ZQ provided interpretation and discussion. All authors contributed and approved the final manuscript.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.