Digitizing Omics Profiles by Divergence from a Baseline

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Nathan seilytcnqe odtrieweea mc rfiefalls profile omics an where determine are variation, to genome-wide population techniques quantify the to especially in Methods data (4). omics underway Notably, now large-scale modalities. monitor data to omics Similar for efforts selection. needed low- therapy now assessment, and are for deviation approaches risk prognosis, clinically of and biomarkers, used use diagnosis and widely disease The already tests disease. is laboratory complex range dimensional normal a a dysregulation of from molecular indicative reflects is profile that omics part, determining single In toward a progress introduced. whether impeding variation, when normal of expected characterization genome-wide was insufficient from arises as limitation this care, of standard clinical impact to (1–3). beginning diseases complex organism also for are practice and mechanisms data organization, substan- molecular These tissue have development. of functioning, we understanding cell conditions, underlying our healthy organisms, enhanced and model and tially diseased different individuals from from across data com- tissues, and these and complex of types Through analyses cell data. statistical omics and as putational are to and samples referred of thousands high-dimensional collectively on These made been domains. have molecular measurements distinct sys- over biological complex tems of molecular characterization unknown the enabling previously features, other and metabolites, proteins, I baseline a from Dinalankara divergence Wikum by profiles omics Digitizing www.pnas.org/cgi/doi/10.1073/pnas.1721628115 eateto nooy on okn nvriySho fMdcn,Blioe D21205; MD Baltimore, Medicine, of School University Hopkins Johns Oncology, of Department rfiigo eei ains N pce,eieei marks, epigenetic species, RNA variants, global genetic enabled have of advances profiling technological decades, recent n eetees mc aahv e osgicnl nomthe inform significantly to yet have data omics Nevertheless, | digitization c arn Younes Laurent , a,1 | dysregulation inKe Qian , b,1 b | ia Xu Yiran , ug Marchionni Luigi , cancer c nttt o ytm ilg,Sate A98109 WA Seattle, Biology, Systems for Institute | rcso medicine precision b alnJi Lanlan , a,2 b n oadGeman Donald and , ioePagane Nicole , 1073/pnas.1721628115/-/DCSupplemental etrso eerglto npriua nra)tsu,the tissue, (normal) particular in unique the regulation determine gene to samples is of normal goal the is features if population However, tissue. “baseline” particular that a natural from of a cancers lies are then it For interest tissue, if distribution. of phenotypes “divergent” probability the baseline be if of the to instance, subset of said this support is sam- the of profile outside that realization population sample the any a also in Then from Suppose features baseline. available set. the are get as subset declared functional this of a are observations or cases ple special the gene important example genes; single for of a features, set any omics of of we levels subset suppose expression particular Briefly, a neces- interest. given of information are phenotypes biological characterize important to the intended divergence sary is of on representation most based data capture simplified states to This discrete baseline. a characterize few high-resolution from to a quantizing framework into by unified variation measurements nonnormal a and present normal we tests, clinical diseases. and pro- complex diagnosis of for they prevention approaches clinical principle, inform can in which disease-specific variation, and assays; normal poten- of quantification the omics unprecedented realize vide high-dimensional fully to of essential be tial may baseline, a to relative ulse nieArl1,2018. 16, April online Published at online information supporting contains article This 2 under distributed 1 4528. page is on QnAs See article BY-NC-ND). (CC 4.0 access License NonCommercial-NoDerivatives open This interest. of conflict no declare Berkeley. authors California, The of University B.Y., and data; University; curated paper. Columbia N.P.T., Y.X., the and Reviewers: wrote collected Q.K., D.G. L.M. W.D., and and results; L.M., T.M., L.Y., E.J.F., the A.L., W.D., N.P., interpreted and L.Y., W.D., N.D.P., and research; E.J.F., performed experiments Q.K., L.J. the W.D., and designed project; the D.G. initiated and D.G. L.M., and L.M. contributions: Author owo orsodnemyb drse.Eal [email protected] rmarchion@jhu. or [email protected] Email: edu. addressed. be may correspondence whom work. To this to equally contributed Q.K. and W.D. ue oaysbe ffaue rae sasnl entity, cancer single various and a Experi- tissues pathway. subtypes. as human fea- a healthy in treated both single levels involve features expression ments from data gene of granularity, the any example subset of for to any level applied to any be tures at can and method modality our Moreover, normal from variables levels. selected of departures diag- landscapes for clinical of testing basic nostic range mimicking thereby the population, to baseline individual a in an of com- profile by the representation paring data develop personalized simplified, We heterogeneity. highly the a person-to-person and data of these degree diseases. of complexity high complex the of are obstacles data treatment major these Two personalized and the cells, inform of landscapes can comprehensive molecular the increasingly of profiling enable advances Technological Significance odvlpti ihdmninlaaou olow-dimensional to analogue high-dimensional this develop To a nhn Lien Anching , b eateto ple ahmtc n ttsis Johns Statistics, and Mathematics Applied of Department b,2 PNAS | a ,2018 1, May a eav Matam Tejasvi , . | o.115 vol. www.pnas.org/lookup/suppl/doi:10. raieCmosAttribution- Commons Creative a | ln .Fertig J. Elana , o 18 no. | 4545–4552 a ,

BIOPHYSICS AND STATISTICS INAUGURAL ARTICLE COMPUTATIONAL BIOLOGY baseline could be all other tissues types combined. The novelty Divergence. Suppose we are given a reference joint probability distribution of our framework is that it applies to any subset of features and P0 of (Xj, j ∈ J ) associated with some baseline phenotype. Of course, in prac- to any new profile by itself. tice, we only observe samples from this distribution where each sample is Divergence is a sample property. In contrast, putative discrim- an omics profile from a single data modality, and the choice of reference inating sets of features, such as “differentially expressed genes,” phenotype depends on the particular study and data modality. As will be seen in Results, the baseline phenotype need not be a single tissue type or a are defined in terms of population-level distributions. For exam- disease-free phenotype and can be any population that is meaningful under ple, given two classes or phenotypes, a gene is differentially the analysis carried out. expressed if its distribution differs in the two classes. However, Given an omics profile X and a subset S ⊂ J , we will apply the defi- S S knowing the status of a molecular feature in the context of pop- nition of divergence to the random vector U = Q (X) = (Qj(X), j ∈ S). We ulations may say very little about the status of that feature for will focus on single features S = {j} (e.g., single genes) and on sets S a particular individual. Divergence then enables the prospec- with 1 |S| |J | (e.g., gene pathways). Let m = |S|, so that US takes m S m S tive analysis of individual samples, as required for applications values in [0, 1] . We describe below how the support U0 ⊂ [0, 1] of U can be estimated from the data, resulting in a set UˆS that is explic- to precision and personalized medicine. It also enables a proba- 0 bilistic analysis at the phenotype level, where the probability that itly defined. When presented with a new sample X, we define a binary variable by a specific feature (say gene), or set of features (say pathway), is ( S S 1 if Q (X) ∈/ Uˆ , divergent is a well-defined concept. Phenotypes can then be char- Z = 0 acterized based on the corresponding divergence probabilities 0 otherwise . and predicted for unlabeled samples by examining the diver- We will then say that the set S is divergent (for the considered sample) if gent set. This approach can be applied to any high-dimensional and only if Z = 1. omics dataset, yielding simplified matrices of digitized values that can be analyzed statistically at the feature, sample, or Support Estimation. The support of the random vector US = QS(X) under phenotype level. P0 is estimated by a “covering” of the observed baseline samples. Cover- Deviation from a “normal” range is a form of “outlier analy- ing methods for the estimation of the support of a multivariate density sis” and as such is related to other work in statistics and genomics were introduced in ref. 13, with a goal similar to ours of detecting abnor- ˆS (5–12). All of these previous studies involve single features and mal behavior. The estimator U0 that we propose is a variation of the m require standardization of a new sample with respect to multi- one proposed in that paper. Let d be a metric on [0, 1] (we will use ple other samples from the same class. In contrast, divergence the Euclidean distance). Assume that n0 independent and identically distributed (i.i.d.) samples of X are observed under P0, resulting in i.i.d. samples applies to any subprofile, and its determination does not require S S m U (1), ... , U (n0); these are now n0 points in [0, 1] . We will define an the availability of other samples from the same class (e.g., disease increasing sequence of empirical supports indexed by a “smoothing” param- type); in fact, a new sample profile need not even be labeled. The eter γ ∈ [0, 1]; the determination of γ is described later. Let l = l(γ) = [γn0], only normalization is within-sample ranking, and all comparisons the greatest integer less than or equal to γn0. For each k ∈ {1, ... , n0}, let rk with baseline samples occur in the rank space. denote the distance between US(k) and its lth nearest neighbor. We define

Methods n0 ˆS m [ S The notion of “divergence” is general: Let U be a random variable assuming U0 = [0, 1] ∩ B(U (k), rk), [2] k=1 values in a space U and let P0 be a reference or baseline distribution on U with support supp(P0) ⊂ U. Then a value u ∈ U is divergent if u ∈/ supp(P0). By definition, if U has distribution P0, then the probability of observing a S S where B(U (k), rk) denotes the closed ball of center U (k) and radius rk. In divergent value is zero; the interest is in quantizing U when it follows alter- m ˆS other terms, a point u ∈ [0, 1] belongs to U0 if and only if there exists native distributions and to compare quantized values among alternatives. S k ∈ {1, ... , n0} such that U (k) is closer to u than to its lth nearest neighbor. We are going to apply this to functions U constructed from rank-normalized The smoothing parameter γ corresponds to the “bandwidth” in multivariate individual omics profiles. The supports will be estimated from data. density estimation (14), and our estimator, based on nearest neighbor dis- An omics profile is a vector X = (Xj, j ∈ J ) where the index set J and tances, implements what is commonly referred as an “adaptive bandwidth” the values assumed by individual features Xj depend on the particular data method in this literature. modality determined by the measurement technology. Typically, the fea- The estimate of the support in Eq. 2 is very conservative: Every sub- tures are states, counts, or concentrations of biomolecular species. In this profile S is nondivergent for every sample from the baseline population. paper, we consider gene expression (microarray and RNA-sequencing) and This is unrealistic, not only in view of possible outliers among the base- DNA methylation (microarray). This framework is directly applicable to other line samples, but also because these baseline samples may in fact contain types of measurements, including single-cell data and other omics modali- a small proportion of nonbaseline cell types. Another drawback is that γ ties (e.g., microRNA and protein expression). Here, for gene expression, J is yet to be specified. We address these two issues simultaneously. Again, is the set of genes and Xj is either a microarray value or RNA-seq count for ... let r1, , rn0 be the radii of the balls centered at baseline samples. Now gene j (“bulk” mRNA); in the case of methylation, J is a set of CpG sites and let ¯r be the 95th percentile of these radii. Instead of covering all of Xj is the “β value” (see SI Appendix). the baseline samples, we remove the 5% for which rk >¯r before computing the support. That is, the support is constructed as in Eq. 2 but Quantiles. In our experiments, each individual profile is transformed into with the union over all k = 1, ..., n0 replaced by the union over {k : rk ≤ quantile space: Each element of the profile is replaced by its normalized ¯r}. Notice that some “left-out” samples may still lie in the support, but rank with respect to the other elements in the same profile. Define in general some will not, and therefore, some features will be declared divergent for some baseline samples. The smaller is γ, the more baseline divergence. |{i ∈ J : 0 < Xi ≤ Xj}| Q = Q (X) = . [1] For m = 1, say S = {j}, the estimated support is simply a union of inter- j j |{ ∈ J : < }| i 0 Xi vals centered at the quantile values of a subset of baseline samples with interval lengths determined by γ. In nearly all cases of interest, this union is itself an interval; that is, there are no “gaps.” Making this assumption (or This definition returns a value 0 ≤ Qj ≤ 1 and implies a separate treatment replacing the support by its convex hull), it suffices to compute its upper and of ties at 0 (e.g., “nonexpressed genes”). If Xj = 0, then Qj = 0, and positive quantiles are accrued only from positive X s. This is important for some data lower bounds. This can be done by first computing the smallest and largest j j ¯ modalities, especially gene expression from RNA-seq, for which the many values among the (retained) baseline samples U (k) for rk ≤ r, denoted ties that typically occur at 0 may offset standard definitions of quantiles, respectively by mj and Mj, and then estimating the baseline support of ˆj 0 0 0 assigning large quantile values to possibly small, nonzero, expression num- feature j by U0 = [lj , uj ] ⊂ [0, 1], where lj = max(0, mj − d1) [respectively, 0 bers. Importantly, this definition of quantiles is sample-based (i.e., they are uj = min(1, Mj + d2)] and d1 (respectively, d2) is the distance from the small- computed across variables measured for a single subject), not population- est sample (respectively, the largest) to its lth nearest neighbor among the or cohort-based. retained baseline samples.

4546 | www.pnas.org/cgi/doi/10.1073/pnas.1721628115 Dinalankara et al. 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BIOPHYSICS AND STATISTICS INAUGURAL ARTICLE COMPUTATIONAL BIOLOGY Fig. 2. Degree of divergence in tumor and normal sample populations. (A) Single features for RNA-seq gene expression, CpG methylation, and microarray expression. (B) The 50 FGSs in MSigDB “Hallmark” collection. Note that, by design, the level of divergence in the normal populations is extremely small. In both cases, divergence is learned from a separate normal population.

computed a multitissue baseline profile from 50% of available ties estimated as described above. This reverse approach also non-T samples, and then, estimated divergence probabilities identified phenotype-specific genes (see SI Appendix, Figs. S2 separately from all of the T samples and the remaining half and S3). of non-T samples. For example, to identify prostate-specific For gene sets, we considered the breast cancer subtypes and genes, we used half of nonprostate samples pooled together the 50 Hallmark gene sets. Since we are estimating only a limited to estimate a nonprostate baseline. This analysis yielded 433 number of supports, we imposed virtually zero normal diver- prostate-specific significant genes (Bonferroni-adjusted P value gence by selecting α = .001, with the risk of underestimating ≤ 0.05), encompassing many known prostate markers including cancer subtype divergence probabilities if in fact the normal KLK3 (a.k.a. the prostate specific antigen; see Fig. 3B). Then, breast samples are contaminated. Fig. 4 shows a heat map of the we reversed the roles: We took tissue T as the baseline, with probabilities of divergence for each of the 50 pathways. The full support estimated from 50% of the T samples and probabili- table of values is in SI Appendix, Table S1.

A B 1.00 1.00 PCAT4T KLK3 PRAC1 LOCO 10005080808046 CDC454 KLKL P1 KLK2 NKX3−1−1 MCM100 RAD51AP1P KLK4 TRPPM8 POLLQQ NCAPG HOXB13 PCGCGEM1 EPR1R1 DIAPH3H3H PAGE4 LOC1019279 4828 0.75 GSG2 TCFC 19 0.75 TGM4 FAM7M72A ZNF761 SOCS2−2 AS1 SLCL 45A3 PRCAT474477 TPMM3P99 LSAS MP−−AS1 0.50 0.50 C20orff103 Prostate Luminal B FGD3

0.25 0.25

NEFE L 0.00 0.00

0.00 0.25 0.50 0.75 1.00 0.00 0.02 0.04 0.06 Luminal A Other

Fig. 3. Differentially divergent genes between tumor phenotypes. (A) Scatter plot of divergence probabilities of genes for 231 luminal A and 127 luminal B breast cancer samples in TCGA. After Bonferroni correction, 941 genes (blue triangles) have an adjusted P value ≤ 0.05 for a χ2 test comparing the two subtypes using ternary digitized data. Among these, the genes with the smallest P values are labeled. (B) Scatter plot of divergence probabilities of genes for 106 prostate and 4216 nonprostate samples in Genotype-Tissue Expression (GTEx) with respect to a nonprostate baseline. After Bonferroni correction, 433 genes (blue triangles) have an adjusted P value ≤ 0.05 in a χ2 test comparing the two groups. The 20 genes with the smallest P values and higher divergence probability in prostate are labeled.

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GTEx divergence; S4 of Fig. degrees varying different the displayed from baseline samples subtypes the the while as divergence, declared least the subtype showed tissue the of samples remaining ye.Frec niiulsml,tease oteqeto s“E”if “NO” “YES” and is divergent question is the CHECKPOINT”) “G2M to otherwise. (e.g., answer pathway the indicated sample, the individual each For types. 5. Fig. utemr,w oprddvrec rfie cosmulti- across profiles divergence compared we Furthermore, the related how on depend divergence of degrees relative The eiinte o eaaiglmnlAadBbes acrsub- cancer breast B and A luminal separating for tree Decision hw diinlrslswt h ulls favailable of list full the with results additional shows α = .01 i.S5). Fig. Appendix, SI PNAS eaiet h nietasrpoe Sam- transcriptome. entire the to relative | a ,2018 1, May | o.115 vol. | IAppendix, SI o 18 no. χ 2 etfor test | 4549

BIOPHYSICS AND STATISTICS INAUGURAL ARTICLE COMPUTATIONAL BIOLOGY A B and batch effects. This is also the first step in “quantile normal- 10000 ization” (24) and in the work on “relative expression analysis.” The latter includes the top scoring pair (TSP) and k-TSP algo- 6000 7500 rithms for distinguishing between two disease phenotypes based on the expression ordering between two genes (25) and between 4000 5000 k pairs of genes (26, 27), as well as the “RankComp” approach (28), which aims at identifying dysregulated gene pairs starting 2000 2500 from a pool of stable pairs precompiled using normal samples. Within-profile rank normalization was extended to “differen- size of divergent set of divergent size size of divergent set of divergent size tial variation” between pathways using Hamming distance to a 0 0 phenotype-specific rank template (29) and using the Kendall-tau or “swap” distance between rank vectors (30, 31). Concerning “outliers” in omics data, an early example is “Can- cer Outlier Profile Analysis” (COPA) (6, 7) for gene expression: Brain−Cortex Brain−Cortex First, each gene is standardized across samples from a phenotype of interest, and then those genes that display an outlier Adipose−Visceral Adipose−Visceral Brain−Cerebellum Brain−Cerebellum Skin−Sun Exposed Skin−Sun Exposed profile over a subset of samples are selected. This COPA frame- Skin−Not Sun Exposed Skin−Not Sun Exposed work can also be extended to model heterogeneity in integrated Adipose−Subcutaneous Adipose−Subcutaneous Breast−Mammary Tissue Breast−Mammary Tissue tissue tissue data modalities, and separate gene set statistics can be applied to determine whether outliers are enriched in multidimensional Fig. 6. Divergence profile comparison between normal tissue types. Using gene sets (11). (A) breast and (B) sun-exposed skin tissue as baselines, divergence sets All of the current methods to quantify dysregulation require are computed for multiple normal tissue subtypes available in GTEx with prior knowledge of the sample phenotypes and a population RNA-seq gene expression profiles. Half of the available samples were used of samples from each phenotype. Divergence is a sample-level to estimate a normal subtype-specific profile, and the divergence of all property that can be applied to a single sample without knowl- remaining samples was computed with respect to this baseline. edge of phenotype and thus is not directly comparable to these methods. The approach perhaps most closely related to diver- For instance, gene expression divergence increased in prostate gence is the “Anti-Profiles” method introduced in refs. 10, 32 cancer with increasing Gleason grade: the average size of the and applied to cancer diagnosis; genes that deviate from a nor- divergent set was 883 for tumors with a primary Gleason grade of mal profile are identified after standardization across samples 3, 1241 for a primary Gleason grade of 4, and 1490 for a primary and after prefiltering for hypervariability with respect to normal Gleason grade of 5. Similar results were also observed in breast samples. In contrast, our method requires no across-sample stan- cancer with increasing grade, in the progression from colon ade- dardization or prefiltering, making it amenable to the analysis of noma to carcinoma, and for DNA CpG methylation divergence individual samples prospectively collected. Moreover, similar to in lung cancer, which increased with tobacco-smoke exposure outlier analysis, the antiprofile approach requires subsequent set (see SI Appendix, Fig. S6 A–D). Notably, the level of devia- statistics for pathway analysis. Divergence applies to any subset tion from the normal baseline is strongly correlated between of features treated as a single entity using the same statistical the- distinct omics modalities derived from the same biological sam- ory. Both of these differences further limit direct comparison of ples, suggesting global genome-wide dysregulation in both DNA divergence to the antiprofile approach. and RNA in cancer; see, for example, the case of prostate Enormous biological variation is another barrier. It has been cancer gene expression and methylation in SI Appendix, Fig. consistently observed that in many disease conditions there is a S7A (Spearman correlation 0.59). Repeating this analysis with high degree of variation in the omics profiles from sample to sam- breast cancer data yielded a Spearman correlation of 0.51 (SI ple. Observations of individual samples and genomic features at Appendix, Fig. S7B). Overall, these results suggest that the diver- high-resolution may obscure patterns of dysregulation. Further- gence framework is capturing a sparse yet biologically meaning- more, omics studies usually focus on population-level statistics ful snapshot of an omics sample and its dysregulation from a healthy state, which can be observed across distinct omics data modalities. AB Discussion 2 Combing through large datasets to make meaningful biologi- 1 cal inferences has become the raison d’être for much of the current work in bioinformatics. Often the goal is the discovery 0 of biomarkers or molecular signatures (e.g., gene sets) of clin- PC2 −1 ical utility, and a great many of these based primarily on gene expression have been proposed over the last 15 years. However, −2 many high-throughput omics classifiers are learned from rela- −3 tively small cohorts of samples using population-level estimates −3 −2 −1 0 1 2 −4 0 4 8 that are highly sensitive to variability, and therefore, the resulting PC1 biomarkers are often difficult to reproduce and validate (21– 23). As a result, leveraging such “gene signatures” has proven subtype cluster difficult; in fact, the majority of such biomarkers are unable Basal−like 1 Non−basal−like 2 to meet the rigorous criteria required to be admitted to clinical use. The work here is aimed at ameliorating some of these Fig. 7. Clustering digitized data. Results of spectral clustering of Breast barriers. TCGA RNA-seq data in (A) quantile form and (B) digitized form. The Sensitivity to platform and preprocessing is one important bar- resulting clusters discriminate basal-like samples from all other subgroups. rier. Divergence coding begins with the initial conversion of raw Digitized data preserve biological information separating the different feature values to within-sample ranks to minimize preprocessing subtypes similarly to quantile data.

4550 | www.pnas.org/cgi/doi/10.1073/pnas.1721628115 Dinalankara et al. Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 2 faiB ea ,Fri J(04 erigdseuae ahasi acr from cancers in pathways dysregulated Learning (2014) EJ Fertig pair D, scoring Geman B, top Afsari with 12. combined analysis set gene Outlier (2013) al. et can MF, variability on Ochs based signatures 11. expression Gene (2015) HC Bravo W, Dinalankara 10. iaakr tal. et Dinalankara a to given interest profile. 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RA Irizarry microarray MJ, for Zilliox HA, code Jaffee K, bar Uppal MN, expression McCall gene 9. A analysis. (2007) expression RA gene Irizarry differential MJ, for Zilliox sums 8. Outlier factor (2006) transcription T ETS Hastie and R, TMPRSS2 Tibshirani of fusion 7. Recurrent (2005) al. statistical et A SA, (2002) Tomlins E 6. dense, Gabrielson personal, R, using Anbazhagan individuals ES, 108 Garrett-Mayer of study G, wellness Parmigiani preterm A 5. (2017) spontaneous al. a et of ND, validation Price and 4. 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E, pp Formenti Berlin), Heidelberg, A, Ngom eds activity. ics, pathway of biomarkers robust provides prognosis. and progression tumor predict robustly 4:911–913. 8:2–8. uli cd Res Acids Nucleic 64:717–736. a Biotechnol Nat Science c rnlMed Transl Sci acrInform Cancer 310:644–648. 39:D1011–D1015. 35:747–756. 5:207ra142. 13:61–67. mJOse Gynecol Obstet J Am atr eonto nBioinformat- in Recognition Pattern acrInform Cancer 14:71–81. ttScSer Soc Stat R J 214:633.e1– u Urol Eur Nat 4 osa M rzryR,AtadM pe P(03 oprsno normalization of Recurrent comparison A work: (2003) TP never Speed M, may Astrand RA, biomarker Irizarry BM, cancer Bolstad 24. new your Why (2012) SE DNA of Kern use the 23. in Pitfalls (2003) LM McShane KK, Dobbin MD, Radmacher R, in Simon Outcomes regional 22. Patient and Predicting local for Tests of Omics-Based risk of the Review and the on subtypes Committee cancer Breast 21. (2010) al. et KD, Voduc 20. sur- breast to on according outcome treatment and recurrence and of Patterns (2013) subtypes al. et cancer O, Metzger-Filho breast 19. of Impact (2012) al. project. et (gtex) R, expression genotype-tissue Haque The 18. (2013) al. set et project. analysis J, gene pan-cancer Lonsdale atlas hallmark genome 17. database cancer The signatures (2013) al. molecular et JN, The Weinstein (2015) 16. al. et A, Liberzon 15. (2015) DW Scott 14. 3 ery ,Ws L(90 eeto fanra eairvanonparametric via behavior abnormal of Detection (1980) GL Wise L, Devroye 13. yNHNtoa acrIsiueGatR01CA200859. supported Grant was Institute research Cancer National This NIH discussions. by helpful for Pereira-Lobo Francisco ACKNOWLEDGMENTS. from from obtained available are here divergence.html analyzed data All Availability Code and Data individualized to profile challenge with divergence major care. associated patient-level patient the often a on is center adapting will divergence of work of perva- Future extent are gravity. the relative divergence that epigenomic cancer, and and with sive genomic here that shown expect As lev- high we heterogeneity. display interpersonal that diseases of human els complex other of study the and meaningful directly presence is risk. sample the assessing disease for quantify appropriate a the can in that of dysregulation method of many a with cell Therefore, normal along of can- behavior. dysregulation of collectively, considered be hallmarks can hallmarks, be remaining and to progression considered are cer pathways, instability metabolic genome in and abnormality inducing proliferation, Stimulating heterogeneity. cell is molecular Cancer such pathogenesis. to of subject paradigms particularly accepted many with is line dysregulation molecular in patient-level on emphasis The cancer. important of deal preserved. great is diver- information a biological the nonetheless, We that, complexity. construction, in demonstrated reduction by have massive a metric-based cases, provides a both transform gence using a In support in argument. the genes all values covering estimated sup- of assuming have expression baseline vectors we the the (e.g., random pathway), space since for m-dimensional interval; gene) an features an in single omics simply a individual is of as port expression the such the variables of no (e.g., random straightforward samples is requires scalar support other baseline sample for the to of given Estimation respect class. any same with of standardization Impor- labeling further distribution. probability divergence baseline sup- tantly, the underlying estimate the to population, of ability baseline the port to suitable down boils a divergence from defining samples many sufficiently h nltclfaeokpeetdhr ol eapidto applied be could here presented framework analytical The primarily disease, human to applications on focused have We ehd o ihdniyoiouloiearydt ae nvrac n bias. and variance on based data array oligonucleotide density Bioinformatics high for failures. methods biomarker in diversity remarkable 6101. and patterns classification. prognostic and 14–18. diagnostic for data microarray Forward Path (2012) al. et Trials, Clinical relapse. breast international IX. from and Results VIII trials disease: group node-negative study cancer lymph in subtypes cancer decades. 1855. two spanning analysis An vival: 45:580–585. Genet collection. NJ). Hoboken, Sons, & Wiley (John siaino h support. the of estimation 45:1113–1120. lnOncol Clin J elSyst Cell Ntoa cdme rs,Wsigo,DC). Washington, Press, Academies (National 19:185–193. github.com/wikum/DivergenceAnalysis utvraeDniyEtmto:Ter,Patc,adVisualization and Practice, Theory, Estimation: Density Multivariate n h eesr akgsadcd a be can code and packages necessary the and , 1:417–425. 28:1684–1691. etakKatnBaov di ud-md,and Luidy-Imada, Eddie Blagoev, Krastan thank We PNAS vlto fTasainlOis esn ere n the and Learned Lessons Omics: Translational of Evolution IMJAp Math Appl J SIAM | a ,2018 1, May lnOncol Clin J acrEieilBoakr Prev Biomarkers Epidemiol Cancer 38:480–488. 31:3083–3090. | o.115 vol. luigimarchionni.org/ alCne Inst Cancer Natl J acrRes Cancer | o 18 no. . a Genet Nat 21:1848– 72:6097– | 4551 Nat 95:

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