A Hidden Markov Model for Identifying Differentially

A Hidden Markov Model for Identifying Di↵erentially Methylated Sites in Bisulfite Sequencing Data


Farhad Shokoohi Department of Epidemiology, Biostatistics and Occupational Health, McGill University, and Lady Davis Institute for Medical Research, Montreal, QC, Canada email: [email protected]

and David A. Stephens Department of Mathematics and Statistics, McGill University, Montreal, QC, Canada email: [email protected]

and Guillaume Bourque Department of Human Genetics, McGill University, Montreal, QC, Canada email: [email protected]

and Tomi Pastinen Department of Human Genetics, McGill University, Montreal, QC, Canada email: [email protected]

and

1

Biometrics 63, 1–19

DOI: 10.1111/biom.12965

August 2018 Celia M.T. Greenwood Lady Davis Institute for Medical Research, Montreal, QC, Canada email: [email protected]

and Aur´ elie Labbe Department of Decision Sciences, HEC Montreal, QC, Canada email: [email protected]

Summary:

DNA methylation studies have enabled researchers to understand methylation patterns and their regu-

latory roles in biological processes and disease. However, only a limited number of statistical approaches have been developed to provide formal quantitative analysis. Specifically, a few available methods do identify di↵erentially methylated CpG (DMC) sites or regions (DMR), but they su↵er from limitations that arise mostly due to challenges inherent in bisulfite sequencing data. These challenges include: (1) that read-depths vary considerably among genomic positions and are often low; (2) both methylation and autocorrelation patterns change as regions change; and (3) CpG sites are distributed unevenly. Furthermore, there are several methodological limitations: almost none of these tools is capable of comparing multiple groups and/or working with missing values, and only a few allow continuous or multiple covariates. The last of these is of great interest among researchers, as the goal is often to find which regions of the genome are associated with several exposures and traits. To tackle these issues, we have developed an efficient DMC identification method based on Hidden Markov Models (HMMs) called “DMCHMM” which is a three-step approach (model selection, prediction, testing) aiming to address the aforementioned drawbacks. Our proposed method is di↵erent from other HMM methods since it profiles methylation of each sample separately, hence exploiting interCpG autocorrelation within samples, and it is more flexible than previous approaches by allowing multiple hidden states. Using simulations, we show that DMCHMM has the best performance among several competing methods. An analysis of cell-separated blood methylation profiles is also provided. Key words: Depth.

Blood Cell-Separated Data; Di↵erentially Methylated Region; Next-Generation Sequencing; Read-


1

1. Introduction Although DNA methylation is the most studied type of epigenetic mark, its patterns and the regulatory roles of these patterns in biological processes and diseases (i.e., tissue development, cell di↵erentiation, cancer, embryonic development, genomic imprinting, aging) are not yet fully understood; see Pacchierotti and Spanò (2015), for instance, and the references therein. Several experimental methods to map methylation patterns have been proposed; one in particular, bisulfite sequencing (BS-Seq) is widely used as it can, in some of its forms, detect DNA methylation patterns at single-site resolution (Behjati and Tarpey, 2013). Although BS-Seq can su↵er from limitations such as producing variable read-depth data, an inability to discriminate between 5-mc and 5-hmc, and incomplete conversion or degradation of DNA during bisulfite treatment, it remains one of the most accurate methods for generating highdimensional methylation data. Variants of BS-Seq include RRBS, BSPP, BC-seq and whole genome bisulfite sequencing (WGBS) (Harris et al., 2010). WGBS provides methylation information for the entire genome at single-site resolution, making it theoretically possible to analyze methylation patterns and their roles more precisely (Stevens et al., 2013). Methylation data have complex patterns due to several factors: (1) read-depths are variable among genomic positions (Sims et al., 2014); (2) local autocorrelation patterns between CpGs change drastically along short distances in the genome; (3) patterns of both methylation and autocorrelation di↵er in regions with di↵erent functions (e.g., promoter regions, intra-gene regions, introns and enhancers are expected to show di↵erent methylation patterns, and di↵erent strengths and extent of correlations); (4) CpGs are unevenly distributed across the genome. These features have noticeable e↵ects on the spatial correlation across CpGs and the amount of hyper-/hypo-methylation. Less dense CpG regions, for instance, are predominantly hyper-methylated (Lövkvist et al., 2016). Strong correlations have been detected across both CpGs (Eckhardt et al., 2006) and samples (Gallego-Fabrega et al., 2015).

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Biometrics, August 2018

In addition, methylation is known to vary as a function of various factors such as age, disease (i.e., cancer), exposures (i.e., smoking), treatment, tissue, and other biological factors (Lam et al., 2012). The dependence of methylation on such factors has created a growing interest among researchers to find which regions of the genome are associated with several exposures and traits. As a result, di↵erentially methylated CpG (DMC) sites and di↵erentially methylated regions (DMR), including those that are aging-specific, tissue-specific, cancer-specific, reprogramming-specific and imprinting-specific (Rakyan et al., 2011), have been the centre of many studies and observed to play important roles in the human genome. In contrast to the paucity of methods that profile methylation patterns across genomic structure, there are a growing number of methods that compare methylation patterns between groups, either at single sites or in small regions. To that end, several statistical approaches based on a wide range of statistical methods have been proposed; a comprehensive list is given in Supplementary Material (SM) S1. We focus here on developing a DMC identification method since it is shown that the methods based on DMCs perform better than those that directly identify DMRs (Klein and Hebestreit, 2015). Existing methods are deficient in several ways: very few are capable of comparing multiple groups; none of the methods (except bsseq) can handle low read-depth data, and none can utilize data containing missing values. Although BiSeq can include positions containing some missing values and imputes these using a (kernel) smoothing technique, it filters the data prior to smoothing to include most covered positions. This option will remove many positions with missing values. Most methods do not allow both continuous and discrete covariates, or combinations. In this paper, we develop a new method, ‘DMCHMM’, for identifying DMCs based on a hidden Markov model (HMM) which addresses the drawbacks mentioned above. DMCHMM exploits inter-CpG autocorrelations within samples, and is more flexible than existing HMM methods for methylation by allowing multiple hidden states. DMCHMM then uses between- and within-


3

group variations to identify DMCs at each position. In general, HMM-based methods can identify DMRs of variable size, in contrast to the approaches that use a fixed window size, and can also identify independent DMCs or short DMRs, or abrupt methylation changes among the CpGs (Shafi et al., 2017). In some applications, researchers are interested in clustering the groups/samples into a finite number of clusters (Backman et al., 2017). In these situations, a discrete number of methylation levels (e.g., using HMMs) is more interpretable than continuous modeling such as beta-binomial regression. The discrete state HMM could potentially be extended to include continuous state components to give a more general latent beta prior structure, although the inclusion of continuously-varying states increases computational and modeling complexity, and lessens the capability of the model to perform site “classification”. Additionally, DMCHMM can consider multiple covariates simultaneously, correct for multiple testing, and utilize sequencing coverage in analysis. The rest of this manuscript is organized as follows. In Section 2, our motivating dataset, consisting of methylation data from three cell types extracted from whole blood, is described. Section 3 provides the proposed method. Comparative simulation studies inspired by the real data structure are presented in Section 4. An analysis of the blood cell-separated data is given in Section 5. Finally, Section 6 summarizes the findings and gives concluding remarks.

2. Methylation data from three cell types in whole blood Developments in this paper were motivated by examining data near the BLK gene on human chromosome 8, which is known to be hypo-methylated in B-cells compared to other cell types (Kulis et al., 2015). WGBS data were derived from whole blood collected on a cohort of healthy individuals from Sweden. Data were obtained from the Genome Québec Innovation Centre and are part of the Canadian Epigenetics, Environment and Health Research Consortium (Cheung et al., 2017). Cell lines were separated into T-cells (19 samples), monocytes (13 samples) and B-cells (8 samples). Sequencing was performed on the Illumina HiSeq2000/2500

4

Biometrics, August 2018

system for each of the 40 samples, separately. The analyzed region is located near the BLK gene on chromosome 8. This region contains 30,440 CpG sites spanning 2 MB (10,352,236 12,422,082) in which approximately 23.39% of positions have missing information in at least one of the samples and 21% of positions have at least 20 missing values across all samples. [Figure 1 about here.] An illustration of the data is given in Figure 1. For each cell type in addition to showing the variability in methylation (gray dots), smooth curves were obtained by first aggregating the data (summing read-depths and summing methylation counts over cell type samples at each position), second, imputing the missing values (using a naive method), and third, fitting a smooth curve. We use part of the information in these data as a baseline to compare the performance of our proposed method (DMCHMM) with several other analytical tools (Section 4).

3. Methods We propose a three-step analysis pipeline to identify DMCs based on HMM profiling. In Step 1, the HMM order (K +2) is estimated for each sample using a model selection approach: similar to its use in other methylation applications, the role of the HMM is to capture large scale structure resulting in dimension reduction. In Step 2, an HMM of the estimated order is fitted to each sample using a Bayesian approach to collect a Markov Chain Monte Carlo (MCMC) posterior sample for the site-specific methylation levels for each biological sample. In Step 3, regressions are carried out at each position using the logit–transformed sampled posterior methylation levels and covariates. The results of the regression dictate whether a position is called as a DMC. Full details are given in the following sections. 3.1 A hidden Markov model for methylation data Let {(yt , nt ) , t = 1, . . . , T } be the vector of methylation read counts and read-depth at the tth genomic position of a given sample. In the simplest version of our model, we assume that


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the sample is generated from an HMM (Xt , Yt ) of order K + 2 where Xt is the hidden path and Yt is emission generated according to the hidden path; Xt represents the propensity for site t to be methylated in each read instance. There are many sites in the genome that are completely methylated or unmethylated (Ehrlich et al., 1982). As such, two hidden states (i.e., 0 and 1) are reserved. Let Xt 2 {0, 1, . . . , K + 1} denote the state sequence where the states correspond to discrete methylation levels

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Mono TC

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Mono vs TC

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Mono < TC

25

(d)

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− log10(p )

20 15 10 5 0

BLK Gene ● ●

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11321495

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11338139

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CpG Position

Figure 6: (a)-(c) Identified DMCs for three pairwise comparisons of cell-type methylation data near the BLK gene. Black vertical lines indicate CpG sites where one cell type was significantly di↵erent from the other, at q-value < 1e 7. In the promoter region of BLK, a set of closely spaced probes demonstrate lower methylation (hypo-methylation) in B-cells. For each panel the average predicted methylation levels of each cell type are also plotted. (d) Manhattan plot for p-values. Results are based on non-weighted likelihood.

WEB-BASED SUPPLEMENTARY MATERIALS FOR ‘A HIDDEN MARKOV MODEL FOR IDENTIFYING DIFFERENTIALLY METHYLATED SITES IN BISULFITE SEQUENCING DATA’ BY F. SHOKOOHI, D.A. STEPHENS, C. GREENWOOD, G. BOURQUE, T. PASTINEN AND A. LABBE The following notation and acronyms are used in the main paper and the supplementary materials. • • • • • • • • • • • • • • • • •

Plat. : Platform based on which methylation data are generated; r.NA : Missing values are removed; filter : Data are filtered to remove positions with low read-depth; Clustering : Clustering sites prior to analysis; Smoothing : Smoothing methylation levels; Parallel : Parallel Computing; Stats. : Providing Statistics such as p-values, FDRs, etc; BSS : BS-Seq technology for retrieving methylation status; bump : The bumphunter method DMRca : The DMRcaller method HMMDM : The HMM-DM method HMMW : The DMCHMM method with weights HMM : The DMCHMM method without weights HFish : The HMM-Fisher method bed : A file type for storing methylation data NDMR : Non-DMR STR, END : Start and end position of a DMR S1. List of References

Many statistical methods have been used to perform inference for methylation data, including binomial models (Hansen et al., 2012), beta-binomial models (Wu et al., 2013; Dolzhenko and Smith, 2014; Wu et al., 2015), logistic regression (Akalin et al., 2012), beta regression (Hebestreit et al., 2013), linear mixed models (Ja↵e et al., 2012), Hidden Markov Models (HMMs) (Hodges et al., 2011; Song et al., 2013; Saito et al., 2014; Saito and Mituyama, 2015; Yu and Sun, 2016b; Sun and Yu, 2016), and Bayesian models (Wu et al., 2013), among others. To identify DMCs/DMRs, a few tools, such as bsseq(BSmooth) (Hansen et al., 2012), BiSeq (Hebestreit et al., 2013), HMM-DM (Yu and Sun, 2016b), HMM-Fisher (Sun and Yu, 2016), DMRcaller (Zabet and Tsang, 2015), bumphunter (Ja↵e et al., 2012) and DSS (Feng et al., 2014), among others, are developed. These methods are reviewed or compared to each other several times (Bock, 2012; Robinson et al., 2014; Klein and Hebestreit, 2015; Yu and Sun, 2016a; Wreczycka et al., 2017; Chen et al., 2016) including Shafi et al. (2017) who provided a comprehensive review of 22 approaches to DNA methylation analysis. Some of these tools report DMCs and others report DMRs.

1

2

WEB-BASED SUPPLEMENTARY MATERIALS: A HMM FOR IDENTIFYING DMCS IN BS-SEQ DATA

S2. List of Selected Methods for Comparison Performance of our method is compared to bsseq (Hansen et al., 2012), BiSeq (Hebestreit et al., 2013), HMM-DM (Yu and Sun, 2016b), DMRcaller (Zabet and Tsang, 2015), bumphunter (Ja↵e et al., 2012), DSS (Feng et al., 2014), DSS-Single (Wu et al., 2015) and HMM-Fisher (Sun and Yu, 2016). We tried di↵erent input parameters for each tool in order to optimize performance. The overall performance of DSS-Single, for instance in terms of accuracy, was close to that of DSS, although di↵erent for individual DMRs and non-DMRs, thus its results are omitted. The chosen methods are presented in Table S1. For bumphunter, all positions, even those with missing values, were included in the analysis to improve the performance. In bsseq, we have removed positions where all the samples in one group were missing at that position. In HMM-Fisher, distances between CpGs were set to 1 to avoid crashing. See Table S1 for the remaining modifications. We were not able to include some methods in simulation studies such as MethPipe (Song et al., 2013), methylPipe (Kishore et al., 2015), methylKit (Akalin et al., 2012), and ComMet (Saito and Mituyama, 2015)) since these tools require specific file formats, or we were unable to implement them. Nonetheless, these methods have been reported to be inferior to some of the methods that were used for comparison in this study (Klein and Hebestreit, 2015; Wu et al., 2015; Yu and Sun, 2016a). Table S1. List of DNA methylation analytical tools with their features and capabilities, and the changes made to increase their efficiency Package

Model beta BiSeq regression binomial bsseq & linear linear mixed bumphunter model Kernel DMRcaller smoothing Bayesian DSS hierarchical Bayesian DSS-Single hierarchical

Platform Data

r.NA filter Clustering Smoothing Parallel

BSS

bed

X

BSS

bed

X

BSS

reads

X

BSS BSS RNA BSS RNA

custom

HMM

HHM-Fisher HMM

HMM-DM

X

X

X

partly

X

X

X

X

X

X

X

X

X

X

X

X

reads

X

X

reads

X

X

X

BSS

reads

X

X

X

BSS

reads

Notes The default “perc.samples” is changed from 4/5 to 1/2. “abs(meanDi↵)” is changed to 0.1. “cuto↵” is changed to 0.05. No position was removed. “minProportionDi↵erence” is changed from 0.4 to 0.1. “p.threshold” is set to 0.05.

X

“p.threshold” is set to 0.05. “meanDi↵.cut”, “max.distance” and “max.EM” are changed from 0.3, 100 and 1 to 0.1, 1000 and 300, respectively. Distances between CpGs are set to 1 to avoid crashing .


3

S3. Numerical methods The discrete HMM is a well-known model in the literature, and researchers have developed many methods and algorithms to perform estimation in such models, mostly based on the forward-backward algorithm for calculation of the marginal log-likelihood. For estimation, the EM and MCMC algorithms are often used. In the following, we provide brief introductions and implementations of these algorithms in our proposed method. S3.A1. Forward-Backward Algorithm. The process of writing the likelihood function is done by first writing the filtering, then prediction and finally smoothing functions (probabilities) as follows. S3.A1.1. Filtering, smoothing and prediction. Filtering, smoothing and prediction functions are denoted as • Filtering: ⇡ t|t for t = 1, . . . , T , • Smoothing: ⇡ st|t⇤ = ⇡ t|t⇤ for t < t⇤ , • Prediction: ⇡ t|t⇤ for t > t⇤ , where ⇡ t|t0 = p(Xt |Y0 , . . . , Yt0 ). Since (x0 , y0 ) is unobserved, the estimation and inference are done conditionally. In numerical implementation, we use the equilibrium distribution to estimate the predicted initial distribution based on a stationarity assumption; i.e., b 0 = P M [1, ], ⇡ where P is the transition matrix, and M can be any fairly large integer. Then the filtering distribution ⇡ 1|1 is computed as b0 {f (y1 |x1 ; 0 ), . . . , f (y1 |x1 ; K+1 )}> ⇡ b 1|1 = ⇡ > b0 1> {f (y1 |x1 ; 0 ), . . . , f (y1 |x1 ; K+1 )} ⇡ > where 1 is the transpose of the vector of 1s with length K + 2 and f (.) is the probability mass function of the binomial distribution. The prediction then is computed as where the filter at position t is computed as b t|t = ⇡

b t|t ⇡

{f (yt |xt ;

1> {f (yt |xt ;

1

bt = P >⇡

1|t 1 ,

> b t|t 1 K+1 )} ⇡ . > b t|t 1 0 ), . . . , f (yt |xt ; K+1 )} ⇡

0 ), . . . , f (yt |xt ;

The log-likelihood contribution for position t is calculated as n `t = log 1> {f (yt |xt ; 0 ), . . . , f (yt |xt ; therefore, the full incomplete data log-likelihood is (S3.1)

`=

XT

t=1

K+1 )}

>

b t|t ⇡

1

o

;

`t .

In terms of calculating the smoothing functions, one should first calculate the filtering probabilities and then calculate the smoothing at position t as follows, ( b ⇤t|t+1 = ⇡ b t|t , ⇡ t=T ⇤ ⇤ b t 1|t = ⇡ b t 1|t 1 P ⇡ b t|t+1 /b ⇡ ⇡ t|t 1 , t = T 1, . . . , 1.

4


S3.A2. EM Algorithm. The EM algorithm is widely used for inference in HMMs. In order to apply the EM algorithm in HMM, some assumptions and regulatory conditions should hold. Thanks to the nature of finite state space HMM, these assumptions and conditions hold, hence the EM algorithm can be used in finite state space HMM. The complete data weighted log-likelihood is (S3.2)

`

⌧ ,c

(✓; x, y, ⌧ ) =

K+1 X

1k (x1 ) log ⇡k k=0 T K+1 X X K+1 X

+

1

+

k0 ,k

K+1 X

⌧1 1k (x1 )b(y1 ;

k)

k=0

(xt

1 , xt ) log p

k0 ,k

+

t=2 k0 =0 k=0

T K+1 X X

⌧t 1k (xt )b(yt ;

k ),

t=2 k=0

where b(yt ; k ) is the logarithm of the binomial mass functions of yt , and 1 is the indicator function. To obtain ✓ = ( , ⇡, P ) using the EM algorithm, we compute the recursion (M-step) ✓ (r+1) = arg max Q(✓, ✓ (r) ), ✓

where in the E-step, we have Q(✓, ✓

(r)

) = EX,Y ` +

⌧ ,c

(✓; x, y)|y; ✓

T K+1 X X K+1 X

(r)

=

r $t,k = Pr Xt 0 ,k

1

log ⇡k +

r $t,k 0 ,k log pk 0 ,k +

K+1 X

r ⌧1 !1,k b(y1 ;

k)

k=0

r ⌧t !t,k b(yt ;

k ),

t=2 k=0

r !t,k = Pr Xt = k, ✓ (r) |y, n ,

and

r !1,k

k=0 T K+1 X X

t=2 k0 =0 k=0

in which

K+1 X

8k = 0, . . . , K + 1,

= k 0 , Xt = k, ✓ (r) |y, n ,

8k 0 , k = 0, . . . , K + 1.

The above probabilities on the right-hand side are the ‘smoothed’ probabilities arising from the Kalman filter calculation. Only the second one is not available from standard smoothing; however, the calculation is feasible using the forward-backward algorithm described in Supplementary Material S3.A1. The M-step is obtained as follows; 8 (r) r r > !1,K+1 r < @Q(✓, ✓ ) !1,k !1,k PK = ⇡k =0 (r+1) 1 ) ⇡k = PK+1 , k = 0, . . . , K + 1; @⇡k j=0 ⇡j r > :PK+1 ⇡ = 1 j=0 !1,j j j=0 for k 0 = 0, . . . , K + 1, 8 ✓ r (r) > $ 0 < @Q(✓, ✓ ) PT = t=2 pt,k0 ,k k ,k @pk0 ,k > :PK+1 p 0 = 1 j=0 k ,j

r $t,k 0 ,K+1 PK 1 j=0 pk0 ,j

◆

=0

)

(r+1) pk0 ,k

and

T

@Q(✓, ✓ (r) ) X r = ⌧t !t,k (yt @ k t=1

nt

k)

=0)

(r+1) k

=

r t=2 $t,k0 ,k PK+1 r , k = 0, . . . , K + 1; t=2 j=0 $t,k0 ,j

= PT

PT

PT

r t=1 ⌧t yt !t,k , PT r t=1 ⌧t nt !t,k

k = 1, . . . , K + 1.


5

S3.A3. MCMC Algorithm. Markov Chain Monte Carlo (MCMC) techniques are sampling methods from complex probability distributions in data modeling for Bayesian inference. Made into mainstream statistics and engineering via the articles Gelfand and Smith (1990) and Gelfand et al. (1990) MCMC became a gold standard in Bayesian analysis. Since then many new sampling techniques have been developed. It our proposed method for analyzing DNA methylation data we use the following MCMC algorithm to perform Bayesian inference. Having estimated K from the EM algorithm, we use the estimated parameters in Step 1 (i.e. MLEs) as initial values for the MCMC algorithm. We set bMLE . ✓ (0) = ✓ The likelihood function is calculated as in (S3.2). The prior distribution is given by (S3.3)

P (✓) = P ( , ⇡, P ) = P ( )P (⇡)P (P ),

where P ( ) ⇠ Beta(., .), P (⇡) ⇠ Dir(.) and P (P k ) ⇠ Dir(.) for k = 0, . . . , K + 1 and P k = (pk,0 , . . . , pk,K+1 ). In this regard, the posterior distribution is given by ⇣ ⌘ (S3.4) P ✓, X y, n / `⌧ ,c (✓; X, y, ⌧ )P (✓). In each step of the MCMC algorithm we update the parameters as follows: ⇣ ⌘ ⇣ ⌘ ✓ (r+1) , X (r+1) (y, n) ⇠ P ✓, X y, n; ✓ (r) , X (r) .

Assume position t(= 1, . . . , T ) for one of the biological samples in the data. In the rth run of MCMC, the posterior (r) (r) probabilities that the position Xt belongs to states 0, 1, . . . , K + 1 are obtained and denoted as p0,t , . . . , pK+1,t , (r) (r) along with the estimated values b , . . . , b . The “Maximum A Posterior” (MAP) is given as 0,t

(S3.5)

K+1,t

b(r) = b(r) t k,t

(r)

if

pk,t is the maximum.

Alternatively, the weighted estimator is given by (S3.6)

(r)

b

t

=

K+1 X k=0

b(r) p(r) . k,t k,t

In our proposed method we use the weighted estimator (S3.6) since better results were obtained. The MCMC algorithms give R samples (after convergence of the posterior distribution). Finally, the estimator of t , the methylation level of position t in the path, is obtained by averaging over R MCMC samples as follows. R

(S3.7)

X (r) bt = 1 b . t R r=1

The estimated variance of the estimator in (S3.7) is then obtained as ◆ R ✓ 1 X b(r) b 2 2 (S3.8) bb = . t t t R r=1

6


S4. Variants of the DMCHMM method In the following table we present variants of our proposed method. All of these variants are studied using simulations. Most of the results are presented in this document. Table S2. Variants of the DMCHMM method used in simulation studies

Name EM EMW EMW2 EMW3 EMW4 EMW5 EMW6 EMW7 EMW8 HMM HMMW HMMW2 HMMW3 HMMW4 HMMW5 HMMW6 HMMW7 HMMW8

Step 1 EM Weight X — X log(nt + 2) X (nt + 1)0.5 X nt + 1 X (nt + 1)1.5 X (nt + 1) 0.5 1 X (nt + P1) X log(nt + 2)/ t P log(nt + 2) X log(nt + 2)/ log t (nt + 2) X X X X X X X X X

— log(nt + 2) (nt + 1)0.5 nt + 1 (nt + 1)1.5 (nt + 1) 0.5 1 (nt + P1) log(nt + 2)/ t P log(nt + 2) log(nt + 2)/ log t (nt + 2)

Step 2 Weight — — — — — — — — —

MCMC

X X X X X X X X X

Step 3 Regression/Test X X X X X X X X X

— log(nt + 2) (nt + 1)0.5 nt + 1 (nt + 1)1.5 (nt + 1) 0.5 1 (nt + P1) log(nt + 2)/ t P log(nt + 2) log(nt + 2)/ log t (nt + 2)

X X X X X X X X X

Here we emphasize that the weights in Table S4 give di↵erent results since these weights are put on the emission part of the likelihood rather that on the whole likelihood. To clarify this matter observer that there are two approaches to introduce weights ⌧t into the likelihood. Recall the log-likelihood formula in (S3.1) of the web-based supplement: T T h i X X b t|t 1 `(✓; x, y) = `t = log 1> {f (yt |xt ; 0 ), . . . , f (yt |xt ; K+1 )}> ⇡ t=1

t=1

Case 1: If we opt to introduce the weights on the whole likelihood we would have `(✓; x, y, ⌧ ) =

T X

⌧t ` t .

t=1

Case 2: If we opt to introduce the weights on the emission part of the likelihood we would have T h i X b t|t 1 `(✓; x, y, ⌧ ) = log 1> {f (yt |xt ; 0 )⌧t , . . . , f (yt |xt ; K+1 )⌧t }> ⇡ t=1

To observe that Case 1 and 2 give di↵erent results having proportial weights assume the following choices of weights: (a) ⌧t = log(nt + 2). P (b) ⌧t = log(nt + 2)/ log{ Tt=1 (nt + 2)} = C log(nt + 2). For these two weights and Case 1 we have the following formulae for the log-likelihood:


7

Case 1 & (a): `(1) = `(✓; x, y, ⌧ ) =

(S4.1)

T X

log(nt + 2)`t

t=1

Case 1 & (b):

`(2) = `(✓; x, y, ⌧ ) = C

(S4.2)

T X

log(nt + 2)`t .

t=1

Clearly (S4.1) and (S4.2) lead to the same estimation and prediction since `(2) is proportional to `(1) , that is `(2) = C`(1) , and the estimates and predictions will not depend on the constant C. However assume Case 2 with weights (a) and (b): Case 2 & (a): T h i X b t|t 1 (S4.3) `(3) = log 1> {f (yt |xt ; 0 )log(nt +2) , . . . , f (yt |xt ; K+1 )log(nt +2) }> ⇡ t=1

Case 2 & (b): (S4.4)

`(4) =

T X t=1

h log 1> {f (yt |xt ;

C log(nt +2) , . . . , f (yt |xt ; 0)

C log(nt +2) > b t|t } ⇡ K+1 )

1

i

Clearly `(4) in (S4.4) is not proportional to `(3) in (S4.3), hence `(4) leads to di↵erent estimation and prediction than `(3) and the results are undoubtably di↵erent for these two weighted log-likelihoods.

8


S5. Simulation Study from the Main Paper S5.A1. Simulation set-up. In the following table we present the simulation set-up for the first scenario. Table S3. Simulation setup parameters for the first scenario

DMR DMR1 DMR2 DMR3 DMR4 DMR5 DMR6 DMR7 DMR8 DMR9 DMR10

Chromosome 8 8 8 8 8 8 8 8 8 8

Start position 10431839 10671903 10911977 11231950 11631990 11671950 11714976 11783926 11870570 12084164

End position 10511940 10751815 10991817 11311987 11641956 11681894 11725000 11793886 11924576 12191969

E↵ective size Length 0.1 1620 0.2 1094 0.3 1205 0.4 1267 0.4 126 0.3 102 0.2 223 0.1 224 0.05 924 0.02 433


9

S5.A2. Figures and Tables of Simulation Setup Scenario 1, = 0.18. In this section we provide the results of simulation study for the first scenario when errors are generated from a normal distribution with standard deviation = 0.18. (a) SRT1

SRT2

SRT3

SRT4

SRT5

SRT6

SRT7

SRT8

SRT9

SRT10

END1

END2

END3

END4

END5

END6

END7

END8

END9

END10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b)

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Method:

HMMW

HMM

HMMDM

bump

Method

bsseq

DMRca

DSS

BiSeq

HFish

Figure S1. Average number of times that the start (a) and end (b) positions of DMRs are identified as DMCs for di↵erent weighted methods and the first scenario with errors generated from N (µ = 0, = 0.18).

10


K=2

K=3

K=4

K=5

K=6

K=7

K=8

K=9

1.00

Group 0.75

1

Proportion

2

0.50

0.25

0.00 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21

Sample Figure S2. Histograms of the distribution of estimated intermediate hidden states, K, over 500 simulations, for each sample from the model with weights in the first scenario with errors generated from N (µ = 0, = 0.18).


K=2

K=3

K=4

K=5

K=6

K=7

11

K=8

1.00

Group 1

Proportion

0.75

2

0.50

0.25

0.00 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21

Sample Figure S3. Histograms of the distribution of estimated intermediate hidden states, K, over 500 simulations, for each sample from the model without weights in the first scenario with errors generated from N (µ = 0, = 0.18).

12

WEB-BASED SUPPLEMENTARY MATERIALS: A HMM FOR IDENTIFYING DMCS IN BS-SEQ DATA (a) SRT1

SRT2

SRT3

SRT4

SRT5

SRT6

SRT7

SRT8

SRT9

SRT10

END1

END2

END3

END4

END5

END6

END7

END8

END9

END10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b)

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Method:

HMMW

HMM

HMMDM

bump

Method

bsseq

DMRca

DSS

BiSeq

HFish

Figure S4. Average number of times that the start (a) and end (b) positions of DMRs are identified as DMCs for the first scenario with errors generated from N (µ = 0, = 0.18).


13

Table S4. Average proportion of the times that CpG sites were correctly identified as DMCs (Sensitivity) for the first scenario with errors generated from N (µ = 0, = 0.18). Method

DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

HMMW

0.477 0.167 0.661 0.291 0.109 0.109 0.002 0.000 0.004 0.103 0.006

0.985 0.970 0.989 0.976 0.456 0.376 0.042 0.001 0.170 0.571 0.075

0.996 0.994 0.998 0.998 0.730 0.572 0.077 0.223 0.650 0.876 0.308

0.996 0.997 0.998 0.999 0.808 0.562 0.113 0.669 0.890 0.880 0.509

0.998 0.999 0.999 1.000 0.912 0.420 0.818 0.425 0.935 0.872 0.646

0.989 0.976 0.990 0.977 0.553 0.107 0.857 0.005 0.500 0.036 0.226

0.911 0.747 0.940 0.847 0.262 0.420 0.769 0.000 0.162 0.000 0.048

0.263 0.305 0.427 0.569 0.003 0.000 0.092 0.000 0.001 0.000 0.001

0.057 0.080 0.092 0.155 0.000 0.003 0.005 0.000 0.000 0.000 0.000

0.025 0.013 0.055 0.035 0.000 0.000 0.000 0.000 0.000 0.000 0.000

HMM EMW EM HMMDM bump bsseq DMRca DSS BiSeq HFish

Table S5. Average proportion of the times that CpG sites were incorrectly identified as DMCs (1Specificity) for the first scenario with errors generated from N (µ = 0, = 0.18). Method

NDMR1

NDMR2

NDMR3

NDMR4

NDMR5

NDMR6

NDMR7

NDMR8

NDMR9

NDMR10

NDMR11

HMMW

0.000 0.000 0.000 0.000 0.017 0.025 0.000 0.000 0.000 0.000 0.001

0.012 0.005 0.011 0.005 0.020 0.037 0.000 0.000 0.000 0.001 0.001

0.011 0.009 0.011 0.010 0.023 0.026 0.001 0.000 0.000 0.002 0.001

0.005 0.004 0.005 0.003 0.008 0.005 0.000 0.000 0.000 0.002 0.001

0.003 0.005 0.003 0.005 0.031 0.025 0.000 0.000 0.000 0.001 0.001

0.002 0.001 0.001 0.001 0.004 0.024 0.000 0.000 0.000 0.003 0.002

0.011 0.005 0.007 0.003 0.003 0.021 0.000 0.000 0.000 0.001 0.001

0.002 0.002 0.003 0.003 0.004 0.013 0.005 0.000 0.000 0.000 0.001

0.005 0.005 0.009 0.011 0.019 0.019 0.000 0.000 0.000 0.000 0.001

0.011 0.009 0.014 0.023 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.012 0.007 0.024 0.015 0.000 0.002 0.000 0.000 0.000 0.000 0.000


Table S6. Average Accuracy (ACC) and Modified Accuracy (MACC) with their standard deviations (sd) for the first scenario with errors generated from N (µ = 0, = 0.18). Method

ACC

sdACC

MACC

sdMACC

HMMW

0.917 0.901 0.929 0.912 0.843 0.819 0.785 0.801 0.839 0.863 0.802

0.009 0.008 0.008 0.008 0.001 0.003 0.002 0.002 0.002 0.003 0.001

0.958 0.940 0.967 0.946 0.887 0.864 0.829 0.846 0.883 0.908 0.847

0.009 0.009 0.009 0.011 0.001 0.003 0.002 0.002 0.002 0.003 0.001


14


Table S7. Average number of times that the exact start position of a simulated DMR was correctly identified as being a DMC for the first scenario with errors generated from N (µ = 0, = 0.18). Method

STR1

STR2

STR3

STR4

STR5

STR6

STR7

STR8

STR9

STR10

HMMW

0.002 0.000 0.002 0.000 0.004 0.000 0.004 0.000 0.000 0.000 0.000

0.878 0.778 0.888 0.762 0.060 0.130 0.000 0.000 0.058 0.034 0.000

0.996 1.000 1.000 1.000 0.966 0.000 0.000 0.000 0.454 0.638 0.696

0.972 0.982 0.980 0.992 0.946 0.000 0.000 0.000 0.922 0.896 0.816

0.976 0.964 0.996 0.982 0.876 0.000 0.000 0.000 0.912 0.000 0.644

0.948 0.734 0.922 0.722 0.742 0.000 0.000 0.000 0.522 0.000 0.464

0.662 0.274 0.802 0.376 0.342 0.000 0.000 0.000 0.034 0.000 0.138

0.012 0.020 0.010 0.032 0.000 0.000 0.000 0.000 0.004 0.000 0.000

0.000 0.002 0.000 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.016 0.014 0.062 0.060 0.000 0.000 0.000 0.000 0.000 0.000 0.000


Table S8. Average number of times that the exact end position of a simulated DMR was correctly identified as being a DMC for the first scenario with errors generated from N (µ = 0, = 0.18). Method

END1

END2

END3

END4

END5

END6

END7

END8

END9

END10

HMMW

0.020 0.000 0.110 0.000 0.034 0.000 0.000 0.000 0.000 0.000 0.000

0.656 0.552 0.732 0.608 0.616 0.000 0.006 0.002 0.110 0.074 0.000

0.972 0.922 0.972 0.930 0.494 0.000 0.000 0.000 0.366 0.766 0.086

0.928 0.738 0.960 0.818 0.906 0.000 0.000 0.000 0.352 0.582 0.000

0.940 0.996 0.968 0.998 0.988 0.000 0.952 0.070 0.962 0.908 0.000

0.766 0.464 0.782 0.424 0.316 0.000 0.000 0.020 0.526 0.136 0.000

0.558 0.152 0.608 0.222 0.198 0.128 0.000 0.000 0.042 0.000 0.144

0.338 0.158 0.584 0.410 0.000 0.000 0.048 0.000 0.000 0.000 0.000

0.066 0.104 0.122 0.214 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.014 0.004 0.024 0.016 0.000 0.000 0.000 0.000 0.000 0.000 0.000



Table S9. Distribution of the number of hidden intermediate methylation states (K) for each sample, from the two groups G1 and G2, and over 500 simulations from the HMM with weights for the first scenario with errors generated from N (µ = 0, = 0.18). Group

G1

G2

K

Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2 0 0 262 478 0 361 0 32 0 0 0 0 0 0 0 4 148 0 247 0 0

3 292 80 134 21 10 111 299 411 5 292 2 207 2 0 0 358 287 0 174 0 1

4 124 294 96 1 183 20 132 48 467 190 443 261 403 449 278 131 57 289 77 357 273

5 80 123 7 0 253 8 68 7 17 18 45 26 68 50 123 6 8 73 2 94 214

6 4 3 1 0 44 0 1 2 8 0 9 1 23 1 62 1 0 94 0 39 12

7 8 9 0 0 0 0 0 0 0 0 0 0 0 0 10 0 0 0 0 0 0 0 0 0 0 0 3 0 0 0 0 0 1 0 0 5 0 0 4 0 0 0 0 0 34 2 1 0 0 0 0 0 0 34 10 0 0 0 0 10 0 0 0 0 0

15

16


Table S10. Distribution of the number of hidden intermediate methylation states (K) for each sample, from the two groups G1 and G2, and over 500 simulations from the HMM without weights for the first scenario with errors generated from N (µ = 0, = 0.18). Group

G1

G2

K

Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2 0 0 500 500 0 496 0 491 0 0 0 0 0 0 0 0 116 0 18 0 0

3 390 182 0 0 0 4 489 6 0 42 5 1 5 160 26 97 299 0 229 19 50

4 5 6 7 8 109 1 0 0 0 305 13 0 0 0 0 0 0 0 0 0 0 0 0 0 429 71 0 0 0 0 0 0 0 0 11 0 0 0 0 3 0 0 0 0 478 20 2 0 0 372 86 0 0 0 462 31 2 0 0 473 26 0 0 0 478 16 1 0 0 282 58 0 0 0 147 261 63 2 1 311 92 0 0 0 85 0 0 0 0 2 298 141 58 1 253 0 0 0 0 413 60 8 0 0 2 438 10 0 0


17

S5.A3. Figures and Tables of Simulation Setup Scenario 1, = 0.24. We have used a second set of simulation scenarios to assess the performance of our proposed method. In this section we present the results of the first scenario in which errors were generated from N (0, 0.24) to add to methylation levels of G1 and G2 curves. (a) DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b) NDMR1

NDMR2

NDMR3

NDMR4

NDMR5

NDMR6

NDMR7

NDMR8

NDMR9

NDMR10

DSS

BiSeq

NDMR11

1 − Specificity

0.04 0.03 0.02 0.01 0.00

Method:

HMMW

HMM

HMMDM

bump

bsseq

DMRca

HFish

Figure S5. (a) Average proportion of correctly identified DMCs (Sensitivity) for each method separated by DMRs (sd error bars are added); (b) Average proportion of incorrectly identified as DMCs (1 - Specificity) for each method separated by Non-DMR regions. (The axis is truncated). The results are based on the first scenario with errors generated from N (µ = 0, = 0.24).

18

WEB-BASED SUPPLEMENTARY MATERIALS: A HMM FOR IDENTIFYING DMCS IN BS-SEQ DATA ACC

MACC

1.000

0.975

0.950

0.925

Accuracy

0.900

0.875

0.850

0.825

0.800

0.775

Method:

HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

HFish

Figure S6. Average overall accuracy (ACC) and average overall modified accuracy (MACC) for di↵erent methods in simulated data. The results are based on the first scenario with errors generated from N (µ = 0, = 0.24).


1.0

0.9

0.8

0.7 ● ● ● ● ● ●

●

● ● ●

●

CKappa

0.6

0.5

0.4 ● ●

●

● ● ● ● ● ●

● ●

●

0.3 ● ●

0.2

● ● ●

● ●

●

0.1

0.0 HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

HFish

Method Figure S7. The boxplots of Cohen’s Kappa for di↵erent methods. The results are based on the first scenario with errors generated from N (µ = 0, = 0.24).

19

20


SRT2

SRT3

SRT4

SRT5

SRT6

SRT7

SRT8

SRT9

SRT10

END1

END2

END3

END4

END5

END6

END7

END8

END9

END10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b)

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Method:

HMMW

HMM

HMMDM

bump

Method

bsseq

DMRca

DSS

BiSeq

HFish

Figure S8. Average number of times that the start (a) and end (b) positions of DMRs are identified as DMCs. The results are based on the first scenario with errors generated from N (µ = 0, = 0.24).


K=1

K=2

K=3

K=4

K=5

K=6

K=7

K=8

K=9

K=10

21

K=11

1.00

Group 1

Proportion

0.75

2

0.50

0.25

0.00 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621 1 6111621

Sample Figure S9. Histograms of the distribution of estimated intermediate hidden states, K, over 500 simulations, for each sample from the model with weights. The results are based on the first scenario with errors generated from N (µ = 0, = 0.24).

22


K=1

K=2

K=3

K=4

K=5

K=6

K=7

K=8

1.00

Group 1

Proportion

0.75

2

0.50

0.25

0.00 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21

Sample Figure S10. Histograms of the distribution of estimated intermediate hidden states, K, over 500 simulations, for each sample from the model without weights. The results are based on the first scenario with errors generated from N (µ = 0, = 0.24).


23

Table S11. Average proportion of the times that CpG sites were correctly identified as DMCs (Sensitivity) for the first scenario with errors generated from N (µ = 0, = 0.24). Method

DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

HMMW

0.114 0.265 0.036 0.007 0.029 0.092 0.004 0.000 0.007 0.048 0.008

0.879 0.918 0.571 0.645 0.265 0.297 0.041 0.002 0.112 0.449 0.073

0.980 0.987 0.914 0.958 0.526 0.519 0.078 0.152 0.478 0.829 0.268

0.997 0.999 0.907 0.971 0.635 0.551 0.106 0.583 0.806 0.881 0.469

0.987 0.992 0.636 0.845 0.780 0.417 0.799 0.320 0.853 0.864 0.595

0.840 0.888 0.544 0.371 0.322 0.095 0.826 0.006 0.347 0.011 0.195

0.521 0.717 0.228 0.158 0.092 0.358 0.712 0.000 0.127 0.000 0.046

0.342 0.404 0.407 0.111 0.001 0.000 0.096 0.000 0.002 0.000 0.001

0.054 0.165 0.180 0.089 0.000 0.004 0.008 0.000 0.000 0.000 0.000

0.019 0.056 0.180 0.039 0.000 0.000 0.000 0.000 0.000 0.000 0.000


Table S12. Average proportion of the times that CpG sites were incorrectly identified as DMCs (1Specificity) for the first scenario with errors generated from N (µ = 0, = 0.24). Method

NDMR1

NDMR2

NDMR3

NDMR4

NDMR5

NDMR6

NDMR7

NDMR8

NDMR9

NDMR10

NDMR11

HMMW

0.000 0.000 0.000 0.000 0.003 0.022 0.000 0.000 0.000 0.000 0.002

0.004 0.004 0.001 0.001 0.008 0.040 0.000 0.000 0.000 0.001 0.002

0.010 0.010 0.010 0.009 0.005 0.028 0.001 0.000 0.000 0.002 0.002

0.005 0.006 0.004 0.005 0.002 0.006 0.000 0.000 0.000 0.002 0.001

0.001 0.001 0.001 0.000 0.011 0.029 0.001 0.000 0.001 0.000 0.002

0.002 0.002 0.002 0.002 0.001 0.026 0.000 0.000 0.000 0.003 0.002

0.007 0.009 0.005 0.002 0.001 0.024 0.000 0.000 0.000 0.001 0.002

0.003 0.005 0.007 0.001 0.001 0.013 0.004 0.000 0.001 0.000 0.001

0.006 0.007 0.012 0.001 0.004 0.020 0.000 0.000 0.001 0.000 0.002

0.010 0.028 0.086 0.022 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.011 0.029 0.082 0.017 0.000 0.002 0.001 0.000 0.001 0.000 0.000


Table S13. Average Accuracy (ACC) and Modified Accuracy (MACC) with their standard deviations (sd) for the first scenario with errors generated from N (µ = 0, = 0.24). Method

ACC

sdACC

MACC

sdMACC

HMMW

0.892 0.905 0.863 0.871 0.822 0.811 0.784 0.795 0.825 0.854 0.798

0.006 0.008 0.010 0.006 0.001 0.004 0.002 0.002 0.002 0.003 0.001

0.933 0.938 0.891 0.909 0.867 0.855 0.828 0.839 0.870 0.899 0.843

0.006 0.009 0.014 0.008 0.001 0.004 0.002 0.002 0.002 0.003 0.001


24


Table S14. Average number of times that the exact start position of a simulated DMR was correctly identified as being a DMC for the first scenario with errors generated from N (µ = 0, = 0.24). Method

STR1

STR2

STR3

STR4

STR5

STR6

STR7

STR8

STR9

STR10

HMMW

0.002 0.010 0.000 0.000 0.002 0.000 0.004 0.000 0.000 0.000 0.000

0.520 0.574 0.158 0.200 0.050 0.088 0.000 0.000 0.032 0.032 0.000

0.984 0.986 0.912 0.890 0.662 0.000 0.000 0.000 0.280 0.464 0.582

0.906 0.862 0.522 0.718 0.874 0.000 0.002 0.000 0.712 0.746 0.702

0.918 0.874 0.400 0.492 0.526 0.000 0.000 0.000 0.738 0.000 0.568

0.522 0.630 0.194 0.182 0.512 0.000 0.000 0.000 0.274 0.000 0.408

0.228 0.366 0.056 0.042 0.092 0.000 0.000 0.000 0.028 0.000 0.108

0.032 0.064 0.116 0.016 0.000 0.000 0.000 0.000 0.002 0.000 0.000

0.002 0.002 0.010 0.002 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.026 0.040 0.258 0.064 0.000 0.000 0.000 0.000 0.000 0.000 0.000


Table S15. Average number of times that the exact end position of a simulated DMR was correctly identified as being a DMC for the first scenario with errors generated from N (µ = 0, = 0.24). Method

END1

END2

END3

END4

END5

END6

END7

END8

END9

END10

HMMW

0.002 0.004 0.000 0.000 0.004 0.000 0.000 0.000 0.002 0.000 0.000

0.296 0.394 0.092 0.096 0.388 0.000 0.014 0.008 0.036 0.024 0.000

0.802 0.810 0.526 0.592 0.394 0.000 0.000 0.000 0.186 0.494 0.054

0.814 0.774 0.320 0.258 0.584 0.000 0.000 0.002 0.290 0.404 0.000

0.958 0.890 0.480 0.460 0.816 0.000 0.868 0.052 0.808 0.754 0.000

0.322 0.396 0.054 0.062 0.226 0.000 0.000 0.012 0.240 0.042 0.000

0.348 0.550 0.168 0.052 0.072 0.146 0.000 0.000 0.048 0.000 0.104

0.294 0.356 0.294 0.038 0.000 0.000 0.056 0.000 0.000 0.000 0.000

0.072 0.236 0.228 0.140 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.006 0.018 0.068 0.006 0.000 0.000 0.000 0.000 0.000 0.000 0.000



Table S16. Distribution of the number of hidden intermediate methylation states (K) for each sample, from the two groups G1 and G2, and over 500 simulations from the HMM with weights for the first scenario with errors generated from N (µ = 0, = 0.24). Group

G1

G2

K

Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

1 2 3 4 0 0 483 15 0 0 496 2 0 55 445 0 52 30 258 145 0 0 34 440 0 85 202 198 0 0 495 3 0 91 409 0 0 0 356 29 0 0 384 36 0 0 361 25 0 0 372 32 0 0 370 22 0 0 431 44 0 0 169 100 0 0 409 34 0 256 232 11 0 0 93 188 0 155 341 3 0 0 388 27 0 0 289 183

5 6 7 8 9 1 0 1 0 0 2 0 0 0 0 0 0 0 0 0 14 1 0 0 0 5 9 9 0 3 12 2 1 0 0 2 0 0 0 0 0 0 0 0 0 45 63 7 0 0 63 15 2 0 0 39 70 5 0 0 51 43 1 1 0 51 51 6 0 0 16 4 5 0 0 49 137 39 4 2 42 14 1 0 0 1 0 0 0 0 35 133 37 4 10 1 0 0 0 0 29 46 7 3 0 19 7 1 1 0

25

26


Table S17. Distribution of the number of hidden intermediate methylation states (K) for each sample, from the two groups G1 and G2, and over 500 simulations from the HMM without weights for the first scenario with errors generated from N (µ = 0, = 0.24). Group

G1

G2

K

Sample

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

2 0 0 6 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3 429 464 453 499 1 499 306 452 62 0 79 18 129 52 105 0 441 1 473 129 0

4 44 16 36 0 311 0 117 47 175 175 162 338 149 21 82 122 55 28 27 75 8

5 6 7 8 24 3 0 0 20 0 0 0 5 0 0 0 0 0 0 0 162 26 0 0 0 0 0 0 76 1 0 0 1 0 0 0 241 21 1 0 292 33 0 0 231 28 0 0 134 10 0 0 210 11 1 0 195 228 4 0 282 31 0 0 316 62 0 0 4 0 0 0 385 72 12 2 0 0 0 0 262 34 0 0 229 259 4 0


27

S5.A4. Figures and Tables of Simulation Setup Scenario 2, = 0.18. In this scenario we have considered alternative DMRs with di↵erent lengths and locations to further assess the performance of our method. Table S18 provides the details of this scenario. Figure S11(a) depicts the curves G1 and G2 on which the simulated data were generated. Table S18. Simulation setup parameters for the second scenario

DMR Chr. Start position End position Length E↵ective size DMR1 8 10431839 10441997 172 0.1 DMR2 8 10701967 10729553 363 0.2 DMR3 8 10951995 10971974 274 0.3 DMR4 8 11281992 11301965 358 0.4 DMR5 8 11584972 11629892 903 0.4 DMR6 8 11679894 11714976 643 0.3 DMR7 8 11764964 11794972 703 0.2 DMR8 8 11874947 11924642 669 0.1 DMR9 8 12044993 12061576 307 0.05 DMR10 8 12204790 12232000 120 0.02 We have generated R = 500 data sets and have run all analytical tools on each simulated dataset separately, and quantified the accuracy of the results using the Specificity (SP), Sensitivity (SE), and Accuracy (ACC). The results are provided as follows. Figure S12 depicts ‘Sensitivity’ and ‘1 - Specificity’ (and their standard deviation error-bar) for each method separated by DMRs and non-DMRs, respectively. The overall ACC and MACC are depicted in Figure S13. The results are provided for the model with (HMMW) and without weights (HMM). The results are aligned with what we observed for the first simulation setup. Our proposed method is the dominant one in terms of sensitivity and accuracy. In terms of ‘1 - Specificity’, our method is either comparable to or better than the existing methods.

28


Figure S11. Methylation profiles illustrating simulation design of the second scenario. (a) The smooth methylation profile (G1), obtained by smoothing aggregated B-cell samples, from which the true mean methylation levels in the simulated data sets are generated. (b) Actual methylation proportions for one B-cell sample. (c) A set of simulated methylation data for the same sample as in (b), with error generated from a normal distribution with standard deviation = 0.18.


29

(a) DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b) NDMR1

NDMR2

NDMR3

NDMR4

NDMR5

NDMR6

NDMR7

NDMR8

NDMR9

NDMR10

DSS

BiSeq

NDMR11

1 − Specificity

0.04 0.03 0.02 0.01 0.00

Method:

HMMW

HMM

HMMDM

bump

bsseq

DMRca

HFish

Figure S12. (a) Average proportion of correctly identified DMCs (Sensitivity) for each method separated by DMRs (standard deviation error-bars are added); (b) Average proportion of incorrectly identified as DMCs (1 - Specificity) for each method separated by NDMR regions (The vertical axis is truncated). The results are based on the second scenario with errors generated from N (µ = 0, = 0.18).

30


MACC

1.000

0.975

0.950

0.925

Accuracy

0.900

0.875

0.850

0.825

0.800

0.775

Method:

HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

HFish

Figure S13. Average overall accuracy (ACC) and average overall modified accuracy (MACC) for di↵erent methods in simulated data. The results are based on the second scenario with errors generated from N (µ = 0, = 0.18).


1.0

0.9

● ●

0.8 ● ● ●

● ● ●

0.7

CKappa

0.6 ●

●

0.5

●

0.4

● ●

● ●

0.3

● ● ● ● ● ● ●

0.2

0.1

0.0 HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

HFish

Method Figure S14. The boxplots of Cohen’s Kappa for di↵erent methods. The results are based on the second scenario with errors generated from N (µ = 0, = 0.18).

31

32


SRT2

SRT3

SRT4

SRT5

SRT6

SRT7

SRT8

SRT9

SRT10

END1

END2

END3

END4

END5

END6

END7

END8

END9

END10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b)

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

Method:

HMMW

HMM

HMMDM

bump

Method

bsseq

DMRca

DSS

BiSeq

HFish

Figure S15. Average number of times that the start (a) and end (b) positions of DMRs are identified as DMCs. The results are based on the second scenario with errors generated from N (µ = 0, = 0.18).


K=2

K=3

K=4

K=5

K=6

K=7

K=8

33

K=9

1.00

Group 1

Proportion

0.75

2

0.50

0.25

0.00 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21

Sample

Figure S16. Histograms of the distribution of estimated intermediate hidden states, K, over 500 simulations, for each sample from the model with weights. The results are based on the second scenario with errors generated from N (µ = 0, = 0.18).

34


K=1

K=2

K=3

K=4

K=5

K=6

K=7

1.00

Group 1

0.75

Proportion

2

0.50

0.25

0.00 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21 1 6 11 16 21

Sample

Figure S17. Histograms of the distribution of estimated intermediate hidden states, K, over 500 simulations, for each sample from the model without weights. The results are based on the second scenario with errors generated from N (µ = 0, = 0.18).


35

S5.A5. Empirical FDR for Di↵erent Methods and Simulation Scenarios. At the request of the Associate Editor we have calculated and compared the Empirical FDR of di↵erent methods for di↵erent simulation setups in the following table. Table S19. Empirical FDR (in percent) of di↵erent methods for two di↵erent simulation scenarios (SC) and variation for FDR threshold of 5%. Scenario

1 2

HMMW

0.18 0.24 0.18

HMM

3.146 2.790 2.595 1.985 5.120 5.425

HMMDM

bumph

bsseq

11.852 16.880 1.615 5.853 19.757 1.969 17.482 26.570 1.155

DMRca

DSS

BiSeq

HFish

0.000 0.143 0.641 1.675 0.002 0.596 0.627 2.929 0.000 0.150 4.347 2.437

36


S6. Performance of weighted likelihood approach using different weights In this section we present the results of comparing the performance of di↵erent weights listed in Table S2. (a) DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b) NDMR1

NDMR2

NDMR3

NDMR4

NDMR5

NDMR6

NDMR7

NDMR8

NDMR9

NDMR10

NDMR11

1 − Specificity

0.06

0.04

0.02

0.00

Method: HMMW

HMMW2

HMMW3

HMMW4

HMMW5

HMMW6

HMMW7

HMMW8

HMM

Figure S18. (a) The average proportion of correctly identified DMCs (Sensitivity) for each weighted method separated by DMRs (sd error bars are added); (b) The average proportion of incorrectly identified as DMCs (1 - Specificity) for each weighted method separated by NDMR regions (The axis is truncated). The considered weights are presented in Table S2. Errors were generated from a normal distribution with standard deviation = 0.18 for the first simulation scenario.


MACC

1.000

0.975

0.950

0.925

Accuracy

0.900

0.875

0.850

0.825

0.800

0.775

Method: HMMW

HMMW2

HMMW3

HMMW4

HMMW5

HMMW6

HMMW7

HMMW8

HMM

Figure S19. The average overall accuracy (ACC) and average overall modified accuracy (MACC) for di↵erent weighted methods in simulated data (sd error bars are added). The considered weights are presented in Table S2. Errors were generated from a normal distribution with standard deviation = 0.18 for the first simulation scenario.

37

38


1.0

0.9 ● ●

●

0.8

● ●

● ● ● ●

0.7 ● ● ●

● ● ● ● ● ●

● ● ● ● ●

●

0.6

CKappa

●

● ● ● ● ● ●

● ● ● ●

● ● ● ●

0.5

0.4 ● ● ● ●

0.3

● ● ●

0.2

0.1

0.0 HMMW

HMMW2

HMMW3

HMMW4

HMMW5

HMMW6

HMMW7

HMMW8

HMM

Method Figure S20. The boxplots of Cohen’s Kappa for di↵erent weighted methods. The considered weights are presented in Table S2. The results are based on the first simulation scenario with errors generated from N (µ = 0, = 0.18).


39

S7. Performance of Fully Bayesian Analysis Versus Other Methods In this section we present the results of comparing the performance of fully Bayesian Analysis with other method. (a) DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

1.0

0.9

0.8

0.7

Sensitivity

0.6

0.5

0.4

0.3

0.2

0.1

0.0

(b) NDMR1

NDMR2

NDMR3

NDMR4

NDMR5

NDMR6

NDMR7

NDMR8

NDMR9

NDMR10

NDMR11

1 − Specificity

0.04 0.03 0.02 0.01 0.00

Method: HMMW−B

HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

Figure S21. (a) The average proportion of correctly identified DMCs (Sensitivity) for each di↵erent method separated by DMRs (sd error bars are added); (b) The average proportion of incorrectly identified as DMCs (1 - Specificity) for each weighted method separated by NDMR regions (The axis is truncated). Errors were generated from a normal distribution with standard deviation = 0.18 for the first simulation scenario. The full Bayesian analysis HMMW-B is shown on the left (pink color).

40


MACC

1.000

0.975

0.950

0.925

Accuracy

0.900

0.875

0.850

0.825

0.800

0.775

Method:

HMMW−B

HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

Figure S22. The average overall accuracy (ACC) and average overall modified accuracy (MACC) for di↵erent methods in simulated data (sd error bars are added). Errors were generated from a normal distribution with standard deviation = 0.18 for the first simulation scenario. The full Bayesian analysis HMMW-B is presented on the left (pink color).


1.0

0.9

●

● ●

0.8 ●

0.7 ●

● ●

CKappa

0.6

●

0.5 ● ● ●

0.4

● ●

● ●

● ●

●

● ●

0.3

●

0.2

● ● ● ● ● ● ●

0.1

0.0 HMMW−B

HMMW

HMM

HMMDM

bump

bsseq

DMRca

DSS

BiSeq

Method Figure S23. The boxplots of Cohen’s Kappa for di↵erent methods. Errors were generated from N (µ = 0, = 0.18) for the first simulation scenario. The full Bayesian analysis HMMW-B is presented on the left.

41

42


S8. FDR control producer and dependency Following a comment from the Associate Editor, we have studied FDR accounting for further dependency in Step 3 of our method. In a simulation study we compared two methods: (i) the FDR control procedure accounting for some dependency proposed in Sun and Cai (2009); here it is denoted by ‘dependent’ procedure; (ii) the FDR control procedure that does not account for any dependency proposed in Benjamini and Hochberg (1995); here it is denoted by ‘independent’ procedure. The results of this (limited) simulation study are provided in Table S20. From this table we observe that, although the sensitivity values for DMRs are very large for the ‘dependent’ procedure, the false positive (‘1specificity’) are also very large for the ‘dependent’ procedure, specially for non-DMRs 2, 5, 10 and 11. Furthermore, we observe from Table S20(C) that the empirical FDR (denoted by ‘eFDR’) is very large for the ‘dependent’ procedure and it is not comparable to either the ‘independent’ procedure or the nominal value of 5%. These results imply that further accounting for dependence in Step 3 of our proposed method, while we have already accounted for autocorrelation in Step 2, does not perform better and even worsens the performance. More investigation should be done on this issue. Table S20. Comparison of FDR using independence and dependence procedure; values are in percent.

(A) 1-specificity Procedure independent dependent

nDMR1

nDMR2

nDMR3

nDMR4

nDMR5

nDMR6

nDMR7

nDMR8

nDMR9

nDMR10

nDMR11

0.000 0.425

0.065 9.918

0.128 1.107

0.073 1.559

0.027 6.361

0.044 0.676

0.113 1.475

0.018 2.335

0.129 3.168

0.264 11.448

0.026 9.561

(B) sensitivity Procedure independent dependent

DMR1

DMR2

DMR3

DMR4

DMR5

DMR6

DMR7

DMR8

DMR9

DMR10

40.854 86.145

95.185 98.549

98.297 99.910

99.381 99.962

99.078 100.00

97.049 99.990

87.507 99.031

10.692 63.722

2.196 31.609

0.219 23.972

Accuracy

eFDR

91.38 91.70

0.35 17.30

(C) other criteria Procedure independent dependent

In this table smaller values of ‘1-specificity’ and empirical FDR (‘eFDR’), and larger values of ’sensitivity’ and ’Accuracy’ are preferred.


43

S9. Real Data Analysis Results In the main paper, we have compared methylation profiles between cell types near the BLK gene described in Section 2 of the main paper using DMCHMM. The whole analysis took less than 3 hours on a MacBook pro (15-inch, Mid 2015). In the following, we have presented tables and figures that are discussed in the main paper. b for the samples in BLK data Table S21. The estimated intermediary states (K)

Group

1 BC 7 monocyte 7 TC 9

Sample

2 7 5 7

3 4 5 9

4 3 4 9

5 9 6 7

6 6 4 8

7 7 6 8

8 9 10 11 12 13 14 15 16 17 18 19 3 3 3 5 3 7 5 9 7 8 5 7 7 5 5 7 6 7 3

44


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0.4

BC vs Mono

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Figure S24. (a)-(c) Ident fied DMCs for three pa rw se compar sons of ce -type methy at on data near the BLK gene B ack vert ca nes nd cate CpG s tes where one ce type was s gn ficant y d ↵erent from the other at q-va ue < 1e 7 In the promoter reg on of BLK a set of c ose y spaced probes demonstrate ower methy at on (hypo-methy at on) n B-ce s For each pane the average pred cted methy at on eve s of each ce type are a so p otted (d) Manhattan p ot for p-va ues Resu ts are based on non-we ghted ke hood


45

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BC < Mono

(b) 0.8

1

BC > TC ● ● ● ● ● ●● ● ●● ●● ● ● ●●● ● ●● ● ●● ●● ● ●● ●● ● ● ● ●● ● ●● ●● ●● ● ● ● ● ●●● ● ●●● ● ● ● ●

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0.4

BC vs TC

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BC TC

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BC < TC

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0.6

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Mono TC

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0.2

Mono vs TC

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− log10(p )

20 15 10 5 0

(d)

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