Original: M ULTI - RESOLUTION MODELING OF A DISCRETE STOCHASTIC PROCESS IDENTIFIES CAUSES OF CANCER
Revised: M ULTI - RESOLUTION MODELING OF A DISCRETE STOCHASTIC PROCESS IDENTIFIES CAUSES OF CANCER

Original: Anonymous authors
Revised: Anonymous authors

Original: Paper under double-blind review
Revised: Paper under double-blind review

Original: A BSTRACT
Revised: A BSTRACT

Original: Detection of cancer-causing mutations within the vast and mostly unexplored human genome is a major challenge.
Revised: Detection of cancer-causing mutations within the vast and mostly unexplored human genome is a major challenge.

Original: Doing so requires modeling the background mutation rate, a highly non-stationary stochastic process, across regions of interest varying in size from one to millions of positions.
Revised: Doing so requires modeling the background mutation rate, a highly non-stationary stochastic process, across regions of interest varying in size from one to millions of positions.

Original: Here, we present the split-PoissonGamma (SPG) distribution, an extension of  classical Poisson-Gamma formulation, to model a discrete stochastic process at multiple resolutions.
Revised: Here, we present the splitPoisson-Gamma (SPG) distribution, an extension of the classical Poisson-Gamma formulation, to model a discrete stochastic process at multiple resolutions.

Original: We demonstrate that the probability model has a closed-form posterior, enabling efﬁcient and accurate linear-time prediction over any length scale after the parameters of the model have been inferred a single time.
Revised: We demonstrate that the probability model has a closed-form posterior, enabling efﬁcient and accurate linear-time prediction over any length scale after the parameters of the model have been inferred a single time.

Original: We apply our framework to model mutation rates in tumors and show that model parameters can be accurately inferred from high-dimensional epigentic data using a convolutional neural network, Gaussian process, and maximum-likelihood estimation.
Revised: We apply our framework to model mutation rates in tumors and show that model parameters can be accurately inferred from high-dimensional epigenetic data using a convolutional neural network, Gaussian process, and maximum-likelihood estimation.

Original: Our method is both more accurate and more efﬁcient than existing models over a large range of length scales.
Revised: Our method is both more accurate and more efﬁcient than existing models over a large range of length scales.

Original: We demonstrate the usefulness of multi-resolution modeling by detecting genomic elementsof vastly differing sizes that drive tumor emergenceincluding genes, regulatory structures, and individual base-pairs.
Revised: We demonstrate the usefulness of multi-resolution modeling by detecting genomic elements that drive tumor emergence and are of vastly differing sizes.

Original:  I NTRODUCTION
Revised:  I NTRODUCTION

Original: Numerous domains involve modeling highly non-stationary discrete-time and integer-valued stochastic processes where event counts vary dramatically over time or space.
Revised: Numerous domains involve modeling highly non-stationary discrete-time and integer-valued stochastic processes where event counts vary dramatically over time or space.

Original: An important open problem of this nature in biology is understanding the stochastic process by which mutations arise across the genome.
Revised: An important open problem of this nature in biology is understanding the stochastic process by which mutations arise across the genome.

Original: This is central to identifying mutations that drive cancer emergence (Lawrence et al., 2013).
Revised: This is central to identifying mutations that drive cancer emergence (Lawrence et al., 2013).

Original: Tumor drivers provide a cellular growth advantage to cells by altering the function of a genomic element such as a gene or regulatory feature (e g. promoter).
Revised: Tumor drivers provide a cellular growth advantage to cells by altering the function of a genomic element such as a gene or regulatory feature (e.g. promoter).

Original: Drivers are identiﬁable because they reoccur across tumors, but there are two major challenges to detecting such recurrence.
Revised: Drivers are identiﬁable because they reoccur across tumors, but there are two major challenges to detecting such recurrence.

Original: First, driver mutations are rare and their signal is hidden by the thousands of passenger mutations that passively and stochastically accumulate in tumors (Stratton et al., 2009; Martincorena & Campbell, 2015).
Revised: First, driver mutations are rare and their signal is hidden by the thousands of passenger mutations that passively and stochastically accumulate in tumors (Stratton et al., 2009; Martincorena & Campbell, 2015).

Original: Second, because functional elements vary dramatically in size (genes: 10 3 -10 6 bases; regulatory elements: 10 1 -10 3 bases; and single positions), driver mutations accumulate across regions that vary many orders of magnitude.
Revised: Second, because functional elements vary dramatically in size (genes: 10 3 -10 6 bases; regulatory elements: 10 1 -10 3 bases; and single positions), driver mutations accumulate across regions that vary many orders of magnitude.

Original: Accurately predicting the stochastic accumulation of passenger mutations at multiple scales is necessary to reveal the subtle recurrence of driver mutations across the genome.
Revised: Accurately predicting the stochastic accumulation of passenger mutations at multiple scales is necessary to reveal the subtle recurrence of driver mutations across the genome.

Original: Here, we introduce the split-Poisson Gamma (SPG) process, an extension of the Poisson-Gamma distribution, to efﬁciently model a non-stationary discrete stochastic process at numerous length scales.
Revised: Here, we introduce the split-Poisson Gamma (SPG) process, an extension of the Poisson-Gamma distribution, to efﬁciently model a non-stationary discrete stochastic process at numerous length scales.

Original: The model ﬁrst approximates quasi-stationary regional rate parameters within small windows; it then projects these estimates to arbitrary regions in linear time (10-15 minutes for genome-wide inference).
Revised: The model ﬁrst approximates quasi-stationary regional rate parameters within small windows; it then projects these estimates to arbitrary regions in linear time (10-15 minutes for genome-wide inference).

Original: This approach is in contrast to existing efforts that model ﬁxed regions and require computationally expensive retraining (e.g. >5 hours) to predict over multiple scales of interest (Nik-Zainal et al., 2016; Martincorena et al., 2017).
Revised: This approach is in contrast to existing efforts that model ﬁxed regions and require computationally expensive retraining (e.g. over 5 hours) to predict over multiple scales of interest (Nik-Zainal et al., 2016; Martincorena et al., 2017).

Original: We apply our framework to model cancer-speciﬁc mutation patterns (ﬁg. 1).
Revised: We apply our framework to model cancer-speciﬁc mutation patterns (ﬁg. 1).

Original: We perform data-driven training of our model’s parameterswith standard machine-learning methods and show that it more accurately captures mutation patterns than existing methods on simulated and real data.
Revised: We perform data-driven training of our model’s parameters and show that it more accurately captures mutation patterns than existing methods on simulated and real data.

Original: We demonstrate the power of our multi-resolution approach by identifying drivers across functional elements: genes, regulatory features, and single base mutations with one training of the model.
Revised: We demonstrate the power of our multi-resolution approach by identifying drivers across functional elements: genes, regulatory features, and single base mutations.

Original: Despite the method having no knowledge of genome structure, it detects nearly all gene drivers present in >5% of samples while making no false discoveries and detects all previously characterized regulatory drivers.
Revised: Despite the method having no knowledge of genome structure, it detects nearly all gene drivers present in over 5% of samples while making no false discoveries and detects all previously characterized regulatory drivers.

Original: Detected events also include novel candidate drivers, providing promising targets for future investigation, especially in the largely unexplored non-coding genome.
Revised: Detected events also include novel candidate drivers, providing promising targets for future investigation.

Original: 1. P REVIOUS WORK
Revised: 1. P REVIOUS WORK

Original: Numerous methods exist for modeling stationary stochastic processes (Lindsey, 2004).
Revised: Numerous methods exist for modeling stationary stochastic processes (Lindsey, 2004).

Original: However, far fewer exist for non-stationary processes because they are difﬁcult to capture with the covariance functions of parametric models (Risser, 2016).
Revised: Far fewer exist for non-stationary processes because they are difﬁcult to capture with the covariance functions of parametric models (Risser, 2016).

Original: Non-stationary kernels have been introduced for Gaussian processes (Paciorek & Schervish, 2004), but these may not be tractable on large datasets due to their computational complexity.
Revised: Non-stationary kernels have been introduced for Gaussian processes (Paciorek & Schervish, 2004) but these may not be tractable on large datasets due to their computational complexity.

Original: There hasmore recently been work developing Poissongamma models for dynamical systems (Schein et al., 2016; Guo et al., 2018), but these methods have focused on learning relationships between count variables, not predicting counts based on continuous covariates.
Revised: More recently, there has been work developing Poisson-gamma models for dynamical systems (Schein et al., 2016; Guo et al., 2018) but these methods have focused on learning relationships between count variables, not predicting counts based on continuous covariates.

Original: In the particular case of modeling mutation patterns across the cancer genome, numerous computational methods exist to model mutation rates within well-understood genomic contexts such as genes (Lawrence et al., 2013; Martincorena et al., 2017; Wadi et al., 2017; Mularoni et al., 2016;
Revised: In the particular case of modeling mutation patterns across the cancer genome, numerous computational methods exist to model mutation rates within well-understood genomic contexts such as genes (Lawrence et al., 2013; Martincorena et al., 2017; Wadi et al., 2017; Mularoni et al., 2016;

Original: Juul et al.).
Revised: Juul et al.).

Original: These models account for < 4% of the genome (Rheinbay et al., 2020).
Revised: These models account for < 4% of the genome (Rheinbay et al., 2020).

Original: They are not applicable in non-coding regions, where the majority of mutations occur (Gloss & Dinger, 2018).
Revised: They are not applicable in non-coding regions, where the majority of mutations occur (Gloss & Dinger, 2018).

Original: A handful of methods to model genome-wide mutation rates have been introduced (Polak et al., 2015;
Revised: A handful of methods to model genome-wide mutation rates have been introduced (Polak et al., 2015;

Original: Nik-Zainal et al., 2016; Bertl et al., 2018).
Revised: Nik-Zainal et al., 2016; Bertl et al., 2018).

Original: However, they operate on a single length-scale or set of regions and require computationally expensive retraining to predict over each new length-scale.
Revised: However, they operate on a single length-scale or set of regions and require computationally expensive retraining to predict over each new length-scale.

Original: Several methods rely on Poisson or binomial regression; however, previous work has extensively documented that mutation counts data are over-dispersed, leading these models tosystematically underestimate variance and yield numerous false-positive driver predictions (Lochovsky et al., 2015; Martincorena et al., 2017; Juul et al., 2019).
Revised: Several methods rely on Poisson or binomial regression; however, previous work has extensively documented that mutation counts data are over-dispersed, leading these models to underestimate variance and yield numerous false-positive driver predictions (Lochovsky et al., 2015; Martincorena et al., 2017; Juul et al., 2019).

Original: Negative binomial regression has recently been used to account for over-dispersion (Nik-Zainal et al., 2016) and perform genome-wide mutation modeling and driver detection.
Revised: Negative binomial regression has recently been used to account for over-dispersion (Nik-Zainal et al., 2016) and perform genome-wide mutation modeling and driver detection.

Original: However, resolution was coarse, and it only found a few, highly recurrent driver mutations.
Revised: However, resolution was coarse, and it only found a few, highly recurrent driver mutations.

Original: 1. O UR CONTRIBUTIONS
Revised: 1. O UR CONTRIBUTIONS

Original: This work makes three key contributions: 1) we introduce an extension of the Poisson-Gamma distribution to model non-stationary discrete stochastic processes at any arbitrary length scale without retraining; 2) we apply the framework to capture cancer-speciﬁc mutation rates with unprecedented accuracy, resolution, and efﬁciency; and 3) we perform a, multi-scale search for cancer driver mutations genome-wide, including the ﬁrst-ever base-resolution scan of the whole genome.
Revised: This work makes three key contributions: 1) we introduce an extension of the Poisson-Gamma distribution to model non-stationary discrete stochastic processes at any arbitrary length scale without retraining; 2) we apply the framework to capture cancer-speciﬁc mutation rates with unprecedented accuracy, resolution, and efﬁciency; and 3) we perform a multi-scale search for cancer driver mutations genome-wide, including the ﬁrst-ever base-resolution scan of the whole genome.

Original: This search yields several new candidate driver events in the largely unexplored non-coding genome, which we are working on validating with experimental collaborators.
Revised: This search yields several new candidate driver events in the largely unexplored non-coding genome, which we are working on validating with experimental collaborators.

Revised: Crucially, our approach allows fast, efﬁcient, and accurate searches for driver elements and mutations anywhere in the genome without requiring arduous retraining of a model, a feat which is not possible with existing approaches.

Original:  M ULTI - RESOLUTION MODELING OF A NON - STATIONARY DISCRETE STOCHASTIC PROCESS
Revised:  M ULTI - RESOLUTION MODELING OF A NON - STATIONARY DISCRETE STOCHASTIC PROCESS

Original: We consider a non-stationary  stochastic process  where discrete events occur at  a discrete set of units (discrete-time and integer-valued).
Revised: We consider a non-stationary discrete stochastic process { M i ; i = 1 , 2 ,... } where M i is the integervalued event count at position i . Associated with each position i is a real-valued, L-dimensional feature vector η i that determines the instantaneous event rate λ i via an unknown function. Thus a region R = { i , i + 1 , ..., i + N } of N contiguous positions is characterized by an L × N feature matrix η R and an event count X R = ∑ i ∈ R M i .

Revised: As training data, η R , X R , and M i are observed for some set of regions { R ∈ T } .

Revised: Then given a set of feature matrices from unobserved regions { η R ; R ∈ H } , the challenge is to predict the distribution of event counts over any arbitrary set I of unseen positions that may or may not be contiguous.

Original: Examples includetransactions by traders in a stock market, packets delivered to routers in a network, and mutations accumulating at positions in the genome.
Revised: Real-world examples include traders in a stock market, packets delivered to routers in a network, and mutations accumulating at positions in the genome.

Original: We assume  the process is near-stationary within a small enough set of contiguous units aka region, R  ,  and  the stationary distribution is associated in some (unknown) way with observed covariates ( η R ).
Revised: We assume that the process is near-stationary within a small enough region R = { i , i + 1 , ..., i + N } and that the L × N covariate matrix η R is observed.

Original: Thus the rate of eventswithin R ( λ R ) is approximately constant, albeit unknown.
Revised: Thus the rate of events λ R within R is approximately constant and associated with η R , albeit in an unknown way.

Original: A number of events ( X R ) may occur within R dependent on λ R and are then stochastically distributed to individual units, implying a  factorization over λ R , X R , and M i, the number of events at unit i ∈ R (ﬁg. 1d)
Revised: A number of events ( X R ) may occur within R dependent on λ R and are then stochastically distributed to individual positions within R , implying a hierarchical factorization of the scalar random variables λ R , X R , and M i (ﬁg. 1e) as [Equation]

Original: [Equation]
Revised: [Equation]

Original: X R and λ R are unknown nuisance variables and are marginalized in general as
Revised: X R and λ R are unknown nuisance variables and are marginalized in general as

Original: [Equation]
Revised: [Equation]

Original: Since applications often require many posterior predictions over regions of varying sizes, we propose a prior parameterizationsuch that builds on the success and ﬂexibility of the classical Poisson-Gamma distribution while ensuring the marginalization has an easy-to-compute posterior distribution:
Revised: Since applications often require many posterior predictions over regions of varying sizes, we propose a prior parameterization that builds on the success and ﬂexibility of the classical Poisson-Gamma distribution while ensuring the marginalization has an easy-to-compute posterior distribution:

Original: [Equation]
Revised: [Equation]

Original: where α R and θ R are shape and scale parameters dependent on η R , p i is the time-averaged probability of an event at i and ˜ p
Revised: where α R and θ R are shape and scale parameters dependent on η R , p i is the time-averaged probability of an event at i and ˜ p

Original: i
Revised: i

Original: = p i
Revised: = p i

Original: ∑ j ∈ R p j , the normalized probability within R .
Revised: ∑ j ∈ R p j , the normalized probability within R .

Revised: A plate diagram of the hierarchical model is presented in ﬁg. 1e.

Original: The above formulation provides a simple, closed form solution to eq.
Revised: The above formulation provides a simple, closed form solution to eq.

Original: (2)  (See Appendix for details):
Revised: (2) as a negative binomial (NB) distribution (See Appendix for details):

Original: [Equation]
Revised: [Equation]

Original: Eq.
Revised: Eq.

Original: 5 implicitly assumes that events are distributed independently to units within R .
Revised: 5 implicitly assumes that events are distributed independently to units within R .

Original: Exploiting this assumption, eq.
Revised: Exploiting this assumption, eq.

Original: (6) immediately generalizes to consider any set of units I ⊆
Revised: (6) immediately generalizes to consider any set of units I ⊆

Original: R as
Revised: R as

Original: [Equation]

Original: An equivalent parameterization for λ R is based on the mean mutation rate µ R and the associated variance (or estimation uncertainty), σ 2 R because the gamma parameters are uniquely determined as

Original: [Equation]
Revised: [Equation]

Original: The above formulation is an extension of the classical Poisson-Gamma distribution whereby the
Revised: The above formulation is an extension of the classical Poisson-Gamma distribution whereby the

Original: Poisson is randomly split by a binomial.
Revised: Poisson is randomly split by a binomial.

Original: We term this a split-Poisson-Gamma (SPG) process.
Revised: We term this a split-Poisson-Gamma (SPG) process.

Original: While the derivation of the SPG solution makes simplifying assumptions, the beneﬁt is that the parameters µ R and σ 2 R need to be estimated only once for each non-overlapping region R .
Revised: While the derivation of the SPG solution makes simplifying assumptions, the beneﬁt is that the parameters α R and θ R need to be estimated only once for each non-overlapping region R .

Original: Estimates for a region of any other size can then be computed in constant time from eq.
Revised: Estimates for a region of any other size can then be computed in constant time from eq.

Original: If a new region R (cid:48) is larger than R , we approximate the gamma distribution in a super-region containing R (cid:48) from the sum of the previously inferred 
Revised: If a new region R (cid:48) is larger than R , we approximate the gamma distribution in a super-region containing R (cid:48) as a superposition of the previously inferred parameters of each region of size R within the super-region (see section 2.2).

Original: F ITTING PARAMETERSTO PREDICT CANCER MUTATION PATTERNS 3. D ATA
Revised: 2. I NFERRING REGIONAL RATE PARAMETERS

Revised: The statistical power of SPG depends on accurate estimation of the regional gamma rate parameters α R and θ R . We propose a variational approach to enable ﬂexible, accurate, and non-linear inference of these parameters from a set of covariates. Let G ( α , θ ) be a gamma distribution.

Revised: By the central limit theorem, lim α → ∞ G ( α , θ ) =

Revised: N ( µ , σ 2 ) where µ = αθ and σ 2 = αθ 2 .

Revised: We thus use a Gaussian process (GP) to non-linearly map covariates to regional estimates for µ R and σ 2 R .

Revised: The variational estimates for the gamma parameters are then

Revised: [Equation]

Revised: For a super-region R (cid:48)

Revised: =

Revised: R i +

Revised: [Equation]

Revised: [Equation]

Revised: A limitation of this approach is that GPs can only operate on vectors of covariates.

Revised: Thus a dimensionality reduction method must be applied to the input matrix η R .

Revised: In cases where η R includes spatial relationships, a convolutional neural network can be a powerful approach to dimension-reduction; however, other approaches are feasible (see section 3.2 and section 5.1).

Revised: 2. I NFERRING TIME - AVERAGED EVENT PROBABILITIES

Revised: The time-averaged parameters { p i ; i = 1 , 2 ,... } must also be inferred.

Revised: Crucially, as seen in eq.

Revised: (5), these parameters are never used directly; instead, they are always renormalized to sum to one within a region of interest.

Revised: Thus, estimates do not need to reﬂect the absolute probability of an event at i but merely the relative rate of events between positions.

Revised: Indeed, because of the renormalization procedure, the estimates need not even be a true probability distribution.

Revised: Estimating p i can thus be accomplished by clustering units with similar relative rates of events.

Revised: How this clustering should be performed will depend on the application of interest (see section 3.3 for a concrete example).

Original: We obtained publicly available mutation counts from four cancer cohorts previously characterized by the Pan-Cancer Analysis of Whole Genomes Consortium (PCAWG) (Campbell et al., 2020): esophageal adenocarcinoma (N = 98 tumors; n ≈ 2.7M mutations), skin melanoma (N = 70 tumors; n ≈ 7.8M mutations), stomach adenocarcinoma (N = 37 tumors; n ≈ 480k mutations), and liver hepatocellular carcinoma (N = 264 tumors; n ≈ 3.3M mutations).
Revised: We obtained publicly available mutation counts from four cancer cohorts previously characterized by the Pan-Cancer Analysis of Whole Genomes Consortium (PCAWG) (Campbell et al., 2020): esophageal adenocarcinoma (N = 98 tumors; n ≈ 2.7M mutations), skin melanoma (N = 70 tumors; n ≈ 7.8M mutations), stomach adenocarcinoma (N = 37 tumors; n ≈ 480k mutations), and liver hepatocellular carcinoma (N = 264 tumors; n ≈ 3.3M mutations).

Original: Crucially, these data contain only the total number of mutations at each position in the genome.
Revised: Crucially, these data contain only the total number of mutations at each position in the genome.

Original: We do not know a priori which mutations are background mutations and which are driver mutations.
Revised: We do not know a priori which mutations are background mutations and which are driver mutations.

Original: We also do not know the true mean and variance of the underlying mutation rate in any region.
Revised: We also do not know the true mean and variance of the underlying mutation rate in any region.

Original: We do know that the mutation rate is highly associated with chemical modiﬁcations of the DNA that set the way it is processed in a cell, collectively termed the epigenome (Schuster-Böckler & Lehner, 2012; Polak et al., 2015).
Revised: We do know that the mutation rate is highly associated with chemical modiﬁcations of the DNA that set the way it is processed in a cell, collectively termed the epigenome (Schuster-Böckler & Lehner, 2012; Polak et al., 2015).

Original: We obtained 733 tracks characterizing the epigenome of  111 human tissues from Roadmap Epigenomics (Roadmap Epigenomics Consortium et al., 2015).
Revised: We obtained 733 datasets characterizing the patterns of these chemical modiﬁcations in 111 human tissues from Roadmap Epigenomics (Roadmap Epigenomics Consortium et al., 2015).

Revised: These data are the largest compendium of uniformly processed human epigenome sequencing currently available.

Original: Each track provides the -log 10 P-value that a particular modiﬁcation is present at each location of the genome in a given tissue type.
Revised: Each track provides the -log 10 P-value that a particular modiﬁcation is present at each location of the genome in a given tissue type.

Original: We additionally created two tracks that provide the average nucleotide and GC content in a region based on the human reference genome build GRCh37.
Revised: We additionally created two tracks that provide the average nucleotide and GC content in a region based on the human reference genome build GRCh37.

Original: Additional information on the epigenetic features are in Appendix and a full list of tracks is provided as supplementary data.
Revised: Additional information on the epigenetic features are in Appendix and a full list of tracks is provided as supplementary data.

Revised: The input matrix for each region η R thus has 735 rows.

Revised: We ﬁxed the number of columns to be 100 irrespective of the size of R , where each column is the mean across R / 100 adjacent positions.

Original: In order to evaluate the ability of SPG and other models to estimate the unknown mean and variance of regional rates, we created simulated datasetsfor esophageal, skin, and stomach cancer with known mean and variance parameters.
Revised: In order to evaluate the ability of SPG and other models to estimate the unknown mean and variance of regional rates, we created simulated datasets with known mean and variance parameters dependent on the observed input matrix (ﬁg. 2a).

Original: For each non-overlapping 50kb region in thegenome we used a k-nearest-neighbors strategy to identify 500 regions with similar epigenetic features.
Revised: We created input matrices of size 735 × 100 from the epigenetic tracks (described above) for non-overlapping regions of 50,000 positions.

Original: We deﬁned µ R and σ 2 R for that region as the mean and variance of the observed mutation counts across theregions.
Revised: To deﬁne non-stationary mean and variance of mutation rate dependent on the each region’s input matrix, we reduced η R to an feature vector of size 735 by taking the mean across columns and used a k-nearest-neighbors (KNN) strategy to identify 500 regions with similar epigenetic feature vectors; we then deﬁned µ R and σ 2 R for each region as the mean and variance of the observed event counts across its 500 neighbouring regions.

Original: The number of "observed" passenger mutations for that region was then randomly drawn from a negative binomial distribution deﬁned by those parameters (full technical details in Appendix).
Revised: The number of observed events for that region was then randomly drawn from a negative binomial distribution deﬁned by those parameters (full technical details in Appendix).

Original: Models were trained on the randomly drawn counts and evaluated on their ability to accurately infer the true mean and variance.
Revised: Models were trained on the randomly drawn counts and evaluated on their ability to accurately infer the true mean and variance.

Original: We created two versions of the simulated data, one in which all regions in the genome were used to estimate the rate parameters and one in which rate parameters were estimated separately within independent train and test subsets.

Original: We chose to simulate within 50kb regions following previous work (Rheinbay et al., 2020)),
Revised: We simulated 50kb regions following previous work (Rheinbay et al., 2020).

Original: 3. E STIMATING TIME - AVERAGED EVENT PROBABILITIES
Revised: 3. E STIMATING DYNAMIC REGIONAL RATES WITH UNCERTAINTY

Revised: The input matrices η R ∈ R 735 × 100 required signiﬁcant dimension reduction before we could employ our GP-based variational strategy to infer SPG regional rate parameters.

Revised: Columns encode the highresolution spatial organization of the epigenome which have recently been shown to be important determinants of local mutation rate (Gonzalez-Perez et al., 2019; Akdemir et al., 2020).

Revised: Therefore, we hypothesized that a convolutional neural network (CNN) would provide a powerful approach to produce a low-dimensional embedding that retrains information about this local structure; the supervised nature of a CNN further enables the resulting embedding to be optimized for the cancer of interest, which is crucial to performance since the epigenetic determinants of mutation rate vary drastically between cancer types (Polak et al., 2015).

Revised: We constructed a 1D CNN model withresidual blocks and 3 fully-connected layers to map mutation-rate-associated local epigenetic patterns to regional mutation rates.

Original: As described in section 2, p i is the long-term, time-averaged rate of events for unit i .
Revised: The CNN non-linearly reduces η R ∈ R 735 × 100 to a 16 dimensional feature vector in its last feature layer. The CNN was trained to minimize the mean squared error between observed and predicted mutation counts. Due to the interchangeable nature of the rows, the 1D kernels allow the network to identify arbitrary inter-track interactions. The ﬁnal 16-dimension feature vector was then passed as input to a sparse GP (Titsias), ﬁt to maximize the likelihood of the observed mutation counts using 2000 inducing points and a radial basis function kernel (ﬁg. 1b).

Original: In most applications, there are too many discrete units to make direct inference feasible. Estimating p i thus requires clustering units with similar event patterns.
Revised: We found that results were robust to the particular choice of kernel and hyperpriors placed over kernel parameters. While end-to-end training is possible (Bradshaw et al., 2017), we did not ﬁnd it necessary to achieve high accuracy in this particular application. A CNN is not the only method available to reduce dimensionality prior to GP inference; we investigated numerous other methods, but found the

Original: How this clustering should be performed will depend on the application of interest.
Revised: CNN+GP to produce the most accurate results (see Appendix).

Original: In the case of cancer mutation patterns, previous work showed that the mutation rate at any position i is heavily inﬂuenced by the nucleotide at i and the two nucleotides directly adjacent to i ; positions with this same “trinucleotide context” will have similar mutation patterns (Alexandrov et al., 2013).
Revised: In the case of cancer mutation patterns, previous work showed that the mutation rate at any position i is heavily inﬂuenced by the nucleotide at i and the two nucleotides directly adjacent to i ; positions with this same “trinucleotide context” will have similar mutation patterns (Alexandrov et al., 2013).

Original: Following previous works (Mularoni et al., 2016; Wadi et al., 2017; Martincorena et al., 2017; Weghorn & Sunyaev, 2017), we used trinucleotide context to estimate p i .
Revised: Following previous works (Mularoni et al., 2016; Wadi et al., 2017; Martincorena et al., 2017; Weghorn & Sunyaev, 2017), we used trinucleotide context to estimate p i .

Original: Let ntn (cid:48) be the trinucleotide context centered at position i .
Revised: Let ntn (cid:48) be the trinucleotide context centered at position i .

Original: We estimate the probability that i is mutated using the ensemble maximum-likelihood estimate of its cluster
Revised: We estimate the probability that i is mutated using the ensemble maximum-likelihood estimate of its cluster

Original: [Equation]
Revised: [Equation]

Original: (8) where N n , t , n (cid:48) is the number of ntn (cid:48) trinucleotides in the genome and v n , t , n (cid:48) is the number of times t is mutated within ntn (cid:48) .
Revised: where N n , t , n (cid:48) is the number of ntn (cid:48) trinucleotides in the genome and v n , t , n (cid:48) is the number of times t is mutated within ntn (cid:48) .

Original: This approach alone explains little variance in sub-megabase regions (see
Revised: This approach alone explains little variance in sub-megabase regions (see

Original: Appendix) because it does not account for regional mutation rates.
Revised: Appendix) because it does not account for regional mutation rates.

Original: Estimates are highly accurate when used within SPG (see Results).
Revised: Estimates are highly accurate when used within SPG (see Results).

Original: 3. E STIMATING DYNAMIC REGIONAL RATES WITH UNCERTAINTY

Original: Gaussian process (GP) regression is a powerful method to perform estimation while quantifying uncertainty.

Original: We thus pursue a GP-based strategy to estimate the unknown mean and variance of the mutation rate using the epigenetic data as covariates.

Original: In order to reduce the high-dimensional input data (see section 3.4) to a size tractable for GP models (Rasmussen & Williams, 2005), we ﬁrst reduce the dimensionality of the epigenetic feature matrix via a convolutional neural network.

Original: We chose a CNN in order to retain information about local structures effecting mutation rate within the low-dimensional embedding.

Original: We constructed a 1D CNN model with 4 residual blocks andfully-connected layers to map mutation-rate-associated local epigenetic patterns to regional mutation rates.

Original: It accepts an input matrix of size 735 epigenetic tracks by 100, where each column is the mean of R / 100 adjacent positions.

Original: The CNN non-linearly reduces the 73,500-dimension to adimensional feature vector in its last feature layer.

Original: The CNN was trained to minimize the mean squared error between observed and predicted mutation counts.

Original: Due to the interchangeable nature of the rows, the 1D kernels allow the network to identify arbitrary inter-track interactions.

Original: The ﬁnal 16-dimension feature vector was then passed as input to a sparse GP (Titsias), ﬁt to maximize the likelihood of the observed mutation counts using 2000 inducing points (Fig. 1b).

Original: While end-to-end training is possible (Bradshaw et al., 2017), we did not ﬁnd it necessary to achieve high accuracy in this particular application.

Original: We investigated numerous other methods for estimating the mean and variance of the rate parameter, but found the CNN+GP to produce the most accurate results (see

Original: Appendix).

Original: 3.3. C OMPARING TO BENCHMARK MODELS
Revised: 3.3. C OMPARING TO BENCHMARK MODELS

Original: We compared SPG to three alternative approaches that have been previously used to learn both the mean and/and variance of regional mutation patterns genome-wide.
Revised: We compared SPG to three alternative approaches that have been previously used to learn both the mean and variance of regional mutation patterns genome-wide.

Original: The alternative models are random forest (RF) regression (Polak et al., 2015), binomial regression (BR) (Bertl et al., 2018), and negative binomial regression (NBR) (Nik-Zainal et al., 2016; Martincorena et al., 2017).
Revised: The alternative models are random forest (RF) regression (Polak et al., 2015), binomial regression (BR) (Bertl et al., 2018), and negative binomial regression (NBR) (Nik-Zainal et al., 2016; Martincorena et al., 2017).

Original: For the RF, we used a jackknife method (Wager et al.)
Revised: For the RF, we used the Jackknife method (Wager et al.)

Original: to estimate the variance; this method requires
Revised: to estimate the variance; this method requires

Original: O ( n ) trees where n is the number of samples in the training set.
Revised: O ( n ) trees where n is the number of samples in the training set.

Original: BR and NBR directly specify the variance as function of the mean: BR as σ 2 R = µ − µ 2 / n and NBR as σ 2 R = µ R ( 1 + βµ R ) , where β is an overdispersion parameter.
Revised: BR and NBR directly specify the variance as function of the mean: BR as σ 2 R = µ − µ 2 / n and NBR as σ 2 R = µ R ( 1 + βµ R ) , where β is an overdispersion parameter.

Original: Benchmarking comparisons were performed on the skin melanoma, esophageal adenocarcinoma, and stomach adenocarcinoma cohorts.
Revised: Benchmarking comparisons were performed on the skin melanoma, esophageal adenocarcinoma, and stomach adenocarcinoma cohorts.

Original: 3. M ODEL TRAINING
Revised: 3. M ODEL TRAINING

Original: For every region of size R , epigenetic features were extracted into matrices of size 735 tracks bybinned position columns, where each column was the mean across R / 100 adjacent base-pairs.
Revised: For every region of size R , epigenetic features were extracted into matrices of size 735 tracks by 100 binned position columns, where each column was the mean across R / 100 adjacent base-pairs.

Original: We excluded from training regions with highly repetitive DNA sequence (<70% of 36mer sub-sequences being unique)  to ensure high data quality as in previous analyses (Polak et al., 2015).
Revised: Regions with highly repetitive DNA sequence (<70% of 36mer sub-sequences being unique) where excluded from the training set to ensure high data quality as in previous analyses (Polak et al., 2015).

Original: Before training, high-quality data regions were strictly split into train (64%), validation (16%) and test (20%) sets.
Revised: Before training, high-quality data regions were strictly split into train (64%), validation (16%) and test (20%) sets.

Original: Predictions for excluded regions and held-out test sets were obtained after model training.
Revised: Predictions for excluded regions and held-out test sets were obtained after model training.

Original: Genomewide predictions were generated using 5-fold cross-validation.
Revised: Genome-wide predictions were generated using 5-fold cross-validation.

Original: Methods designed to ingest matrices (CNN) received the full 735x100 matrices as input.
Revised: Methods designed to ingest matrices (CNN) received the full 735x100 matrices as input.

Original: Vector-based methods (RF, NBR, BR) received the 735-dimension vector of epigentic values averaged across position columns.
Revised: Vector-based methods (RF, NBR, BR) received the 735-dimension vector of epigenetic values averaged across position columns.

Original: Following previous work, we also included the expected number of mutations based on the trinucleotide composition of a region as an offset term in NBR and BR when predicting mutation counts Nik-Zainal et al.
Revised: Following previous work, we also included the expected number of mutations based on the trinucleotide composition of a region as an offset term in NBR and BR when predicting mutation counts (Nik-Zainal et al., 2016).

Original: Additional details on training (e.g. number of epochs) are in Appendix.
Revised: Additional details on training (e.g. number of epochs) are in Appendix.

Original:  I DENTIFYING GENETIC DRIVERS OF CANCER
Revised:  I DENTIFYING GENETIC DRIVERS OF CANCER

Original: Because cancer drivers reoccur across tumors, driver elements (genes, regulatory structures, and individual base-pairs) will contain an excess of mutations relative to the of expected background mutations.
Revised: Because cancer drivers reoccur across tumors, driver elements (genes, regulatory structures, and individual base-pairs) will contain an excess of mutations relative to the of expected background mutations.

Original: The SPG model provides a simple, efﬁcient, and accurate method to search for this recurrence.
Revised: The SPG model provides a simple, efﬁcient, and accurate method to search for this recurrence.

Original: We ﬁrst estimate mean and variance of the background mutation rate using the CNN+GP estimation method.
Revised: We ﬁrst estimate mean and variance of the background mutation rate using the CNN+GP estimation method.

Original: We then apply eq.
Revised: We then apply eq.

Original: (7) to search for statistical evidence that the number of observed mutations, k , exceeds expectation within every gene, known regulatory structure, andbp window in the genome by changing the set of tested positions I .
Revised: (7) to search for statistical evidence that the number of observed mutations, k , exceeds expectation within every gene, known regulatory structure, andbp window in the genome by changing the set of tested positions I .

Original: For a gene, k is the number of observed missense or nonsense mutations and I is the set of all possible mutations in the gene.
Revised: For a gene, k is the number of observed missense or nonsense mutations and I is the set of all possible mutations in the gene.

Original: For both a regulatory element and window of ﬁxed size, k is the number of mutations observed in the element / window and I is the set of all positions within the element / window.
Revised: For both a regulatory element and window of ﬁxed size, k is the number of mutations observed in the element / window and I is the set of all positions within the element / window.

Original: If an element overlaps multiple 10kb regions, we merge the mean and variance estimates for overlapped regions as described in section 2.
Revised: If an element overlaps multiple 10kb regions, we merge the mean and variance estimates for overlapped regions as described in section 2.

Original: To maintain strict train-test separation, both the rate parameters and p i are estimated excluding the element being tested.
Revised: To maintain strict train-test separation, both the rate parameters and p i are estimated excluding the element being tested.

Original: We controlled family-wise error rate at the α = 0 .
Revised: We controlled family-wise error rate at the α = 0 .

Original: 5 level using a Bonferroni correction for the total number of tests in genes, regulatory elements, or 50bp windows.
Revised: 05 level using a Bonferroni correction for the total number of tests in genes, regulatory elements, or 50bp windows.

Original: Gene information was obtained from Martincorena et al.
Revised: Gene information was obtained from Martincorena et al.

Original: and regulatory element information from Rheinbay et al.
Revised: and regulatory element information from Rheinbay et al.

Original: Driver detection was performed in all four cancer cohorts.
Revised: Driver detection was performed in all four cancer cohorts.

Original:  R ESULTS 5. A CCURACY OF REGIONAL RATE PARAMETER ESTIMATION
Revised:  R ESULTS 5. A CCURACY OF REGIONAL RATE PARAMETER ESTIMATION

Original: We ﬁrst evaluated various methods’ abilities to infer regional rate parameters, considering both new (CNN+GP) and existing (RF, NBR, BR) methods.
Revised: We ﬁrst evaluated various methods’ abilities to infer regional rate parameters, considering both new (CNN+GP) and existing (RF, NBR, BR) methods.

Original: We assessed each method’s ability to learn the expected mutation rate by directly assessing the amount of variance (Pearson R 2 ) it explained over observed mutation counts in 50kb  windows (ﬁg. 2 top), and found thethe CNN+GP estimation method performed the best, although random forest was a close second (results were similar when estimating the mean in simulated data, Appendix).
Revised: We assessed each method’s ability to learn the expected mutation rate by directly assessing the amount of variance (Pearson R 2 ) it explained over observed mutation counts in 50kb real data windows (ﬁg. 2b top), and found the CNN+GP estimation method performed the best, although random forest was a close second (results were similar when estimating the mean in simulated data; ﬁg. 2b middle).

Original: We then evaluated each method’s ability to capture the variance σ 2 R in the simulated data, quantiﬁed as the Pearson R 2 to the true variance.
Revised: We then evaluated each method’s ability to capture the variance σ 2 R in the simulated data, quantiﬁed as the Pearson R 2 to the true variance.

Original: The CNN+GP method again outperformed the others (ﬁg. 2 bottom).
Revised: The CNN+GP method again outperformed the others (ﬁg. 2b bottom).

Original: Notably, RF was unable to infer the variance beyond chance level, and thus we did not consider this method further because its inability to infer variance precludes accurate driver detection.
Revised: Notably, RF was unable to infer the variance beyond chance level, and thus we did not consider this method further because its inability to infer variance precludes accurate driver detection.

Original: We also considered other dimensionality reduction techniques, both  supervised and unsupervised, as an alternative to the CNN; no other approach achieved accuracy comparable to the CNN+GP over both mean and variance (see Appendix).
Revised: We also considered other dimensionality reduction techniques including both non-neural and neural approaches as well as supervised and unsupervised approaches, as an alternative to the CNN; no other approach achieved accuracy comparable to the CNN+GP over both mean and variance (see Appendix).

Original: Moreover, we validated the necessity of the GP by directly optimizing the CNN to predict  both parameters and found it signiﬁcantly reduced model performance (7% decrease over mutation counts and 13% over σ 2 R within 10kb windows in melanoma).
Revised: Moreover, we validated the necessity of the GP by directly optimizing the CNN to predict [Figure] both parameters and found it signiﬁcantly reduced model performance (7% decrease over mutation counts and 13% over σ 2 R within 10kb windows in melanoma) [Figure]

Original:  5. A CCURACY AND EFFICIENCY OF MUTATION RATE PREDICTION
Revised: [Table] 5. A CCURACY AND EFFICIENCY OF MUTATION RATE PREDICTION

Original: To further compared theaccuracy of SPG  to existing methods, we evaluated the accuracy and efﬁciency of each method over length scales ranging over 5 orders of magnitude (10-10 6 positions).
Revised: To further compare the SPG performance to existing methods we evaluated the accuracy and efﬁciency of each method over length scales ranging over 5 orders of magnitude (10-10 6 positions).

Original: To evaluate SPG, we estimated the background mutation rate parameters, µ R and σ 2 R , in 10kb regions genome-wide using the CNN+GP estimation strategy; we then applied the SPG distribution to estimate mutation count distributions over all other region sizes.
Revised: To evaluate SPG we estimated the background mutation rate parameters, µ R and σ 2 R , in 10kb regions genome-wide using the CNN+GP estimation strategy; we then applied the SPG distribution to estimate mutation count distributions over all other region sizes.

Original: The existing methods with reasonable performance on both mean and variance prediction (BR and NBR) were trained to directly predict the count distribution in each region for each length scale genome-wide.[Figure]
Revised: The existing methods with reasonable performance on both mean and variance prediction (BR and NBR) were trained to directly predict the count distribution in each region for each length scale genome-wide.

Original: [Table] Across all tested window sizes and cancers, SPG outperformed existing methods, with performance particularly improved in esophageal adenocarcinoma and skin melanoma (ﬁg. 3a), crucial for highaccuracy driver detection downstream.
Revised: Across all tested window sizes and cancers, SPG outperformed existing methods, with performance particularly improved in esophageal adenocarcinoma and skin melanoma (ﬁg. 3a), crucial for highaccuracy driver detection downstream.

Original: Across 1Mb windows, SPG explains > 95% of the variance in mutation density across all three cancers (ﬁg. 3a,b); this is >15% more variance than both existing methods (Fig. 3a), highlighting the ability of SPG to accurately capture regional distribution parameters and project them upwards.
Revised: Across 1Mb windows, SPG explains > 95% of the variance in mutation density across all three cancers (ﬁg. 3a,b); this is >15% more variance than both existing methods (Fig. 3a), highlighting the ability of SPG to accurately capture regional distribution parameters and project them upwards.

Original: The decrease in variance explained in smaller window sizes is expected because observed mutation counts become increasingly stochastic relative to the expected number of mutations predicted by each method.
Revised: The decrease in variance explained in smaller window sizes is expected because observed mutation counts become increasingly stochastic relative to the expected number of mutations predicted by each method.

Revised: The theoretical foundations of negative binomial regression and SPG are similar, both built upon the classical Poisson-gamma model.

Revised: SPG differs from NBR in three key ways that help explain its improved performance: 1) SPG models mutation patterns over arbitrary sets of positions enabling it to dynamically pool information across positions after a single training; in contrast, NBR operates on ﬁxed regions and must be retrained for every new region size.

Revised: 2) SPG’s variational inference method estimates the gamma parameters for each region independently; NBR estimates only the shape parameter independently for each window and uses a single scale parameter for all windows.

Revised: 3) SPG’s CNN data reduction enables non-linear mapping of spatial covariate information to mutation rate, whereas NBR can perform only linear inference and disregards the spatial organization of the genome.

Original: SPG is also the most efﬁcient method for multi-resolution search (appendix D.3).
Revised: SPG is also the most efﬁcient method for multi-resolution search (appendix D.3).

Original: Initial training of parameters using the CNN+GP method for one fold of 10kb regions required 36 minutes using 1 GPU.
Revised: Initial training of parameters using the CNN+GP method for one fold of 10kb regions required 36 minutes using 1 GPU.

Original: Projection to each additional scale using 8 CPUs required at most 4 minutes.
Revised: Projection to each additional scale using 8 CPUs required at most 4 minutes (table 1).

Original: In contrast, training time for BR and NBR increases considerably as the resolution decreases.
Revised: In contrast, training time for BR and NBR increases considerably as the resolution decreases.

Original: Performing a search across resolutions of 50bp, 100bp, 500bp, 1kb, and 10kb would require >5h for negative binomial, >2h for BR, and only 52 minutes for SPG (Appendix).
Revised: Performing a search across resolutions of 50bp, 100bp, 500bp, 1kb, and 10kb would require >5h for negative binomial, >2h for BR, and only 52 minutes for SPG (Appendix).

Original: We have also found that parameter estimation on windows as large as 100kb does not signiﬁcantly reduce accuracy across scale (Appendix), allowing
Revised: We have also found that parameter estimation on windows as large as 100kb does not signiﬁcantly reduce accuracy across scale (Appendix), allowing

Original: SPG parameter estimation in considerably shorter time (e.g. only 8 minutes for 50kb).
Revised: SPG parameter estimation in considerably shorter time (e.g. only 8 minutes for 50kb).

Original: 5. I DENTIFICATION OF CANCER DRIVER MUTATIONS
Revised: 5. I DENTIFICATION OF CANCER DRIVER MUTATIONS

Original: We leveraged SPG’s ability to model multiple resolutions to search the whole genome of each of the four cancer cohorts for gene drivers, non-coding regulatory drivers, and 50bp windows that may harbor a driver mutation.
Revised: We leveraged SPG’s ability to model multiple resolutions to search the whole genome of each of the four cancer cohorts for gene drivers, non-coding regulatory drivers, and 50bp windows that may harbor a driver mutation.

Original: All signiﬁcant results are provided as supplementary data tables.
Revised: All signiﬁcant results are provided as supplementary data tables.

Original: We compared our results to those obtained from a previous comprehensive characterization of these cohorts by Campbell et al.
Revised: We compared our results to those obtained from a previous comprehensive characterization of these cohorts by Campbell et al.

Original: (2020), who used 13 different methods to identify drivers.
Revised: (2020), who used 13 different methods to identify drivers.

Original: Our model did not have access to information about gene structure or function unlike the methods used in the previous characterization.
Revised: Our model did not have access to information about gene structure or function unlike the methods used in the previous characterization.

Original: Nonetheless, the model’s p-values were well calibrated (ﬁg. 3c), and we identiﬁed 19 genes with a signiﬁcant excess of missense or nonsense mutations.
Revised: Nonetheless, the model’s p-values were well calibrated (ﬁg. 3c), and we identiﬁed 19 genes with a signiﬁcant excess of missense or nonsense mutations.

Original: All 19 genes were previously reported as drivers by Campbell et al.
Revised: All 19 genes were previously reported as drivers by Campbell et al.

Original: We failed to detect only two known driver genes present in >5% of samples.
Revised: We failed to detect only two known driver genes present in >5% of samples.

Original: This performance is on par with state-of-the-art methods speciﬁcally designed for driver gene identiﬁcation (Rheinbay et al., 2020).
Revised: This performance is on par with state-of-the-art methods speciﬁcally designed for driver gene identiﬁcation (Rheinbay et al., 2020).

Original: When analyzing non-coding regulatory elements, SPG’s p-values were again well calibrated (ﬁg. 3c), and it identiﬁed all non-coding drivers (n=11) identiﬁed by Campbell et al.
Revised: When analyzing non-coding regulatory elements, SPG’s p-values were again well calibrated (ﬁg. 3c), and it identiﬁed all non-coding drivers (n=11) identiﬁed by Campbell et al.

Original: Moreover,
Revised: Moreover,

Original: SPG implicated several additional putative non-coding driver elements that had not been previously reported.
Revised: SPG implicated several additional putative non-coding driver elements that had not been previously reported.

Original: Examples include 1) the promoter of the gene MTERFD1 in esophageal cancer (P = 3 .
Revised: Examples include 1) the promoter of the gene MTERFD1 in esophageal cancer (P = 3 .

Original: 1 × 10 − 8 ), whose over-expression has been observed in numerous cancers, has been shown to promote cell growth, and decrease clinical survival (Zhang et al., 2014); 2) an enhancer of DHX33 in liver cancer (P = 4 .
Revised: 1 × 10 − 8 ), whose over-expression has been observed in numerous cancers, has been shown to promote cell growth, and decrease clinical survival (Zhang et al., 2014); 2) an enhancer of DHX33 in liver cancer (P = 4 .

Original: 8 × 10 − 11 ), whose over-expression has been shown to promote cancer development (Wang et al., 2019); and 3) the 5’ UTR of ERN1 in melanoma, which has been linked to cancer therapy resistance (Šušti ´ c et al., 2018).
Revised: 8 × 10 − 11 ), whose over-expression has been shown to promote cancer development (Wang et al., 2019); and 3) the 5’ UTR of ERN1 in melanoma, which has been linked to cancer therapy resistance (Šušti´c et al., 2018).

Original: Finally, we performed, to our knowledge,the ﬁrst genome-wide search for individual driver mutations.
Revised: Finally, we performed the ﬁrst, to our knowledge, genome-wide search for individual driver mutations.

Original: All signiﬁcant genic hits fell within known driver genes whose functions have been experimentally validated including TPF3 , BRAF , KRAS , PIK3CA , and CTNNB1 .
Revised: All signiﬁcant genic hits fell within known driver genes whose functions have been experimentally validated including TPF3 , BRAF , KRAS , PIK3CA , and CTNNB1 .

Original: In addition, SPG identiﬁed two recurrent mutations in the genes GPR98 and KLB that had not been previously identiﬁed in Campbell et al.
Revised: In addition, SPG identiﬁed two recurrent mutations in the genes GPR98 and KLB that had not been previously identiﬁed in Campbell et al.

Original: (2020)’s analysis of the data .
Revised: (2020)’s analysis of the data .

Original: These mutations are listed as driver mutations in the Catalogue of
Revised: These mutations are listed as driver mutations in the Catalogue of

Original: Somatic Mutations in Cancer Tate et al.
Revised: Somatic Mutations in Cancer Tate et al.

Original: SPG implicated numerous hotspots in the mostly unexplored non-coding genome, including the well-known TERT promoter mutation (ﬁg. 3d).
Revised: SPG implicated numerous hotspots in the mostly unexplored non-coding genome, including the well-known TERT promoter mutation (ﬁg. 3d).

Original: These results are promising targets for future studies of non-coding drivers in cancer cell lines and organoids.
Revised: These results are promising targets for future studies of non-coding drivers in cancer cell lines and organoids.

Original:  D ISCUSSION
Revised:  D ISCUSSION

Original: Here, we introduced an extension of theclassical Poisson-Gamma distribution to model discrete-time, integer-valued stochastic processes at multiple scales.
Revised: We introduced an extension of the Poisson-Gamma distribution to model discrete-time, integervalued stochastic processes at multiple scales.

Original: The split-Poisson-Gamme (SPG) model makes several simplifying assumptions including: 1) that the process is quasi-stationary in a small enough region; 2) events are distributed among the discrete units approximately independently; and 3) the behavior of the random variables can be captured by particular parametric distributions.
Revised: The split-Poisson-Gamma (SPG) model makes several simplifying assumptions including: 1) that the process is quasi-stationary in a small enough region; 2) events are distributed among the discrete units approximately independently; and 3) the behavior of the random variables can be captured by particular parametric distributions.

Original: While these assumptions may be restrictive, they enable the derivation of a closed-form posterior distribution.
Revised: The assumptions are necessary to derive a closed-form posterior distribution.

Original: This enables efﬁcient prediction over multiple length-scales without having to re-estimate the parameters of the model.
Revised: This enables efﬁcient prediction over multiple length-scales without having to re-estimate the model parameters.

Revised: We additionally proposed a variational inference strategy to estimate the parameters of the model using a Gaussian process.

Original: To demonstrate the utility of the SPG, we applied it to model mutation rates in cancer and identify genomic elements that drive tumor emergence.
Revised: To demonstrate the utility of the SPG, we applied it to model mutation rates in cancer and identify genomic elements that drive tumor emergence.

Original: We leveraged a convolutional neural network and 
Revised: In the case of this application, previous work has established the validity of the above assumptions, demonstrating that the mutation rate is approximately constant within 50kb regions (Rheinbay et al., 2020) and that mutations occur approximately independently given each position’s trinucloeotide context (Martincorena et al., 2017).

Original: Gaussian process to infer themodel’s parameters from high-dimensional epigenentic data, and demonstrated that the approach is more accurate than other methods on both real and synthetic data.
Revised: We leveraged a convolutional neural network to reduce the high-dimensional epigenentic data prior to GP-based variational inference, and demonstrated that the approach is more accurate than other methods on both real and synthetic data.

Original: We also demonstrated that multi-resolution prediction enables identiﬁcation of both known and novel putative drivers of cancer, including in the non-coding genome, a crucial open problem in genomics (Khurana et al., 2016; Rheinbay et al., 2020).
Revised: We also demonstrated that multi-resolution prediction enables identiﬁcation of both known and novel putative drivers of cancer, including in the non-coding genome, a crucial open problem in genomics (Khurana et al., 2016; Rheinbay et al., 2020).

Original: SPG should also be applicable in other domains.
Revised: SPG should also be applicable in other domains that model integer-valued discrete stochastic processes.

Original: Example uses include identifying hotspots of infectious disease outbreaks and detecting anomalous transaction patterns within a computer network.
Revised: Example uses could include: identifying hotspots of infectious disease outbreaks, detecting anomalous transmission patterns within a computer network, and modeling the ﬁring pattern of neurons in brain recordings.

Original: We note, however, that method(s) to infer the parameters of the model will vary depending on the application.
Revised: We note, however, that method(s) to infer the parameters of the model will vary depending on the application.

Original: Additionally, the assumptions of the model maynot be valid in particular settings.
Revised: For example, when modeling infectious disease outbreaks, regional rate parameters would reﬂect geographic infection rates while individual people would be the "positions" of the stochastic process. The task would be to identify groups of people (e.g. schools, neighborhoods, cities, etc.) with more cases than expected. A 2-dimensional autoregressive method may be a useful method to infer regional infection rates while an individual’s demographics could provide the clustering criteria to estimate p i .

Original: In such cases, the model could be adapted but would likely preclude a closed-form solution.
Revised: Such an application could be particularly useful for early identiﬁcation of hotspot outbreaks of COVID-19, enabling cities, states, and countries to quickly implement containment strategies.

Original: Nonetheless, as collections of time- andspace-varying data grow increasingly large, we expectmethods that  model non-stationarity will become increasingly important.
Revised: Similar adaptions to parameter estimation schemes are possible for other applications, and we expect that methods which model non-stationarity will become increasingly important as collections of time- and space-varying data grow increasingly large.

Original: A A PPENDIX
Revised: A A PPENDIX

Original: In this appendix we provide detailed information on:
Revised: In this appendix, we provide detailed information on:

Original: The data used in this work including its origin and all preprocessing steps. 2.
Revised: The data used in this work including its origin and all preprocessing steps. 2.

Original: Additional method details including: • A derivation of the closed-form marginal distribution of the graphical model presented in main text.
Revised: Additional method details including: • A derivation of the closed-form marginal distribution of the graphical model presented in main text.

Original: • Architecture and training details of all models.
Revised: • Architecture and training details of all models.

Original:  How the genome-wide search for driver mutations was performed.
Revised: • How the genome-wide search for driver mutations was performed.

Original: The appendix also includes an analysis of the sensitivity of negative binomial regression to detect well-known drivers genome-wide and additional ﬁgures that provide context to results presented in the main paper.
Revised: The appendix also includes an analysis of the sensitivity of negative binomial regression to detect well-known drivers genome-wide and additional ﬁgures that provide context to results presented in the main paper.

Original: B D ATA B. E PIGENETIC TRACKS
Revised: B D ATA B. E PIGENETIC TRACKS

Original: We obtained 733 − log 10 (P-value) chromatin tracks representing the epigenetic organization of 111 human tissues from Roadmap Epigenomics Roadmap Epigenomics Consortium et al.
Revised: We obtained 733 − log 10 (P-value) chromatin tracks representing the epigenetic organization of 111 human tissues from Roadmap Epigenomics Roadmap Epigenomics Consortium et al.

Original: (2015) (see appendix table “predictor_track_descriptions.csv”).
Revised: (see Appendix table “predictor_track_descriptions.csv”).

Original: These tracks measure the abundance of a particular chromatin mark genome-wide, with smaller (more signiﬁcant) p-values reﬂecting a greater abundance of the chromatin mark at a genomic position.
Revised: These tracks measure the abundance of a particular chromatin mark genome-wide, with smaller (more signiﬁcant) p-values reﬂecting a greater abundance of the chromatin mark at a genomic position.

Original: Chromatin marks are chemical modiﬁcations of histones, the proteins used to package DNA within a cell.
Revised: Chromatin marks are chemical modiﬁcations of histones, the proteins used to package DNA within a cell.

Original: We additionally obtained 10 replication timing tracks from the ENCODE consortium.
Revised: We additionally obtained 10 replication timing tracks from the ENCODE consortium.

Original: Replication timing assays measure the relative time at which each position in the genome is replicated during cell division.
Revised: Replication timing assays measure the relative time at which each position in the genome is replicated during cell division.

Original: For non-overlapping regions
Revised: For non-overlapping regions

Original: R of predeﬁned size and location (see main text for more details), we extracted the signal for each epigenetic track using 100 bins per region with pybbi Abdennur (2018).
Revised: R of predeﬁned size and location (see main text for more details), we extracted the signal for each epigenetic track using 100 bins per region with pybbi Abdennur (2018).

Original: We additionally calculated the average nucleotide content in each window by assigning each nucleotide a numeric value between 1 and 5 and taking the average across a bin (N [unspeciﬁed nucleotide] = 1, A = 2, C = 3, G = 4, T = 5), and we calculated the GC content as the percent of G and C nucleotides in a bin, resulting in a total of 735 epigenome tracks per region.
Revised: We additionally calculated the average nucleotide content in each window by assigning each nucleotide a numeric value between 1 and 5 and taking the average across a bin (N [unspeciﬁed nucleotide] = 1, A = 2, C = 3, G = 4, T = 5), and we calculated the GC content as the percent of G and C nucleotides in a bin, resulting in a total of 735 epigenome tracks per region.

Original: The mean values for each region were calculated as themean chromatin signal for each track in the region.
Revised: The mean values for each region were calculated as the

Original:  B. M UTATION COUNT DATA
Revised: . B. M UTATION COUNT DATA

Original: We downloaded somatic single-base substitution mutations identiﬁed in the ICGC subset of the
Revised: We downloaded somatic single-base substitution mutations identiﬁed in the ICGC subset of the

Original: Pan-Cancer Analysis of Whole Genomes Consortium cohorts of esophageal adenocarcinoma, skin melanoma, stomach adenocarcinoma, and liver hepatocellular carcinoma.
Revised: Pan-Cancer Analysis of Whole Genomes Consortium cohorts of esophageal adenocarcinoma, skin melanoma, stomach adenocarcinoma, and liver hepatocellular carcinoma.

Original: These data are freely available for download from the International Cancer Genomics Consortium data portal (see ﬁg. 4).
Revised: These data are freely available for download from the International Cancer Genomics Consortium data portal (see ﬁg. 4).

Original: We excluded mutations on the sex chromosomes (X and Y) because males and females carry different sets of these chromosomes, leading to differential mutation patterns.
Revised: We excluded mutations on the sex chromosomes (X and Y) because males and females carry different sets of these chromosomes, leading to differential mutation patterns.

Original: We summarized the data as mutation counts per window for window sizes of 50bp, 100bp, 500bp, 1kb, 5kb, 10kb, 25kb, 50kb, 100kb, and 1Mb.
Revised: We summarized the data as mutation counts per window for window sizes of 50bp, 100bp, 500bp, 1kb, 5kb, 10kb, 25kb, 50kb, 100kb, and 1Mb.

Original: B. R ESTRICTION TO REGIONS OF HIGH MAPPABILITY
Revised: B. R ESTRICTION TO REGIONS OF HIGH MAPPABILITY

Original: High-throughput genome sequencing works by randomly reading millions of short sequences of nucleotides (36-150 bases in length) from a target genome.
Revised: High-throughput genome sequencing works by randomly reading millions of short sequences of nucleotides (36-150 bases in length) from a target genome.

Original: These “reads” are then mapped to the human reference genome to reconstruct the target.
Revised: These “reads” are then mapped to the human reference genome to reconstruct the target.

Original: A challenge is that short sequences of k nucleotides (kmers) can occur multiple times in the genome.
Revised: A challenge is that short sequences of k nucleotides (kmers) can occur multiple times in the genome.

Original: This results in ambiguous mappings for some reads and thus a degradation of data quality in regions composed of many kmers that occur multiple times across the genome.
Revised: This results in ambiguous mappings for some reads and thus a degradation of data quality in regions composed of many kmers that occur multiple times across the genome.

Original: Following previous work Polak et al.
Revised: Following previous work Polak et al.

Original: , we removed regions of the genome with low quality data by calculating a mappability score for each region.
Revised: , we removed regions of the genome with low quality data by calculating a mappability score for each region.

Original: Mappability scores reﬂect how many times a particular kmer occurs in the genome and have been pre-computed for the human reference genome GRCh37.
Revised: Mappability scores reﬂect how many times a particular kmer occurs in the genome and have been pre-computed for the human reference genome GRCh37.

Original: We required that a region’s average mappability score based on 36mers (e.g. average across all sequences of 36 nucleotides in the region) be >70%, reﬂecting that all 36mers in the region be >70% unique.
Revised: We required that a region’s average mappability score based on 36mers (e.g. average across all sequences of 36 nucleotides in the region) be >70%, reﬂecting that all 36mers in the region be >70% unique.

Original: The majority of the genome passed this threshold; for 10kb regions, for example, >75% of the genome passed this threshold.
Revised: The majority of the genome passed this threshold; for 10kb regions, for example, >75% of the genome passed this threshold.

Original: We chose to measure mappability with 36mers because this was the length of read used to generated the Roadmap Epigenomics sequencing data.
Revised: We chose to measure mappability with 36mers because this was the length of read used to generate the Roadmap Epigenomics sequencing data.

Original: [Figure]
Revised: [Figure]

Original: B. S YNTHETIC DATA SIMULATION
Revised: B. S YNTHETIC DATA SIMULATION

Original: We generated synthetic datasets for each of the cancers in order to have datasets with known mean and variance rate parameters.
Revised: We generated synthetic datasets for each of the cancers in order to have datasets with known mean and variance rate parameters.

Original: To generate the datasets [Figure] we used a k-nearest-neighbors strategy to identify the 500 nearest neighbors for each region.
Revised: To generate the datasets, we used a k-nearest-neighbors strategy to identify the 500 nearest neighbors for each region.

Original: The mean and variance for that region were then taken to be the empirical mean and variance calculated from the 500 nearest neighbors.
Revised: The mean and variance for that region were then taken to be the empirical mean and variance calculated from the 500 nearest neighbors.

Original: The number of "observed" mutation was then randomly sampled from a binomial deﬁned by the mean and variance parameters.
Revised: The number of "observed" mutations was then randomly sampled from a binomial deﬁned by the mean and variance parameters.

Original: It is important to note that these datasets are purely derived for the purpose of comparing methods over datasets with a known ground-truth.
Revised: It is important to note that these datasets are purely derived for the purpose of comparing methods over datasets with a known ground-truth.

Original: They do not reﬂect mutation patterns in the real datasets.
Revised: They do not reﬂect mutation patterns in the real datasets.

Original: The speciﬁc steps to generate the simulated data were:
Revised: The speciﬁc steps to generate the simulated data were:

Original: Generate vectors of the mean values for each of the 735 tracks (733 epigenetic tracks, GC content track, and average nucleotide content track) in 50kb regions of the genome with 36mer uniqueness >70%.
Revised: Generate vectors of the mean values for each of the 735 tracks (733 epigenetic tracks, GC content track, and average nucleotide content track) in 50kb regions of the genome with 36mer uniqueness >70%.

Original: Perform ordinary least-squares (OLS) regression of the mean vectors against the observed number of mutations in each 50kb window for that cancer.
Revised: Perform ordinary least-squares (OLS) regression of the mean vectors against the observed number of mutations in each 50kb window for that cancer.

Original: Scale each value in the feature vectors by its corresponding coefﬁcient from OLS and compress the weighted mean vectors to 50 components using Principal Components Analysis (capturing >94% of the variance for each cancer).
Revised: Scale each value in the feature vectors by its corresponding coefﬁcient from OLS and compress the weighted mean vectors to 50 components using Principal Components Analysis (capturing >94% of the variance for each cancer).

Original: For each region R , perform k-nearest-neighbor clustering with Euclidean distance to identify its 500 nearest neighbors in the PC space.
Revised: For each region R , perform k-nearest-neighbor clustering with Euclidean distance to identify its 500 nearest neighbors in the PC space.

Original: Deﬁne the mean µ R and variance σ 2 R of the mutation rate in R to be the mean and variance of the KNN cluster.
Revised: Deﬁne the mean µ R and variance σ 2 R of the mutation rate in R to be the mean and variance of the KNN cluster.

Original: For region R , randomly draw a new “observed” number of mutations from a negative binomial distribution deﬁned using the associated mean and variance.
Revised: For region R , randomly draw a new “observed” number of mutations from a negative binomial distribution deﬁned using the associated mean and variance.

Original: Speciﬁcally, X R ∼ NB ( α , 1 / ( θ + 1 )) where α = µ 2 R / σ 2 R
Revised: Speciﬁcally, X R ∼ NB ( α , 1 / ( θ + 1 )) where α = µ 2 R / σ 2 R

Original: and θ =
Revised: and θ =

Original: σ 2 R / µ R
Revised: σ 2 R / µ R

Revised: We created two versions of the simulated data, one in which all regions in the genome were used to estimate the rate parameters and one in which rate parameters were estimated separately within independent train and test subsets.

Revised: Results were qualitatively indistinguishable.

Original: C M ETHODS C. G RAPHICAL MODEL
Revised: C M ETHODS C. G RAPHICAL MODEL

Original: Here we derive the closed form negative binomial distribution presented in the main text as the graphical model marginal distribution over events at some unit i in a region R .
Revised: Here we derive the closed form negative binomial distribution presented in the main text as the graphical model marginal distribution over events at some unit i in a region R .

Original: We use the following notation:
Revised: We use the following notation:

Original: • M i : # mutations observed at pos i (observed) • p
Revised: • M i : # mutations observed at pos i (observed) • p

Original: i : genome-wide probability of observing a mutation at the nucleotide context of i (inferred) • ˜ p
Revised: i : genome-wide probability of observing a mutation at the nucleotide context of i (inferred) • ˜ p

Original: i : normalized probability of observing a mutation at i in region R (inferred) • λ R : the background mutation rate in region R (unobserved) • X R : # background mutations in region R (unobserved) • µ R : the expected background mutation rate in region R (inferred) • σ 2 R : the variance of background mutation rate in region R (inferred).
Revised: i : normalized probability of observing a mutation at i in region R (inferred) • λ R : the background mutation rate in region R (unobserved) • X R : # background mutations in region R (unobserved) • µ R : the expected background mutation rate in region R (inferred) • σ 2 R : the variance of background mutation rate in region R (inferred).

Original: • η R : covariates associated with the behavior of the stochastic process within R (observed)
Revised: • η R : covariates associated with the behavior of the stochastic process within R (observed)

Original: As presented in the main text and main Figure 1, the graphical model implies the factorization 
Revised: As presented in the main text and main Figure 1, the graphical model implies the factorization Pr

Original: [Equation]
Revised: [Equation]

Original: where
Revised: where

Original: [Equation]
Revised: [Equation]

Original: Since η R is a given in each equation, we suppress it for notational ease.
Revised: Since η R is a given in each equation, we suppress it for notational ease.

Original: To marginalize out X R , we note that
Revised: To marginalize out X R , we note that

Original: [Equation]
Revised: [Equation]

Original: is equivalent to a split Poisson process (Gallager, 2013).
Revised: is equivalent to a split Poisson process (Gallager, 2013).

Original: Thus
Revised: Thus

Original: [Equation]
Revised: [Equation]

Original: We now marginalize out the unknown rate parameter λ R .
Revised: We now marginalize out the unknown rate parameter λ R .

Original: [Equation]
Revised: [Equation]

Original: Making the substitution t = λ ( ˜ p
Revised: Making the substitution t = λ ( ˜ p

Original: i + 1 / θ R ) and noting that the resulting integrand is an unnormalized gamma distribution, we have:
Revised: i + 1 / θ R ) and noting that the resulting integrand is an unnormalized gamma distribution [Equation] we have:

Original: [Equation]
Revised: [Equation]

Revised: C. O VERVIEW OF PARAMETER ESTIMATION PROCEDURE

Revised: Estimation of regional rate parameters : As training data, we use a set of input matrices { η R ; R ∈ T } and associated mutation counts { X R ; R ∈ T } .

Revised: First, a CNN is trained to take η R as input and predict

Revised: X R as output, using mean squared error loss.

Revised: The ﬁnal 16-dimension feature vector of the trained

Revised: CNN is then used as input to train a Gaussian process to predict the mutation count X R and the associated estimation uncertainty by maximizing the likelihood of the observed data.

Revised: The mean and variance output by the GP were used as estimates for µ R and σ 2 R .

Revised: Estimation of time-averaged event probabilities : the time-average probability of an event at p i was estimated based on it’s trinucleotide composition, n , t , n (cid:48) where n is the nucleotide at i − 1, t is the nucleotide at i and n (cid:48) is the nucleotide at i + 1 in the reference genome.

Revised: We ﬁrst counted every occurrence of n , t , n (cid:48) in the human genome and then counted the number of times the middle nucleotide of the 3mer was mutated across the genome.

Revised: The maximum likelihood estimate of p i is then the ratio of the number of observed mutations of the 3mer divided by the total occurrences of the 3mer.

Original: C. R EGIONAL PARAMETERS ESTIMATION METHODS
Revised: C. R EGIONAL PARAMETERS ESTIMATION METHODS

Original: To compute a model’s R 2 accuracy to µ R and σ 2 R for regions R of size S , the genome was divided into non-overlapping contiguous segments of size S .
Revised: To compute a model’s R 2 accuracy to µ R and σ 2 R for regions R of size S , the genome was divided into non-overlapping contiguous segments of size S .

Original: To assure high data quality, any region with mappability score < 70% was excluded from further analysis.
Revised: To assure high data quality, any region with mappability score < 70% was excluded from further analysis.

Original: The remaining windows (accounting for more than 75% of the genome) were randomly divided into train and test sets in an 80–20 split respectively.
Revised: The remaining windows (accounting for more than 75% of the genome) were randomly divided into train and test sets in an 80–20 split respectively.

Original: The test set was held-out and served solely for evaluation purposes.
Revised: The test set was held-out and served solely for evaluation purposes.

Original: The train set was then divided into train and validation sets by another 80–20 split respectively (train set = 64%, validation = 16%, and test =
Revised: The train set was then divided into train and validation sets by another 80–20 split respectively (train set = 64%, validation = 16%, and test =

Original: 20% of the considered regions with mappability score < 70%, see appendix B.3).
Revised: 20% of the considered regions with mappability score < 70%, see appendix B.3).

Original: C.2. G AUSSIAN PROCESS FEATURE VECTOR GENERATION
Revised: C.3. G AUSSIAN PROCESS FEATURE VECTOR GENERATION

Original: All networks were independently trained for 20 epochs with a batch size of 128 samples and using the Adam optimizer to minimize mean squared error loss to either the true mutation count (CNN and
Revised: All networks were independently trained for 20 epochs with a batch size of 128 samples and using the Adam optimizer to minimize mean squared error loss to either the true mutation count (CNN and

Original: FCNN) or input tensor (AE).
Revised: FCNN) or input tensor (AE).

Original: After training the model parameters using the train set, predictions over the held-out test set were computed by 1) extracting the last 16-dimensional feature layer (middle feature layer for AE) for all sets over the best performing model over the validation set across all epochs (according to the validation accuracy); 2) training multiple GPs (typically 10) to predict mutation counts using the 16 dimension feature vectors of the train set as input (see appendix C.2.
Revised: After training the model parameters using the train set, predictions over the held-out test set were computed by 1) extracting the last 16-dimensional feature layer (middle feature layer for AE) for all sets over the best performing model over the validation set across all epochs (according to the validation accuracy); 2) training multiple GPs (typically 10) to predict mutation counts using the 16 dimension feature vectors of the train set as input (see appendix C.3.

Original: for details); 3) taking the mean µ R and σ 2 R of all 10 runs over the test set as the ensemble prediction of the model.
Revised: for details); 3) taking the mean µ R and σ 2 R of all 10 runs over the test set as the ensemble prediction of the model.

Original: All neural network models were implemented in Pytorch Paszke et al.
Revised: All neural network models were implemented in Pytorch Paszke et al.

Original: Convolutional neural network (CNN): The CNN contains 4 convolutional blocks withbatch normalized convolutional layers and ReLU activation.
Revised: Convolutional neural network (CNN): The CNN contains 4 convolutional blocks withbatch normalized convolutional layers and ReLU activation.

Original: The ﬁrst block transformed the input tesor from 735 × 100 to 256 × 50 with 256 channels and a double stride.
Revised: The ﬁrst block transformed the input tesor from 735 × 100 to 256 × 50 with 256 channels and a double stride.

Original: The other blocks are ResNet-style residual blocks that maintain their input dimension to facilitate residual connections, with 256, 512, and 1024 channels respectively.
Revised: The other blocks are ResNet-style residual blocks that maintain their input dimension to facilitate residual connections, with 256, 512, and 1024 channels respectively.

Original: Between each of theresidual blocks there is a double stride (ReLU activated and batch normalized) convolutional layer, which divide the tensor length by two and doubles its height with additional channels.
Revised: Between each of theresidual blocks there is a double stride (ReLU activated and batch normalized) convolutional layer, which divides the tensor length by two and doubles its height with additional channels.

Original: The output of the last residual block is ﬂattened and passed through 3 fully-connected layers.
Revised: The output of the last residual block is ﬂattened and passed through 3 fully-connected layers.

Original: The ﬁrst two are ReLU activated and reduce the dimensionality of the tensor to 128 anddimensions respectively.
Revised: The ﬁrst two are ReLU activated and reduce the dimensionality of the tensor to 128 anddimensions respectively.

Original: The last uses linear functions to reduce the tensor to a single cell holding the output of the regression.
Revised: The last uses linear functions to reduce the tensor to a single cell holding the output of the regression.

Original: This forces a linear relation between the regression output and the last feature layer, thus simplifying the function the GP needs to learn, which we found empirically improves the GP’s accuracy.
Revised: This forces a linear relation between the regression output and the last feature layer, thus simplifying the function the GP needs to learn, which we found empirically improves the GP’s accuracy.

Original: Fully-connected neural network (FCNN): The FCNN has an architecture similar to the CNN’s 3 fully-connected layers but with an input space of the mean epigenetic vector ( dimensions).
Revised: Fully-connected neural network (FCNN): The FCNN has an architecture similar to the CNN’s 3 fully-connected layers but with an input space of the mean epigenetic vector ( dimensions).

Original: Thus, the FCNN is computationally similar to the CNN, but operates on the mean vector instead of the full matrix as an input.
Revised: Thus, the FCNN is computationally similar to the CNN, but operates on the mean vector instead of the full matrix as an input.

Original: The FCNN is designed to demonstrate maximum performance possible when reducing the input tensor to an averaged feature vector.
Revised: The FCNN is designed to demonstrate maximum performance possible when reducing the input tensor to an averaged feature vector.

Original: Autoencoder neural network (AE): The encoder of the AE used the same architecture as the CNN, excluding the last linear fully connected layer.
Revised: Autoencoder neural network (AE): The encoder of the AE used the same architecture as the CNN, excluding the last linear fully connected layer.

Original: The decoder has a mirror architecture with the same number of parameters but differs in the internal design of the convolutional blocks.
Revised: The decoder has a mirror architecture with the same number of parameters but differs in the internal design of the convolutional blocks.

Original: Convolutional layers were replaced by 1D transpose convolutional layers with no batch normalization and no residual connections.
Revised: Convolutional layers were replaced by 1D transpose convolutional layers with no batch normalization and no residual connections.

Original: The AE was designed to demonstrate the predictive power of a feature embedding that was not optimized to a speciﬁc task but produced in a way comparable to the CNN.
Revised: The AE was designed to demonstrate the predictive power of a feature embedding that was not optimized to a speciﬁc task but produced in a way comparable to the CNN.

Original: Other dimensionality reduction methods: PCA was computed using the Python Scikitlearn package with default settings and UMAP was computed via Python’s umap-learn package McInnes et al.
Revised: Other dimensionality reduction methods: PCA was computed using the Python Scikitlearn package with default settings and UMAP was computed via Python’s umap-learn package McInnes et al.

Original: (2018) with 20 nearest neighbours and Euclidean distance.
Revised: (2018) with 20 nearest neighbours and Euclidean distance.

Original: Both methods were computed over the entire training set (80%) with no validation set and reduced the mean epigenetic vector dimensionality (735 dimensions) to 16, just like all other models.
Revised: Both methods were computed over the entire training set (80%) with no validation set and reduced the mean epigenetic vector dimensionality (735 dimensions) to 16, just like all other models.

Original: Prior to processing, we log-transformed the epigenetic data as we found this improved prediction accuracy downstream.
Revised: Prior to processing, we log-transformed the epigenetic data as we found this improved prediction accuracy downstream.

Original: C.2. G AUSSIAN PROCESS
Revised: C.3. G AUSSIAN PROCESS

Original: We implemented a sparse, inducing-point Gaussian process Titsias with a radial basis function kernel using Python’s GPyTorch package Gardner et al.
Revised: We implemented a sparse, inducing-point Gaussian process Titsias with a radial basis function kernel using Python’s GPyTorch package Gardner et al.

Original: The GP was optimized with 2000 inducing points using the Adam optimizer for 100 steps.
Revised: The GP was optimized with 2000 inducing points using the Adam optimizer for 100 steps.

Original: All features were mean-centered and standardized to unit variance prior to training.
Revised: All features were mean-centered and standardized to unit variance prior to training.

Original: For each dataset, we ran the GP ten independent times and calculated the ensemble mean of the mean and variance predictions from each of the individual runs.
Revised: For each dataset, we ran the GP ten independent times and calculated the ensemble mean of the mean and variance predictions from each of the individual runs.

Original: We took these ensemble predictions as the mean and variance for each region.
Revised: We took these ensemble predictions as the mean and variance for each region.

Original: C.2. A LTERNATIVE MODELS
Revised: C.3. A LTERNATIVE MODELS

Original: We implemented previously proposed alternative methods Polak et al. (2015); Nik-Zainal et al.
Revised: We implemented previously proposed alternative methods Polak et al. (2015); Nik-Zainal et al.

Original: Martincorena et al.
Revised: Martincorena et al.

Original: (2017) for the estimation of µ R and σ 2 R without the use of GP.
Revised: (2017) for the estimation of µ R and σ 2 R without the use of GP.

Original: These methods use the mean epigenetic vector as an input.
Revised: These methods use the mean epigenetic vector as an input.

Original: Random forest (RF): RF regression was implemented via the Ensemble Methods module in the Python Scikit-learn package, with a maximum depth of 50 trees.
Revised: Random forest (RF): RF regression was implemented via the Ensemble Methods module in the Python Scikit-learn package, with a maximum depth of 50 trees.

Original: Since RF does not directly compute a variance, we implemented the Jackknife method as described in Wager et al.
Revised: Since RF does not directly compute a variance, we implemented the Jackknife method as described in Wager et al.

Original: (we have compared our implementation to Polimis et al.
Revised: (we have compared our implementation to Polimis et al.

Original: and found them highly correlated).
Revised: and found them highly correlated).

Original: Wager et al.
Revised: Wager et al.

Original: suggests that the number of estimators, i.e., trees, must be linearly related to the number of samples to obtain reasonable estimates of the variance.
Revised: suggests that the number of estimators, i.e., trees, must be linearly related to the number of samples to obtain reasonable estimates of the variance.

Original: We chose to have one tenth as many estimators as samples in an attempt to keep running time within reasonable limit for datasets of smaller region sizes.
Revised: We chose to have one tenth as many estimators as samples in an attempt to keep running time within reasonable limit for datasets of smaller region sizes.

Original: Even so, for 10kb regions (containing approximately 300K regions), RF required >24 hours to train. 
Revised: Even so, for 10kb regions (containing approximately 300K regions), RF required >24 hours to train. Negative binomial regression (NBR): As

Original: Negative binomial regression (NBR): As described in section 3.3.2 of the main text, NBR directly speciﬁes the variance as σ 2 R = µ R ( 1 + βµ R ) , where β is an overdispersion parameter.
Revised: described in section 3.3.2 of the main text, NBR directly speciﬁes the variance as σ 2 R = µ R ( 1 + βµ R ) , where β is an overdispersion parameter.

Original: When β = 0 NBR reduces to Poisson regression, also widely used in the community.
Revised: When β = 0 NBR reduces to Poisson regression, also widely used in the community.

Original: NBR was implemented via the discrete module in the Python statsmodels package Seabold & Perktold (2010) with the Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and 1k maximum iterations.
Revised: NBR was implemented via the discrete module in the Python statsmodels package Seabold & Perktold (2010) with the Broyden–Fletcher–Goldfarb–Shanno optimization algorithm and 1k maximum iterations.

Original: Epigenetic predictors were log-transformed and reduced toprinciple components, following the ﬁeld-standard Martincorena et al.
Revised: Epigenetic predictors were log-transformed and reduced toprinciple components, following the ﬁeld-standard Martincorena et al.

Original: (2017) in both train and test sets.
Revised: (2017) in both train and test sets.

Original: When used to compare against the GM we also included the expected number of mutations based on the sequence context model (see main paper section 3.2) as an exposure term in the model as in previous work Nik-Zainal et al.
Revised: When used to compare against the GM we also included the expected number of mutations based on the sequence context model (see main paper section 3.2) as an exposure term in the model as in previous work Nik-Zainal et al.

Original: ; Martincorena et al.
Revised: ; Martincorena et al.

Original: Binomial regression (BR): Following a previous study Bertl et al.
Revised: Binomial regression (BR) : Following a previous study Bertl et al.

Original: (2018) that suggested multinomial regression to model multiple types of mutations, we also considered binomial regression (as the binary version of multinomial regression applicable to our simple counts data) as a method to model mutation rates at high resolution.
Revised: (2018) that suggested multinomial regression to model multiple types of mutations, we also considered binomial regression (as the binary version of multinomial regression applicable to our simple counts data) as a method to model mutation rates at high resolution.

Original: BR was implemented via the generalized linear module in the Python statsmodels package Seabold & Perktold (2010).
Revised: BR was implemented via the generalized linear module in the Python statsmodels package Seabold & Perktold (2010).

Original: As in previous work Nik-Zainal et al.
Revised: As in previous work Nik-Zainal et al.

Original: ; Martincorena et al.
Revised: ; Martincorena et al.

Original: , we included the expected number of mutations based on the sequence context model (see main paper section 3. ) as an exposure term in the model.
Revised: (2017), we included the expected number of mutations based on the sequence context model (see main paper section 3.2) as an exposure term in the model.

Original: As with NBR, the epigenetic predictors were log-transformed and reduced to 20 principle components for both train and test sets following state-of-the-art recommendations Martincorena et al.
Revised: As with NBR, the epigenetic predictors were log-transformed and reduced to 20 principle components for both train and test sets following state-of-the-art recommendations Martincorena et al.

Original: C. E MPIRICAL VARIANCE ESTIMATION
Revised: C. E MPIRICAL VARIANCE ESTIMATION

Original: For real data, the true variance in mutation counts of a region is unknown.
Revised: For real data, the true variance in mutation counts of a region is unknown.

Original: Thus to estimate variance empirically for a given model, we used the following approach:
Revised: Thus to estimate variance empirically for a given model, we used the following approach

Original: For a region in the test set, perform k-nearest neighbors clustering with Euclidean distance to identify the 500 regions in the train set that are most similar to the region of interest based on the model’s feature embedding.
Revised: For a region in the test set, perform k-nearest neighbors clustering with Euclidean distance to identify the 500 regions in the train set that are most similar to the region of interest based on the model’s feature embedding.

Original: For all models, a feature embedding of 16 dimensions was used.
Revised: For all models, a feature embedding of 16 dimensions was used.

Original: Calculate the empirical variance as the variance of the KNN cluster.
Revised: Calculate the empirical variance as the variance of the KNN cluster.

Original: Since feature embeddings are model-speciﬁc, we calculated an empirical variance estimate per model.
Revised: Since feature embeddings are model-speciﬁc, we calculated an empirical variance estimate per model.

Original: The feature-vector embeddings for models speciﬁed in section C.2.1 were the feature vectors used as input to the GP.
Revised: The feature-vector embeddings for models speciﬁed in section C.3.1 were the feature vectors used as input to the GP.

Original: Models speciﬁed in section C.2.3 do not create or require comparable feature vectors and therefore were not considered in the main paper results.
Revised: Models speciﬁed in section C.3.3 do not create or require comparable feature vectors and therefore were not considered in the main paper results.

Original: However, to measure the ability of these methods to estimate empirical variance (Fig. 7), we computed their feature vectors by 1) taking the dot product of the model parameters and the input data mean vectors and 2) reduced these scaled vectors to 16 dimensions via a PCA reduction (explaining 80%-95% of the variance across the different region scales).
Revised: However, to measure the ability of these methods to estimate empirical variance (Fig. 7), we computed their feature vectors by 1) taking the dot product of the model parameters and the input data mean vectors and 2) reduced these scaled vectors to 16 dimensions via a PCA reduction (explaining 80%-95% of the variance across the different region scales).

Original: For RF, we took the model parameters to be the feature importance weights derived from the trained forest and for NBR, we used the model coefﬁcients as the parameters.
Revised: For RF, we took the model parameters to be the feature importance weights derived from the trained forest and for NBR, we used the model coefﬁcients as the parameters.

Original: C. P ERFORMING A GENOME - WIDE SEARCH FOR CANCER DRIVER MUTATIONS
Revised: C

Revised: P ERFORMING A GENOME - WIDE SEARCH FOR CANCER DRIVER MUTATIONS

Original: For each cancer, the background mutation rate parameters were estimated across the genome using 5-fold cross validation in 10kb, 25kb and 50kb regions.
Revised: For each cancer, the background mutation rate parameters were estimated across the genome using 5-fold cross validation in 10kb, 25kb and 50kb regions.

Original: While the model is robust to choice of 10kb, 25kb or 50kb region size (ﬁg. 5), the 25kb and 50kb models include some additional regions of the genome due to the the mappability threshold (see section B.3).
Revised: While the model is robust to choice of 10kb, 25kb or 50kb region size (ﬁg. 5), the 25kb and 50kb models include some additional regions of the genome due to the the mappability threshold (see section B.3).

Original: To analyze the largest possible subset of the genome, we preformed our analysis iteratively: we ﬁrst searched for drivers using regions accessible via the 10kb model; we then searched additional regions not accessible by the 10kb model in the 25kb model and then in the 50kb model.
Revised: To analyze the largest possible subset of the genome, we performed our analysis iteratively: we ﬁrst searched for drivers using regions accessible via the 10kb model; we then searched additional regions not accessible by the 10kb model in the 25kb model and then in the 50kb model.

Original: To search for drivers, we applied our probabilistic model to estimate the mutation count distributions in 50bp regions across the genome, and we then searched for 50bp regions with signiﬁcantly more observed mutations than expected under the null distribution of our model.
Revised: To search for drivers, we applied our probabilistic model to estimate the mutation count distributions in 50bp regions across the genome, and we then searched for 50bp regions with signiﬁcantly more observed mutations than expected under the null distribution of our model.

Original: We controlled false-discovery rate at the 0.05 level using a
Revised: We controlled false-discovery rate at the 0.05 level using a

Original: Bonferroni-corrected p-value threshold of P<1e-9.
Revised: Bonferroni-corrected p-value threshold of P<1e-9.

Original: To compare our hits with known cancer drivers, we tabulated the recurrent driver mutations reported by PCAWG that were present in our dataset, including in the TERT promoter , a well known noncoding driver.
Revised: To compare our hits with known cancer drivers, we tabulated the recurrent driver mutations reported by PCAWG that were present in our dataset, including in the TERT promoter , a well known noncoding driver.

Original: While most recurrent driver mutations are activating mutations (e.g. cause a gain of cellular function), we also found recurrent mutations in the tumor suppressor genes TP53 and
Revised: While most recurrent driver mutations are activating mutations (e.g. cause a gain of cellular function), we also found recurrent mutations in the tumor suppressor genes TP53 and

Original: SMAD4 .
Revised: SMAD4 .

Original: Recurrent mutations in a single position are far less likely in tumor suppressor genes because any deleterious mutation can act as a potential cancer-causing mutation.
Revised: Recurrent mutations in a single position are far less likely in tumor suppressor genes because any deleterious mutation can act as a potential cancer-causing mutation.

Original: For example, TP53 hadgenome-wide signiﬁcant 50bp regions, consistent with its status as a crucial tumor suppressor that can be knocked-out with many different mutations (see table 2).
Revised: For example, TP53 hadgenome-wide signiﬁcant 50bp regions, consistent with its status as a crucial tumor suppressor that can be knocked-out with many different mutations (see table 2).

Original: Methods specialized to discover driver genes are necessary to ﬁnd tumor suppressor genes in general Lawrence et al.
Revised: Methods specialized to discover driver genes are necessary to ﬁnd tumor suppressor genes in general Lawrence et al.

Original: ; Mularoni et al.
Revised: ; Mularoni et al.

Original: ; Martincorena et al.
Revised: ; Martincorena et al.

Original: [Figure]
Revised: [Figure]

Original: C. E NVIRONMENT AND COMPUTE TIME
Revised: C. E NVIRONMENT AND COMPUTE TIME

Original: A benchmark run at 10kb scale with 10 GP reruns takes 2-3 hours on a single 24 Gb Nvidia RTX
Revised: A benchmark run at 10kb scale with 10 GP reruns takes 2-3 hours on a single 24 Gb Nvidia RTX

Original: GPU, with 8 CPU cores and 756GB RAM.
Revised: GPU, with 8 CPU cores and 756GB RAM.

Original: Thus, a full 5-fold of the entire genome takes 10-15 hours.
Revised: Thus, a full 5-fold of the entire genome takes 10-15 hours.

Original: Due to the model’s robustness to scale [Figure] this time may be signiﬁcantly reduced without drastic loss of accuracy by using larger region scales (e.g. only 30-40 minutes for 50Kb regions, ﬁg. 5).
Revised: Due to the model’s robustness to scale, this time may be signiﬁcantly reduced without drastic loss of accuracy by using larger region scales (e.g. only 30-40 minutes for 50Kb regions, ﬁg. 5).

Original: Importantly, after completing the CNN+GP training, projections to lower or higher scales via the GM require no additional training [Figure]
Revised: Importantly, after completing the CNN+GP training, projections to lower or higher scales via the GM require no additional training.

Original: D A PPENDIX RESULTS D. N EGATIVE B INOMIAL R EGRESSION DOES NOT DETECT WELL - KNOWN DRIVERS GENOME - WIDE
Revised: D A PPENDIX RESULTS D. N EGATIVE B INOMIAL R EGRESSION DOES NOT DETECT WELL - KNOWN DRIVERS GENOME - WIDE

Original: Negative binomial regression is the only other method that has been used to perform an unbiased genome-wide search for driver mutations Nik-Zainal et al.
Revised: Negative binomial regression is the only other method that has been used to perform an unbiased genome-wide search for driver mutations Nik-Zainal et al.

Original: ; Rheinbay et al.
Revised: ; Rheinbay et al.

Original: We thus evaluated how the sensitivity of NBR to detect driver mutations genome-wide compared with the sensitivity of our method.
Revised: We thus evaluated how the sensitivity of NBR to detect driver mutations genome-wide compared with the sensitivity of our method.

Original: While all known melanoma drivers present in >3 samples were found by the
Revised: While all known melanoma drivers present in >3 samples were found by the

Original: GM by projecting down to only 1kb scale, NBR at 1kb fails to detect TERT , the only known common non-coding driver mutation, yielding a p-value that was an order of magnitude less signiﬁcant than the genome-wide signiﬁcance for this scale.
Revised: GM by projecting down to only 1kb scale, NBR at 1kb fails to detect TERT , the only known common non-coding driver mutation, yielding a p-value that was an order of magnitude less signiﬁcant than the genome-wide signiﬁcance for this scale.

Original: Similarly, while the GM detects all known esophageal adenocarcinoma drivers by projecting down to 100bp, NBR over 100bp fails to detect KRAS , an important genic driver of esophageal cancers, again yielding a p-value that was an order of magnitude less signiﬁcant than the genome-wide signiﬁcance threshold for 100bp.
Revised: Similarly, while the GM detects all known esophageal adenocarcinoma drivers by projecting down to 100bp, NBR over 100bp fails to detect KRAS , an important genic driver of esophageal cancers, again yielding a p-value that was an order of magnitude less signiﬁcant than the genome-wide signiﬁcance threshold for 100bp.

Original: Note: we presented results at 50bp in the text to highlight our model’s ability to search in arbitrarily small regions, but all known drivers for esophageal adenocarcinoma are also detected in a search over regions of 100bp.
Revised: Note: we presented results at 50bp in the text to highlight our model’s ability to search in arbitrarily small regions, but all known drivers for esophageal adenocarcinoma are also detected in a search over regions of 100bp.

Original: D. C ONVOLUTIONAL NEURAL NETWORK OUTPERFORMS OTHER DIMENSIONALITY REDUCTION ALTERNATIVES FOR A G AUSSIAN PROCESS
Revised: D. C ONVOLUTIONAL NEURAL NETWORK OUTPERFORMS OTHER DIMENSIONALITY REDUCTION ALTERNATIVES FOR A G AUSSIAN PROCESS

Original: We ﬁrst evaluated the methods for regional rate ﬁrst and second moment inference, µ R and σ 2 R , using our simulated datasets.
Revised: We ﬁrst evaluated the methods for regional rate ﬁrst and second moment inference, µ R and σ 2 R , using our simulated datasets.

Original: We calculated accuracy as the Pearson R 2 of the estimated mean and variance to the simulated ground-truth mean and variance.
Revised: We calculated accuracy as the Pearson R 2 of the estimated mean and variance to the simulated ground-truth mean and variance.

Original: CNN+GP, FCNN+GP, NBR and RF accurately inferred µ R , with R 2 µ R > 0 .
Revised: CNN+GP, FCNN+GP, NBR and RF accurately inferred µ R , with R 2 µ R > 0 .

Original: 95 for all three datasets (ﬁg. 6a).
Revised: 95 for all three datasets (ﬁg. 6a).

Original: However, PCA+GP, UMAP+GP, and
Revised: However, PCA+GP, UMAP+GP, and

Original: AE+GP consistently under-performed (ﬁg. 6a left), suggesting supervision when creating feature vectors is critical for the GP downstream performance.
Revised: AE+GP consistently under-performed (ﬁg. 6a left), suggesting supervision when creating feature vectors is critical for the GP downstream performance.

Original: The CNN+GP and FCNN+GP outperformed the other models when estimating the simulated variance (ﬁg. 6a, right), suggesting the ability to represent arbitrary functions is important for learning uncertainty in a complex dataset.
Revised: The CNN+GP and FCNN+GP outperformed the other models when estimating the simulated variance (ﬁg. 6a, right), suggesting the ability to represent arbitrary functions is important for learning uncertainty in a complex dataset.

Original: This conclusion is strengthened by the observation that UMAP and AE enabled relatively accurate variance estimation despite mediocre performance over the mean.
Revised: This conclusion is strengthened by the observation that UMAP and AE enabled relatively accurate variance estimation despite mediocre performance over the mean.

Original: Importantly, the clusters used for the simulated data were computed from mean epigenetic vectors; thus our CNN architecture (receiving a input in matrix form) was at a disadvantage.
Revised: Importantly, the clusters used for the simulated data were computed from mean epigenetic vectors; thus our CNN architecture (receiving an input in matrix form) was at a disadvantage.

Original: Nonetheless, the CNN+GP most accurately learned both µ R and σ 2 R across all three simulated datasets (Fig. 6a), with slight improvement over the FCNN+GP.
Revised: Nonetheless, the CNN+GP most accurately learned both µ R and σ 2 R across all three simulated datasets (Fig. 6a), with slight improvement over the FCNN+GP.

Original: To further compare the approaches, we applied the GP coupled models to estimate real mutation counts from the three cancers on multiple scales.
Revised: To further compare the approaches, we applied the GP coupled models to estimate real mutation counts from the three cancers on multiple scales.

Original: Models were compared by their R 2 to the observed mutations over the test set and to an empirical variance based on the model’s own feature vectors (ﬁg. 6b) (see Appendix).
Revised: Models were compared by their R 2 to the observed mutations over the test set and to an empirical variance based on the model’s own feature vectors (ﬁg. 6b) (see Appendix).

Original: The CNN+GP outperformed the FCNN+GP model over observed mutation counts and empirical variance estimation for all three cancer types.
Revised: The CNN+GP outperformed the FCNN+GP model over observed mutation counts and empirical variance estimation for all three cancer types.

Original: Additionally, the performance advantage of the CNN appeared to grow as window size and observed mutation counts increased.
Revised: Additionally, the performance advantage of the CNN appeared to grow as window size and observed mutation counts increased.

Original: This suggests that local epigenetic patterns play an appreciable role in setting mutational processes and indicates that our model is well-designed to leverage the recent growth in genomics corpus sizes.
Revised: This suggests that local epigenetic patterns play an appreciable role in setting mutational processes and indicates that our model is well-designed to leverage the recent growth in genomics corpus sizes.

Original: [Figure]
Revised: [Figure]

Original: D. E XISTING WHOLE - GENOME REGRESSION MODELS ARE TIME INEFFICIENT AT MULTI - RESOLUTION SEARCH
Revised: D. E XISTING WHOLE - GENOME REGRESSION MODELS ARE TIME INEFFICIENT AT MULTI - RESOLUTION SEARCH

Original: All existing regression models (RF, NBR, BR) require retraining for each desired scale.
Revised: All existing regression models (RF, NBR, BR) require retraining for each desired scale.

Original: A requirement that becomes computationally challenging at ﬁner resolutions (e.g. >1.5h for NBR at 100bp).
Revised: A requirement that becomes computationally challenging at ﬁner resolutions (e [Figure] g. >1.5h for NBR at 100bp).

Original: To provide an estimate of the differences between existing methods and our SPG, we performed a multi-scale time analysis presented in .
Revised: To provide an estimate of the differences between existing methods and our SPG, we performed a multi-scale time analysis presented in .

Original: However, it does not include scales <100bp, such as 50bp used in this work to detect driver hot-spots.
Revised: However, it does not include scales <100bp, such as 50bp used in this work to detect driver hot-spots.

Original: A log-log transform of the scale against the run-time () exposes a polynomial relation between the the window size and time (for small enough scales where the compute power is not governed by the machine’s memory and system operations).
Revised: A log-log transform of the scale against the run-time () exposes a polynomial relation between the the window size and time (for small enough scales where the compute power is not governed by the machine’s memory and system operations).

Original: Extending this relation to a scale as small as 50bpwe get that the run-time  as high as 1.5h for BR and 2.5h for NBR.
Revised: Extending this relation to a scale as small as 50bp run-time is as high as 1.5h for BR and 2.5h for NBR.

Original: Making the overall run-time for a typical multi-resolution scan of 50bp, 100bp, 500bp, 1kb, 10kb to be >2h for BR and >4h for NBR, while the SPG run-time remains <1h.
Revised: Making the overall run-time for a typical multi-resolution scan of 50bp, 100bp, 500bp, 1kb, 10kb over 2h for BR and over 4h for NBR, while the SPG run-time remains under 1h.

Original: [Table]
Revised: [Table]