 Research
 Open Access
 Published:
Mining statisticallysolid kmers for accurate NGS error correction
BMC Genomics volume 19, Article number: 912 (2018)
Abstract
Background
NGS data contains many machineinduced errors. The most advanced methods for the error correction heavily depend on the selection of solid kmers. A solid kmer is a kmer frequently occurring in NGS reads. The other kmers are called weak kmers. A solid kmer does not likely contain errors, while a weak kmer most likely contains errors. An intensively investigated problem is to find a good frequency cutoff f_{0} to balance the numbers of solid and weak kmers. Once the cutoff is determined, a more challenging but lessstudied problem is to: (i) remove a small subset of solid kmers that are likely to contain errors, and (ii) add a small subset of weak kmers, that are likely to contain no errors, into the remaining set of solid kmers. Identification of these two subsets of kmers can improve the correction performance.
Results
We propose to use a Gamma distribution to model the frequencies of erroneous kmers and a mixture of Gaussian distributions to model correct kmers, and combine them to determine f_{0}. To identify the two special subsets of kmers, we use the zscore of kmers which measures the number of standard deviations a kmer’s frequency is from the mean. Then these statisticallysolid kmers are used to construct a Bloom filter for error correction. Our method is markedly superior to the stateofart methods, tested on both real and synthetic NGS data sets.
Conclusion
The zscore is adequate to distinguish solid kmers from weak kmers, particularly useful for pinpointing out solid kmers having very low frequency. Applying zscore on kmer can markedly improve the error correction accuracy.
Background
The massively parallel nextgeneration sequencing (NGS) technology is revolutionizing a wide range of medical and biological research areas as well as their application domains, such as medical diagnosis, biotechnologies, virology, etc [1]. It has been shown that the NGS data is so informative and powerful that some ever thorny problems can be effectively tackled through this technology, e.g., the genome wide association study [2].
The information contained in NGS data is deep and broad, but the raw data is still error prone. Various kinds of errors exist in the raw sequencing data, including substitution, insertion and deletion. The substitution error rate can be as high as 1 to 2.5% for the data produced by the Illumina platform [3]; and the collective insertion and deletion error rate can be as high as 10 to 40% for the PacBio and Oxford Nanopore platforms [4, 5]. It has been widely recognized that correcting these sequencing errors is the first and critical step for many downstream data analyses, such as de novo genome assembly [6], variants calling from genome resequencing [7], identification of single nucleotide polymorphism as well as sequence mapping [3, 8]. For instance, the number of nodes of the De Bruijn graph generated from the HapMap sample NA12878 (https://www.ncbi.nlm.nih.gov/sra/ERR091571/) is 6.92 billion; however, this number can be reduced to only 1.98 billion after error correction. This reduction significantly alleviates the burden of graph manipulation.
Owing to the importance of error correction, dozens of approaches have been proposed to cope with various types of errors. Depending on the key ideas that have been used, existing approaches can be categorized into three major approaches: (i) the kspectrumbased approach, including Quake [3], Reptile [9], DecGPU [10], SGA [11], RACER [12], Musket [13], Lighter [14], Blue [15], BFC [16], BLESS2 [17], MECAT [18] (ii) the suffix tree/arraybased approach, including SHREC [19], HSHREC [20], HiTEC [21], Fiona [22] and; (iii) the multiple sequence alignmentbased approach, including ECHO [23], Coral [8], CloudRS [24], MEC [25]. Among these approaches, the most advanced ones are the kspectrumbased. It provides a very good scalability and competitive performance. Scalability is crucial for NGS data analysis since the input volume is usually huge.
The performance of kspectrumbased approach heavily depends on the selection of solid kmers. A solid kmer is a kmer frequently occurring in NGS reads. The other kmers are called weak kmers. A solid kmer often does not contain any sequencing error, but a weak kmer often contains sequencing errors. An intensively investigated problem is to find a good frequency cutoff f_{0} to balance the numbers of solid and weak kmers, cf. Fig. 1. It is clear that even a very carefully determined f_{0} cannot tidily differentiate erroneous kmers from those kmers that do not contain any error bases. The reason is that there are very often a small portion of solid kmers that contain errors and there are very often a tiny portion of weak kmers that do not have errors, cf. the shaded part in Fig. 1. This discrepancy is caused by the skewed distribution of the coverage of the sequencing reads. For instance, Ross et al. [26] has reported that the coverage of GC rich and poor regions is markedly lower than the average coverage. That is, the kmers from these regions very likely have low frequency, even lower than f_{0}.
In this research, we focus on a more challenging but lessstudied problem: (i) remove a small subset of solid kmers that are likely to contain errors, and (ii) add a small subset of weak kmers that are likely to contain no errors, into the set of solid kmers. This is achieved by using f_{0} as well as zscore of kmer, z(κ). With the purified set of solid kmers, the correction performance can be much improved.
Our approach starts with counting kmer frequencies by using KMC2 [27], then calculates the zscores of kmers. Later, the statisticallysolid kmers are mined by considering both frequency and zscore. After that, the Bloom filter is constructed by the statisticallysolid kmers, and the weak kmers are corrected. The newly proposed approach is named as ZEC, short for zscorebased error corrector.
Algorithm: mining statisticallysolid kmers
A solid kmer is conventionally defined as a kmer which occurs in a data set of NGS reads with high frequency. A solid kmer is usually considered errorfree, and taken as the template for error correction. If a kmer is not solid, then it is defined as a weak kmer considered as errorcontaining. Existing kmerbased approaches use a frequency cutoff, f_{0}, to identify solid and weak kmers from NGS reads, e.g., BLESS2 [17], Musket [13], and BFC [16]. The main difference of these methods is how the f_{0} is determined.
In fact, a solid kmer is not definitely errorfree. Sometimes, it may contain errors with a small chance. It is also true for the weak kmers — a weak kmer can be absolutely errorfree. The reason that a solid kmer is not always correct is that the coverage is not under uniform distribution. Thus the cutoff f_{0} itself is unable to perfectly distinct correct kmers from erroneous kmers; cf. the part labeled as α and β in Fig. 1. However, the purpose of the research is to obtain correct kmers as many as possible.
In this study, we present a time and memory efficient algorithm to purify the solid kmer set as well as the weak kmer set, so that more correct kmers can be identified.
Let R be the input set of NGS reads, and K be the set of kmers contained in R. To determine whether a kmer, say κ, of K is correct or not, the following metrics are examined:

f(κ), the frequency of κ;

z(κ), the zscore of κ.
Calculating f(κ)
The straightforward approach to determine f(κ) is as follows: (i) scan each read r of R from the beginning to the end; (ii) sum over the occurrence that κ appears. Then the summation is f(κ). This approach works for one kmer, but it cannot be applied to all the kmers simultaneously as the number of kmers can be very large, demanding a huge size of memory.
In this study, we make use of the kmer counting algorithm, KMC2 [27], to solve this problem. KMC2 can remarkably reduce the memory usage because: (i) it is diskbased; (ii) it uses (k,x)mer; and (iii) it applies the minimizer idea to deal with kmer.
Computing z(κ)
Given a kmer κ, we define the neighbor of κ, N(κ), as
where D(κ,κ^{′}) is the edit distance between κ and κ^{′}, and the d_{0} is the predefined maximum distance. The default value of d_{0} is 1 as used in this study, but user can adjust this value to any reasonable integer.
The kmer cluster centered at κ is defined as
and the set of frequencies associated with these kmers is defined as
The zscore of κ, z(κ), is computed by
where μ is the averaged frequency of F(κ) and σ is the standard deviation of F(κ).
It is straightforward to calculate the zscore of each kmer given the frequency of the kmer as well as that of its neighbor that have been determined by the aforementioned approach.
Determining f _{0}
Unlike existing approaches that determining solid kmers based on their frequency only, we examine their zscores as well.
Traditionally, an optimal f_{0} is used to distinct weak and solid kmers, which is determined as the count minimizing misclassification rates (see misclassified parts labeled as α and β in Fig. 1). To learn the optimal value, we model the frequency of erroneous kmers by a Gamma distribution P_{G}(X), and those correct ones by a mixture of Gaussian distributions P_{N}(X). A Gamma distribution is defined as:
where k accounts for the shape of the distribution, θ is for the scale of the distribution, i.e., how the data spread out, Γ(k) is the Gamma function evaluated at k; cf. the dash skyblue line in Fig. 1. While a mixture of Gaussian distributions is
where π_{i} is the mixture parameter, μ_{i} and σ_{i} represent the mean and standard deviation of the component i, and K is the number of Gaussian components. In this study, K is set as 2, with one accounting for kmers that are from GC rich or poor regions, and the other for the rest correct kmers.
The two distributions are estimated by using EM algorithm based on the frequencies of kmers. An example of the two distributions are shown in Fig. 1, i.e., the skyblue dash line and the orange dash line. Based on the two distributions, we can determine the threshold f_{0}, such that it can minimize the area marked as α and β. Note that, the threshold f_{0} determined in this way may not be the intersection point of the two density functions.
Mining solid kmers
It is clear that the optimal f_{0} cannot perfectly distinct the solid kmers from the weak kmers. Taking Fig. 1, the kmers marked by α will be wrongly corrected although they do not have errors but just because their frequencies are lower than f_{0}; likely, the ones marked by β will keep unchanged although they have errors because they have high frequency. To further refine the purity as well as the completeness of solid kmers, we borrow the statistical idea of using zscore to solve the problem. The purity is defined as
where p_{correct} is the proportion of correct kmers in the solid kmers, and p_{erroneous} is the proportion of erroneous kmers in the solid kmers. The completeness is calculated as
where \(N^{\text {solid}}_{\text {correct}}\) is the number of correct kmers in the solid kmers, and N_{correct} is the total number of correct kmers.
The zscore as well as the frequency are collectively incorporated into solid kmer identification through the following two situations:

If f(κ)<f_{0} and z(κ)≥z_{0}, then κ is removed from the weak kmers and added to the solid kmers, i.e., increases the completeness.

If f(κ)≥f_{0} and \(z(\kappa) < z^{'}_{0}\), then κ is removed from the solid kmers and added to the weak kmers, i.e., improves the purity.
The f_{0} is the minimum frequency that has been determined, while the z_{0} and \(z^{'}_{0}\) are the maximum zscore and minimum zscore for weak kmers and solid kmers, respectively.
The z_{0} and \(z^{'}_{0}\) are learned from the zscore distribution automatically. To obtain the optimal z_{0}, the zscores of the kmers having frequency less than f_{0} are collected. Later, the distribution of these zscores is estimated and z_{0} is set as the value having the lowest density between two peaks (viz. the trough of the bimodal; see results for more details). Analogously, \(z^{'}_{0}\) is determined on the zscores of kmers having frequency greater than f_{0}.
Methods
Our error correction model contains two main steps: (i) build Bloom filter from solid kmers and; (ii) correct errors in weak kmers by the Bloom filter.
Build bloom filter
Bloom filter [28] is a probabilistic data structure that can check whether an item is contained in a set of items with very frugal memory consumption. Instead of storing each item as is, the Bloom filter maps the item into several bits of a bit vector. Each bit can be reused by many items, and the mapping is achieved by hash functions. To check whether an item exists in a set of items, one only need to check whether all the mapped bits are “1”s. In case any one of them is “0”, it indicates that the item is definitely not contained in the set. Since each bit can be reused, it is possible that an item is not contained in the set but all of its mapped bits are “1”s. The probability that it happens is false positive rate. The relation between the number of hash function h, the false positive rate p, the size of the bit vector n, and the actual number of elements m is
In our study, m is the number of solid kmers that have been determined from all the kmers by means of the aforementioned algorithm. Per existing approaches, p is set to 1%. One can also tune p, h and n to fit the real hardware limitations.
It has been reported that the Bloom filter has been successfully used to correct NGS errors, such as BLESS2 [17] and BFC [16]. The major difference between our model and the existing models is that we dedicate to efficiently refine the solid kmers that are used to construct Bloom filter, which directly improves the error correction performance in theory. Note that, the solid kmers play the key role in error correction, as all the rest kmers (viz. the weak kmers) are to be corrected based on the solid ones.
Figure 2 illustrates the forward search and backward search.
Correct errors
By using Bloom filter, the errors contained in each read can be correct as follows: (i) check the existence of each kmer of the read from the beginning to the end sequentially. (ii) partition the kmers into groups that each group contains only solid kmers or weak kmers, deemed as solid group G_{s} or weak group G_{w}, respectively. The order of the groups is kept according to their appearance in the read. (iii) correct the errors causing the weak group G_{w} according to the following situations:

1
If G_{w} is the first group and there exists a successive group G_{s} that is solid, we iteratively change the first base of each kmer of G_{w} to its alternatives and check the existence of the kmers against the Bloom filter. Once there exist a solution that makes all the weak kmers solid, the amendment of the bases is accepted, thus the correction of the error. This process is applied to the kmers of G_{w} from the last one to the first one. In case the number of kmers contained in G_{w} is less than a predefined value, say τ, the processive solid kmers that are extended from the corrected kmers will be generated until the total number of kmers in G_{w} is τ. If this criterion cannot be satisfied, the solution is abandoned. On the other hand, if G_{s} does not exist, we will alter the bases to their alternatives of all the kmers iteratively until a solution that make all the kmers solid can be found.

2
If G_{w} has a solid processive group G_{s} and a solid successive group \(G_{s}^{'}\), we substitute the last base of each kmer in G_{w} by its alternatives from the first kmer to the last kmer, namely the forward search. Solutions that make all the kmers solid till the current substitution are recorded. Similarly, the backward search is conducted on the first base of the kmers from the last one to the first one. A solution is accepted if the forward search and the backward search meet and the kmers contained in both of them are solid. In case the number of kmers in G_{w} is less than k, we will only alter the last base of the first kmer.

3
If G_{w} is the last group and there exists a solid processive group G_{s}, we will apply the backward search to obtain the solution. Analogously to the first situation, if the number of kmers of G_{w} is less than τ, we will extend the kmers toward their downstream until the number is satisfied. In case G_{s} does not exist, it is the same as the second part of the first situation, thus the same approach is applied.
Results
Datasets
We collected six data sets to test the performance of our proposed method in comparison with the stateofart methods. Four of the six data sets are the NGS reads produced by the Illumina platform, including Staphylococcus aureus (S. aureus), Rhodobacter sphaeroides (R. sphaeroides), Human Chromosome 14 (H. chromosome 14) and Bombus impatiens (B. impatiens). These data sets are the gold standards used by GAGE [6] for NGS data analysis. Besides these real data sets, two synthesized data sets have been generated by using ART [29] based on the genomes H. chromosome 14 and B. impatiens. The two synthetic data sets contain exactly the same number of reads as the real ones. They are included because the ground truth of the synthesized errors are known, i.e., the positions of the errors as well as their bases are available. On the contrary, such information is unavailable for the real data sets. Typically, the raw reads of the real data sets are mapped to the corresponding reference, and those mapped are kept for performance evaluation. Although this is arguable as various deleterious situations can emerge from the mapping, e.g., unmapped reads, multimapped reads, wrongly mapped reads, it is necessary to carry out the mapping as only in this way can we perform the evaluation directly. This is another reason that the synthetic data should be included. Details of these data sets are shown in Table 1.
Performance evaluation
The error correction performance is evaluated through the widely accepted procedure implemented by [30]. Metrics that are considered include gain, recall, precision and per base error rate (pber). Gain is defined as (TP−FP)/(TP+FN), recall is TP/(TP+FN), precision is TP/(TP+FP) and pber is N^{e}/N, where TP stands for the number of corrected bases that are truly erroneous bases, FP represents the number of corrected bases that are not sequencing errors intrinsically, FN is the number of erroneous bases that remain untouched, N^{e} is the number of erroneous bases and N is the total number of bases. Among these metrics, gain is the most informative.
All experiments are carried out on a cluster having eight Intel Xeon E7 CPUs and 1Tb RAM. Each CPU has eight cores.
Overall Performance of ZEC. The experimental results of ZEC are presented in Table 2. ZEC performs well on both of the real data sets and the synthetic data sets. Comparing the performance on H. chromosome 14 and B. impatiens, ZEC has a much better performance on S. aueus and R. sphaeroides. This is consistent with our understanding that the genomes of the former two data sets are much more complicated than the latter two, where the errors introduced in complicated genomes are more difficult to correct.
Relation with GCcontent. A previous study by Ross et al. [26] shows that the GCcontent (GC poor and GC rich) regions have direct influence on the low sequencing coverage of NGS data. Hence, the kmers obtained from the reads sequenced from these regions are more likely to be treated as weak. Figure 3 highlights an example of the relation between GCcontent (GC poor and GC rich) and kmer frequency derived from H. chromosome 14. It can be seen that the kmers having a low frequency can spread out wider than those having a high frequency, and the wide range is coincident with the GC content. This result is in accordance with the performance shown in Table 2, meanwhile it also consolidates our intuition that refining the set of solid kmers is necessary, particularly for the subset of kmers that have a low frequency. More importantly, it empirically supports our idea of using mixture model to treat solid kmers and weak kmers separately.
Comparison with Stateoftheart. The performance of ZEC is much superior to the stateoftheart methods, including Lighter [14], Racer [12], BLESS2 [17], Musket [13], SGA [11], BFC [16]. See Table 2. ZEC markedly outperforms the existing error correctors in terms of the most informative evaluation metric—gain. For instance, on the dataset R4, the gain of ZEC is 0.746, while the best performance produced by the other methods is 0.705. For the synthetic datasets, ZEC also has higher gain than other methods. For example, on the dataset S2, the gain of ZEC is 0.853, while the best and worst gain generated by the other methods are 0.849 and 0.058, respectively. The lowest average perbase error rate of ZEC also consolidates its effectiveness.
Distinguishbility of zscore
The key to the performance improvement is the idea of using zscore for identifying the two special subsets of kmers from the sets of solid kmers and weak kmers. An example of zscore distribution pertaining to kmer frequency is shown in Fig. 4, which is derived from B. impatiens. The highlighted kmers shown in the figure have relatively low frequencies—less than 9, while the zscores are pretty high—greater than 1. Interestingly, almost all the solid kmers (the top right region) have the similar level of zscores comparing to these highlighted ones. These observations indicate that the highlighted kmers are very likely to be correct kmers instead of erroneous kmers although their frequencies are very low. The zscore distribution pertaining to the other three real data sets has similar patterns compared to the one shown here.
By exploring the four real data sets, we found that the proportion of kmers that can be refined comparing to the solely frequency determined kmers are 12.3%, 14.2%, 11.4%, 7.1% for the real data R1, R2, R3 and R4, respectively; see Fig. 5. These refinement are the major contributions of the performance improvement.
Efficiency of zscore calculation
Calculating zscore of kmers is not trivial for very large data sets, as the kmers and their frequencies are usually too large to be hold by a main memory of a moderate computer. We designed a novel algorithm and solved this problem. The efficiency of the algorithm in terms of the memory usage and running speed are studied.
Figure 6 shows the relation between memory saving ratio and the percentage of input (kmers as well as their frequencies) that can be held by only one bit vector. The memory saving ratio is calibrated as the ratio between the real memory allocation and the input data volume. For instance, the ratio of 0.01 pertaining to the data R4 means that the allocated memory is one percent of the input size of R4. That is, 70Mb memory is allocated for holding the 6.97Gb data. It is promising that, with one percent memory allocation, around 22 percent input data can be hold by only one bit vector. When the memory allocation increased to 2.5 percent, 30 percent input data and even more can be held by one bit vector. Obviously, keep increasing the size of allocated memory does not guarantee the linear scale of holding the input data. Based on the experiments, we set the memory allocation ratio to 2.5 percent through the whole study. Typically, three bit vectors are constructed for holding all the input. Note that, the size of bit vector decreases along with the reduced size of input. The ratios between the input and the total allocated memory are 20.0, 13.4, 7.0 and 7.9 for the four real datasets, respectively.
Regarding the running speed, this algorithm is linearly scaled. Since locating each kmer in a bit vector is O(1) pertaining to time complexity by using hash, this algorithm is pretty fast. For instance, based on our computing power, it only takes 387 s to construct the bit vectors and calculate the zscores of all the kmers of R4—the largest data set.
Since a Bloom Filter has false positives, this may cause the zscore of a kmer different from its genuine value. However, the false positive rate is pretty small, usually less than 1%, thus this impact can be neglected.
Discussion
Our model effectively pinpoints out correct kmers having low frequency, achieving an improvement of 11.25% on weak kmers. However, some issues still remain further exploration, including neighbor inclusion and neighbor retrieval.
Neighbor inclusion means how neighbor kmers are determined given a kmer of interest, say κ. Our current approach takes kmers having edit distance of 1 as neighbors of κ, but there still has a small chance that a true neighbor having edit distance larger than 1. Suppose the error rate is e, the probability of a kmer having exactly one error is k·e(1−e)^{k−1}/k·e=(1−e)^{k−1}. When e=1% and k=1, the probability is (1−0.01)^{31−1}=73.97%. That been said, about 26% real neighbors are excluded. However, even extending the minimum edit distance from 1 to 2 significantly elongates running time. This is because the number of candidate kmers increases from 3∗k to 3∗k∗3∗(k−1).
Neighbor retrieval is another issue to be considered. Usually, the size of counted kmers is too large to fit into a main memory. Hence, a more sophisticated approach is required to solve this problem. We use Bloom Filter to overcome the limitation. For kmers having small count, say 5, we use classical Bloom Filters to save them, each Bloom Filter saves kmers having the same count. For kmers having large count, we use coupledBloom Filter to save them. One Bloom Filter for kmer encoding, while the other is for count representation. This approach significantly reduces memory usage while achieving constant time complexity of kmer retrieval. However, it may cause false positives although the probability is small. Hence, more effort is required to handle this problem.
Conclusions
We have proposed a novel method for correcting the NGS errors. The novel idea is the use of statisticallysolid kmers to construct the Bloom filter. These kmers are mined from all the kmers of a NGS data set by considering both their frequency and zscore, particular the latter one that can effectively fishing out the solid kmers having low frequency. Pinpointing out such kmers has been a very challenging problem. The experimental results show that our approach markedly outperforms the existing stateoftheart methods in terms of error correction performance.
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Acknowledgments
We thank the anonymous reviewers for their valuable comments and insightful suggestions.
Funding
This study is collectively supported by the National Natural Science Foundation of China (No. 31501070), the Natural Science Foundation of Hubei (No. 2017CFB137) and Guangxi (No. 2016GXNSFCA380006), the Scientific Research Foundation of GuangXi University (No. XGZ150316) and Taihe hospital (No. 2016JZ11), and the Australia Research Council (ARC) Discovery Project 180100120. Publication costs are funded by the National Natural Science Foundation of China (No. 31501070).
Availability of data and materials
The source codes are available at github.com/lzhlab/zec/.
About this supplement
This article has been published as part of BMC Genomics Volume 19 Supplement 10, 2018: Proceedings of the 29th International Conference on Genome Informatics (GIW 2018): genomics. The full contents of the supplement are available online at https://bmcgenomics.biomedcentral.com/articles/supplements/volume19supplement10.
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LZ conceived and designed the experiments, LZ and JL wrote the manuscript. Program coding: LZ, YW and ZZ. Data analyses: LZ, JX, LB, WC, MW and ZZ. All authors read and approved the final manuscript.
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Zhao, L., Xie, J., Bai, L. et al. Mining statisticallysolid kmers for accurate NGS error correction. BMC Genomics 19, 912 (2018). https://doi.org/10.1186/s128640185272y
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Keywords
 Error correction
 Nextgeneration sequencing
 zscore