Comparing K-mer based methods for improved classification of 16S sequences
© Vinje et al. 2015
Received: 13 February 2015
Accepted: 6 June 2015
Published: 1 July 2015
The need for precise and stable taxonomic classification is highly relevant in modern microbiology. Parallel to the explosion in the amount of sequence data accessible, there has also been a shift in focus for classification methods. Previously, alignment-based methods were the most applicable tools. Now, methods based on counting K-mers by sliding windows are the most interesting classification approach with respect to both speed and accuracy. Here, we present a systematic comparison on five different K-mer based classification methods for the 16S rRNA gene. The methods differ from each other both in data usage and modelling strategies. We have based our study on the commonly known and well-used naïve Bayes classifier from the RDP project, and four other methods were implemented and tested on two different data sets, on full-length sequences as well as fragments of typical read-length.
The difference in classification error obtained by the methods seemed to be small, but they were stable and for both data sets tested. The Preprocessed nearest-neighbour (PLSNN) method performed best for full-length 16S rRNA sequences, significantly better than the naïve Bayes RDP method. On fragmented sequences the naïve Bayes Multinomial method performed best, significantly better than all other methods. For both data sets explored, and on both full-length and fragmented sequences, all the five methods reached an error-plateau.
We conclude that no K-mer based method is universally best for classifying both full-length sequences and fragments (reads). All methods approach an error plateau indicating improved training data is needed to improve classification from here. Classification errors occur most frequent for genera with few sequences present. For improving the taxonomy and testing new classification methods, the need for a better and more universal and robust training data set is crucial.
The exploration of microbial communities is now a major focus in microbiology, opening new approaches to the study of microbiomes of humans and other organisms as well as the communities found in natural environments of air, water or soil . Already in the 1980s Carl Woese introduced the rRNA-based phylogenetic comparisons of prokaryotes [2, 3], and the 16S rRNA gene is still the most useful genomic marker for the study of diversity and composition of metagenomes. The classification of 16S sequences obtained from some samples is a classical pattern recognition problem, i.e. recognizing some pattern in a sequence and assign it to one out of several predetermined categories. Whether the sequences are subjected to multiple alignments or, as in this paper, counting of short words, some assignment must be made based on how similar these sequences are to previously classified sequences. Naturally, the methods employed should give as accurate classifications as possible, but in metagenomics time-efficiency is also an issue since the number of sequences to classify may be vast. It should also be noted that with today’s massively parallel sequencing technologies, shorter reads covering only a region of the gene are more accessible [4–6], making classification methods that perform well on sequence fragments essential.
However, classifications based on 16S rRNA sequences do not only have a practical use in metagenomics. In fact, this marker is used to build the entire prokaryotic taxonomy and is considered the gold standard for phylogenetic studies [7–9]. In this perspective the classification of full-length 16S sequences is the issue. It should also be noted that in this context we should make all possible efforts to have the absolute best classifications available, and time-efficiency is no longer important.
A number of different procedures have been used to classify 16S sequences, and several different databases purposely designed as 16S rRNA repositories are available, e.g. Greengenes , RDP  and SILVA . Most procedures for taxonomic studies have been based on alignments and reconstruction of phylogenetic trees, making use of some predefined evolutionary models and relevant algorithms [3, 13, 14]. However, with the enormous increase in data from next generation sequencing technology, these approaches suffer some problems. First, the computational time required to align a large set of sequences increases exponentially by its size. Secondly, greedy algorithms of some kind are required to construct these huge alignments and these sparse, monolithic alignments will most likely contain a substantial number of errors due to the heuristics employed. Finally, the lack of consensus, e.g. on evolutionary model assumptions, has made it impossible to arrive at an official taxonomy for prokaryotes, the most widely accepted taxonomy being the Bergey’s Manual of Systematics of Archaea and Bacteria . Thus, objective pattern recognition algorithms are likely to be valuable tools for building the prokaryotic taxonomy itself.
The most popular pattern recognition methods for 16S sequences are those based on counting K-mers, i.e. overlapping ‘words’ of length K in the sequences [16–19]. Wang et al.  developed the RDP classifier, based on the naïve Bayes principle and a word-length of K=8. The RDP classifier is now close to being a standard in 16S based classification, and was in 2011 selected by Essential Science Indicators as the most-cited paper in a highlighted research area of microbiology . K-mer methods are fast and will not suffer from the same uncertainties as the procedures based on evolutionary models and alignments. This way of converting sequences to numerical data is not as intuitive as evolutionary models, and lack the obvious interpretation given by evolutionary distances, but they are very objective in their mechanism. Also, in a previous study  we found that in order to obtain the best possible classification at the genus level, one has to consider more or less all positions along the full-length 16S sequences (around 1500 bases), not only hyper-variable regions or other subsequences. This is another advantage of the K-mer methods; they use all data in asequence.
However, K-mer based pattern recognition methods are not without model assumptions, and the RDP classifier uses the K-mer counts in one out of a number of alternative ways. Recent suggested improvements of this approach  have made it necessary to make a more systematic investigation on how well other K-mer based methods would perform, and possibly to reveal how and where efforts should be made to improve the objective classification of prokaryotes. In this paper we have compared different classification methods based on K-mer data for 16S sequences. We consider five different methods based on different machine-learning approaches, and we have compared their performance for full-length sequences as well as fragments. In addition to the method comparison, we also try to pinpoint where improvements should be made in order to give us better future methods for the important problem of identifying the majority of species on thisplanet.
To compare methods we used two data sets. The Trainingset9 is the data used to compare 16S classification methods in , and was downloaded from RDP . It consists of 10032 16S rRNA sequences varying from 320 to 2210 bases in length, with the majority around 1400 bases. There are 37 phyla and 1943 genera represented in this set.
The SilvaSet is an extract from the SILVA database , where the largest genera have been ‘pruned’ by random sampling to contain fewer sequences. This set has 29520 sequences, covering 29 phyla and 1533 genera. The main reason for including this data set is that it is a manually curated data set different from Trainingset9, which was used during the development of the RDP-classifier.
In this paper we only consider classification to genus, i.e. the lowest taxonomic level of these data. This is the most challenging and also the most relevant problem for most studies where taxonomic classification is important.
To estimate the model performance we conducted a 10-fold cross validation  for all methods. The data were ordered alphabetically by genus name and split into ten different segments by enumeration from one to ten repeatedly, and then assigned to segments according to this number, i.e. every tenth sequence belongs to the same segment. This ensured a maximum spread of all genera across the segments. Each segment was set aside once as a test set, while the rest were used as training set in each cross-validation iteration.
K-mer based methods
All methods compared here represent a 16S sequence by its overlapping K-mers, i.e. words of length K. There are D=4 K possible words of length K in the DNA (RNA) alphabet, and in our study we tested word lengths from two to eight. The methods tested differ in the way they represent a sequence as K-mers and how this information is utilized in a statistical learning algorithm to achieve best possible classification.
All five methods were implemented in the software environment R . Our implementation of the RDP classifier was tested against the original Java-implementation to ensure consistency. The PLS and nearest-neighbour methods already exist in the R-environment.
The RDP method considers only the presence/absence of a word in a sequence, not its frequency. All words of length K are ordered alphabetically as w 1,w 2,…,w D . For every sequence, we create a vector of D elements where element j is 1 if word w j is present in the sequence, and 0 if not. We have chosen to describe the RDP method in detail below, even if this has been done in , because this method serves as a reference for the other methods described later.
For each of the N sequences in the training set we get a vector of 1’s and 0’s, and these vectors are arranged as rows in the N×D matrix A r d p .
where the added 0.5 and 1 guarantees that no probability is zero or one.
If the training set contains data for G genera, we can arrange the probabilities q g,j in a G×D matrix Q r d p where the element in row g and column j is q g,j , for g=1,…,G, j=1,…,D. This matrix Q r d p is the trained model, with a set of probabilities (a row) for each genus.
where a j is element j in a and p j is from (1). Notice that P r(a is a joint probability of observing the words we see in this sequence. The naïve Bayes approach lies in the assumption that this joint probability can be written as a product of the marginal probabilities, as we have done on the right hand side above. This assumption is correct only if the elements of a are independent, which is a naïve assumption, but often still works in a satisfactory manner.
where the probability on the left hand side is the criterion we use to classify. This is the posterior probability of genus g given the observed sequence a, and we classify to the genus that maximizes this probability. On the right hand side we have the prior probability of genus g, P r(g), in addition to the two probabilities we computed in (3) and (4). It is customary to set the prior probability equal to the proportion of data from genus g in the training data set. In the RDP classifier the prior probabilities are assumed to be equal for all genera, and genera with few sequences are just as likely to be observed as those with many sequences in the training set. In our study we considered both flat priors (RDP) as well as priors proportional to genus abundances.
The posterior probability P r(g|a) is computed for every genus, and we assign the sequence to the genus where we get the largest probability. Notice that the denominator P r(a) in (5) does not depend on genus g. Hence, the g that maximizes P r(g|a) is exactly the same g that maximizes P r(a|g)P r(g), and we can ignore P r(a) altogether. Also, if the prior probabilities P r(g) are identical for all genera, we get the simple relation P r(g|a)=P r(a|g).
as the inner product of L r d p and the column vector a ′. The score vector z has one element for each genus, and we assign to the genus where z has its maximum value. In case of two or more genera obtaining the same maximum value, the sequence is marked as unclassified.
Notice that with flat priors, the terms log2(P r(g)) are identical for all g, i.e. all elements of p are identical, and it can be omitted from (7) since it will add the same to all genera.
The Multinomial method differs from the RDP method by considering the relative frequency of every word instead of presence/absence. The naïve Bayes principle is the same. A similar approach has also been tested by Lui and Wong in their work in .
For each of the N sequences in the training set we get a vector of frequencies, i.e. element j is the number of times we observe w j in the sequence. These vectors are arranged as rows in the N×D matrix A f r q .
The trained model consists of the (G×D) matrix Q m l t where row g contains the multinomial probabilities q g,j for genus g.
where L m l t is the log-transformation of Q m l t from the training step and p are the log-priors just as for the RDP-classifier. The score vector z has one element for each genus, and the sequence is assigned to the genus with maximum score in z. In case of two or more genera obtaining the same maximum value, the sequence is marked as unclassified.
In the present context ordinary Markov models consider word frequencies, but differ from the naïve Bayes principle used by the previous two methods. Markov models have been tested on sequence data with the K-mer approach in earlier studies, e.g. by Davidsen et al. .
The training step corresponds to estimating the transition probabilities of the Markov model. Any word of length K can be split into the pretext consisting of the first K−1 symbols, and the last letter, being A, C, G or T. The transition probabilities are the conditional probabilities of the last letter given the pretext. These probabilities are usually organized in a transition matrix with 4 columns (one for each letter) and one row for each pretext (4 K−1 rows). However, these probabilities can equally well be organised in a single row-vector, where the conditional probabilities of A given the ordered pretexts is found at positions I A =(1,5,9,…), for C given the ordered pretexts in positions I C =(2,6,10,…) and so on. Note that this corresponds to the K-mers in alphabetical order. Each consecutive four positions corresponds to the same pretext, extended by A, C, G and T, respectively.
and similar if the pretext is followed by C, G or T, I A is replaced by the corresponding index set. If we had organized the transition probabilities in a matrix, this value would appear in cell (h,1) since we consider pretext h followed by A (column 1). Instead we arrange these probabilities in a row vector of D elements. Having the transition probabilities for each genus, we arrange the vectors as rows in a (G×D) matrix Q m r k . The latter organization of the transition probabilities is done only to have the same data structure as for the other methods; it does not affect the computations.
where L m r k is the log-transformation of Q m r k . Again we classify to the genus yielding maximum score. In case of a tie, the sequence is marked as unclassified.
In this method we use nearest-neighbour classification based on multinomial probabilities. Nearest-neighbour methods have no specific training step, but use the training data as a database and perform a lookup based on some characteristics of the query sequence. Another 16S nearest-neighbour method, called the Similarity Rank tool, was published by Maidak et al.  for use in The Ribosomal Database Project.
As before we compute the (N×D) matrix A f r q by word counting, where N is the number of sequences in the training set. Then we divide all elements in a row by its row-sum to obtain multinomial probabilities, and these are stored in the (N×D) matrix A m l t . Thus, each training sequence, with its labelled genus, is represented as a row in this matrix.
For every new sequence we also count word frequencies and divide by the number of words in the sequence, producing a vector a similar to a row in A m l t . The Euclidean distance from a to all sequences (rows) in the training set is computed. The new sequence is assigned to the same genus as the nearest neighbour in the training set. In case of a tie, i.e. two or more genera are nearest neighbours, it is left unclassified.
Preprocessed nearest-neighbour (PLSNN)
In this method we extend the nearest-neighbour by combining it with the partial least squares (PLS) method . This is a supervised learning method that has been used in many bioinformatics applications (e.g. [28–32]). A reason for the wide-spread use of PLS is that it is especially applicable when we have many correlated explanatory variables, which is typical for the present K-mer data, especially as K increases.
The idea is to compute a linear mapping from the K-mer frequency space to a much lower dimensional space, and then look for nearest-neighbours in this low-dimensional space. In K-mer space every sequence has D=4 K coordinates, and in the nearest-neighbour method above all coordinates (K-mers) have equal weight. However, it is more than likely that some of these will be more or less important for recognizing a particular genus. Replacing the original D dimensional space by a smaller number of combinations can be seen as a preprocessing of the data before the nearest-neighbour step, hopefully resulting in more ‘correct’ distances between sequences when seeking the nearest neighbour.
From the training data we again compute the (N×D) matrix A m l t as above. This is used as the matrix of explanatory variables in training the PLS-method. The response is the genus for each sequence. This is coded as a row-vector of G elements, with 1 in position g if the sequence comes from genus g and 0 in all other positions. This assembles into an (N×G) matrix Y.
where the (N×C) dimensional matrix S consist of linear combinations of the columns in A m l t , and R is some orthonormal projection matrix. The rows of S are the training sequences represented in the C-dimensional subspace with maximum covariance to genus information. In this representation we have filtered out less important variation in K-mer frequencies, e.g. variation within genera. Distances between sequences in this space should be more sensitive to between-genus variation and less sensitive to within-genus variation. For every word length K we tested 8 different dimensions C. The maximum was set to C max = min(N−1,D−1,2000), and we used C=i C max /8 for i=1,2…,8.
For every new sequence we compute a vector a similar to a row in A m l t . From (14) it follows that A m l t R≈S since R is orthonormal, and thus we can compute s=a R. The vector s is the representation of the new sequence in the subspace spanned by S. The new sequence is finally classified with the nearest-neighbour method as before, where Euclidean distances from s to all rows of S areconsidered.
Results and discussion
We have tested five methods for K-mer based classification of 16S sequences, using a 10-fold cross validation, on two different data sets to compare their performance.
Results from a 10-fold cross validation. Classification errors (% misclassified) for the different methods at their optimal word length and for various data sets. Singleton genera errors are not included since they add the same to all methods
p-values for pairwise comparison of methods. Results from ANOVA on the effect of methods. The RDP is considered our control level and the p-values stated in the table below are the pairwise comparison for the four other methods to RDP
A difference between 4.2 % (full-length PLSNN Trainingset9) and 5.4 % (full-length RDP Trainingset9) error may seem small, but for building the taxonomy itself, there is no excuse for ignoring any improvement in methods. In principle the error should be zero. In a more practical use, where we want to classify a large number of sequences, a difference in 1 % means many misclassified sequences. Computation time is also an issue that should be taken into consideration. The RDP, Multinomial and Markov methods are fast and easy to both train and use for subsequent classification. All nearest-neighbour methods, including NN and PLSNN, are slower since they require distance computations for each new sequence to every sequence in the training set. The PLSNN method requires heavy computations during training, but once this has been done, new sequences are classified faster (and better) than with NN since distances are computed in a smallersub-space.
The Markov method appears to be the clear loser in our tests. Not only does it give poorest best-case results, but we also noticed that the best word length for the Markov classifier changed from four to seven depending on the data set. The uncertainty in word length makes this method unstable and unreliable and it is discarded as a fruitful approach for 16S sequence classification.
In the PLSNN method we employ the PLS method as a preprocessing of the count data, finding linear combinations of the K-mer counts having maximum class information. If we consider word length seven there are 47=16384 different K-mers. A full-length 16S sequence has around 1500 words of this length, which means more than 90 % of these K-mers occur zero times in any given sequence. Not all K-mers of this length can be equally important and a dimension reduction must be possible. We found that for K>6 a reduction to 2000 dimensions gave the best PLS-performance. Thus, for K=7 we reduce the coordinate-space from more than 16000 dimensions to 2000 before computing distances. Still, 2000 dimensions is remarkably large, but of course affected by the fact that we want to classify into a huge number of distinct genera. If the training set includes 1800 different genera, it is perhaps not surprising that we need at least this many dimensions to get a proper resolution to discriminate between them. This huge number of categories, as well as the considerable size variation between them seen in Fig. 1, makes this a rather special classification problem with several methodological challenges worthpursuing.
In  flat priors were used in their RDP-classifier. The results presented above also employ this strategy, assuming all genera are equally likely to occur in a new 16S sequence. If genera with many sequences in the training set are truly more widespread, this should be taken into account, and priors reflecting the abundance of each genus in the training set should produce better classifications. On the other hand, if a small training sample is due to an unexplored or newly discovered genus frequency-weighted priors supplies no further information to the data. We tested the RDP-classifier and the Multinomial method with both prior strategies on Trainingset9. The results were surprisingly similar regardless of priors. For word length eight the misclassified sequences were practically identical for the two cases, both for full-length and fragmented sequences. With this lack of differences we conclude that, unless very good arguments for the opposite can be provided, flat priors should be used. A flat prior means a single parameter (probability) is used for the entire population instead of (Ockham’s razor) favours the simplersolution.
In the results we observed an error-plateau or barrier below which no K-mer based method seemed to reach. Data sets like Trainingset9 and SilvaSet will always contain some proportion of questionable classifications partly since the actual relatedness between various genera is unknown, but also because the 16S gene itself is not a flawless marker. Variability between copies within the same genome as well as recombination events have been reported even for this highly conserved gene [33, 34]. If some sequences have been assigned to an incorrect genus from the beginning, classification errors seems unavoidable. Wang et al. tested their naïve Bayes classifier (RDP-classifier) on two different data sets in their work  from 2007. They reported the classification errors at genus level as 8.6 % and 7.9 % for the Bergey corpus and the NCBI corpus, respectively. The difference from our errors for the same method can be explained by a data set effect, presumably the data sets we have been ‘improved’ by eliminating some obvious mis-assignments since 2007. This emphasizes the importance of training data for classification performance .
We have compared the popular RDP method to four other K-mer based methods with respect to classification of prokaryotes based on 16S sequences. The differences in classification performance are significant, but all methods apart from the Markov method seem to stabilise on a classification error less than 6.6 % for word length bigger than seven for full-length sequences. Small extensions to the RDP method, such as counting the frequencies instead of just present/absent, seem to be an advantage, as also pointed out by . On full-length 16S sequences, the Preprocessed nearest-neighbour method stands out as the best, and should be considered for high-precision jobs. With shorter ‘reads’ as input, the naïve Bayes based Multinomial method proves to be the method with least classification errors and therefore the method, out of the five presented methods, which is the optimal option for rapid taxonomicassignments.
The study also reveals the importance of high-quality data for improving the classifications further. All methods seem to level out at some error which is inherent in the various data sets, and it is not likely that improved methods as such will lower this barrier. We have pointed out the special features of this type of data; a large number of categories (genera) in combination with an extreme skewness in their sizes. A key to improve classification is to obtain gold standard training sets in which all efforts have been made to have as few genera as possible with only a few sequences. Increasing the number of representative sequences from one to three or four can greatly increase the classificationaccuracy.
The K-mer methods examined here ignore the position specific information that is most likely important to discriminate certain genera. For further improvement of classification, pattern-recognition methods that takes into account position specific information through the 16S sequences may be a good place to start.
Hilde Vinjes scholarship has been fully financed by the Norwegian University of Life Sciences.
- Özlem Taştan Bishop. 2014. Bioinformatics and Data Analysis in Microbiology. Rhodes University, South Africa: Caister Academic Press.Google Scholar
- Woese CR, Stackebrand E, Macke TJ, Fox GE. A phylogenetic definition of the major eubacterial taxa. Syst Appl Microbiol. 1985; 6:143–51.View ArticlePubMedGoogle Scholar
- Woese CR. Bacterial evolution. Syst Appl Microbiol. 1987; 51:221–71.Google Scholar
- Caporaso JG, Lauber CL, Walters WA, Berg-Lyons D, Lozupone CA, Turnbaugh PJ, Fierer N, Knight R. Global patterns of 16S rRNA diversity at a depth of millions of sequences per sample. Proc Natl Acad Sci USA. 2011; 108(Suppl 1):4516–22.View ArticlePubMedGoogle Scholar
- Claesson M, Wang Q, O’Sullivan O, Greene-Diniz R, Cole J, Ross R, et al.Comparison of two next-generation sequencing technologies for resolving highly complex microbiota composition using tandem variable 16S rRNA gene regions. Nucleic Acids Res. 2010; 38:e200.View ArticlePubMedPubMed CentralGoogle Scholar
- Tringe S, Hugenholtz P. A renaissance for the pioneering 16S rRNA gene. Curr Opin Microbiol. 2008; 11:442–6.View ArticlePubMedGoogle Scholar
- Woese CR, Fox GE. Phylogenetic structure of the prokaryotic domain: the primary kingdoms. Proc Natl Acad Sci U S A. 1977; 74(11):5088–90.View ArticlePubMedPubMed CentralGoogle Scholar
- Pace NR. A molecular view of microbial diversity and the biosphere. Science. 1997; 276:734–40.View ArticlePubMedGoogle Scholar
- Harmsen D, Karch H. 16S rDNA for diagnosing pathogens: a living tree. ASM News. 2004; 70:19–24.Google Scholar
- Greengenes database. 2015. [http://greengenes.lbl.gov/cgi-bin/nph-index.cgi]. Accessed date May 18, 2015.
- Ribosomal Database Project. 2015. [http://rdp.cme.msu.edu/]. Accessed date May 18, 2015.
- SILVA database. 2015. [http://www.arb-silva.de/]. Accessed date May 18, 2015.
- Ludwig W, Strunk O, Klugbauer S, Klugbauer N, Weizenegger M, Neumaier J, Bachleitner M, Schleifer KH. Bacterial phylogeny based on comparative sequence analysis. Electrophoresis. 1998; 19(4):554–68.View ArticlePubMedGoogle Scholar
- Kolaczkowski B, Thornton JW. Performance of maximum parsimony and likelihood phylogenetics when evolution is heterogeneous. Nature. 2004; 431:980–4.View ArticlePubMedGoogle Scholar
- Bergeys. 2015. [http://www.bergeys.org/]. Accessed date May 28, 2015.
- Rudi K, Zimonja M, Næs T. Alignment-independent bilinear multivariate modelling (AIBIMM) for global analyses of 16S rRNA gene phylogeny. Intl J Syst Evol Microbiol. 2006; 56:1565–75.View ArticleGoogle Scholar
- Liu Z, DeSantis TZ, Andersen GL, Knight R. Accurate taxonomy assignments from 16S rRNA sequences produced by highly parallel pyrosequencers. Nucleic Acids Research. 2008; 36:e120.View ArticlePubMedPubMed CentralGoogle Scholar
- Soergel D, Dey N, Knight R, Brenner S. Selection of primers for optimal taxonomic classification of environmental 16S rRNA gene sequences. ISME J. 2012; 6:1440–4.View ArticlePubMedPubMed CentralGoogle Scholar
- Wang Q, Garrity GM, Tiedje JM, Cole JR. Naïve Bayesian Classifier for Rapid Assignment of rRNA Sequences into the New Bacterial Taxonomy. Appl Enviromental Microbiol. 2007; 73:5261–67.View ArticleGoogle Scholar
- Science Watch. 2015. [http://archive.sciencewatch.com/dr/erf/2011/11decerf/11decerfCole/]. Accessed date May 18, 2015.
- Vinje H, Almøy T, Liland KH, Snipen L. A systematic search for discriminating sites in the 16S ribosomal RNA gene. Microb Inf Experimentation. 2014; 4:2.View ArticleGoogle Scholar
- Liu K, Wong T. Naïve Bayesian Classifiers with Multinomial Models for rRNA Taxonomic Assignment. IEEE/ACM Trans Comput Biol Bioinformatics. 2013; 10(5):1334–9.View ArticleGoogle Scholar
- Stone M. Cross-validatory choice and assesment of statistical predictions. J R Stat Soc Serie B-Methodological. 1974; 36:111–47.Google Scholar
- R. 2015. [http://www.r-project.org/]. Accessed date May 18, 2015.
- Davidsen T, Rødland EA, Lagesen K, Seeberg E, Rognes T, Tønjum T. Biased distribution of DNA uptake sequences towards genome maintenance genes. Nucleic Acids Res. 2004; 32(3):1050–8.View ArticlePubMedPubMed CentralGoogle Scholar
- Maidak BL, Larsen N, McCaughey MJ, Overbeek R, Olsen GJ, Fogel K, Blandy J, R WC. The Ribosomal Database Project. Nucleic Acids Res. 1994; 22(17):3485–7.View ArticlePubMedPubMed CentralGoogle Scholar
- Wold S, Martens H, Wold H. The Multivariate Calibration Problem in Chemistry solved by the PLS Method. Lect Notes Math. 1983; 973:286–93.View ArticleGoogle Scholar
- Nguyen DV, Rocke DM. Tumor classification by partial least squares using microarray gene expression data. Bioinformatics. 2002; 18:39–50.View ArticlePubMedGoogle Scholar
- Nguyen DV, Rocke DM. Multi-class cancer classification via partial least squares with gene expression profiles. Bioinformatics. 2002; 18:1216–26.View ArticlePubMedGoogle Scholar
- Aarøe J, Lindahl T, Dumeaux V, Sæbø S, Tobin D, Hagen N, Skaane P, Lönneborg A, Sharma P, Børresen-Dale A. Gene expression profiling of peripheral blood cells for early detection of breast cancer. Breast Cancer Res. 2010; 12:R7. doi:http://dx.doi.org/10.1186/bcr2472.View ArticlePubMedPubMed CentralGoogle Scholar
- Mehmood T, Martens H, Sæbø S, Warringer J, Snipen L. Mining for genotype-phenotype relations in Saccharomyces using partial least squares. BMC Bioinformatics. 2011; 12(318):318.View ArticlePubMedPubMed CentralGoogle Scholar
- Mehmood T, Bohlin J, Kristoffersen AB, Sæbø S, Warringer J, Snipen L. Exploration of multivariate analysis in microbial coding sequence modeling. BMC Bioinformatics. 2012; 13:97. doi:http://dx.doi.org/10.1186/1471-2105-13-97.View ArticlePubMedPubMed CentralGoogle Scholar
- Vetrovsky T, Baldrian P. The variability of the 16S rRNA gene in bacterial genomes and its consequences for bacterial community analyses. PLoS ONE. 2013; 8(2):e57923. doi:http://dx.doi.org/10.1371/journal.pone.0057923.View ArticlePubMedPubMed CentralGoogle Scholar
- Kitahara K, Miyazaki K. Natural and experimental evidence for horizontal gene transfer of 16S rRNA. Mobile Genet Elem. 2013; 3(1):e24210.View ArticleGoogle Scholar
- Werner JJ, Koren O, Hugenholtz P, DeSantis TZ, Walters WA, Caporaso JG, et al.Impact of training sets on classification of high-throughput bacterial 16s rRNA gene surveys. ISME J. 2012; 6:94–103.View ArticlePubMedGoogle Scholar
This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.