For the comparison of our SVM classifiers to the approaches proposed in 32 36 , we first measure the performance of our methods on the four tasks used in 32 (see Methods for details). The approach in 32 is outperformed by a neural network approach proposed in 15 . However, we do not compare our methods to the latter method, since it already reaches its computational limits for the small datasets with only a few thousand short sequences (cf. 15 , page 138) and hence is not suitable for large-scale genome-wide computations. On each task we trained SVMs with the weighted degree kernel (WD) 28 , and the weighted degree kernel with shifts (WDS) 34 . On the NN269 Acceptor and Donor sets we additionally trained an SVM using the locality improved kernel (LIK) 21 ; as it gives the weakest prediction performance and is computationally most expensive we exclude this model from the following investigations. As a benchmark method we also train higher order Markov Chains (MCs) (e.g. 44 ) of "linear" structure and predict with the posterior log-odds ratio (cf. Methods section). Note that Position Specific Scoring Matrices (PSSM) are recovered as the special case of zeroth-order MCs. A summary of our results showing the auROC and auPRC scores is displayed in Table 1.

We first note that the simple MCs perform already fairly well in comparison to the SVM methods. Surprisingly, we find that the MC-SVM proposed in 32 performs worse than the MCs. (We have reevaluated the results in 32 with the code provided by the authors and found that the stated false positive rate of their method is wrong by a factor of 10. We have contacted the authors for clarification and they published an erratum 45 . The results for MC-SVMs given in Table 1 are based on the corrected performance measurement.) As anticipated, for the two acceptor recognition tasks, EBN and MCs are outperformed by all kernel models which are performing all at a similar level. However, we were intrigued to observe that for the DGSplicer Donor recognition task, the MC based predictions outperform the kernel methods. For NN269 Donor recognition their performance is similar to the performance of the kernel methods.

There are at least two possible explanations for the strong performance of the MCs. First, the DGSplicer data set has been derived from the genome annotation, which in turn might have been obtained using a MC based gene finder. Hence, the test set may contain false predictions easier reproduced by a MC. Second, the window size for the DGSplicer Donor recognition task is very short and has been tuned in 36 to maximize the performance of their method (EBN) and might be suboptimal for SVMs. We investigated these hypotheses with two experiments:

• In the first experiment, we shortened the length of the sequences in DGSplicer Acceptor from 36 to 18 (with consensus AG at 8,9). We retrained the MC and WD models doing a full model selection on the shortened training data. We observe that on the shortened data set the prediction performance drops drastically for both MC and WD (by 60% relative) and that, indeed, the MC outperforms the WD method (to 12.9% and 9% auPRC, respectively).

• In a second experiment, we started with a subset of our new data sets generated from the genomes of worm and human which only uses EST or cDNA confirmed splice sites (see methods section). In the training data we used the same number of true and decoy donor sites as in the DGSplicer data set. For the test data we used the original class ratios (in order to allow a direct comparison to following experiments; cf. Table 2). Training and testing sequences were shortened from 218 nt in steps of 10 nt down to 18 nt (same as in the DGSplicer donor data set). We then trained and tested MCs and WD-SVMs for the sets of sequences of different length. Figure 1 shows the resulting values for the auPRC on the test data for different sequence lengths. For the short sequences, the prediction accuracies of MCs and SVMs are close for both organisms. For human donor sequences of length 18 MCs indeed outperform SVMs. With increasing sequence length, however, the auPRC of SVMs rapidly improves while it degrades for MCs. Recall that the short sequence length in the DGSplicer data was tuned through model selection for EBN, and thus the performance of the EBN method will degrade for longer sequences 36 , so that we can safely infer that our methods would also outperform EBN for longer training sequences.

The results do not support our first hypothesis that the test data sets are enriched with MC predictions. However, the results confirm our second hypothesis that the poor performance of the kernel methods on the NN269 and DGSplicer donor tasks is due to the shortness of sequences. We also conclude that discriminative information between true and decoy donor sequences lies not only in the close vicinity of the splice site but also further away (see also the illustrations using k-mer scoring matrices below). Therefore, the careful choice of features is crucial for building accurate splice site detectors and if an appropriate window size is chosen, the WD kernel based SVM classifiers easily outperform previously proposed methods.

In this section we compare SpliceMachine 29 with the WD kernel based SVMs. SpliceMachine 46 is the current state-of-the art splice site detector. It is based on a linear SVM and outperforms the freely available GeneSplicer 47 12 by a large margin 29 . We therefore perform an extended comparison of our methods to SpliceMachine on subsets of the genome-wide datasets (cf. the results and methods sections). One fifth and one twenty-fifth of the data set was used each for training and for independent testing for cress and human, respectively. We downloaded the SpliceMachine feature extractor 48 to generate train and test data sets. Similar to the WD kernel, SpliceMachine makes use of positional information around the splice site. As it explicitly extracts these features it is however limited to a low order context (small d). In addition, SpliceMachine explicitly models coding-frame specific compositional context using tri- to hexamers. Note that this compositional information is also available to a gene finding system for which we are targeting our splicing detector. Therefore, in order to avoid redundancy, compositional information should ideally not be used to detect the splicing signal. Nevertheless, for comparative evaluation of the potential of our method, we augment our WD kernel based methods with 6 spectrum kernels 49 (order 3, 4, 5, each up- and downstream of splice site) and use the same large window sizes as were found out to be optimal in 29 . For cress acceptor [-85, +86], donor [-87, +124], and for human acceptor [-105, +146], donor [-107, +104]. For the WD kernel based SVMs, we fixed the model parameters C = 1 and d = 22, and for WDS we additionaly fixed the shift parameter σ = 0.5. For the SpliceMachine we performed an extensive model selection and found C = 10^-3 to be consistently optimal. We trained with C ∈ {10⁰, 10^-1, 10^-2, 10^-3, 5·10^-4, 10^-4, 10^-5, 10^-6, 10^-7, 10^-8}. Using these parameter settings we trained SVMs a) on the SpliceMachine features (SM), b) using the WD kernel (WD) c) using the WD kernel augmented by the 6 spectrum kernels (WDSP) d) using the WDS kernel (WDS) and e) using the WDS and spectrum kernels (WDSSP). Table 3 shows the area under the ROC and precision recall curve obtained in this comparison. Note that SpliceMachine always outperforms the WD kernel, but is in most cases inferior to the WDS kernel. Furthermore, complementing the WD kernels with spectrum kernels (methods WDSP and WDSSP) always improves precision beyond that of SpliceMachine. As this work is targeted at producing a splicing signal detector to be used in a gene finder, we will omit compositional information in the following genome-wide evaluations. To be fair, one can note that a WDS kernel using a very large shift is able to capture compositional information, and the same holds to some extend for the WD kernel when it has seen many training examples. It is therefore impossible to draw strong conclusions on whether window size and (ab)use of compositional features will prove beneficial when the splice site predictor is used as a module in a gene finder, which we hope is enabled by our work providing genome wide predictions.

Figure 2 shows the prediction performance in terms of the auROC and auPRC of SVMs using the MC and the WD kernel on the human acceptor and donor splice data that we generated for this work (see the methods section) for varying training set sizes. For training we use up to 80% of all examples and the remaining examples for testing. MCs and SVMs were trained on sets of size varying between 1000 and 8.5 million examples. Here we sub-sampled the negative examples by a factor of five. We observe that the performance steadily increases when using more data for training. For SVMs, over a wide range, the auPRC increases by about 5% (absolute) when the amount of data is multiplied by a factor of 2.7. In the last step, when increasing from 3.3 million to 8.5 million examples, the gain is slightly smaller (3.2 – 3.5%), indicating the start of a plateau. Similarly MCs improve with growing training set sizes. As MCs are computationally a lot less demanding, we performed a full model selection over the model order and pseudo counts for each training set size. For the WD-SVM the parameters were fixed to the ones found optimal in the results section. Nevertheless MCs did constantly perform inferior to WD-SVMs. We may conclude that one should train using all available data to obtain the best results. If this is infeasible, then we suggest to only sub-sample the negatives examples in the training set, until training becomes computationally tractable. The class distribution in the test set, however, should never be changed unless explicitly taken into account in evaluation.

For the pilot study we use the NN269 and the DGSplicer data sets originating from 9 and 32 , respectively. The data originates from 56 and the training and test splits can be downloaded from 46 . The data sets only include sequences with the canonical splice site dimers AG and GT. We use the same split for training and test sets as used in 32 . A description of the properties of the data set is given in Table 5.

We collected all known ESTs from dbEST 57 (as of February 28, 2007; 346,064 sequences for worm, 514,613 sequences for fly, 1,276,130 sequences for cress, 1,168,572 sequences for fish, and 7,915,689 sequences for human). We additionally used EST and cDNA sequences available from wormbase 58 for worm, (file confirmed_genes.WS170) 59 for fly, (files na_EST.dros and na_dbEST.same.dmel) 60 for cress, (files cDNA_flanking_050524.txt and cDNA_full_reading_050524.txt) 61 for fish, (file Danio_rerio.ZFISH6.43.cdna.known.?? and 62 for fish and human (file dr_mgc_mrna.fasta for fish and hs_mgc_mrna.fasta for human). Using blat 63 we aligned ESTs and cDNA sequences against the genomic DNA (releases WS170, dm5, ath1, zv6, and hg18, respectively). If the sequence could not be unambiguously matched, we only considered the best hit. The alignment was used to confirm exons and introns. We refined the alignment by correcting typical sequencing errors, for instance by removing minor insertions and deletions. If an intron did not exhibit the consensus GT/AG or GC/AG at the 5' and 3' ends, we tried to achieve this by shifting the boundaries up to two base pairs (bp). If this still did not lead to the consensus, then we split the sequence into two parts and considered each subsequence separately. Then, we merged alignments if they did not disagree and if they shared at least one complete exon or intron.

In a next step, we clustered the alignments: In the beginning, each of the above EST and cDNA alignments were in a separate cluster. We iteratively joined clusters, if any two sequences from distinct clusters match to the same genomic location (this includes many forms of alternative splicing).

From the clustered alignments we obtained a compact splicing graph representation 64 , which can be easily used to generate a list of positions of true acceptor and donor splice sites. Within the boundaries of the alignments (we cut out 10 nt at both ends of the alignments to exclude potentially undetected splice sites), we identified all positions exhibiting the AG, GT or GC dimer and which were not in the list of confirmed splice sites. The lists of true and decoy splice site positions were used to extract the disjoint training, validation and test sets consisting of sequences in a window around these positions. Additionally, we divided the whole genome into regions, which are disjoint contiguous sequences containing at least two complete genes; if an adjacent gene is less than 250 base pairs away, we merge the adjacent genes into the region. Genes in the same region are also assigned to the same cross-validation split. The splitting was implemented by defining a linkage graph over the regions and by using single linkage clustering. The splits were defined by randomly assigning clusters of regions to the split.

The training and model selection of our methods for each of the four tasks was done separately by partial 10-fold cross-validation on the training data. For this, the training sets for each task are divided into 10 equally sized data splits, each containing the same number of splice sequences and the same proportion of true versus decoy sequences. For each parameter combination, we use only 3 out of the 10 folds, that is we train 3 times by using 9 out of the 10 training data splits and evaluate on the remaining training data split. Since the data is highly unbalanced, we choose the model with the highest average auPRC score on the three evaluation sets. This best model is then trained on the complete training data set. The final evaluation is done on the corresponding independent test sets (same as in 32 ). The supplementary website includes tables with all parameter combinations used in model selection for each task and the chosen parameters.

The training and model selection of our methods for the five organisms on the acceptor and donor recognition tasks was done separately by 5-fold cross-validation. The optimal parameter was chosen by selecting the parameter combination that maximized the auPRC score. This model selection method was nested within 5-fold cross-validation for final evaluation of the performance. The reported auROC and auPRC are averaged scores over the five cross-validation splits. The supplementary website includes tables with all considered parameter combinations and the chosen parameters for each task. All splits were based on the basis of the clusters derived from EST and cDNA alignments, such that different splits come from random draws of the genome.

The sensitivity is defined as the fraction of correctly classified positive examples among the total number of positive examples, i.e. it equals the true positive rate TPR = TP/(TP + FN). Analogously, the fraction FPR = FP/(TN + FP) of negative examples wrongly classified as positive is called the false positive rate. Plotting TPR against FPR results in the Receiver Operator Characteristic Curve (ROC) 41 42 . Plotting the positive predictive value PPV = TP/(FP + TP), i.e. the fraction of correct positive predictions among all positively predicted examples, against the TPR, one obtains the Precision Recall Curve (PRC) (see e.g., 43 ). The area under the ROC and PRC are denoted by auROC and auPRC respectively.

In order to provide an interpretable and comparable confidence score of the SVM predictions, we estimated the conditional likelihood P(y = 1|f( x )) of the true label y being positive for a given SVM output value f( x ). To do this, we applied a piecewise linear function which was determined on the validation set (the same used for the classifier model selection). We used the N = 50 quantiles taken on the SVM output values as supporting points φ _i , i = 1,...,N. For convenience, denote φ ₀ = -∞. For each point φ _i the corresponding π ^ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFapaCgaqcamaaBaaaleaacqWGPbqAaeqaaaaa@3007@ -value, which represents the empirical probability of being a true positive, was computed as π ^ i = n i T P n i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFapaCgaqcamaaBaaaleaacqWGPbqAaeqaaOGaeyypa0ZaaSaaaeaacqWGUbGBdaqhaaWcbaGaemyAaKgabaGaemivaqLaemiuaafaaaGcbaGaemOBa42aaSbaaSqaaiabdMgaPbqabaaaaaaa@3964@ , where n _i (i = 1,...,N) is the number of examples with output values φ _i-1 ≤ f( x ) <φ _i and n i T P MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGUbGBdaqhaaWcbaGaemyAaKgabaGaemivaqLaemiuaafaaaaa@31F3@ is the number of true splice sites in the same output range. Additionally, we determined the empirical cumulative probability as follows π ^ i c = ( ∑ j = i N n j T P ) / ( ∑ j = i N n j ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacuWFapaCgaqcamaaDaaaleaacqWGPbqAaeaacqWGJbWyaaGccqGH9aqpdaqadaqaamaaqadabaGaemOBa42aa0baaSqaaiabdQgaQbqaaiabdsfaujabdcfaqbaaaeaacqWGQbGAcqGH9aqpcqWGPbqAaeaacqWGobGta0GaeyyeIuoaaOGaayjkaiaawMcaaiabc+caVmaabmaabaWaaabmaeaacqWGUbGBdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqWGPbqAaeaacqWGobGta0GaeyyeIuoaaOGaayjkaiaawMcaaaaa@4C5E@ . In order to obtain a smooth and strictly monotonically increasing probability estimate, we solve the following quadratic optimization problem:

min⁡ π, π c ∈ ℝ + N ∑ i=1 N ( s i (π)+ t i ( π c )) s.t. π i ≤ π i c , for all i=1,...,N π i ≤ π i+1 −ε, for all i=1,...,N−1 π i c ≤ π i+1 c −ε, for all i=1,...,N−1, MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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v7aLjabcYcaSiabbccaGiabbAgaMjabb+gaVjabbkhaYjabbccaGiabbggaHjabbYgaSjabbYgaSjabbccaGiabdMgaPjabg2da9iabigdaXiabcYcaSiabc6caUiabc6caUiabc6caUiabcYcaSiabd6eaojabgkHiTiabigdaXaqaaaqaaiab=b8aWnaaDaaaleaacqWGPbqAaeaacqWGJbWyaaGccqGHKjYOcqWFapaCdaqhaaWcbaGaemyAaKMaey4kaSIaeGymaedabaGaem4yamgaaOGaeyOeI0Iae8xTduMaeeilaWIaeeiiaaIaeeOzayMaee4Ba8MaeeOCaiNaeeiiaaIaeeyyaeMaeeiBaWMaeeiBaWMaeeiiaaIaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemOta4KaeyOeI0IaeGymaeJaeiilaWcaaaaa@C8C6@

where ε = 10^-4 is a small constant ensuring that the functions are strictly monotonically increasing and s i ( π ) = n i ∑ j = 1 N n j ( π i − π ^ i ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGZbWCdaWgaaWcbaGaemyAaKgabeaakiabcIcaOGGadiab=b8aWjabcMcaPiabg2da9maalaaabaGaemOBa42aaSbaaSqaaiabdMgaPbqabaaakeaadaaeWaqaaiabd6gaUnaaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabg2da9iabigdaXaqaaiabd6eaobqdcqGHris5aaaakiabcIcaOGGaciab+b8aWnaaBaaaleaacqWGPbqAaeqaaOGaeyOeI0Iaf4hWdaNbaKaadaWgaaWcbaGaemyAaKgabeaakiabcMcaPmaaCaaaleqabaGaeGOmaidaaaaa@4B00@ and t i ( π c ) = ∑ j = i N n j ∑ j = 1 N n j ( π i c − π ^ i c ) 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG0baDdaWgaaWcbaGaemyAaKgabeaakiabcIcaOGGadiab=b8aWnaaCaaaleqabaGaem4yamgaaOGaeiykaKIaeyypa0ZaaSaaaeaadaaeWaqaaiabd6gaUnaaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabg2da9iabdMgaPbqaaiabd6eaobqdcqGHris5aaGcbaWaaabmaeaacqWGUbGBdaWgaaWcbaGaemOAaOgabeaaaeaacqWGQbGAcqGH9aqpcqaIXaqmaeaacqWGobGta0GaeyyeIuoaaaGccqGGOaakiiGacqGFapaCdaqhaaWcbaGaemyAaKgabaGaem4yamgaaOGaeyOeI0Iaf4hWdaNbaKaadaqhaaWcbaGaemyAaKgabaGaem4yamgaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaaaa@5604@ ensuring that big differences between the final and empirical estimates in ranges with many outputs are penalized stronger. Using the newly computed values π ₁,...,π _N , we can compute for any output value f( x ) the corresponding posterior probability estimate P(y = 1|f( x )) by linear interpolation

P ( y = 1 | f ( x ) ) = { π 1 for  f ( x ) < φ 1 r ( φ i , φ i + 1 ) for  φ i ≤ f ( x ) < φ i + 1 , π N for  f ( x ) ≥ φ N MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGqbaucqGGOaakcqWG5bqEcqGH9aqpcqaIXaqmcqGG8baFcqWGMbGzcqGGOaakieWacqWF4baEcqGGPaqkcqGGPaqkcqGH9aqpdaGabeqaauaabaqadiaaaeaaiiGacqGFapaCdaWgaaWcbaGaeGymaedabeaaaOqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdAgaMjabcIcaOiab=Hha4jabcMcaPiabgYda8iab+z8aMnaaBaaaleaacqaIXaqmaeqaaaGcbaGaemOCaiNaeiikaGIae4NXdy2aaSbaaSqaaiabdMgaPbqabaGccqGGSaalcqGFgpGzdaWgaaWcbaGaemyAaKMaey4kaSIaeGymaedabeaakiabcMcaPaqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGiab+z8aMnaaBaaaleaacqWGPbqAaeqaaOGaeyizImQaemOzayMaeiikaGIae8hEaGNaeiykaKIaeyipaWJae4NXdy2aaSbaaSqaaiabdMgaPjabgUcaRiabigdaXaqabaGccqGGSaalaeaacqGFapaCdaWgaaWcbaGaemOta4eabeaaaOqaaiabbAgaMjabb+gaVjabbkhaYjabbccaGiabdAgaMjabcIcaOiab=Hha4jabcMcaPiabgwMiZkab+z8aMnaaBaaaleaacqWGobGtaeqaaaaaaOGaay5Eaaaaaa@7E5D@

where r ( φ i , φ i + 1 ) = π i + 1 ( f ( x ) − φ i ) + π i ( φ i + 1 − f ( x ) ) φ i + 1 − φ i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGYbGCcqGGOaakiiGacqWFgpGzdaWgaaWcbaGaemyAaKgabeaakiabcYcaSiab=z8aMnaaBaaaleaacqWGPbqAcqGHRaWkcqaIXaqmaeqaaOGaeiykaKIaeyypa0ZaaSaaaeaacqWFapaCdaWgaaWcbaGaemyAaKMaey4kaSIaeGymaedabeaakiabcIcaOiabdAgaMjabcIcaOGqadiab+Hha4jabcMcaPiabgkHiTiab=z8aMnaaBaaaleaacqWGPbqAaeqaaOGaeiykaKIaey4kaSIae8hWda3aaSbaaSqaaiabdMgaPbqabaGccqGGOaakcqWFgpGzdaWgaaWcbaGaemyAaKMaey4kaSIaeGymaedabeaakiabgkHiTiabdAgaMjabcIcaOiab+Hha4jabcMcaPiabcMcaPaqaaiab=z8aMnaaBaaaleaacqWGPbqAcqGHRaWkcqaIXaqmaeqaaOGaeyOeI0Iae8NXdy2aaSbaaSqaaiabdMgaPbqabaaaaaaa@634A@ . The cumulative posterior probability P ^c (y = 1|f( x )) is computed analogously. The above estimation procedure was performed separately for every classifier.

The posterior log-odds of a probabilistic model with parameters θ are defined by

g ( x ) : = log ⁡ ( P ( y = + 1 | x , θ ) ) − log ⁡ ( P ( y = − 1 | x , θ ) ) = log ⁡ ( P ( x | θ + ) ) − log ⁡ ( P ( x | θ − ) ) + b , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeGadaaabaGaem4zaCMaeiikaGccbmGae8hEaGNaeiykaKcabaGaeiOoaOJaeyypa0dabaGagiiBaWMaei4Ba8Maei4zaCMaeiikaGIaemiuaaLaeiikaGIaemyEaKNaeyypa0Jaey4kaSIaeGymaeJaeiiFaWNae8hEaGNaeiilaWcccmGae4hUdeNaeiykaKIaeiykaKIaeyOeI0IagiiBaWMaei4Ba8Maei4zaCMaeiikaGIaemiuaaLaeiikaGIaemyEaKNaeyypa0JaeyOeI0IaeGymaeJaeiiFaWNae8hEaGNaeiilaWIae4hUdeNaeiykaKIaeiykaKcabaaabaGaeyypa0dabaGagiiBaWMaei4Ba8Maei4zaCMaeiikaGIaemiuaaLaeiikaGIae8hEaGNaeiiFaWNae4hUde3aaWbaaSqabeaacqGHRaWkaaGccqGGPaqkcqGGPaqkcqGHsislcyGGSbaBcqGGVbWBcqGGNbWzcqGGOaakcqWGqbaucqGGOaakcqWF4baEcqGG8baFcqGF4oqCdaahaaWcbeqaaiabgkHiTaaakiabcMcaPiabcMcaPiabgUcaRiabdkgaIjabcYcaSaaaaaa@7B34@

where b is a bias term. We use f( x ) = sign(g( x )) for classification and Markov chains of order d

P ( x | θ ± ) = P ( x 1 , ... , x N | θ ± ) = P ( x 1 , ... , x d | θ ± ) ∏ i = d + 1 N P ( x i | x i − 1 , ... , x i − d , θ ± ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8056@

as for instance described in 44 . Each factor in this product has to be estimated in model training, i.e. one counts how often each symbol appears at each position in the training data conditioned on every possible x _i-1,...,x _i-d . Then for given model parameters θ we have

P ( x | θ ± ) = θ 0 ± ( x 1 , ... , x d ) ∏ i = d + 1 N θ i ± ( x i , ... , x i − d ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@655E@

where θ 0 ± MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF4oqCdaqhaaWcbaGaeGimaadabaGaeyySaelaaaaa@3172@ is an estimate for P(x ₁,...,x _d ) and θ _i (x _i ,...,x _i-d ) an estimate for P(x _i |x _i-1,...,x _i-d ). As the alphabet has four letters, each model has (N - d + 1)·4 ^d+1 parameters and the maximum likelihood estimate is given by:

θ 0 ( s 1 , ... , s d ) = 1 m + π ( ∑ k = 1 m I ( s 1 = x 1 k ∧ ⋯ ∧ s d = x d k ) + π ) θ i ( s i , ... , s i − d ) = ∑ k = 1 m I ( s i = x i k ∧ ⋯ ∧ s i − d = x i − d k ) + π ∑ k = 1 m I ( s i = x i − 1 k ∧ ⋯ ∧ s i − d = x i − d k ) + 4 π MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeabbaaaaeaaiiGacqWF4oqCdaWgaaWcbaGaeGimaadabeaakiabcIcaOiabdohaZnaaBaaaleaacqaIXaqmaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4Cam3aaSbaaSqaaiabdsgaKbqabaGccqGGPaqkcqGH9aqpaeaacaWLjaWaaSaaaeaacqaIXaqmaeaacqWGTbqBcqGHRaWkcqWFapaCaaWaaeWaaeaadaaeWbqaaGqabiab+LeajjabcIcaOiabdohaZnaaBaaaleaacqaIXaqmaeqaaOGaeyypa0JaemiEaG3aa0baaSqaaiabigdaXaqaaiabdUgaRbaakiabgEIizlabl+UimjabgEIizlabdohaZnaaBaaaleaacqWGKbazaeqaaOGaeyypa0JaemiEaG3aa0baaSqaaiabdsgaKbqaaiabdUgaRbaakiabcMcaPiabgUcaRiab=b8aWbWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdaakiaawIcacaGLPaaaaeaacqWF4oqCdaWgaaWcbaGaemyAaKgabeaakiabcIcaOiabdohaZnaaBaaaleaacqWGPbqAaeqaaOGaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaem4Cam3aaSbaaSqaaiabdMgaPjabgkHiTiabdsgaKbqabaGccqGGPaqkcqGH9aqpaeaacaWLjaWaaSaaaeaadaaeWaqaaiab+LeajjabcIcaOiabdohaZnaaBaaaleaacqWGPbqAaeqaaOGaeyypa0JaemiEaG3aa0baaSqaaiabdMgaPbqaaiabdUgaRbaakiabgEIizlabl+UimjabgEIizlabdohaZnaaBaaaleaacqWGPbqAcqGHsislcqWGKbazaeqaaOGaeyypa0JaemiEaG3aa0baaSqaaiabdMgaPjabgkHiTiabdsgaKbqaaiabdUgaRbaakiabcMcaPiabgUcaRiab=b8aWbWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdaakeaadaaeWaqaaiab+LeajjabcIcaOiabdohaZnaaBaaaleaacqWGPbqAaeqaaOGaeyypa0JaemiEaG3aa0baaSqaaiabdMgaPjabgkHiTiabigdaXaqaaiabdUgaRbaakiabgEIizlabl+UimjabgEIizlabdohaZnaaBaaaleaacqWGPbqAcqGHsislcqWGKbazaeqaaOGaeyypa0JaemiEaG3aa0baaSqaaiabdMgaPjabgkHiTiabdsgaKbqaaiabdUgaRbaakiabcMcaPiabgUcaRiabisda0iab=b8aWbWcbaGaem4AaSMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdaaaaaaaaa@C8DB@

where I(·) is the indicator function, k enumerates over the number of observed sequences m, and π is the commonly used pseudocount (a model parameter, cf. 44 ) which is also tuned within the model selection procedure (cf. the model selection and evaluation section).

As the second method we use SVMs. The generated classification function can be written as

f ( x ) = sign ( ∑ i = 1 m y i α i K ( x i , x ) + b ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzcqGGOaakieWacqWF4baEcqGGPaqkcqGH9aqpcqqGZbWCcqqGPbqAcqqGNbWzcqqGUbGBdaqadaqaamaaqahabaGaemyEaK3aaSbaaSqaaiabdMgaPbqabaacciGccqGFXoqydaWgaaWcbaGaemyAaKgabeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaakiab9Pq8ljabcIcaOiab=Hha4naaBaaaleaacqWGPbqAaeqaaOGaeiilaWIae8hEaGNaeiykaKIaey4kaSIaemOyaigaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGTbqBa0GaeyyeIuoaaOGaayjkaiaawMcaaiabcYcaSaaa@5C25@

where y _i ∈ {-1, +1} (i = 1,...,m) is the label of example x _i . The α _i 's are Lagrange multipliers and b is the usual bias which are the results of SVM training 16 . The kernel K MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFke=saaa@3834@ is the key ingredient for learning with SVMs.

In the following paragraphs we describe the kernels which are used in this study. They are all functions defined on sequences. In the following x = x ₁ x ₂...x _N denotes a sequence of length N.

The locality improved (LI) kernel has been proven useful in the context of translation initiation site (TIS) recognition 21 . Similar to the polynomial kernel of degree d for discrete input data, this kernel considers correlations of matches up to order d. In contrast to polynomial kernels however, the LI kernel only considers local subsequence correlations within a small window of length 2l + 1 around a sequence position:

win p ( x , x ′ ) = ( 1 2 l + 1 ∑ j = − l + l I ( x p + j = x ′ p + j ) ) d , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqG3bWDcqqGPbqAcqqGUbGBdaWgaaWcbaGaemiCaahabeaakiabcIcaOGqadiab=Hha4jabcYcaSiqb=Hha4zaafaGaeiykaKIaeyypa0ZaaeWaaeaadaWcaaqaaiabigdaXaqaaiabikdaYiabdYgaSjabgUcaRiabigdaXaaadaaeWbqaaGqabiab+LeajjabcIcaOiabdIha4naaBaaaleaacqWGWbaCcqGHRaWkcqWGQbGAaeqaaOGaeyypa0JafmiEaGNbauaadaWgaaWcbaGaemiCaaNaey4kaSIaemOAaOgabeaakiabcMcaPaWcbaGaemOAaOMaeyypa0JaeyOeI0IaemiBaWgabaGaey4kaSIaemiBaWganiabggHiLdaakiaawIcacaGLPaaadaahaaWcbeqaaiabdsgaKbaakiabcYcaSaaa@59FE@

where p = l + 1,...,N - l. These window scores are then summed up over the length of the sequence using a weighting w _p which linearly decreases to both ends of the sequence, i.e. w p = { p − l p ≤ N / 2 N − p − l + 1 p > N / 2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWG3bWDdaWgaaWcbaGaemiCaahabeaakiabg2da9maaceqabaqbaeqabiGaaaqaaiabdchaWjabgkHiTiabdYgaSbqaaiabdchaWjabgsMiJkabd6eaojabc+caViabikdaYaqaaiabd6eaojabgkHiTiabdchaWjabgkHiTiabdYgaSjabgUcaRiabigdaXaqaaiabdchaWjabg6da+iabd6eaojabc+caViabikdaYaaaaiaawUhaaaaa@48CE@ . Then we have the following kernel:

K ( x , x ′ ) = ∑ p = l + 1 N − l w p win p ( x , x ′ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFke=scqGGOaakieWacqGF4baEcqGGSaalcuGF4baEgaqbaiabcMcaPiabg2da9maaqahabaGaem4DaC3aaSbaaSqaaiabdchaWbqabaGccqqG3bWDcqqGPbqAcqqGUbGBdaWgaaWcbaGaemiCaahabeaakiabcIcaOiab+Hha4jabcYcaSiqb+Hha4zaafaGaeiykaKcaleaacqWGWbaCcqGH9aqpcqWGSbaBcqGHRaWkcqaIXaqmaeaacqWGobGtcqGHsislcqWGSbaBa0GaeyyeIuoakiabc6caUaaa@597D@

The weighting allows one to emphasize regions of the sequence which are believed to be of higher importance; in our case this is the center, which is the location of the splice site. (Note that the definition of the LI kernel in 21 is slightly different from ours. Previously the weighting was inside the window and was not very effective. Moreover, the version presented here of the kernel can be computed 2l + 1 times faster than the original one.)

The weighted degree (WD) kernel 28 uses a similar approach by counting matching subsequences u _δ,l ( x ) and u _δ,l ( x' ) between two sequences x and x' , with u _δ,l ( x ) = x _l x _l+1...x _l+δ-1 for all l and 1 ≤ δ ≤ d. Here, δ denotes the order (length of the subsequence) to be compared. The WD kernel is defined as

K ( x , x ′ ) = ∑ δ = 1 d w δ ∑ l = 1 N − δ + 1 I ( u δ , l ( x ) = u δ , l ( x ′ ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacqWFke=scqGGOaakieWacqGF4baEcqGGSaalcuGF4baEgaqbaiabcMcaPiabg2da9maaqahabaGaem4DaC3aaSbaaSqaaGGaciab9r7aKbqabaaabaGae0hTdqMaeyypa0JaeGymaedabaGaemizaqganiabggHiLdGcdaaeWbqaaGqabiab8LeajjabcIcaOiab+vha1naaBaaaleaacqqF0oazcqGGSaalcqWGSbaBaeqaaOGaeiikaGIae4hEaGNaeiykaKIaeyypa0Jae4xDau3aaSbaaSqaaiab9r7aKjabcYcaSiabdYgaSbqabaGccqGGOaakcuGF4baEgaqbaiabcMcaPiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOta4KaeyOeI0Iae0hTdqMaey4kaSIaeGymaedaniabggHiLdGccqGGSaalaaa@6A78@

where we choose the weighting to be w _δ = d - δ + 1. This kernel emphasizes position dependent information and the weighting decreases the influence for higher order matches, which would anyway have a higher contribution due to all their matching subsequences. It can be computed very efficiently without even extracting and enumerating all subsequences of the sequences 40 . Note that this kernel is similar to the spectrum kernel as proposed by 49 , with the main difference that the weighted degree kernel uses position specific information.

The weighted degree kernel with shifts (WDS) 34 is defined as

K ( x i , x j ) = ∑ δ = 1 d w δ ∑ l = 1 N − δ + 1 ∑ s = 0 s + l ≤ N S ( l ) δ s μ δ , l , s , x i , x j , μ δ , l , s , x i , x j = I ( u δ , l + s ( x i ) = u δ , l ( x j ) ) + I ( u δ , l ( x i ) = u δ , l + s ( x j ) ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@C3E8@

where w _δ is as before, δ _s = 1/(2(s + 1)) is the weight assigned to shifts (in either direction) of extent s, and S(l) determines the shift range at position l. Here, we choose S(l) = σ|l - l _c |, where l _c is the position of the splice site. An efficient implementation for this kernel allowing large scale computations is described in 40 .

For both the WD and WDS kernel we use the following normalization

K ˜ ( x , x ′ ) = K ( x , x ′ ) K ( x , x ) K ( x ′ , x ′ ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaat0uy0HwzTfgDPnwy1egaryqtHrhAL1wy0L2yHvdaiqaacuWFke=sgaacaiabcIcaOGqadiab+Hha4jabcYcaSiqb+Hha4zaafaGaeiykaKIaeyypa0ZaaSaaaeaacqWFke=scqGGOaakcqGF4baEcqGGSaalcuGF4baEgaqbaiabcMcaPaqaamaakaaabaGae8NcXVKaeiikaGIae4hEaGNaeiilaWIae4hEaGNaeiykaKIae8NcXVKaeiikaGIaf4hEaGNbauaacqGGSaalcuGF4baEgaqbaiabcMcaPaWcbeaaaaGccqGGUaGlaaa@55E7@

Training and evaluation of the SVMs and the MCs were performed using our shogun machine learning toolbox (cf. http://www.shogun-toolbox.org) 38 in which efficient implementations of the aforementioned kernels can be found.

Kernel methods are aimed directly at the classification task which is to discriminate between the true and decoy classes by learning a decision function separating the classes in an associated feature space. In contrast, generative methods like position weight matrices or Markov models are statistical models which represent the data under specific assumptions on the statistical structure and hence it is relatively straightforward to interpret their results. Although kernel methods outperform in many cases generative models, especially when the true statistical structure is more intricate than the assumed one, one of the main criticisms of kernel methods is the difficulty to directly interpret their decision function in a way that allows to gain biologically relevant insight. However, by taking advantage of our specific kernels and of their sparse representation, we are able to efficiently use the decision function of our SVMs in order to understand which k-mers at which positions are contributing the most in discriminating between true and decoy splice sites.

To see how this is possible, recall that, for SVMs, the resulting classifier can be written as a dot product between an α -weighted linear combination of support vectors mapped into the feature space (which is often only implicitly defined via the kernel function) 18 :

f ( x ) = ∑ i = 1 m α i y i Φ ( x i ) ︸ w ⋅ Φ ( x ) = ∑ i = 1 m α i y i K ( x i , x ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGMbGzcqGGOaakieWacqWF4baEcqGGPaqkcqGH9aqpdaagaaqaamaaqahabaacciGae4xSde2aaSbaaSqaaiabdMgaPbqabaGccqWG5bqEdaWgaaWcbaGaemyAaKgabeaakiabfA6agjabcIcaOiab=Hha4naaBaaaleaacqWGPbqAaeqaaOGaeiykaKcaleaacqWGPbqAcqGH9aqpcqaIXaqmaeaacqWGTbqBa0GaeyyeIuoaaSqaaiab=Dha3bGccaGL44pacqGHflY1cqqHMoGrcqGGOaakcqWF4baEcqGGPaqkcqGH9aqpdaaeWbqaaiab+f7aHnaaBaaaleaacqWGPbqAaeqaaOGaemyEaK3aaSbaaSqaaiabdMgaPbqabaWenfgDOvwBHrxAJfwnHbqeg0uy0HwzTfgDPnwy1aaceaGccqqFke=scqGGOaakcqWF4baEdaWgaaWcbaGaemyAaKgabeaakiabcYcaSiab=Hha4jabcMcaPaWcbaGaemyAaKMaeyypa0JaeGymaedabaGaemyBa0ganiabggHiLdGccqGGUaGlaaa@71B5@

In the case of sparse feature spaces, as with string kernels, one can represent w in a sparse form and then efficiently compute dot products between w and Φ( x ) in order to speed up SVM training or testing 40 . This sparse representation comes with the additional benefit of providing us with means to interpret the SVM classifier. For k-mer based string kernels like the spectrum kernel, each dimension w _u in w represents a weight assigned to that k-mer u. From the learned weighting one can thus easily identify the k-mers with highest absolute weight or above a given threshold τ: { u | |w _u | >τ}. Note that the total number of k-mers appearing in the support vectors is bounded by dN _s L where L is the maximum length of the sequences L = max ⁡ i = 1 , ... , m l x i MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGmbatcqGH9aqpcyGGTbqBcqGGHbqycqGG4baEdaWgaaWcbaGaemyAaKMaeyypa0JaeGymaeJaeiilaWIaeiOla4IaeiOla4IaeiOla4IaeiilaWIaemyBa0gabeaakiabdYgaSnaaBaaaleaaieWacqWF4baEdaWgaaadbaGaemyAaKgabeaaaSqabaaaaa@40F0@ . This approach also works for the WD kernel (with and without shifts). Here a weight is assigned to each k-mer with 1 ≤ k ≤ d at each position in the sequence. This allows us to generate the k-mer importance matrices, displayed in Figure 4, associated with our splice classifiers 54 . They display the weight which the SVM assigns to each k-mer at each position in the input sequence, i.e. given a SVM classifier trained with a WD kernel of degree d we extract the k-mers weightings for 1 ≤ k ≤ d ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaacaaaa@2E0C@ starting at position p = 1,...,N, where we used d as selected in model selection and d ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaacaaaa@2E0C@ = 1,...,8. This leads to a weighting for d ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaacaaaa@2E0C@ -mers u for each position in the sequence: W _u,p , which may be summarized by S d ˜ , p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaGafmizaqMbaGaacqGGSaalcqWGWbaCaeqaaaaa@31B0@ = max _u (W _u,p ). We compute this quantity for d ˜ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGKbazgaacaaaa@2E0C@ = 1,...,8 leading to the two 8 × 141 matrices, which are transformed into percentile values and then displayed color-coded in Figure 4. Note that the above computation can be done efficiently using string index data structures implemented in SHOGUN and described in detail in 40 .