Each input data must be mapped to a numerical value. This is a fundamental process of the algorithm because it enables different datasets to be analysed together, regardless of their nature. In this manner, dissimilar passport data, single nucleotide polymorphisms (SNPs), restriction fragment length polymorphisms (RFLPs), geographic information and phenotypical traits can be included in one comprehensive dataset. To consistently represent each data type, reference tables are implemented according to the nature of each particular data: genetic information (originally represented as character elements) is now represented by a numerical vector, and trait variation, simple sequence repeat (SSR) molecular markers and passport data can be represented as either binary or normalized data depending on the quantitative/qualitative nature of the data. The original data and reference tables for this study are available in supplementary material 1. Data transformation for this study rendered a matrix containing the representation of MC samples ( i ₁, i ₂, i ₃, … i _n ) with ( j ₁, j ₂, j ₃, … j _m ) elements each, where n is the total number of samples, and m is the number of included samples characteristics, represented by a numerical value as data _{( i, j)}.

Numerical representations of each jth data element can be treated as frequency values in m data time in such a manner that each ith sample is treated as a discrete signal. The i signal corresponds to the information behaviour from each sample. This perspective will enable the implementation of SPTs such as the discrete Fourier transform and power spectrum comparison. Although SPTs can be implemented on all data available for each sample, not all data elements contain the same informativeness value to discriminate between samples. To overcome the informative difference in each j element of data, a principal component analysis (PCA) can be performed to rearrange data into a new matrix that has the high informative elements of data at the beginning and that arranges subsequent elements according to their informativeness, discarding those whose variance equals 0. This process renders two new matrices: the original characteristics mapped vectors matrix ( x) and rearranged variance value matrix ( X). Matrix X, therefore, contains n samples that are formed by a numerical vector with m= m-(non informative characteristics).

The main objective of Fourier transform is the decomposition of any signal into a complex histogram of frequencies. Signal function is then represented as a vectorial function whose angle and magnitude determine a sampled point in the signal ¹⁹ . The original Fourier model is expressed as follows:

f ^ ( ξ ) = ∫ − ∞ ∞ f ( x ) e − 2 π i x ξ d x           ( 1 )

where x is the temporal variable, ξ it the frequential variable, i is a -1 square root and e is the natural exponent. From equation 1, a derivative can be determined for any point ξ sampled in the signal.

f ( x ) [ cos ( 2 π e ξ ) + i * ​ sin ( 2 π e ξ ) ]           ( 2 )

Fourier transform can be implemented into any complex numerical series, but in a practical sense, the computational cost increases exponentially. Thus, fast Fourier transform (FFT) is more often implemented and can be defined according to Cooley-Tukey algorithm ²⁰ as follows:

X k = ∑ n = 0 N − 1 x n e − i π k n N         ( 3 )

where N is the vector length, x is the temporal variable, i is a -1 square root and e is the natural exponent; in such matter that an euclidean representation - with the angle, magnitude and phase that corresponds to their position in the signal - exists for any signal dot.

Therefore, mapping any signal into a vectorial representation that contains information from every original signal dot is possible. From this complex vector, useful data can be retrieved to establish a comparison between them that indirectly represents the original signal’s juxtaposition ²¹ .

Inspired by the genomic signal processing alignment-free distance (GAFD) model ²² , each signal corresponding to the PCA-mapped accessions data in a set Ŝ _i was converted into its frequency representation by applying discrete Fourier transform. Its power spectrum F ^ i was then computed. Subsequently, the distance d( i, j) for a given pair of comprehensive data signals was calculated by obtaining the mean squared error (MSE) of their respective power spectra:

D ( i , j ) = ∑ x ( F ^ i ( x ) − F ^ j ( x ) ) 2           ( 4 )

Finally, a distance matrix (DM) was created by performing a pairwise comparison of all sequences in the set.

In parallel, we constructed a point-to-point (RAW) DM on the basis of the MSE given to a pair of signals prior to the PCA analysis.

Selecting a CC by this method requires the generation of a DM for each sample of the MC; this provides the interrelations among samples and enables adequate selection. A schematic of the complete workflow is presented in Figure 1.

In the past, several methodological procedures have been implemented to select K elements from an MC on the basis of information provided by its DM; among such procedures, the most frequently used one is the hierarchical clustering method ¹¹ . However, the current algorithm does not rely on hierarchical clustering for CC selection, instead - similar to the least distance stepwise sampling method ²³ - CC elements are selected by an iterative process, where r samples are selected by different criteria (which may be individually implemented) on each iteration.

Selection criteria (based on the DM without hierarchical clustering) for the current algorithm is as follows:

In cases where multiple samples share selection values, an appearance priority will complete the criteria.

An example of selection process is presented in Figure 2 and its final result is presented in Figure 3.

Once the selected samples ( r) are included in the future CC, they (along with others that are identical to them ( s)) are removed from X for the next iteration; then, a DM ₂ with n ₂ = n– r– s is calculated. This process will continue Z times until R >= K, where R = ( r ₁ + r ₂…. r _Z ) and K = predefined CC elements desired.

As discussed previously, the best way to evaluate a CC depends on the purpose of that CC, and even if it can be evaluated from the same dataset from which it was constructed, evaluating it with a different dataset ⁷ is desirable. In this study, we use other datasets for our evaluation whenever possible. The list given below provides the evaluation parameters implemented in this study.

To compare the obtained CCs with an established methodology, we implemented Core Hunter 2 (CH) ¹³ as a reference and used it with the program’s default parameters on the agrological and genomic datasets.

To determine the efficiency of the analysis of data behaviour by a point-to-point direct comparison, a synthetic dataset was constructed using binary data ( Sdata) with manageable n and m elements ( supplementary material 1).

To test the algorithm in real biological-context scenarios, the CCs from different MCs were constructed and evaluated.

To test the algorithm’s CCs versus the scores of the MCs, 780 rice ( Oriza sativa(L.)) accession and 423 foxtail millet ( Setaria italica subsp. italica (L.) P. Beauv.) accession data were retrieved from the National Institute of Agrobiological Sciences (NIAS) http://www.gene.affrc.go.jp/databases_en.php.

According to the available data, different datasets were assembled. The 762 SNPs from the 780 rice accessions retrieved from the NIAS database ( Rdata) were divided arbitrarily into two subsets of 331 SNPs each for constructing two smaller datasets ( RdataI and RdataIII). In addition, ATs were categorized and mapped into the binary data for 273 of the 780 accessions, resulting in 38 variables ( RdataII). The variables from the 423 foxtail millet genotypes with transposon displays ²⁴ were used as a single dataset ( Fdata). For a subset of 141 accessions ( FdataI), 9 ATs were categorized and mapped into binary data, resulting in 28 variables ( FdataII). The substitution tables used during this mapping are presented as supplementary material 1.

All methodological procedures (except for CH, which was implemented according to the software’s default parameters and which is available for download at www.corehunter.org) were performed using FreeMat v4.2 ( www.freemat.sourceforge.net). All original codes are available as supplementary material 1.

The selection criteria were chosen to look for the best possible distribution of selected CC elements within the DM. Although hierarchical clustering has proven to be an effective method for determining collection structure and sampling CC ²⁵ and although it has been implemented in different crops ^{26, 27} and included in various selection algorithms ¹¹ , hierarchical reconstruction presents the challenge of selecting an appropriate model for biological interpretation that can be applied to everything from unweighted pair-group averages to Markov models in Bayesian estimations ²⁸ . To avoid the challenge of selecting a reconstruction model, we decided to work strictly with the DM. By selecting the items described in this methodology, we aimed to retrieve representative elements from among the distributions of collections; however, because of its iterative nature, this methodology may render high redundancy under certain data distributions. Despite this limitation, the methodology has proven to be capable of selecting representative elements of the MC’s diversity.

Evaluation criteria were applied according to Odong et al. ⁷ without excluding the classic criteria used in 9, 10. The selected CCs render proper results in general terms. As expected, selected CCs did not always reach for optimal values for MD and CR, this is due the fact that it is not the aim of the selection method to render a CC with similar distribution to that of the MC, but to make sure to include as much diversity as possible.

It is our belief that scoring the CC sets obtained with these methodologies will enable genetic resource banks to provide clear descriptors of what their CC strengths and limitations are with respect to the MC from which they come and will provide adequate tools for determining the possible purposes of the selected CCs.

Although several representations of genotypic characteristics (particularly those involving DNA sequences ^{29– 31} ) have been proposed, real-number-based mappings have not been discarded; indeed, this type of mapping has been highly studied for signal analysis even when they share two principal problems: the preferential magnitude of some nucleotides and the non-equidistance of all nucleotides ^{32, 33} . The arbitrary values selected for SNPs’s numerical representation of genotypes aim to maintain equidistant relations among purines and among pyrimidines in such a manner that the same distance is also preserved between at least one of them and the undetermined values. ATs are represented as binary data. This representation may prove useful for discrete data but requires a clustering procedure for continuous data. In this study, we arbitrarily generated clusters for the latter and then represented them as the former. Although this implementation may not be the most accurate regarding biological or agronomical significance, it serves as the first approach for testing the feasibility of the use of signal processing techniques when merging several datasets to construct one CC.

The RAW comparison establishes a distance value on the basis of the average distance between each mapped value on each element while the FFT power spectra implementation compares the signals in the frequency domain. Using FFT, establishing a DM on the basis of how data ’shifted’ rather than on the basis of average point-to-point comparisons was possible. The FFT approach provides a different DM, where its compared elements are clustered based on the similarity of the shift among data, regardless of whether the shift is in the opposite phase. We believe that this procedure may reveal additional information about the relations not only between elements but also between the individual components within each element.

FFT comparisons of signals without PCA are a good approach for CC selection. Nevertheless, PCA implementation enables us to avoid possible misleads in random data arrangements, as, for example, palindromic data that could result in the same power spectra. Moreover, through PCA, we could organize data according to their levels of impact on the difference between accessions, which - when their magnitudes were obtained - inherently rendered a representation of informativity relations among values. This ’data behaviour’ was used as the element for pairwise comparisons, and although this approach clusters differently from RAW comparisons, we believe that it will provide a new perspective for CC selection and open the possibility of further data exploration.

Our first approach was to measure the comparisons under different K values. We compared the approach of the RAW signals with the PCA-FFT-treated signals. Results from Sdata, Fdata and Rdata are presented in Table 1– Table 3. As expected, most evaluation criteria improved as K increased.

The use of FFT signals renders better overall scores than the use of RAW signals in Sdata and Fdata; however, this advantage diminishes in Rdata. We speculate that this difference can be explained by the mapping procedures used; further research regarding this matter is encouraged.

Using the CHs’ rendered K values, we used both CH and FFT to generate the CCs on the above datasets. The evaluation of these CCs is summarized in Table 4 and in Figure 4, Figure 5. Both methodologies rendered similar results, yet PCA rendered better results on parameters representing MC distribution; this could be an effect of the selection method’s intrinsic redundancy.

Thus far, the proposed CC selection method and algorithm appear worthy of further exploration. We are aware that two particular fundamental elements require immediate attention. First, a better mapping solution for both genotypic and AT numerical representation needs to be determined. Second, the selection system developed by us is directly based on the DM and is prone to high redundancy in some data distributions. As discussed earlier, this selection system was chosen in order to avoid the problems associated with hierarchical clustering and further allocation selections ^{13, 34} . Both issues should be addressed in the near future.

To demonstrate that FFT-based CC selection can include and analyse data regardless of its origin, we concatenated corresponding signals from FdataI with FdataII as well as RdataI and RdataIII with RdataII to construct MFdata, MRdataI and MRdataIII. The comprehensive sets were used to construct CCs; the sets were then compared with both their original genotype and phenotype MCs. These comparisons are shown in Table 5– Table 8, and their distributions are represented in Figure 6– Figure 9.

These comprehensive CCs showed overall better scores than genotypic-only CCs when compared with genotypic-only data. On the contrary, there was a better overall score in phenotypic-only CCs when compared against phenotypic-only data. In the latter case, it should be kept in mind that comprehensive data also consider genotypic data; this could explain why better selections are made when only phenotypic data are considered because genotypic variations may reduce the impact of some phenotypic traits in the PCA analysis.

The generation of a DM based on signal comparisons originating from mixed data construction enables us to explore one of the most interesting applications of this algorithm. By mapping genotypic and AT data, constructing a single signal with all data available for a particular accession is possible. The possibility of including genotypic data with phenotypic traits, geographical locations, climates, habitats, nutritional requirements, symbiotic relationships and so forth provides an opportunity for determining the best information to be included in the selection process in order to cope with the particular objectives for which that CC is being selected. This concept, in addition to adequate scoring systems, may prove useful in designing tailored CCs that comply with specific research/breeding objectives.