TRIQ: a new method to evaluate triclusters

Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 https://doi.org/10.1186/s13040-018-0177-5 RESEARCH Open Access TRIQ: a new method to evaluate triclusters David Gutiérrez-Avilés1* , Raúl Giráldez1, Francisco Javier Gil-Cumbreras1and Cristina Rubio-Escudero2 *Correspondence: [email protected] 1School of Engineering, Pablo de Olavide University, Seville, Spain Full list of author information is available at the end of the article Abstract Background: Triclustering has shown to be a valuable tool for the analysis of microarray data since its appearance as an improvement of classical clustering and biclustering techniques. The standard for validation of triclustering is based on three different measures: correlation, graphic similarity of the patterns and functional annotations for the genes extracted from the Gene Ontology project (GO). Results: We propose TRIQ, a single evaluation measure that combines the three measures previously described: correlation, graphic validation and functional annotation, providing a single value as result of the validation of a tricluster solution and therefore simplifying the steps inherent to research of comparison and selection of solutions. TRIQ has been applied to three datasets already studied and evaluated with single measures based on correlation, graphic similarity and GO terms. Triclusters have been extracted from this three datasets using two different algorithms: TriGen and OPTricluster. Conclusions: TRIQ has successfully provided the same results as a the three single evaluation measures. Furthermore, we have applied TRIQ to results from another algorithm, OPTRicluster, and we have shown how TRIQ has been a valid tool to compare results from different algorithms in a quantitative straightforward manner. Therefore, it appears as a valid measure to represent and summarize the quality of tricluster solutions. It is also feasible for evaluation of non biological triclusters, due to the parametrization of each component of TRIQ. Keywords: Triclustering, Quality measure, Genetic algorithms, Biological quality, Graphical quality, Correlation Background Analysis of data structured in 3D manner is becoming an essential task in fields such as biomedical research, for instance in experiments studying gene expression data taking time into account. There is a lot of interest in this type of longitudinal experiments because they allow an in-depth analysis of molecular processes in which the time evolution is important, for example, cell cycles, development at the molecular level or evolution of diseases [1]. Therefore, the use of specific tools for data analysis in which genes are evaluated under certain conditions considering the time factor becomes necessary. In this sense, triclustering [2] appears as a valuable tool since it allows for the assessment of genes under a subset of the conditions of the experiment and under a subset of time points. The evaluation of solutions obtained by triclustering algorithms is challenging by the fact that there is no ground truth to describe triclusters present in real 3D data. In © The Author(s). 2018 Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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. Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 2 of 29 literature, the standard measures to evaluate tricluster solutions are based on three areas as can be seen in the triclustering publications [3–7]. First, correlation measures such as Pearson [8]orSpearman[9]. Second, graphic validation of the patterns extracted based on the graphic representation, i.e., how similar the genes from a tricluster are based on the graphic representation of the genes across conditions and time points. Third, functional annotations extracted from the Gene Ontology project (GO) [10] for the genes in the tricluster. However, we consider that providing a single evaluation measure capable of combining the information from the three aforementioned sources of validation is a neccesary task. Therefore, in this work we propose TRIQ, a validation measure which combines the three previously proposed validation mechanisms (correlation, graphic validation and functional annotation of the genes). The application of clustering and biclustering techniques to gene expression data has been broadly studied in the literature [11,12]. Although triclustering is the result from the natural evolution of the clustering and biclustering techniques, is still a very recent concept. However, nowadays, these techniques are arousing a great interest from the scientific community, which has caused a notable increase of the number of researches focused on finding new triclustering approaches. This section is to provide a general overview of triclustering published in literature. We particularly focus on the validation methods applied to assess the quality of the triclusters obtained. In 2005, Zhao and Zaki [3] introduced the triCluster algorithm to extract patterns in 3D gene expression data. They presented a measure to assess triclusters’s quality based on the symmetry property. They validated their triclusters based on their graphical representation and Gene Ontology (GO) results. g-triCluster, an extended and generalized version of Zhao and Zaki’s proposal, was published one year later [4]. The authors claimed that the symmetry property is not suitable for all patterns present in biological data and proposed the Spearman rank correlation [9]asamore appropriate tricluster evaluation measure. They also showed validation results based on GO. An evolutionary computation proposal was made in [13]. The fitness function defined is a multi-objective measure which tries to optimize three conflicting objectives: clusters size, homogeneity and gene-dimension variance of the 3D cluster. The tricluster quality validation was based on GO. LagMiner was introduced in [6] to find timelagged 3D clusters, what allows to find regulatory relationships among genes. It is based on a novel 3D cluster model called S2D3Cluster. They evaluated their triclusters on homogeneity, regulation, minimum gene number, sample subspace size and time periods length. Their validation was based on graphical representation and GO results. Hu et al. presented an approach focusing on the concept of Low-Variance 3-Cluster [5], which obeys the constraint of a low-variance distribution of cell values. This proposal uses a different functional enrichment tool called CLEAN [14], which uses GO as one of their components. The work in [7] was focused on finding Temporal Dependency Association Rules, which relate patterns of behavior among genes. The rules obtained are used to represent regulated relations among genes. They also validated their triclusters based on their graphical representation and GO results. Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 3 of 29 Tchagang et al. [15] proposed OPTricluster, a triclustering algorithm which obtains 3D short time series gene expression datasets by applying a statistical methodology. In this case, the authors carried out an in-depth biological validation based on GO, but they tested the robustness of OPTricluster to noise using the Adjusted Rand Index (ARI) [16], which also was used by aforementioned g-tricluster. In 2013, two new and very interesting approaches were proposed. On the one hand, the δ−TRIMAX algorithm [17], which applies a variant of the MSR adapted to 3D datasets and yields triclusters that have a MSR score below a threshold δ. This algorithm has a version based on evolutionary multi-objective optimization, named EMOA −δ−TRIMAX [18], which aims at optimizing the use of δ−TRIMAX by adding the capabilities of evolutionary algorithms to retrieve overlapping triclusters. On the other hand, OAC- Triclustering was also proposed by Gnatyshak et al. in [19]. In the following years, the authors developed improvements and extensions of this algorithm [20–22]. More recent works have extended the capabilities of the tricluster algorithms by combination of several approaches. Thereby, Liu et al. [23]mixedfuzzyclusteringand fuzzy biclustering algorithms in order to expands them to support 3D data and they used the F-Measure and Entropy as criteria to evaluate the performance. Also, Kakati et al. [24] combined parallel biclustering and distributed triclustering approaches to obtain improvements on the computational cost. In this work, the authors use a quality measure based on shifting and scaling patterns [25] to optimize the triclusters obtained. Most of the methods studied base the quality of the triclusters on the graphic representation or on metrics aimed at measuring diverse characteristics of such representation. From a biological point of view, the standard for validation of triclusters quality is based on GO functional annotations. Methods This section presents the TRIQ (TRIcluster Quality) validation measure [26], a novel method to evaluate the quality of triclusters extracted from gene expression datasets. From an overall perspective, TRIQ takes into account the three principal components of a tricluster, i.e. the genes, experimental conditions and time points, in order to measure its quality from three approaches: the level of biological notoriety of the cluster (biological quality), the graphic quality of the patterns of the genes in the tricluster (graphic quality), and the level of correlation of the genes in the tricluster by means of the Pearson [8] and the Spearman [9] indexes. Therefore, TRIQ is composed by a combination of four indexes: BIOQ (BIOlogical Quality), GRQ (GRaphic Quality), PEQ (PEarson Quality) and SPQ (SPearman Quality). In Eq. 1we define TRIQ as the weighted sum of each of the four aforementioned terms. Therefore, four associated weights must be defined: the weight for BIOQ, denoted as Wbio; the weight for GRQ, denoted as Wgr; the weight for PEQ, denoted as Wpe;andtheweight for SPQ, denoted as Wsp. TRIQ(TRI)=1 Wbio +Wgr +Wpe +Wsp ∗[Wbio ∗BIOQ(TRI) +Wgr ∗GRQ(TRI)+Wpe ∗PEQ(TRI)+Wsp ∗SPQ(TRI) (1) Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 4 of 29 This is a general definition of TRIQ.InordertoobtainaTRIQ index as balanced as possible among the four quality indexes BIOQ,GRQ,PEQ,andSPQ we performed an exhaustive testing procedure with well known datasets. Several combinations of values of BIOQ,GRQ,PEQ,andSPQ were tested, and in Fig. 1we show the results obtained. We see that that the value of TRIQ is slightly directly dependent on the weights related to correlation, PEQ,andSPQ. This is due to the fact that these values rank in the [0-1] interval, being usually high, from 0.7 to 1. The value of TRIQ has a higher level of dependence to the graphical quality, GRQ, and reverse strong dependence to the biological quality, BIOQ, due to the fact that BIOQ ranks in low values, usually around 10−3to 10−5. Based on this experiments, we have configured the TRIQ measure with the weights showed in Eq. 2in order to obtain a balanced value of TRIQ. Wbio =0.5, Wgr =0.4, Wpe =0.05, Wsp =0.05 (2) Next, we describe in depth each of the terms involved in the TRIQ measure. Correlation measures: PEQ and SPQ The correlation measures involved in TRIQ are Pearson’s PEQ [8]andSpearman’sSPQ [9] correlations. They have been chosen since they are the standard correlation measures and they are widely used in literature [4]. The correlation provides a numerical estimation of the dependence among the genes, conditions and times in the tricluster solutions. 0.00 0.25 0.50 0.75 0.00 0.25 0.50 0.75 1.00 WEIGHT VALUE TRIQ Weights Wbio Wgr Wpe Wsp Fig. 1 Representation of BIOQ,GRQ,PEQ,andSPQ influence on TRIQ Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 5 of 29 Given a tricluster TRI,wecomputePEQ and SPQ by the following mechanism. Given the subset of genes (see Eq. 3a), conditions (see Eq. 3b) and time stamps (see Eq. 3c), we obtain a value of expression for each combination gene, condition and time. For instance, for a tricluster consisting of four genes, two conditions and three time points, we have twenty four expression values. We then compute the Pearson correlation for each pair of values, and compute PEQ as the average of the absolute values to avoid negative and positive correlations canceling each other (see Eq. 4). Furthermore, for this measure we do not care if the correlation is positive or negative between values, we only want to know the level of correlation. The SPQ value is the equivalent using the Spearman correlation (see Eq. 5). TRIG=<g0,g1,...,gG>(3a) TRIC=<c0,c1,...,cC>(3b) TRIT=<t0,t1,...,tT>(3c) PEQ(TRI)=#exp i=0,j=0 Pearsoni=jexpi,expj  #pairs of exp (4) SPQ(TRI)=#exp i=0,j=0 Spearmani=jexpi,expj  #pairs of exp (5) with exp representing the expressions in each tricluster TRI. Graphical validation: GRQ The GRQ member of Eq. 1measures the graphical quality of the tricluster. This graphical quality of a tricluster is a quantitative representation of a qualitative measure: how homogeneous the members of the tricluster are. This method is widely used in literature for visual validation of the results by means of graphically representing the triclusters on their three components: genes, conditions and time points [3,6,7]. The GRQ index is described in Eq. 6. This measure is defined based on the normalization of the angle value given by MSL. The Multi SLope (MSL) evaluation function was defined in [27]and,givenatriclusterTRI, provides a numerical value of the similarity among the angles of the slopes formed by each profile shaped by the genes, conditions, and times of the tricluster. GRQ(TRI)=1−MSL(TRI) 2π(6) The MSL measure considers the three graphical views of a tricluster, also defined in [27]: TRIgct,TRIgtc,andTRItgc. These three terms are generally defined as TRIxop,with the expression levels of the tricluster represented in the Y axis, xrepresents the tricluster component in the X axis (genes or time points), orepresents the lines plotted in the graph (genes, conditions or time lines) and pthetypeoffacetsorpanelsrepresented(time points or conditions). We can observe an example of the TRItgc view of a tricluster with the genes g1,g4,g7and g10, the experimental conditions c2,c5and c8and the time points t0,t2,t11 in Fig. 2and see how each line or gene forms a set of angles (two for this particular example) defined by each time point in the X axis for every panel or experimental condition. Thus, MSL measures the differences among the angles formed by every series traced on each of the three graphic representations taking into account TRIgct,TRIgtc, and TRItgc. A near to zero value of MSL implies a better graphical quality of a tricluster Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 6 of 29 Fig. 2 Representation of how the MSL measure is calculated. This figure shows three graphics containing four lines each one with a representation of their slopes therefore, according to GRQ formulation in Eq. 6, a tricluster is graphically better the smaller the value of MSL. Biological validation: BIOQ The BIOQ member of Eq. 1measures the biological quality of the tricluster. Specifically, BIOQ uses the genes (TRIG) of the input tricluster TRI tocomputethisindex.Asyou can see in Eq. 7, the biological quality of a tricluster TRI is defined as the biological significance, SIGbio,ofthesetofgenesTRIGdivided by the Smax value. BIOQ(TRI)=SIGbio (TRIG) Smax (7) The SIGbio and Smax elements of the BIOQ index have been designed in order to represent, by means of a quantitative score, the value of the Gene Ontology analysis of the genes that compose the measured tricluster. The Gene Ontology Project (GO) [10] is a major bioinformatics initiative with the aim of standardizing the representation of gene and gene product attributes across species and databases, besides identifying the annotated terms, performs the statistical analysis for the over-representation of those terms, also providing a statistical significance p-value. However, it is also important to take into account how deep in the ontology the terms are annotated, with the deeper terms being more specific than the superficial ones [28]. The SIGbio and Smax elements are calculated based on the GO analysis that identifies, for a set of genes in a tricluster, the terms listed in each of the three available ontologies: biological processes, cellular components, and molecular functions. This GO analysis is performed with the software Ontologizer [29]. The computation of SIGbio consists on counting how many terms of the annotated genes of the tricluster in the GO analysis are in a particular intervals of p-value. Table 1repre- sents the ah-hoc designed system of intervals of p-value and scoring system. The intervals and the scoring system are defined in Eq. 8where for a given level, Interlis defined by a weight value wlfor the level, and by the lower and upper bounds (infland supl,respectively), being an open-closed p-values interval (Eq. 8a). The set of existing LV consists of all levels with Inflsmaller or equal to a minimum p-value, th. For each interval of each level Interl, the weight value wlis defined in Eq. 8c;Inflis defined in Eq. 8d,andsuplis defined in Eq. 8e. Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 7 of 29 Table 1 Biological significance intervals Level (l) Weight (wl)Interval(interl) 41 401 (0.0E-00,1.0E-40] 40 391 (1.0E-40,1.0E-39] 39 381 (1.0E-39,1.0E-38] 38 371 (1.0E-38,1.0E-37] 37 361 (1.0E-37,1.0E-36] 36 351 (1.0E-36,1.0E-35] 35 341 (1.0E-35,1.0E-34] 34 331 (1.0E-34,1.0E-33] 33 321 (1.0E-33,1.0E-32] 32 311 (1.0E-32,1.0E-31] 31 301 (1.0E-31,1.0E-30] 30 291 (1.0E-30,1.0E-29] 29 281 (1.0E-29,1.0E-28] 28 271 (1.0E-28,1.0E-27] 27 261 (1.0E-27,1.0E-26] 26 251 (1.0E-26,1.0E-25] 25 241 (1.0E-25,1.0E-24] 24 231 (1.0E-24,1.0E-23] 23 221 (1.0E-23,1.0E-22] 22 211 (1.0E-22,1.0E-21] 21 201 (1.0E-21,1.0E-20] 20 191 (1.0E-20,1.0E-19] 19 181 (1.0E-19,1.0E-18] 18 171 (1.0E-18,1.0E-17] 17 161 (1.0E-17,1.0E-16] 16 151 (1.0E-16,1.0E-15] 15 141 (1.0E-15,1.0E-14] 14 131 (1.0E-14,1.0E-13] 13 121 (1.0E-13,1.0E-12] 12 111 (1.0E-12,1.0E-11] 11 101 (1.0E-11,1.0E-10] 10 91 (1.0E-10,1.0E-09] 9 81 (1.0E-09,1.0E-08] 8 71 (1.0E-08,1.0E-07] 7 61 (1.0E-07,1.0E-06] 6 51 (1.0E-06,1.0E-05] 5 41 (1.0E-05,1.0E-04] 4 31 (1.0E-04,1.0E-03] 3 21 (1.0E-03,1.0E-02] 2 11 (1.0E-02,1.0E-01] 1 1 (1.0E-01,1.0E-00] interl=wl,infl,supl (8a) LV =∀l∈N:infl≤th (8b) wl=[(l−1)∗d]+1(8c) infl=s bl(8d) supl=s b(l−1)(8e) Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 8 of 29 This definition is made in order to establish a general interval system dependent on the parameters described above. For our purpose, we have settled these parameters as shown in Eq. 9; this configuration produces the intervals of Table 1, furthermore, it describes all the biological significance intervals for the configuration detailed in Eq. 9.Foreach row, weight (wl) and range (interl)foreachlevel(L) sorted in ascending order are shown. Each interval provides a set of p-values where their significance is directly related to the corresponding level, that is, a p-value is better the higher the level to which it belongs, and a p-value is better the closer to zero it is. th =1.0 ×10−40 (9a) d=10.0 (9b) b=10.0 (9c) s=1.0 (9d) LV ={1, ...,41}(9e) Taking into account each level land each predefined interval interl, the biological significance for the genes of the measured tricluster is defined in Eq. 10a as the addition of all scores for each level lfrom the LV level set Eq. 9e. The score function Sfor a level l (Eq. 10b) is defined by the multiplication of the concentration of terms for this level C(l), defined in Eq. 10c as the number of terms of the level ldivided by the total number of terms, by the weight of the level, and by the level plus a bonus function fbonus,defined in Eq. 10d as the sum of the level plus a bonus value Vbonus if the current level is the maximum level of LV or zero in any another case. SIGbio(TRIG)= l∈LV S(l)(10a) S(l)=C(l)∗wl∗l+fbonus(l)(10b) C(l)=#terms(l) #total terms (10c) fbonus(l)=if (lequaltol max)then l +Vbonus else 0 (10d) Again, this definition is made in order to establish a general system of SIGbio. For our purpose and as a result of an exhaustive testing, the Vbonus parameter has been settled to 0; this fact produces Smax as the maximum achievable score for the interval configuration as you can see in Eq. (11), that has been used to the SIGbio normalization in Eq. 7. Smax =Cmax ∗wl41 ∗l41+fbonus(41)=(1∗401 ∗41)+(41 +0)=16482 (11) Results In this section, we present how TRIQ works in an experimental environment. To reach this goal, we have used the TriGen algorithm [2] and the OPTricluster algorithm [15]in order to analyze the datasets, find triclusters and measure them with TRIQ. TriGen is based on an heuristic, genetic algorithm, and its performance greatly depends on the fitness function used to find the triclusters. There are three fitness functions available in TriGen: Mean Squared Residue 3D (MSR3D)[30], Least Squared Lines (LSL)[31] and Multi SLope Measure (MSL)[27]. OPTricluster identifies triclusters of genes with Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 9 of 29 expression levels having the same direction across the time point experiments in subsets of samples taking into consideration the sequential nature of the time-series. The three datasets analyzed that involve genes and experimental conditions examined under certain time points are: •Delu3D: The yeast cell cycle (Saccharomyces Cerevisiae) [32], in particular, the elutriation experiment. •DGDS45103D: The GDS4510 dataset from an experiment with mice (Mus Musculus) [33]. •DGSD44723D: The GDS4472 dataset from an experiments with humans (Homo Sapiens) [34]. The first dataset is available at the Stanford University website. The last two datasets have been retrieved from Gene Expression Omnibus [35], a repository of high throughput gene expression data. For each dataset, we have performed four algorithm executions: TriGen with MSR3D (hereon MSR3D), TriGen with LSL (hereon LSL), TriGen with MSL and (hereon MSL)and OPTricluster (hereon OPT). For each algorithm execution and dataset, we have yielded 10 triclusters and the TRIQ measure has been used to evaluate their quality. We have found 10 triclusters for each execution in order to have a high number of solutions where TRIQ can show its suitability. In the case of MSR3D,LSL,andMSL executions the number of triclusters has been chosen as one of the TriGen algorithm parameters and for OPT executions, the tricluster have been randomly selected from the wide collection of triclusters yielded. Summarizing, we present three experimental batches (Yeast Elutriation Dataset,Mouse GDS4510 Dataset and, Human GDS4472 Dataset) with four experiments each one: MSR3D,LSL,MSL and OPT. Yeast elutriation dataset This batch corresponds to the yeast (Saccharomyces Cerevisiae) cell cycle problem [32]. The yeast cell cycle analysis project’s goal is to identify all genes whose mRNA levels are regulated by the cell cycle. The resources used are public and available in http://genomewww.stanford.edu/cellcycle/. There, we can find information relative to gene expression values obtained from different experiments using microarrays. For our purpose, we have created a dataset Delu3Dfrom the elutriation experiment with 7744 genes, 13 experimental conditions, and 14 time points. Experimental conditions correspond to different statistical measures of the Cy3 and Cy5 channels while time points represent different moments of taking measures from 0 to 390 min. Delu3Dhas been used as the input of the TriGen and the OPTtricluster algorithminfour experiments: MSR3D,LSL,MSL and, OPT. Elutriation MSR3Dexperiment WecanverifyinTable2how TRI9has the best values of BIOQ,PEQ and SPQ whereas TRI10 has the best value of GRQ.TheGRQ,PEQ and SPQ values are stabilized from TRI2to TRI8until TRI9−TRI10 when these values reach the maximum. Regarding BIOQ values, these vary around 0.0012. Furthermore, TRIQ values are stable for all solutions except TRI9−TRI10 due to the genetic algorithms nature. In conclusion, TRI9is the best solution since it has the best value of TRIQ, closely followed by TRI10. Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 16 of 29 Table 10 LSL GDS4510 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.435171938 0.005902591 0.902662935 0.6949723 0.728137064 TRI20.427168871 0.002676716 0.885320349 0.700788434 0.733259027 TRI30.422560787 0.002909652 0.887599773 0.648196371 0.673124663 TRI40.42987221 0.004218243 0.901813346 0.657798589 0.68295641 TRI50.416008121 0.002006917 0.869706286 0.658784335 0.683658641 TRI60.41490654 0.001678815 0.866767983 0.661174595 0.686024185 TRI70.417068507 0.001262531 0.87574877 0.648950843 0.673803828 TRI80.40861261 0.001179271 0.854772399 0.649739272 0.672541022 TRI90.42454573 0.001683951 0.890635412 0.663209255 0.68578253 TRI10 0.417718487 0.002071672 0.874308872 0.657210052 0.681971994 a central value among the triclusters whereas BIOQ haspeakvaluesinTRI1and TRI4. The TRIQ values oscillates between 0.40 and 0.43 being TRI1,TRI4and TRI9the most outstanding solutions. We can conclude that TRI1is the best solution since it has the best value of TRIQ. GDS4510 MSL experiment For this experiment, we can observe in Table 11 how TRI1has the best value of BIOQ and GRQ meanwhile TRI2has the best value of PEQ and TRI8has the best value of SPQ. The PEQ and SPQ indexes of all solutions vary uniformly around 0.5 whereas all the GRQ values are close to 0.9. The BIOQ values oscillate between 0.0012 and 0.0019 reaching its higher value in the TRI1solution. The TRIQ values are in the [ 0.42, 0.44] interval, therefore we can conclude that they are good results for this experiment. The highest value of TRIQ is reached by TRI1, hence it is the best solution for this experiment. GDS4510 OPT experiment In Table 12 we can see how TRI2has the best value of BIOQ,TRI4has the best value of GRQ and, TRI9and TRI1have the best value of PEQ and SPQ respectively. The BIOQ values vary among [ 0.0012, 0.0016] interval with the exception of TRI2and TRI3whilst the GRQ values vary uniformly around the 0.80 value excepting TRI4.ThePEQ and SPQ values oscillate among the [ 0.5, 0.8] interval.The highest value of TRIQ is reached by TRI4, thus it is the best solution for this experiment. Table 11 MSL GDS4510 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.446289279 0.003624207 0.990551544 0.496833632 0.468297522 TRI20.430638622 0.001399471 0.945717127 0.515568434 0.51747227 TRI30.429698209 0.00149303 0.943951098 0.506740977 0.520684131 TRI40.425844616 0.001388422 0.935696236 0.506485147 0.510953062 TRI50.431185402 0.001224915 0.948344121 0.507510722 0.517195194 TRI60.422692807 0.001367523 0.927464145 0.507112049 0.513355693 TRI70.429129078 0.001401202 0.944156645 0.501640545 0.513675839 TRI80.436192976 0.001999141 0.958438573 0.512285251 0.524074273 TRI90.433173322 0.001604555 0.95182792 0.510885656 0.521911883 TRI10 0.422409162 0.001390319 0.928018397 0.501351072 0.508781791 Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 17 of 29 Table 12 OPT GDS4510 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.42009257 0.00134246 0.83749772 0.8574642 0.8309809 TRI20.42005895 0.00356033 0.85404927 0.82019206 0.71298946 TRI30.42901805 0.00256279 0.87133294 0.85498273 0.72908679 TRI40.44490172 0.00136003 0.93478154 0.77222529 0.63395639 TRI50.37648925 0.00128696 0.81223451 0.51226675 0.50677264 TRI60.37500834 0.00122195 0.80916885 0.50865504 0.50594149 TRI70.37783613 0.00125442 0.81768005 0.50153608 0.50120198 TRI80.37545313 0.00144891 0.80990327 0.50841987 0.50692736 TRI90.43860855 0.00167827 0.89471647 0.86719834 0.73045828 TRI10 0.37115418 0.00120689 0.80002563 0.50727999 0.50352975 GDS4510 summary We can see how the solutions are distributed regarding BIOQ and GRQ in Fig. 7; we observe that all points of all experiments are concentrated in a BIOQ interval of [ 0.0011, 0.0059]. Regarding the GRQ values, the MSR3Dand LSL experiments have all the solutions in the [ 0.83, 0.90] interval, the MSL experiment has all the solutions in the [ 0.92, 0.99] interval and, the OPT experiment has all the solutions in the [ 0.80, 0.95] interval. Regarding the PEQ and SPQ distribution we can see in Fig. 8how the majority of solutions are concentrated around the point PEQ =0.5, SPQ =0.5 in the MSR3Dand MSL experiments, meanwhile the solutions of LSL experiment are concentrated in the interval [ 0.625, 0.75] for PEQ and SPQ values and, the OPT experiment has his solutions dispersed in two groups: one group around the PEQ =0.5, SPQ =0.5 point and the other in an interval of [ 0.60, 0.83] for both PEQ and SPQ values. AglobalTRIQ-based ranking of solutions is shown in Table 13.TheMSL,LSL and a part of OPT solutions are placed alternatively on the first positions and the MSR3Dand the remaining of OPT solutions are in the last positions. We can see in Table 14 how the GDS4510 MSL experiment has the best value of the mean of TRIQ and the four experiments have low values of standard deviation having the MSR3Dexperiment the lowest value but very close to the MSL one. This fact implies MSR3D LSL MSL OPT 0.001 0.002 0.003 0.004 0.005 0.0060.001 0.002 0.003 0.004 0.005 0.0060.001 0.002 0.003 0.004 0.005 0.0060.001 0.002 0.003 0.004 0.005 0.006 0.80 0.85 0.90 0.95 1.00 BIOQ GRQ GDS4510 experiment: BIOQ v GRQ Fig. 7 BIOQ vs GRQ dispersion graph for each GDS4510 solution of each experiment Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 18 of 29 MSR3D LSL MSL OPT 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.5 0.6 0.7 0.8 0.55 0.65 0.75 PEQ SPQ GDS4510 experiment: PEQ v SPQ Fig. 8 PEQ vs SPQ dispersion graph for each GDS4510 solution of each experiment that the four experiments have a low sparse distribution and solutions with high quality. We can see in Fig. 9how the MSR3D,LSL and, MSL points are located on the bottom side of the graph meanwhile the OPT point is located in a high level of the standard deviation axis; on the other hand, LSL and, MSL points are located on the right side of the graph meanwhile the MSR3Dand OPT points are located in a left level of the average axis. Hence, in terms of standard deviation and average, we can conclude that MSL is the best experiment. The most valuable solution of all experiments is the tricluster TRI1of the MSL experiment. We can see in Fig. 10 how this solution depicts very uniform patterns consistent with the GRQ value. Also, we can see in Table 15 that this solution has Gene Ontology terms with low p-value such as sensory perception of chemical stimulus,olfactory receptor activity or detection of chemical stimulus involved in sensory perception of smell. The term olfactory receptor activity is a molecular function that combining with an odorant and transmitting the signal from one side of the membrane to the other to initiate a change in cell activity in response to detection of smell; this function is part of the biological process detection of chemical stimulus involved in sensory perception of smell that is the series of events involved in the perception of smell in which an olfactory chemical stimulus is received and converted into a molecular signal. Finally, that process is framed in a more general biological process called sensory perception of chemical stimulus that is the series of events required for an organism to receive a sensory chemical stimulus, convert it to a molecular signal, and recognize and characterize the signal. Human GDS4472 dataset The dataset, corresponding to this batch, has been obtained from GEO [35]undercode GDS4472 titled Transcription factor oncogene OTX2 silencing effect on D425 medulloblastoma cell line: time course [34]. In this experiment, the effect of doxycycline on medulloblastoma cancerous cells at six times after induction (0, 8, 16, 24, 48 and 96 h) had been studied. Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 19 of 29 Table 13 GDS4510 ranking table EXPERIMENT SOLUTION TRIQ BIOQ GRQ PEQ SPQ MSL TRI10.446289279 0.003624207 0.990551544 0.496833632 0.468297522 OPT TRI40.44490172 0.00136003 0.93478154 0.77222529 0.63395639 OPT TRI90.43860855 0.00167827 0.89471647 0.86719834 0.73045828 MSL TRI80.436192976 0.001999141 0.958438573 0.512285251 0.524074273 LSL TRI10.435171938 0.005902591 0.902662935 0.6949723 0.728137064 MSL TRI90.433173322 0.001604555 0.95182792 0.510885656 0.521911883 MSL TRI50.431185402 0.001224915 0.948344121 0.507510722 0.517195194 MSL TRI20.430638622 0.001399471 0.945717127 0.515568434 0.51747227 LSL TRI40.42987221 0.004218243 0.901813346 0.657798589 0.68295641 MSL TRI30.429698209 0.00149303 0.943951098 0.506740977 0.520684131 MSL TRI70.429129078 0.001401202 0.944156645 0.501640545 0.513675839 OPT TRI30.42901805 0.00256279 0.87133294 0.85498273 0.72908679 LSL TRI20.427168871 0.002676716 0.885320349 0.700788434 0.733259027 MSL TRI40.425844616 0.001388422 0.935696236 0.506485147 0.510953062 LSL TRI90.42454573 0.001683951 0.890635412 0.663209255 0.68578253 MSL TRI60.422692807 0.001367523 0.927464145 0.507112049 0.513355693 LSL TRI30.422560787 0.002909652 0.887599773 0.648196371 0.673124663 MSL TRI10 0.422409162 0.001390319 0.928018397 0.501351072 0.508781791 OPT TRI10.42009257 0.00134246 0.83749772 0.8574642 0.8309809 OPT TRI20.42005895 0.00356033 0.85404927 0.82019206 0.71298946 LSL TRI10 0.417718487 0.002071672 0.874308872 0.657210052 0.681971994 LSL TRI70.417068507 0.001262531 0.87574877 0.648950843 0.673803828 LSL TRI50.416008121 0.002006917 0.869706286 0.658784335 0.683658641 LSL TRI60.41490654 0.001678815 0.866767983 0.661174595 0.686024185 LSL TRI80.40861261 0.001179271 0.854772399 0.649739272 0.672541022 MSR3DTRI70.40677296 0.004479209 0.882477468 0.520330064 0.510517317 MSR3DTRI10.399937853 0.001348086 0.870211819 0.516112583 0.507469069 MSR3DTRI20.397972383 0.001177971 0.866535835 0.511042941 0.504338338 MSR3DTRI40.397028323 0.0014405 0.863853884 0.512208068 0.503122322 MSR3DTRI60.392316477 0.00190466 0.850869722 0.513791033 0.506534134 MSR3DTRI30.391066411 0.001255371 0.849235889 0.508613518 0.506273874 MSR3DTRI90.390891083 0.001281294 0.848296937 0.510926324 0.507706903 MSR3DTRI10 0.390730352 0.001137925 0.8484396 0.50930819 0.506402803 MSR3DTRI50.388644055 0.001187588 0.842929885 0.511734835 0.505831309 MSR3DTRI80.3851186 0.001240227 0.834606686 0.508323861 0.504792392 OPT TRI70.37783613 0.00125442 0.81768005 0.50153608 0.50120198 OPT TRI50.37648925 0.00128696 0.81223451 0.51226675 0.50677264 OPT TRI80.37545313 0.00144891 0.80990327 0.50841987 0.50692736 OPT TRI60.37500834 0.00122195 0.80916885 0.50865504 0.50594149 OPT TRI10 0.37115418 0.00120689 0.80002563 0.50727999 0.50352975 Table 14 GDS4510 summary table EXPERIMENT BEST SOLUTION BEST TRIQ MEAN STDEV MSR3DTRI70.40677296 0.39404785 0.006348192 LSL TRI10.435171938 0.42136338 0.007979308 MSL TRI10.446289279 0.430725347 0.006987671 OPT TRI40.44490172 0.402862087 0.030140772 Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 20 of 29 MSR3D LSL MSL OPT 0.010 0.015 0.020 0.025 0.030 3 4 .0 2 4.01 4. 0 0 4.0 MEAN STDEV GDS4510 experiment: TRIQ summary Fig. 9 MEAN vs STDEV dispersion graph for each GDS4510 experiment Our input dataset DGSD44723Dis composed of 54675 genes, 4 conditions (one for each individual involved) and 6 time points (one per hour) and has been used as the input of the TriGen and the OPTtricluster algorithm in four experiments: MSR3D,LSL,MSL and, OPT. GDS4472 MSR3Dexperiment For this experiment, TRI4has the best value of BIOQ,TRI6has the best value of PEQ, TRI3has the best value of SPQ and TRI5has the best value of GRQ as you can see Table 16. Conditions for each Time genes expression levels 2 3 4 5 6 Lgals12 Svs3b Gpr182 day 2 2 3 4 5 6 day 4 2 3 4 5 6 day 6 2 3 4 5 6 day 8 Times for each Condition genes expression levels 2 3 4 5 6 Lgals12 Svs3b Gpr182 WTREP2 2 3 4 5 6 RD1REP2 2 3 4 5 6 RD1REP3 Genes for each Condition times expression levels 2 3 4 5 6 day 2 day 4 day 6 day 8 WTREP2 2 3 4 5 6 RD1REP2 2 3 4 5 6 RD1REP3 Fig. 10 TRI1graphic views of the GDS4510 MSL experiment Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 21 of 29 Table 15 TRI1GO table of the MSL GDS4510 experiment TERM ID TERM P-VALUE GO:0007606 Sensory perception of chemical stimulus 1.68E-25 GO:0004984 Olfactory receptor activity 6.56E-19 GO:0050911 Detection of chemical stimulus involved in sensory perception of smell 6.56E-19 GO:0050907 Detection of chemical stimulus involved in sensory perception 2.97E-18 GO:0004930 G-protein coupled receptor activity 4.68E-17 GO:0007186 G-protein coupled receptor signaling pathway 4.68E-17 GO:0007608 Sensory perception of smell 6.93E-16 GO:0009593 Detection of chemical stimulus 1.16E-15 GO:0007600 Sensory perception 5.28E-15 GO:0050906 Detection of stimulus involved in sensory perception 8.32E-14 GO:0004872 Receptor activity 9.34E-14 GO:0060089 Molecular transducer activity 6.27E-13 GO:0004888 Transmembrane signaling receptor activity 8.08E-13 GO:0050877 Nervous system process 1.07E-12 GO:0099600 Transmembrane receptor activity 2.01E-12 GO:0038023 Signaling receptor activity 1.43E-11 GO:0004871 Signal transducer activity 2.69E-11 GO:0051606 Detection of stimulus 1.64E-10 GO:0003008 System process 1.09E-09 GO:0005549 Odorant binding 1.85E-08 The PEQ and SPQ values of the solutions oscillate around 0.64 and the GRQ values vary between 0.76 and 0.64; the BIOQ index oscillates around 0.0014 reaching two peaks at TRI4and TRI8. In general, the TRIQ value of solutions are in [ 0.32, 0.37] having TRI3and TRI7as outstanding ones and TRI5as the best solution in this experiment. GDS4472 LSL experiment WecanverifyinTable17 how TRI1has the best values of BIOQ,GRQ,PEQ and SPQ. In general, the GRQ,PEQ and SPQ indexes of the solutions depicts homogeneous values with the exception of TRI1where they reach their maximum; regarding BIOQ values, those reach three peaks at TRI1,TRI4and TRI10.TheTRIQ values vary between 0.39 and 0.44 being TRI1the best solution of this experiment. GDS4472 MSL experiment In Table 18 we can see how TRI9has the best values of BIOQ and GRQ while TRI7has the best value of PEQ and TRI10 has the best value of SPQ.ThePEQ values of the solutions Table 16 MSR3DGDS4472 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.339109333 0.001444791 0.696219908 0.596698979 0.601280513 TRI20.321761941 0.001591523 0.645157719 0.633303534 0.624758294 TRI30.363970471 0.001440455 0.742093089 0.650828401 0.677431755 TRI40.343765956 0.001732664 0.69802844 0.623399523 0.650365438 TRI50.370128492 0.001337649 0.761586904 0.637388072 0.659110049 TRI60.360725206 0.001406179 0.730735981 0.688724917 0.665829566 TRI70.366252916 0.001263468 0.750692098 0.655100071 0.651786783 TRI80.351001074 0.00159526 0.709109493 0.674238924 0.656954002 TRI90.327754495 0.001401494 0.664697595 0.606214508 0.617279679 TRI10 0.360821995 0.001434449 0.743345617 0.631027541 0.624302919 Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 22 of 29 Table 17 LSL GDS4472 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.447346181 0.027287612 0.923852614 0.69377633 0.589450252 TRI20.392576223 0.004031468 0.862302229 0.468881448 0.443910489 TRI30.409737004 0.002294049 0.886097803 0.570674741 0.512342415 TRI40.421749212 0.00967779 0.895313993 0.60768017 0.568014215 TRI50.402016193 0.002568856 0.869691073 0.55912722 0.497979503 TRI60.394065329 0.00467901 0.864596579 0.474398477 0.443345363 TRI70.39497655 0.005644179 0.865341762 0.477395239 0.442959875 TRI80.397055929 0.005748916 0.868777413 0.482543181 0.450866935 TRI90.40510461 0.007434596 0.881069048 0.514114336 0.465079524 TRI10 0.411954946 0.019662416 0.89416486 0.458207225 0.430948663 vary in the [ 0.43, 0.46] interval and the SEQ values are in the [ 0.40, 0.44] interval while all solutions have high GRQ values close to 0.90; the BIOQ valueshavethreepeaksatTRI5, TRI7and TRI9. Regarding TRIQ values, they vary in [ 0.40, 0.42] interval being TRI1,TRI5 and TRI7the outstanding solutions and being TRI9the best solution. GDS4472 OPT experiment For this experiment, TRI5has the best value of BIOQ,TRI10 has the best value of GRQ and SPQ and, TRI8has the best value of PEQ as you can see Table 19.TheBIOQ index oscillates around 0.0015 reaching three peaks at TRI5,TRI9and, TRI10.TheGRQ index vary in the [ 0.6, 07] interval reaching an outstanding value in the TRI10 solution. Regarding the PEQ values they vary in a interval of [ 0.42, 0.86] and the SPQ values in the [ 0.34, 0.76] interval. The TRIQ values vary between 0.28 and 0.44 being TRI10 the best solution of this experiment. GDS4472 summary We can observe in Fig. 11 how the solutions of the four experiments are in a BIOQ interval of [ 0.0012, 0.0272] meanwhile the GRQ values of the solutions of MSR3Dare in the [ 0.6451, 0.7615] interval, the solutions of LSL are in the [ 0.8623, 0.8953] interval, the solutions of MSL are in the [ 0.8964, 0.9238] interval and, the solutions of OPT are in the [ 0.6, 0.7] interval with an outstanding point near to GRQ =0.92. Regarding the PEQ and SPQ solutions distribution we can see in Fig. 12 how the PEQ and SPQ of MSR3Dare concentrated in the [ 0.50, 0.75] interval, the values PEQ and SPQ of LSL are in the [ 0.325, 0.75] interval, the values PEQ and SPQ of MSL are in the [ 0.325, 0.50] Table 18 MSL GDS4472 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.413005918 0.008623332 0.909739803 0.463874665 0.432091958 TRI20.406682712 0.005351847 0.901242812 0.449739986 0.420453301 TRI30.404078935 0.004069221 0.896447319 0.445616204 0.423691724 TRI40.409123273 0.004869646 0.9053715 0.456215467 0.43458153 TRI50.410786658 0.011209144 0.903088937 0.453954127 0.424976095 TRI60.404207143 0.004999521 0.896798986 0.44398627 0.415769491 TRI70.411937377 0.012628523 0.901459314 0.468134175 0.432653616 TRI80.405644251 0.0030952 0.902252364 0.445061054 0.418853066 TRI90.42006885 0.025664213 0.912118818 0.439476488 0.408307841 TRI10 0.41078403 0.006450477 0.90556104 0.465916366 0.440771152 Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 23 of 29 Table 19 OPT GDS4472 solution table SOLUTION TRIQ BIOQ GRQ PEQ SPQ TRI10.361091084 0.001165855 0.728642443 0.841096481 0.539927104 TRI20.302530473 0.001445096 0.649190316 0.42227834 0.420357627 TRI30.298417139 0.001567996 0.639083858 0.421848063 0.418143898 TRI40.290997577 0.0013925 0.620125263 0.423388521 0.421635907 TRI50.353327655 0.00233497 0.7175687 0.832938328 0.469715461 TRI60.298612766 0.001430159 0.640127397 0.421427176 0.415507376 TRI70.282392223 0.0018726 0.610369316 0.397933947 0.348229987 TRI80.35196608 0.00159536 0.707996999 0.865220464 0.49417155 TRI90.328919371 0.001916523 0.649746035 0.838138155 0.523115758 TRI10 0.446233789 0.002289266 0.924944835 0.740548556 0.761675883 interval and, the values PEQ and SPQ of OPT are dispersed in three groups: the first in the [ 0.42, 0.45] interval for PEQ and SPQ, the second in the [ 0.70, 0.85] interval for PEQ and the [ 0.46, 0.54] interval for SPQ and the third, that is a single point, in PEQ =0.74, SPQ =0.76. We can see the global TRIQ-based ranking of solutions in Table 20;theMSL solutions, one OPT solution and, the LSL solutions are placed alternatively on the first positions and the MSR3Dand the remaining of OPT solutions are on the last positions. We can see in Table 21 how the MSL experiment has the best value of the average and standard deviation of TRIQ, however, the LSL experiment has the best tricluster closely followed by the OPT experiment. In Fig. 13 we can see how the MSL is placed in the bottom-right position being the best experiment in terms of standard deviation and average. The most valuable solution of all experiments is the tricluster TRI1of the LSL experiment. This solution depicts very uniform patterns since has a very high GRQ value, we can check this fact in Fig. 14.Also,wecanseeinTable22 that this solution has Gene Ontology terms with very low p-value such as SRP-dependent cotranslational protein targeting to membrane,nuclear-transcribed mRNA catabolic process, nonsense-mediated decay or ribonucleoprotein complex. MSR3D LSL MSL OPT 0.00 0.01 0.02 0.00 0.01 0.02 0.00 0.01 0.02 0.00 0.01 0.02 0.6 0.7 0.8 0.9 BIOQ GRQ GDS4472 experiment: BIOQ v GRQ Fig. 11 BIOQ vs GRQ dispersion graph for each GDS4472 solution of each experiment Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 24 of 29 MSR3D LSL MSL OPT 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 0.8 0.4 0.5 0.6 0.7 PEQ SPQ GDS4472 experiment: PEQ v SPQ Fig. 12 PEQ vs SPQ dispersion graph for each GDS4472 solution of each experiment The SRP-dependent cotranslational protein targeting to membrane processisdescribed as the targeting of proteins to a membrane that occurs during translation and is dependent upon two key components, the signal-recognition particle (SRP) and the SRP receptor. SRP is a cytosolic particle that transiently binds to the endoplasmic reticulum (ER) signal sequence in a nascent protein, to the large ribosomal unit, and to the SRP receptor in the ER membrane; it is a protein targeting process that occurs in the intracellular component and is part of the cellular protein localization process. The nuclear-transcribed mRNA catabolic process, nonsense-mediated decay is a biological process that describes the nonsense-mediated decay pathway for nuclear-transcribed mRNAs degrades mRNAs in which an amino-acid codon has changed to a nonsense codon; this prevents the translation of such mRNAs into truncated, and potentially harmful, proteins; it is a negative regulation of gene expression process that negatively regulates the macromolecule metabolic process. Finally the ribonucleoprotein complex is a cellular component that is defined as a macromolecular complex containing both protein and RNA molecules. Conclusions and discussion Although triclustering has emerged as an essential task to study 3D datasets, there is no consensus on how to evaluate tricluster solutions obtained from each data set. Different authors validate their triclusters on different measures, with correlation, graphic validation and Gene Ontology terms being the most common ones. In this work we have presented a tricluster validation measure, TRIQ,asingleevaluation measure that combines the information from the three aforementioned sources of validation. We have applied TRIQ to three different datasets: the yeast cell cycle (Saccharomyces Cerevisiae), in particular the elutriation experiment, an experiment with mice (Mus Musculus) called GDS4510 and data from an experiments with humans (Homo Sapiens) called GDS4472. We have shown that TRIQ has successfully resumed the three validation measures (correlation, graphic validation and Gene Ontology terms) yielding the same validation results Gutiérrez-Avilés et al. BioData Mining (2018) 11:15 Page 25 of 29 Table 20 GDS4472 ranking table EXPERIMENT SOLUTION TRIQ BIOQ GRQ PEQ SPQ LSL TRI10.447346181 0.027287612 0.923852614 0.69377633 0.589450252 OPT TRI10 0.446233789 0.002289266 0.924944835 0.740548556 0.761675883 LSL TRI40.421749212 0.00967779 0.895313993 0.60768017 0.568014215 MSL TRI90.42006885 0.025664213 0.912118818 0.439476488 0.408307841 MSL TRI10.413005918 0.008623332 0.909739803 0.463874665 0.432091958 LSL TRI10 0.411954946 0.019662416 0.89416486 0.458207225 0.430948663 MSL TRI70.411937377 0.012628523 0.901459314 0.468134175 0.432653616 MSL TRI50.410786658 0.011209144 0.903088937 0.453954127 0.424976095 MSL TRI10 0.41078403 0.006450477 0.90556104 0.465916366 0.440771152 LSL TRI30.409737004 0.002294049 0.886097803 0.570674741 0.512342415 MSL TRI40.409123273 0.004869646 0.9053715 0.456215467 0.43458153 MSL TRI20.406682712 0.005351847 0.901242812 0.449739986 0.420453301 MSL TRI80.405644251 0.0030952 0.902252364 0.445061054 0.418853066 LSL TRI90.40510461 0.007434596 0.881069048 0.514114336 0.465079524 MSL TRI60.404207143 0.004999521 0.896798986 0.44398627 0.415769491 MSL TRI30.404078935 0.004069221 0.896447319 0.445616204 0.423691724 LSL TRI50.402016193 0.002568856 0.869691073 0.55912722 0.497979503 LSL TRI80.397055929 0.005748916 0.868777413 0.482543181 0.450866935 LSL TRI70.39497655 0.005644179 0.865341762 0.477395239 0.442959875 LSL TRI60.394065329 0.00467901 0.864596579 0.474398477 0.443345363 LSL TRI20.392576223 0.004031468 0.862302229 0.468881448 0.443910489 MSR3DTRI50.370128492 0.001337649 0.761586904 0.637388072 0.659110049 MSR3DTRI70.366252916 0.001263468 0.750692098 0.655100071 0.651786783 MSR3DTRI30.363970471 0.001440455 0.742093089 0.650828401 0.677431755 OPT TRI10.361091084 0.001165855 0.728642443 0.841096481 0.539927104 MSR3DTRI10 0.360821995 0.001434449 0.743345617 0.631027541 0.624302919 MSR3DTRI60.360725206 0.001406179 0.730735981 0.688724917 0.665829566 OPT TRI50.353327655 0.00233497 0.7175687 0.832938328 0.469715461 OPT TRI80.35196608 0.00159536 0.707996999 0.865220464 0.49417155 MSR3DTRI80.351001074 0.00159526 0.709109493 0.674238924 0.656954002 MSR3DTRI40.343765956 0.001732664 0.69802844 0.623399523 0.650365438 MSR3DTRI10.339109333 0.001444791 0.696219908 0.596698979 0.601280513 OPT TRI90.328919371 0.001916523 0.649746035 0.838138155 0.523115758 MSR3DTRI90.327754495 0.001401494 0.664697595 0.606214508 0.617279679 MSR3DTRI20.321761941 0.001591523 0.645157719 0.633303534 0.624758294 OPT TRI20.302530473 0.001445096 0.649190316 0.42227834 0.420357627 OPT TRI60.298612766 0.001430159 0.640127397 0.421427176 0.415507376 OPT TRI30.298417139 0.001567996 0.639083858 0.421848063 0.418143898 OPT TRI40.290997577 0.0013925 0.620125263 0.423388521 0.421635907 OPT TRI70.282392223 0.0018726 0.610369316 0.397933947 0.348229987 Table 21 GDS4472 summary table EXPERIMENT BEST SOLUTION BEST TRIQ MEAN STDEV MSR3DTRI50.370128492 0.350529188 0.016814529 LSL TRI10.447346181 0.407658218 0.016734175 MSL TRI90.42006885 0.409631915 0.004869533 OPT TRI10 0.446233789 0.331448816 0.049451114