TRIQ: a new method to evaluate triclusters

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.


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 literature, the standard measures to evaluate tricluster solutions are based on three areas as can be seen in the triclustering publications [3][4][5][6][7]. First, correlation measures such as Pearson [8] or Spearman [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] as a more 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 S 2 D 3 Cluster. 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.
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][21][22].
More recent works have extended the capabilities of the tricluster algorithms by combination of several approaches. Thereby, Liu et al. [23] mixed fuzzy clustering and 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. 1 we 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 W bio ; the weight for GRQ, denoted as W gr ; the weight for PEQ, denoted as W pe ; and the weight for SPQ, denoted as W sp .
This is a general definition of TRIQ. In order to obtain a TRIQ index as balanced as possible among the four quality indexes BIOQ, GRQ, PEQ, and SPQ we performed an exhaustive testing procedure with well known datasets. Several combinations of values of BIOQ, GRQ, PEQ, and SPQ were tested, and in Fig. 1 we show the results obtained.
We see that that the value of TRIQ is slightly directly dependent on the weights related to correlation, PEQ, and SPQ. 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 −3 to 10 −5 . Based on this experiments, we have configured the TRIQ measure with the weights showed in Eq. 2 in order to obtain a balanced value of TRIQ.
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] and Spearman's SPQ [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. Given a tricluster TRI, we compute PEQ 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).
with exp representing the expressions in each tricluster TRI.

Graphical validation: GRQ
The GRQ member of Eq. 1 measures 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, given a tricluster TRI, 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.
The MSL measure considers the three graphical views of a tricluster, also defined in [27]: TRI gct , TRI gtc , and TRI tgc . These three terms are generally defined as TRI xop , with the expression levels of the tricluster represented in the Y axis, x represents the tricluster component in the X axis (genes or time points), o represents the lines plotted in the graph (genes, conditions or time lines) and p the type of facets or panels represented (time points or conditions). We can observe an example of the TRI tgc view of a tricluster with the genes g 1 , g 4 , g 7 and g 10 , the experimental conditions c 2 , c 5 and c 8 and the time points t 0 , t 2 , t 11 in Fig. 2 and 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 TRI gct , TRI gtc , and TRI tgc . A near to zero value of MSL implies a better graphical quality of a tricluster

Biological validation: BIOQ
The BIOQ member of Eq. 1 measures the biological quality of the tricluster. Specifically, BIOQ uses the genes (TRI G ) of the input tricluster TRI to compute this index. As you can see in Eq. 7, the biological quality of a tricluster TRI is defined as the biological significance, SIG bio , of the set of genes TRI G divided by the S max value.
The SIG bio and S max 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 SIG bio and S max 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 SIG bio 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 1 represents the ah-hoc designed system of intervals of p-value and scoring system. The intervals and the scoring system are defined in Eq. 8 where for a given level, Inter l is defined by a weight value w l for the level, and by the lower and upper bounds (inf l and sup l , respectively), being an open-closed p-values interval (Eq. 8a). The set of existing LV consists of all levels with Inf l smaller or equal to a minimum p-value, th. For each interval of each level Inter l , the weight value w l is defined in Eq. 8c; Inf l is defined in Eq. 8d, and sup l is defined in Eq. 8e. inter l = w l , inf l , sup l (8a) 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. For each row, weight (wl) and range (inter l ) for each level (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.
Taking into account each level l and each predefined interval inter l , the biological significance for the genes of the measured tricluster is defined in Eq. 10a as the addition of all scores for each level l from the LV level set Eq. 9e. The score function S for 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 l divided by the total number of terms, by the weight of the level, and by the level plus a bonus function f bonus , defined in Eq. 10d as the sum of the level plus a bonus value V bonus if the current level is the maximum level of LV or zero in any another case.
Again, this definition is made in order to establish a general system of SIG bio . For our purpose and as a result of an exhaustive testing, the V bonus parameter has been settled to 0; this fact produces S max as the maximum achievable score for the interval configuration as you can see in Eq. (11), that has been used to the SIG bio normalization in Eq. 7.

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 (MSR 3D ) [30], Least Squared Lines (LSL) [31] and Multi SLope Measure (MSL) [27]. OPTricluster identifies triclusters of genes with 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: • D elu 3D : The yeast cell cycle (Saccharomyces Cerevisiae) [32], in particular, the elutriation experiment.
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 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 MSR 3D , LSL, and MSL 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: MSR 3D , 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 D elu 3D from 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.
D elu 3D has been used as the input of the TriGen and the OPTtricluster algorithm in four experiments: MSR 3D , LSL, MSL and, OPT.

Elutriation MSR 3D experiment
We can verify in Table 2 how TRI 9 has the best values of BIOQ, PEQ and SPQ whereas TRI 10 has the best value of GRQ. The GRQ, PEQ and SPQ values are stabilized from TRI 2 to TRI 8 until TRI 9 − TRI 10 when these values reach the maximum. Regarding BIOQ values, these vary around 0.0012. Furthermore, TRIQ values are stable for all solutions except TRI 9 − TRI 10 due to the genetic algorithms nature. In conclusion, TRI 9 is the best solution since it has the best value of TRIQ, closely followed by TRI 10 .

Elutriation LSL experiment
In Table 3 you can see how TRI 3 has the best value of BIOQ, TRI 2 has the best value of GRQ, TRI 6 has the best value of PEQ and, TRI 1 has the best value of SPQ. In general, the GRQ, PEQ and SPQ values vary around an average value from TRI 1 until TRI 8 . Then, these values decrease in TRI 9 − TRI 10 solutions due to the fact that the algorithm reached a local minimum in this two solutions; the BIOQ values fluctuate around 0.0012 value reaching a maximum in TRI 3 and a minimum in TRI 4 . The values of TRIQ reach the maximum values at the first two solutions, then remain stable and finally fall in local minimum in the last two solutions. In conclusion, TRI 1 is the best solution since it has the best value of TRIQ.

Elutriation MSL experiment
We can observe in Table 4 how TRI 2 has the best value of BIOQ, PEQ and SPQ whereas TRI 1 has the best value of GRQ. The GRQ, PEQ and SPQ have a stable fluctuation throughout the solutions whilst BIOQ varying around the central value 0.0011. The TRIQ values reach their maximum value at TRI 2 , the minimum at TRI 3 and the rest are stabilized. In conclusion, TRI 2 is the best solution since it has the best value of TRIQ.

Elutriation OPT experiment
We can verify in Table 5    The global TRIQ-based ranking of solutions is showed in Table 6; we can see how the solutions of the MSL experiment are placed on the first positions followed by two outstanding solutions of the MSR 3D experiment, all solutions of the LSL experiment, all solutions of the OPT experiment and, the remaining solutions of the MSR 3D experiment.
The MSL experiment has the best average values of TRIQ and the lowest standard deviation of TRIQ as seen in Table 7. This fact is reflected in Fig. 5 wherein the MSL point is located on the bottom-right side of the graph which implies that the MSL experiment has the highest values of TRIQ and a sparsely dispersed distribution, thus this is a high-quality experiment.
The most valuable solution of all experiments is the tricluster TRI 2 of the MSL experiment. We can see in Fig. 6 its three graphic views showing that its high value of GRQ is reflected in the patterns depicted. Furthermore, in Table 8 we observe terms with moderately low p-value as fermentation, vesicle fusion to plasma membrane and exocytosis. Fermentation is a biological process that is part of the process called energy derivation by oxidation of organic compounds and, in turn, belongs to the generation of precursor metabolites and energy process and the oxidation-reduction process; Vesicle fusion to plasma membrane is a biological process that is part of the exocytosis proccess; the first term is a process of cellular component organization whereas the second is an establishment of localization process.

Mouse GDS4510 dataset
This batch corresponds to the mouse GDS4510 dataset. This dataset was obtained from GEO [35] with accession code GDS4510 and title rd1 model of retinal degeneration: time course [33]. In this experiment, the degeneration of retinal cells in different individuals of home mouse (Mus musculus) is analyzed over 4 days just after birth, specifically on days 2, 4, 6 and 8.

GDS4510 MSR 3D experiment
We can verify in Table 9 how TRI 7 has the best value of BIOQ, GRQ, PEQ, SPQ. The GRQ, PEQ and SPQ indexes vary uniformly among all the solutions. BIOQ has a peak of TRI 7 which has the maximum value. The TRIQ values oscillate between 0.385 and 0.4 with the exception of TRI 7 , therefore this is the best solution since it has the best value of TRIQ.

GDS4510 LSL experiment
In Table 10 we can see how TRI 1 has the best value of BIOQ and GRQ meanwhile TRI 2 has the best values of PEQ and SPQ. The GRQ, PEQ and SPQ values vary uniformly around  The TRIQ values oscillates between 0.40 and 0.43 being TRI 1 , TRI 4 and TRI 9 the most outstanding solutions. We can conclude that TRI 1 is the best solution since it has the best value of TRIQ.

GDS4510 MSL experiment
For this experiment, we can observe in Table 11 how TRI 1 has the best value of BIOQ and GRQ meanwhile TRI 2 has the best value of PEQ and TRI 8 has the best value of SPQ.

GDS4510 OPT experiment
In Table 12 we can see how TRI 2 has the best value of BIOQ, TRI 4 has the best value of GRQ and, TRI 9

GDS4510 summary
We can see how the solutions are distributed regarding BIOQ and GRQ in Fig. 7  We can see in Table 14 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] under code 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.

GDS4472 MSR 3D experiment
For this experiment, TRI 4 has the best value of BIOQ, TRI 6 has the best value of PEQ, TRI 3 has the best value of SPQ and TRI 5 has the best value of GRQ as you can see Table 16.

GDS4472 LSL experiment
We can verify in Table 17 how TRI 1 has 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 TRI 1 where they reach their maximum; regarding BIOQ values, those reach three peaks at TRI 1 , TRI 4 and TRI 10 . The TRIQ values vary between 0.39 and 0.44 being TRI 1 the best solution of this experiment.

GDS4472 MSL experiment
In Table 18 we can see how TRI 9 has the best values of BIOQ and GRQ while TRI 7 has the best value of PEQ and TRI 10 has the best value of SPQ. The PEQ values of the solutions

GDS4472 OPT experiment
For this experiment, TRI 5 has the best value of BIOQ, TRI 10 has the best value of GRQ and SPQ and, TRI 8 has the best value of PEQ as you can see Table 19. The BIOQ index oscillates around 0.0015 reaching three peaks at TRI 5 , TRI 9 and, TRI 10

GDS4472 summary
We can observe in Fig. 11 Table 20; the MSL solutions, one OPT solution and, the LSL solutions are placed alternatively on the first positions and the MSR 3D and 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 TRI 1 of 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, we can see in Table 22 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. The SRP-dependent cotranslational protein targeting to membrane process is described 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, a single evaluation 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   as in [27] where each of the components of TRIQ (BIOQ, GRQ, PEQ, and SPQ) where applied separately. In that publication we presented the MSL measure, comparing it to MSR 3D and LSL, with the same datasets used in this article. We concluded that MSL was the best fitness function. In this publication, we have seen how MSL has obtained the best general results, with high values of TRIQ and low standard deviation for all solutions presented. Therefore, we can conclude that TRIQ has been successful in representing

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