Renan Dos Reis
email: renandosreiss@usp.br

Luciano Da Fontoura Costa

Data-Driven Metamodels: A Systematic Approach to Modeling Scientific Data

Keywords: metamodel, classification, modeling frameworks, pattern recognition, food webs

The growing availability of data has motivated continued efforts toward respective analysis and integration into models. The latter endeavor is particularly challenging because the sheer amount of data compounds the human-assisted identification of possible models. The present work reports the development of a systematic metamodeling framework that can provide subsidies for characterizing and automating the construction of models. The proposed metamodel framework involves three main components: a universe set that contains all accessible data for scientific experimentation, a data environment with organized datasets, and a modeling framework capable of explaining the data environment. The metamodel includes a bijective mapping between datasets and their respective models in the modeling framework, which secures a strong consistency between available datasets and constructed models, a procedure summarized in the concept of cartouche. Additionally, the relationship between datasets and models can be visualized using weighted connections, whose weights indicate the level of explanation of each alternative models over the elements of each available dataset. These features enable the metamodel to be used for the quantification of model specificity, thus addressing the quality assessment of modeling frameworks, and the construction of new models through the logical composition matched by set operations between datasets, which is evaluated by using a coincidence-based metric. The analysis of simulated metamodels allowed the recognition of sources of unspecificity in modeling frameworks and the identification of modeling conditions for the reduction of unspecificity. It was shown that the distribution of repetitive elements in the data environment is the main factor leading to higher unspecificity in modeling frameworks. Furthermore, we traced the two optimal attributes of modeling frameworks for the construction of models through logical composition, either for explaining highly unspecific or specific datasets. To illustrate the potential applications of the proposed approach, we analyzed two real-case examples: the metamodeling of symmetries in 3-by-3 pixel patterns and species in a food web. In the first case, we derived an accurate model for a dataset using logical compositions of alternative models within the metamodel. In the second case, we evaluate the association of mutually exclusive models with highly unspecific ones to explain a data environment, and the characterization of this modeling framework provided in-depth information about the original food web data. The developed metamodel approach has potential for assisting the characterization and improvement of database and knowledgebase systems in multiple scientific fields.

Introduction

Modeling real-life objects and phenomena constitutes a central activity in science [START_REF] Da | An ample approach to data and modeling[END_REF][START_REF] Frigg | Models in science[END_REF][START_REF] Montáns | Data-driven modeling and learning in science and engineering[END_REF]. Through modeling, scientists can study a whole class of phenomena using just a single construct, the model, helping them to identify and explain essential features and behaviors of the modeled objects [START_REF] Da | An ample approach to data and modeling[END_REF][START_REF] Frigg | Models in science[END_REF][START_REF] Montáns | Data-driven modeling and learning in science and engineering[END_REF]. Scientific modeling is widespread among all scientific areas, particularly in natural sciences [START_REF] Da | An ample approach to data and modeling[END_REF][START_REF] Montáns | Data-driven modeling and learning in science and engineering[END_REF][START_REF] Peierls | Model-making in physics[END_REF][START_REF] Hinchliffe | Chemical modelling[END_REF][START_REF] Allman | Mathematical models in biology[END_REF].

The availability of experimentally acquired data is the main requirement for model construction. In the XXI century, with the development of new data generation techniques such as high-throughput experimentation, there is an accelerating growth of available data in multiple scientific fields [START_REF] Marx | Biology: The big challenges of big data[END_REF][START_REF] Argenio | The high-throughput analyses era: Are we ready for the data struggle?[END_REF]. For example, massive datasets that are usually challenging to capture, store, manage and analyze -referred to as big data [START_REF] Sagiroglu | Big data: A review[END_REF] -are currently applied for studies and development of models in geology [START_REF] Chen | Review of the application of big data and artificial intelligence in geology[END_REF], chemistry [START_REF] Chiang | Big data analytics in chemical engineering[END_REF], healthcare [START_REF] Belle | Big data analytics in healthcare[END_REF], ecology [START_REF] Kelling | Data-intensive science: A new paradigm for biodiversity studies[END_REF], molecular biology [START_REF] Perakakis | Omics, big data and machine learning as tools to propel understanding of biological mechanisms and to discover novel diagnostics and therapeutics[END_REF] and several other sciences [START_REF] Sagiroglu | Big data: A review[END_REF][START_REF] Wamba | How 'big data' can make big impact: Findings from a systematic review and a longitudinal case study[END_REF]. However, interpreting big data is a difficultand often humanly impossible -task, thus imposing major challenges for traditional modeling approaches. Several methods have been developed to face the challenges of dealing with big data, such as machine learning techniques [START_REF] Al-Jarrah | Efficient machine learning for big data: A review[END_REF], but there is still room to devise new approaches aimed at scientific modeling.

Usually, modeling approaches starts with the empirical observations of a phenomenon of interest. Then, based on that observations, a simplified and useful emulation of that phenomenon, called a model, is conceived [START_REF] Montáns | Data-driven modeling and learning in science and engineering[END_REF][START_REF] Frantz | A taxonomy of model abstraction techniques[END_REF]. When starting from qualitative and quantitative measurements of physical reality, the modeling approach can be understood as being data-driven [START_REF] Montáns | Data-driven modeling and learning in science and engineering[END_REF]. There are several components that can be constructed and combined into a data-driven approach to form a model, such as mathematical equations, descriptions of observed data, categories and rules [START_REF] Da | An ample approach to data and modeling[END_REF][START_REF] Frigg | Models in science[END_REF][START_REF] Frantz | A taxonomy of model abstraction techniques[END_REF].

Figure 1 shows a diagram with the three main components of data-driven scientific modeling. The first component, represented as Ω, is the universe that contains all accessible data related to the studied phenomenon. While observing the phenomenon of interest within its specific universe, it becomes possible to collect data of particular interest, which are progressively organized as datasets into the available data environment E. Based on the latter, preliminary models can be constructed and interactively combined, resulting in a possible explanation of the phenomenon of interest. The set of models developed along the modeling cycles constitutes the modeling framework of the studied phenomenon M .

Examples of modeling problems include:

• Modeling the identification of objects in images, considering color, contrast, shape, and The main components of the data-driven modeling approach considered in this work. In order to model a phenomenon of interest, one starts by observing the phenomenon, collecting datasets, and creating respective models. The set Ω corresponds to the respective universe, containing all the accessible data related to the studied phenomenon. Particular data of interest are collected from Ω and organized into datasets that become available to the data environment E. In this way models can be constructed and combined into a single modeling framework, M , which can be used to explain the studied phenomenon.

texture. In this case, the modeling framework include categories of identifiable objects, such as wheels, saddle, handlebar, frame, and their eventual combination into a bicycle.

• Modeling the role of species within food webs in diverse ecosystems by using graphs, taking into account the involved trophic levels [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF][START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF]. In this case, the modeling framework could include rules and equations for the different trophic levels of the species, the number of prays and predators.

• Modeling the intermolecular interactions between chemical substances, taking into account complementary laws of physics. For this problem, the modeling framework could incorporate different components including the Lennard-Jones potential, the Buckingham potential as well as the Mie potential [START_REF] Kaplan | Intermolecular interactions[END_REF].

The possibility of coupling models into frameworks facilitates the understanding of multiple classes of objects. However, modeling frameworks have several aspects deserving further consideration. The first one is that a data-driven modeling framework may require an enormous amount of data to be collected from the different studied objects [START_REF] Da | An ample approach to data and modeling[END_REF], which may turn out to be unfeasible [START_REF] Da | An ample approach to data and modeling[END_REF]. Another constraint of modeling frameworks is that even the limited data collected from nature might be prone to error and noise, ultimately undermining the construction of more reliable models [START_REF] Da | An ample approach to data and modeling[END_REF]. These limitations can ultimately lead to a narrow and biased modeling framework that may imply misleading conclusions, especially when the given framework is faced with new data.

Consequently, it becomes of great interest to conceive methods capable of evaluating the quality of a modeling framework while consistently classifying and explaining its supporting datasets. In order to do so, we can resort to the construction of a metamodel : a model of how models are developed as well as their relationships with the available datasets.

Metamodels are commonly used in engineering and information systems to describe how respectively modeled portions of a system relate to each other [START_REF] Aßmann | Ontologies, meta-models, and the model-driven paradigm[END_REF][START_REF] Henderson-Sellers | Bridging metamodels and ontologies in software engineering[END_REF]. A metamodel will thus provide a complete and consistent structure for the construction of a system that covers all constraints and application demands [START_REF] Henderson-Sellers | Bridging metamodels and ontologies in software engineering[END_REF]. Additionally, metamodels have also been applied in computational science to describe how a single agent-based model behaves in simulations under multiple different conditions [START_REF] Pietzsch | Metamodels for evaluating, calibrating and applying agent-based models: A review[END_REF]. However, for more general data-driven scientific approaches, metamodeling the relationship between different models might not be sufficient to reveal the scientific specificity implied by each scientific problem.

For this reason, we propose to construct and analyze metamodels while emphasizing the available datasets and their implications for the respective modeling frameworks, further developing the methodology previously proposed in [START_REF] Da | An ample approach to data and modeling[END_REF]. The relevance given to the datasets in the proposed metamodel is the reason why we call our approach as being data-driven.

In order to develop our metamodeling approach, we start by formally describing how to build and represent data-driven metamodels. Then, we present possible metrics for quantifying the quality and specificity of the obtained modeling structures based on their relationship with their respective datasets. Additionally, we will deepen this analysis by evaluating the relationship between different models in the modeling frameworks and defining resources for deriving new models through their successively logical compositions. In order to demonstrate the potential of the described methodology, we illustrate its application in two real-world problems: a metamodel describing 3-by-3 pixel patterns, and a metamodel for the classification of species in a food web.

Concepts and Methods

In this section, we describe the approach to constructing metamodels that integrate data environments and modeling frameworks and subsequently propose metrics that can be used to characterize the specificity of models within the metamodel. Additionally, we elaborate on a procedure to create and improve models using logical compositions within an available modeling framework.

Definition of the Data-Driven Metamodel

In this work, we will describe the M * metamodeling approach, a tool to represent how a scientific modeling framework integrates datasets and models [START_REF] Da | An ample approach to data and modeling[END_REF]. The M * metamodel is constructed by considering four aspects of modeling frameworks -namely (i) its representation based on domains and maps; (ii) the underlying relationship between available data and models; (iii) the possibility of consolidating modeling structures to account for new data; and (iv) the possibility to combine models within a modeling framework -each of which is discussed below. Although it is possible to build modeling frameworks through data-or model-driven approaches, the present work focuses on the former possibility.

Domains, Maps and Metamodel Representations

Scientific models can be related to a collection of datasets whose data elements represent real-world objects. These datasets contain elements interrelated to one another through some common and observable characteristics of interest. Each assembled dataset is a candidate to be explained in terms of a scientific model. For example, a dataset can contain all relevant information on different examples of spoons observed by a scientist. The scientist can then construct a spoon model to represent this data collection.

Let us illustrate and develop these ideas by analyzing the metamodel of a hypothetical collection of cutlery present in a kitchen drawer that will be modeled through a data-driven modeling approach, depicted in Figure 2(a).

In the M * metamodel, the universe domain is the Ω set, and it contains virtually all data elements in the real world that can be drawn to form datasets of scientific interest. In Figure 2(a) left, the elements of Ω stand for the finite number of cutlery in a kitchen drawer. Two datasets are constructed from the Ω domain: the forks dataset by picking 6 elements, and the spoons dataset by selecting 5 elements. Observe that the datasets available will be, by definition, subsets of the universe Ω.

The set of all datasets collected constitutes the environment domain E. Therefore, E represents the set of subsets of Ω available for scientific analysis. In that way, the largest possible E is the power set of Ω, containing 2 N elements, where N is the number of data elements in Ω [START_REF] Da | An ample approach to data and modeling[END_REF].

The modeling framework is progressively built based on the datasets available in E, which will support the construction of different models. In the case shown in Figure 2(a), for each dataset available in E, a specific tag model was constructed: the forks dataset is modeled by a tag "Fork", while the spoons dataset is modeled by a tag "Spoon". The set composed of the constructed models defines the modeling framework domain M .

One way to depict the interactions between the three domains involved in the metamodel, Ω, E, and M , is by considering two mapping operations between them. First, the data elements in Ω map into the datasets in E. Second, the datasets available in E map into their respective models in M . This second mapping operation, represented as (E, M ), is the base of the construction of a modeling framework and will represent the current theory of the objects in study [START_REF] Da | An ample approach to data and modeling[END_REF]. It highlights the available datasets gathered from the universe and the models that could be built upon them, thus capturing the idea that a modeling framework depends on the datasets available.

Being that important, it is desirable that the mapping between the E and M domains have special mathematical properties. In that sense, we impose that the (E, M ) mapping must be a bijective association. Mathematically, a bijective association between E and M enables us to identify not only models through datasets but also datasets through models without loss of information [START_REF] Da | An ample approach to data and modeling[END_REF]. Through setting the (E, M ) mapping as a bijective association, the M * metamodel can represent modeling frameworks with both consistency and simplicity by avoiding the depiction of models that are unverified (models without a dataset), redundant (two models connecting in the same dataset) or ambiguous (models with two distinct datasets with different characteristics).

In Figure 2(a), the mapping of elements from Ω into datasets in E is represented by onedirectional arrows. The bijective association between datasets in E and models in M is represented with bidirectional arrows. This representation is henceforth called the domain mapping representation.

Relationship between Data Elements and Models

In the metamodel approach, each dataset in the E domain is connected to its respective model in the modeling framework M . Because each data element drawn from Ω will be a part of a dataset in E, and each dataset in E is connected to a respective model in M , we can indirectly connect each data element, through its dataset, straight to a model.

As mentioned, the datasets are composed of data elements that represent real-world objects. What unites different data elements in the same dataset are common characteristics shared by the real-world objects they represent. However, the same real-world object represented by a data element may share characteristics with objects belonging to different datasets. In this case, this object is part of more than one dataset, thus underlying the construction of more than one model. Consequently, a non-injective relationship is established between data elements and models. For example, in the cutlery metamodel of Figure 2(a), the orange sporks -spoon-like cutlery with fork-like tines -are mapped into both the spoons and forks datasets, resulting in unavoidable overlapping between the two respective models.

This non-injective relationship highly increases the complexity of the relationship between datasets and models. To visualize this relationship more directly, a complementary representation of the metamodel, depicted in Figure 2(b), shows weighted connections between the E and M domains. The weight of the connection between a dataset and a model is equal to the number of dataset elements that fit into that model. This representation is henceforth called weighted bipartite representation.

The weighted bipartite representation highlights how different models can partially explain the same dataset, thus showing possible redundancies of the modeling framework. These redundancies could be seen as undesirable for some scientific modeling approaches, as it discloses a lack of specificity of the built models. However, as the domain mapping implies, data elements satisfying multiple models will not affect the consistency of modeling frameworks, as the correct association between a dataset and its respective model is still maintained.

In order to systematically connect the two representations of the metamodel, as well as to properly evaluate the specificity of a modeling framework over its dataset, we can define an approach in which a dataset, its weighted connection, and its respective model are jointly analyzed as a part of what we hereafter call a cartouche [START_REF] Da | An ample approach to data and modeling[END_REF].

The cartouche corresponds to the collection of all data elements involved in a given dataset ω i , its respective model m i , and the weighted connection w i,i between ω i and m i that must be equal to the number of elements of ω i [START_REF] Da | An ample approach to data and modeling[END_REF]. Although the data elements can be indirectly connected to any models in the M domain, the cartouche becomes satisfied only when all the ω i data elements connect to m i . At the same time, although the model m i can receive weighted connections of multiple datasets, the cartouche is only validated when every element in the ω i dataset maps to m i . Both criteria are necessary conditions for the bijective association (ω i , m i ) of the domain mapping representation. This approach conciliates the non-injective mappings from data elements to models and will ensure the consistency of the relationship between the E and the M domains.

Let us further illustrate the utilization of the cartouche approach. In Figure 3(a), there is a metamodel of polygons with only two datasets: the squares and the rectangles datasets. Because squares are a particular case of rectangles, the two datasets share elements that are explained by both models, as depicted in Figure 3(b). While two data elements of the rectangles dataset are mapped into the squares model, the entirety of the rectangles dataset is associated to the rectangles model. Therefore, the cartouche between the rectangles dataset and the rectangles model is satisfied, justifying the bijective association in Figure 3

(a).

There is also the possibility that a dataset is fully mapped into two or more models. As illustrated in the weighted bipartite representation in Figure3(b), this happens with the squares dataset, which is entirely contained within the rectangles dataset. Consequently, the squares dataset can be explained by the two available models, although with different degrees of particularity: the squares model, a more particular model, and the rectangles model, a more general one.

When the E domain includes datasets that are subsets of others, such as the squares dataset in the previous example, the translation from the weighted bipartite representation back into the domain mapping representation becomes ambiguous. This happens because the cartouche of the squares dataset can be mistakenly satisfied when considering the rectangles model as its respective model. The solution to this problem is to bijectively associate models only to the largest dataset it can fully explain. By following this rule, we can reconstruct the Figure 3(a) from Figure 3(b) by connecting the rectangles model to the rectangles dataset (5 elements -the largest dataset it can fully explain) and the squares model to the squares dataset (2 elementsthe only dataset it can fully explain).

Methods for Consolidating Modeling Frameworks

Data-driven modeling approaches typically starts with preliminary observations and discovery of first principles models, then develop through the collection and analysis of new data [START_REF] Da | An ample approach to data and modeling[END_REF][START_REF] Montáns | Data-driven modeling and learning in science and engineering[END_REF]. The appearance of a new dataset that cannot be sufficiently explained by the initial basic models should prompt the construction of new improved models, thus expanding and consolidating Although all the elements on the squares dataset are explained by both the rectangles and the squares models, the squares dataset is mapped only to the second. This follows the rule that, in order to ensure the bijective mapping between E and M , a model is bijectively associated with the largest dataset that it can entirely explain. Therefore, being the smallest of the two datasets that satisfy the rectangles model, the squares dataset is mapped only to the squares model.

the modeling framework to that new circumstance. This represents the incorporation of new knowledge into a modeling framework, a pivotal characteristic of scientific modeling [START_REF] Da | An ample approach to data and modeling[END_REF].

In the M * metamodel, the E domain represents all the initially available data, while the M domain encompasses initial models [START_REF] Da | An ample approach to data and modeling[END_REF]. Collecting further data can be depicted as a new mapping between elements of Ω into a new dataset in E, which should map into a newly created model in M . This procedure conveys the idea that the modeling frameworks can expand, which is suitable for portraying data-driven approaches in scientific modeling.

While building new models and adding them into the M * metamodel can happen in numerous forms, there are only two ways in which new datasets can be incorporated.

The first possibility applies to new datasets not explained by any available models. In that case, a new model can be built specifically for that dataset, thus ensuring the bijective association between the E and M domains. As model building can be difficult, this new model can be a black box model or a provisional label for the respective dataset, which can be later developed into a more informative model.

Let us return to our cutlery example. Figure 4 depicts different forms of incorporating new datasets into the E domain. If a new dataset composed of knives is collected from Ω, then the modeling framework could be consolidated by incorporating a knife label model in M (Fig. 4(a)).

The second way a dataset can be incorporated into the M * metamodel applies to new datasets that are fully explained by an available model. For example, suppose that a new dataset ω i is entirely explained by a model m k already associated with a dataset ω k . In that case, we should unite the ω i and ω k datasets into a single dataset ω ∪ = ω i ∪ ω k , which will be bijectively mapped into m k . Although the prior procedure, creating a new model for the new dataset, could also be used, a new m i model for the ω i dataset may share several characteristics with the available m k model. Hence, the latter procedure, combining ω i and ω k , might be preferred. Back to the cutlery example, if a new dataset composed of gray forks is collected from Ω, any of the two approaches can be adopted depending on the decision to take or not the color of the forks into account. If the color is taken into account, the first method should be used, giving rise to a new model for gray forks. Otherwise, the second procedure should be taken, with the gray forks dataset being united with the already existing forks dataset (Fig. 4(b)).

In order to ensure the consistency of both the proposed procedures, all datasets available in E must be updated after each expansion of the M * metamodel [START_REF] Da | An ample approach to data and modeling[END_REF]. Every data element in E must be continuously checked against every pair of bijectively associated datasets and models and should be added to every dataset whose model is satisfied. This approach ensures consistency by guaranteeing that a dataset respective to a model always gathers all data elements that satisfy it.

A consequence of this procedure is that the weight of the connection between a dataset ω i and a model m k should always be equal to the weight of the connection between ω k and m i . This stands because all elements of ω i that satisfy m i and m k must also be in ω k , just as the elements of ω k that satisfy m k and m i must also be in ω i .

Paired Algebra of Set and Logical Operations in the Metamodel

As discussed previously, some new datasets drawn from Ω might not be fully explained by preexisting models, thus prompting the construction of new ones. However, these new datasets may be explained by the composition of multiple available models. For example, the data elements of a new dataset can satisfy either the m 1 or m 2 models, thus satisfying the logical composition of models m 1 ∨ m 2 . This motivates a formal procedure to explain new datasets solely through the logical composition of the preexisting models.

The bijective association built into the M * metamodel will be useful for this procedure, as it guarantees a bidirectional bridge between model composition and dataset combination. For example, let us consider the case of sporks again. A dataset composed of sporks can be understood as the set combination spoon dataset ∩ fork dataset. Because of the bijective association between the E and M domains, this new dataset immediately implies a respective model corresponding to the logical conjunction spoons model ∧ forks model. This composition, illustrated in Figure 4(b), reflects a paired algebra of set operations between datasets and logical operation between models [START_REF] Da | An ample approach to data and modeling[END_REF]. Now, we have the opportunity to model a new dataset by simply expressing it as a set combination of previous datasets, followed by translating that expression into the respective logical composition of models already in use [START_REF] Da | An ample approach to data and modeling[END_REF]. Furthermore, this procedure can be used to verify how already available models are associated with the other models of the modeling framework. For example, given a dataset ω i and its model m i , it is possible to combine alternative models m j̸ =i into a new model mi and verify how much its power of explanation overlaps to the model m i . Therefore, this approach provides a way to construct new models and characterize models already available in a modeling framework.

In the following, we formalize the methods involved in characterizing the specificity of a modeling framework, as well as the methods for evaluating the potential of the M * metamodel to construct new models.

Quantifying the Specificity of Modeling Frameworks

In general, data elements drawn from Ω may be part of more than one dataset, each explained by a respective model. Therefore, a model can be related, through its associated dataset, to elements that also belong to datasets of other models, which indicates a lack of specificity between the modeled data elements and the modeling framework. Although it would be desirable that the modeling frameworks are composed only of specific models, this is not necessarily always the case (such as in Figure 2).

To quantify the specificity of models in a modeling framework, we can consider three metrics: the element multiplicity, the connections specificity, and the connections diversity. Each of these metrics focuses on a different aspect of the specificity of models, thus helping to obtain further insights into the behavior of a modeling framework.

The multiplicity of an element is the number of times that the element repeats across different datasets. Because each dataset has a respective specific model, if the same data element appears in multiple datasets, that element will fit into different models. In other words, the element multiplicity will indicate the number of models indirectly connected to that element. The mean element multiplicity is the avarage of how many models are used to explain each data element available in E, thus being a metric of overall specificity of the modeling framework M .

If a given metamodel has N unique data elements distributed into n datasets, the mean element multiplicity will equal the sum of the cardinality C j of each dataset ω j divided by N . This is calculated with the Equation 1:

Ū (M ) = 1 N n i=1 C i (1)
This metric has a lower bound equal to 1, where each data element belongs to just one dataset, and an upper bound equal to n, corresponding to the extreme case where all elements belong to all datasets.

Distinct to the mean element multiplicity, the connections specificity and the connections diversity are calculated over each individual model in M . To estimate these metrics, we will use the weighted bipartite representation of the metamodel, where the weights of the connections equal the number of data elements in a dataset fitting a target model. This feature facilitates the visualization and quantification of the specificity of a model.

The connections specificity of a individual model m i is the fraction of the total incoming weights that comes from its respective dataset ω i . Let n be the number of datasets available in E, and w j,i be the weight of the connection between any dataset ω j and the model m i , the connections specificity of the model m i can be calculated by using the Equation 2:

S(m i ) = w i,i n j=1 w j,i (2)
with 0 < S(m i ) ≤ 1. The value of S(m i ) will be maximum if all weights w j̸ =i,i equals 0. If a dataset ω i accumulates data elements with high multiplicity, then the data elements of ω i will also be present in datasets ω j̸ =i . Having some of the same elements of ω i , these alternative datasets will also connect with m i with weights w j̸ =i,i ̸ = 0. This will lower S(m i ), which indicates that the m i model is unspecific to its dataset. Alternatively, if a dataset ω i accumulates data elements with low multiplicity, more w j̸ =i,i will be null, and a higher S(m i ) can be expected, thus representing a more specific model.

Another metric suitable to characterize the specificity of models is the exponential of the entropy of the weights of the connections, hereafter called connections diversity. To estimate this property, we treat the weights of the connections received by a model as a discrete distribution of values, and then we compute its exponential entropy. The entropy of the connecting weights of a model m i is calculated by using Equation 3, while the connections diversity is obtained from Equation 4:

H(m i ) = - n k=1 w k,i n j=1 w j,i log w i,i n j=1 w j,i (3)
D(m i ) = exp(H(m i )) (4)
with 0 ≤ H(m i ) denoting the entropy and 1 ≤ D(m i ) denoting the connections diversity.

Because the entropy of a distribution is a measure of uncertainty, the entropy of the weighted connections received by a model from different datasets shows the level of uncertainty of that model over the available datasets. This principle can be exemplified by the models m a and m b of the two metamodels depicted in Figure 5. The model m a has only one weighted connection coming from only one dataset, the entropy of its weights distribution is equal to 0, and it has no uncertainty over its connection to the datasets (Fig. 5(a)). Alternatively, the m b model has multiple weighted connections, thus yielding a non-zero entropy distribution, which reveals a higher connection uncertainty to the datasets (Fig. 5(c)).

The exponential entropy equals x when applied to a uniform distribution of x classes. This property can be used to interpret the value of the exponential entropy of a random distribution as the number of classes of an equivalent uniform distribution. For these reasons, the exponential entropy is often adopted as a measure of the diversity of distributions: the higher the value of the exponential entropy, the higher the uncertainty and the more diverse its equivalent uniform distribution is [START_REF] Jost | Entropy and diversity[END_REF][START_REF] Leinster | Measuring diversity: the importance of species similarity[END_REF][START_REF] Viana | Effective number of accessed nodes in complex networks[END_REF].

When applied to our case, the exponential entropy of the weighted connections will supply the connections diversity of a model m i . Using the property of the exponential entropy, the connections diversity indicates the number of categories (connections) of an equivalent uniform distribution of weights. We can deepen this interpretation by considering that the connections diversity of a model m i equals the number of different datasets that this model could explain with the same level of uncertainty, as illustrated in Figure 5.

In Figure 5(a), the model m a with only one weighted connection has its exponential entropy equal to 1, i.e., this model has the uncertainty equivalent to connecting to only one datasetsuch as the arbitrary model ma in Figure 5 connections has its exponential entropy approximately equal to 2, i.e., it has the uncertainty equivalent to uniformly connecting to two different datasets of the same size -such as the arbitrary model mb in Figure 5(d).

We now illustrate the preceding metrics respectively to the characterization of the cutlery metamodel in Figure 2. The whole metamodel has a mean element multiplicity of approximately 1.22, showing that most elements are quite specific to their datasets. The connections specificity of the forks model and the spoons model is equal to 0.75 and 0.71, respectively. Therefore, the forks model is slightly more specific than the spoons model, as it has more specific elements in its dataset. Their connections diversity is 1.75 and 1.82, respectively. Both datasets are quite unspecific, as they have a diversity almost as high as being equally connected to two different datasets.

Applying a Paired Algebra for Building Models

The connection between set theory, logic, and Boolean algebra is applied in many mathematical fields, such as computer [START_REF] Gregg | Ones and Zeros: Understanding Boolean Algebra, Digital Circuits, and the Logic of Sets[END_REF] and social sciences [START_REF] Levitz | Logic and Boolean Algebra[END_REF]. The application of this theoretical repertoire to the M * metamodel makes it possible to pair the assemble of datasets through set operations to a new method of building models: the construction of models through logical operations. In addition to the example in Figure 4(b), Table 1 illustrates some examples of how to construct a model m k to its respective dataset ω k [START_REF] Da | An ample approach to data and modeling[END_REF].

Table 1: Examples of the connection between the set operation in the environment domain E and the logic operations in the modeling framework domain M [START_REF] Da | An ample approach to data and modeling[END_REF].

E M ω k = ω i m k = m i ω k = ω i ∪ ω j m k = m i ∨ m j ω k = ω i ∩ ω j m k = m i ∧ m j ω k = ω C i m k = ¬m i ω k = (ω i ∪ ω j ) ∩ ω l m k = (m i ∨ m j ) ∧ m l ω k = (ω i ∪ ω j ) C ∪ ω l m k = (m i ∨ m j ) C ∨ m l ... ...
Where the complement set operation, [] C , considers the universe as the union of all data elements available in E, including elements on ω k .

When a new dataset ω k is identical to the result of a set operation between other datasets, such as in the examples of Table 1, the model m k constructed via logical operations will necessarily explain the whole ω k . However, the most common situation is that ω k is just partially explained through the combination of other datasets, thus being possible to achieve just an approximated dataset ω k(a) and an approximated model m k(a) through logical compositions.

To quantify how much an approximated model m k(a) is close to the desired exact model m k respective to the dataset ω k , we adopt the Coincidence index [START_REF] Da | On similarity[END_REF]. The Coincidence index compares two sets with a combination of the Jaccard index and the Interiority index, thus incorporating information about both the relative intersection and the relative interiority between them [START_REF] Da | On similarity[END_REF]. In our case, we will compare the models m k and the model m k(a) trough the coincidence index between its respective exact datasets, the set ω k and the set ω k(a) .

The Jaccard index is used to quantify the similarity between two datasets, A and B, based on the number of elements they have in common in front of the number of elements available on both of them [START_REF] Da | On similarity[END_REF][START_REF] Jaccard | Étude comparative de la distribution florale dans une portion des alpes et des jura[END_REF][START_REF] Ramos-Guajardo | Testing the degree of overlap for the expected value of random intervals[END_REF][START_REF] Brusco | A comparison of 71 binary similarity coefficients: The effect of base rates[END_REF]:

J (A, B) = |A ∩ B| |A ∪ B| (5)
The Interiority index [START_REF] Da | On similarity[END_REF], also called Szymkiewicz-Simpson coefficient or overlap index [START_REF] Ramos-Guajardo | Testing the degree of overlap for the expected value of random intervals[END_REF][START_REF] Szymkiewicz | Une contribution statistique à la géographie floristique[END_REF][START_REF] Vijaymeena | A survey on similarity measures in text mining[END_REF], quantifies how much one dataset is interior to the other:

I(A, B) = |A ∩ B| min{|A|, |B|} (6)
The Coincidence Index between two datasets, A and B, can be calculated by using the Equation 7 [START_REF] Da | On similarity[END_REF]:

C(A, B) = J (A, B) * I(A, B) (7)
with 0 ≤ C(A, B) ≤ 1. This Coincidence Index allows us to compare different approximated models obtained for a ω k dataset, therefore redefining the problem of model building as an optimization problem aimed at finding the dataset combination of the available datasets in E that gives the partial model with the highest possible coincidence index [START_REF] Da | An ample approach to data and modeling[END_REF].

Furthermore, the Coincidence Index can be used to quantify how much of a dataset ω k is explained by the logical composition of the alternative models m j̸ =k available in the modeling framework. Using the paired algebra between logic and sets operations, this quantification is performed by comparing set combinations of other datasets ω j̸ =k with the ω k . If a dataset ω k and a given logical composition of models m j̸ =k have a coincidence value equal to 1, then the model m k can be seen as redundant, after all its respective dataset, ω k , is sufficiently explained with the help of other models. This analysis follows the specificity characterization by focusing on accessing the level of redundancy of modeling frameworks. This procedure can also indicate models that could be decomposed and simplified in the framework, thus increasing its simplicity.

Verifying Concepts and Methods

In order to verify the concepts and methods presented, we performed several experiments, which are divided into three sections. First, we characterize simulated metamodels by considering their modeling framework specificity. Second, we used logical compositions of models within simulated modeling frameworks to quantify the relationship between the available models. Third, we applied the developed methods to two real-world examples of metamodels.

Characterization of Modeling Frameworks

In order to study the characterization of metamodels and access the specificity of their models, we resorted to the simulation of random structures.

First, we selected the number N of data elements drawn from a hypothetical universe Ω to form n different datasets ω and their respective models m. Also, we selected the max multiplicity of an element in the metamodel as u, i.e., an element will be a part of at least 1 and at most u different datasets. For all simulations, we set n = 6 datasets and u = 3. The simulations were performed using four different number of elements: N = 100, 500, 2500 and 10000.

Second, we distributed the N elements into the n datasets, enumerated from 1 to n, following the algorithm below. For each element e:

1. Draw the integer number A between 1 and u following the probability distribution P robA, which corresponds to the number of datasets that the element e can be part of. If A = 1, the element e is specific to a single dataset. If A ̸ = 1, the element e is unspecific.

2. For the specific elements, draw the integer number B between 1 and n following the probability distribution P robB, which corresponds to the dataset that the specific element e will be a part of;

3. For the unspecific elements, draw A different times the integer number C between 1 and n, without repetition, following the probabilities distribution P rob C. The sequence of the C different numbers corresponds to the datasets that the unspecific element e will be a part of.

The probability distribution P rob A gives the probability for each element to have a multiplicity of 1,2,... or u. As we set u = 3, the probability distribution P rob A are distributions of three numbers. The following five distributions were used in the simulations:

• P rob A 1 : (90, 8, 2)/100, very low chance of elements being a part of two or three datasets;

• P rob A 2 : (70, 24, 6)/100, low chance of elements being a part of two or three datasets;

• P rob A 3 : (50, 40, 10)/100, medium chance of elements being a part of two or three datasets;

• P rob A 4 : (30, 56, 14)/100, high chance of elements being a part of two or three datasets;

• P rob A 5 : (10, 72, 18)/100, very high chance of elements being a part of two or three datasets.

The probability distribution P rob B provides the probability for each specific element (elements of multiplicity 1) to be allocated into the datasets 1 to n. For our experiments, we set n = 6. The following three distributions were used in the simulations:

• P rob B 1 : (1, 1, 1, 1, 1, 1)/6, uniform distribution of elements between the six available datasets;

• P rob B 2 : (1, 2, 3, 4, 5, 6)/21, linear distribution of elements between the six available datasets;

• P rob B 3 : (6, 3, 2, 10, 8, 1)/30, irregular distribution of elements between the six available datasets;

The probability distribution P rob C provides the probability of each unspecific element (elements of multiplicity different than 1) to be allocated into the datasets 1 to n. The following three distributions were used in the simulations:

• P rob C 1 : (1, 1, 1, 1, 1, 1)/6
, uniform distribution of elements between the six available datasets;

• P rob C 2 : (1, 2, 3, 4, 5, 6)
/21, linear distribution of elements between the six available datasets;

• P rob C 3 : (7, 6, 9, 2, 1, 5)/30, irregular distribution of elements between the six available datasets.

We performed 180 simulations of metamodels, each considering a different combination of the four parameters defined above: N , P rob A, P rob B and P rob C. For each simulation, we obtained the weighted bipartite representation (see Figure 2) with all the respective weights w j,i between any ω j dataset and m i model.

All the simulations were characterized using four general metrics:

• Mean element multiplicity: Mean number of repetitions of an element among the available datasets in E, given by the Equation:

Ū (M ) = 1 N n i=1 C i (8)
with 1 ≤ Ū (M ) ≤ n, where N is the number of elements, and C i is the cardinality of the dataset ω i . This metric represents how many times, on average, the elements repeat among the datasets.

• Mean connections specificity: Mean value of the connections specificity of the models within a modeling framework M , given by the Equation:

S(M ) = 1 n n i=1 w i,i n j=1 w j,i (9)
with 0 < S(M ) ≤ 1, where w j,i is the weight between the dataset ω j and the model m i , n is the number of datasets, and M is the modeling framework domain. This is the mean value of the connections specificity of the models available (Eq. 2). This measure represents the average fraction of weights received by the models that come from their respective dataset.

• Total connections diversity: Sum of the diversity of the connections of all models, estimated with the Equation:

D T (M ) = n i=1 exp   - n j=1 w i,i n j=1 w j,i log w i,i n j=1 w j,i   (10)
with n ≤ D T (M ). The connections diversity of a model m i , estimated by using Equation 4, will indicate the number of different datasets that m i could explain with the same level of uncertainty (see Fig. 5). The total connections diversity is the sum of the diversity of each model, which indicates the total number of disjoint datasets that could be entirely connected to the available models with the same level of uncertainty. When D T (M ) = n, the models of M are highly specific, and each one have the uncertainty equivalent to having only one connected dataset. When D T (M ) > n, the models of M are unspecific, and at least one of them has the uncertainty equivalent to having more than one connected dataset.

• Mean dataset cardinality: Mean number of elements in each dataset, obtained from the following Equation:

C(M ) = 1 n n i=1 C i (11)
This measurement will provide further information on the simulated metamodel structure.

These metrics were analyzed in generalized pairs plots [START_REF] Emerson | The generalized pairs plot[END_REF]. The Pearson correlation coefficient [START_REF] Johnson | Applied multivariate statistical analysis[END_REF] between each pair of metrics was estimated considering all 180 simulations together or divided into groups based on their simulation parameters (N , P rob A, P rob B and P rob C).

Composing Models within Modeling Frameworks

After we evaluated the specificity of metamodels, we proceed to use the logical composition of models within a modeling framework in order to analyze the composition of models within frameworks.

Again, we resorted to the simulation of metamodels, following the algorithm presented previously (3.1). Now, for all simulations, we set n = 6 datasets, u = 3 and N = 100 elements.

Ten probability distributions for the probability P robA were used in the simulations, varying from lower to higher chances of elements being assigned to take part in two or three datasets:

• P rob A 1 to P rob A 5, as previously; The distributions for P rob B (P rob B 1, P rob B 2 and P rob B 3) and P rob C (P rob C 1, P rob C 2 and P rob C 3) were used as previously. We performed 90 simulations of metamodels, each considering a different combination of values of three parameters: P rob A, P rob B and P rob C.

All individual models of each simulated modeling framework were compared with models m X that were built through logical compositions of the other five models m j̸ =k available in the same framework. The extent to which each model m k can be explained by the logical composition m X of the other models m j̸ =k is quantified using the Coincidence Index (Eq. 7) between the respective dataset ω k and the respective dataset ω X , assembled through set combinations of the datasets ω j̸ =k .

Each model m X is built following the logical equation:

m X (m k ) =   i̸ =k ¬m i   ∨ (m O (m k )) (12)
Using the metamodeling approach, we can translate the logical operations between models in the Equation 12to set operations between their respective datasets (see Table 1), thus obtaining:

ω X (ω k ) =   i̸ =k ω C i   ∪ (ω O (ω k )) (13)
The first term accounts for elements specific to the ω k dataset, i.e., that are not explained by any other models but m k . These cases are the majority in modeling frameworks formed by mutually exclusive models, meaning that we can construct one model by simply combing the negative of the others. The second term, ω O , is any set combination of O alternative datasets using any of the three basic set operations: union -∪; intersection -∩; and complement -() C .

We obtained all possible ω O datasets using O = 1 to O = 4, following the iterative construction: [START_REF] Frantz | A taxonomy of model abstraction techniques[END_REF] where () □ might be the operation () C , and ■ is either ∪ or ∩. Notice that more complex equations, such as those between operations in parentheses, are not included -e.g., the equation ((

ω O=1 (ω k ) = ω □ j̸ =k (14) ω O=2 (ω k ) = ω O=1 (ω k )■ ω □ j̸ =k (15) ω O=3 (ω k ) = ω O=2 (ω k )■ ω □ j̸ =k (16) ω O=4 (ω k ) = ω O=3 (ω k )■ ω □ j̸ =k
ω 1 ∪ ω 2 ) ∩ (ω 3 ∩ ω 4 )) is not evaluated, but the equation (((ω 1 ∪ ω 2 ) ∩ ω 3 ) ∩ ω 4 ) is.
The maximum Coincidence Index between ω k and each possible ω X represents the maximum coincidence between the model m k and any logical composition of models m j̸ =k . If this number is close to 1, then m k is sufficiently explained by the combination of alternative models available, which indicates that this model is redundant within its modeling framework. If this number is low, then m k is a unique model that cannot be acquired with the information embedded in the other models.

Furthermore, the maximum Coincidence Index achieved for any of the six ω k in a metamodel shows the maximum level of inter-dependency between models in the modeling framework. Therefore, the top coincidence (TC) value of a metamodel reveals its redundancy level, and it can be calculated using the Equation 18:

T C(M * ) = max (ω k ,ω X (ω k )) C(ω k , ω X (ω k )) (18)
for any possible pair (ω k , ω X (k)). The C denotes the function for the Coincidence Index, shown in Equation 7.

The 90 simulations of metamodels were characterized using the TC value (Eq. 18) and the results were analysed considering the simulation parameters.

Analysis of Two Real-World Case Examples

We constructed and analyzed the M * metamodels of two real-world case examples of modeling frameworks: (i) a framework that describes symmetries in 3-by-3 pixel patterns; and (ii) a framework that classifies species in a food web based on their topological features.

In each case example, we started using a predefined initial configuration of the environment domain E and the modeling framework M . We then constructed the metamodel to analyze how the available models relate to each organized dataset. First, we organize the data initially available in the universe domain Ω into six datasets constituting the environment domain E. Each dataset was chosen to follow a respective predefined rule. Based on each adopted dataset, we created a respective model that was included in the modeling framework M . These models can be understood as the formalization of the predefined rules used to assemble the datasets. In the two case examples under analysis, the pre-defined rules used to organize the data sets are well understood, and the respective models for each dataset are clearly defined. However, this is not the case for most modeling approaches, which usually will have incomplete and poorly defined models.

After the initial configuration of the metamodel was built, as explained above, we characterized the respective modeling framework using the specificity metrics presented in section 2.2. We also quantified the level of relationship between the models associated with each dataset using the coincidence index following the data-driven approach presented in section 2.3. These characterizations enabled the identification of models with low specificity and high redundancy within the proposed modeling framework. Based on this analysis, it was possible to logically combine models in order to obtain more specific and accurate models.

The first case example addresses the modeling of six datasets of 3-by-3 pixel patterns filled with black and white pixels. We then generated all 512 possible figures using only black and white pixels and divided them into six datasets obeying respective models defined by the rules in Table 2.

Table 2: The six rules defining the datasets and respective models of the 3-by-3 pixel patterns.

Dataset Rule and Model 1. Patterns that have rotational symmetry around their central cell 2. Patterns that have reflection symmetry respectively to their main diagonal (lower left corner to upper right corner) 3. Patterns that have reflection symmetry respectively to their antidiagonal (upper left corner to lower right corner) 4. Patterns that have a black pixel in at least one corner 5. Patterns that have a black pixel in at least one of north, south, west, and east cells 6. Patterns that have more than half of their cells with black pixels Figure 6 depicts four instances of patterns classified using the previous modeling framework. Each one of the depicted pattern samples belongs to more than one dataset, as they satisfy more than one of the rules presented in Table 2. For example, Pattern 495 in Figure 6 belongs to four different datasets: dataset 2, because it has reflection symmetry respectively to its main diagonal; dataset 4, given that it has at least one corner with a black pixel; dataset 5, as it has at least one of north, south, west and east cells with a black pixel; and dataset 6, because it has more than half of its cells with black pixels.

The second case example addresses the modeling of six datasets of aquatic insects, algae, and fish species interacting in a food web. A food web consists of a network that describes the trophic interactions within an ecological system, in which species are linked with directed connections to others that predate them [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF][START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF] Figure 6: Four 3-by-3 pixel pattern samples filled with black and white pixels. The patterns are numbered in the order they are created using a permutation algorithm. The shown patterns were included in different datasets according to the rules presented in Table 2. Pattern 5 is included in datasets 3 and 4 as it has reflection symmetry respectively to its antidiagonal and has at least one corner with a black pixel. Pattern 11 is contained in datasets 2 and 5, since it has reflection symmetry respectively to its main diagonal and has the north and east cells with black pixels. Pattern 41 has rotational symmetry, and both west and east cells with black painted, thus being included in datasets 1 and 5. Pattern 495 has reflection symmetry respectively to its main diagonal, has three corners with black pixels, has north, east, south, and west cells with black pixels, and has more than half of its cells with black pixels, so it is included in datasets 2, 4, 5, and 6.

The Thompson and Townsend food web data, collected in a series of works from 1998 to 2005 [START_REF] Jaarsma | Characterising food-webs in two new zealand streams[END_REF][START_REF] Townsend | Disturbance, resource supply, and food-web architecture in streams[END_REF][START_REF] Thompson | The effect of seasonal variation on the community structure and food-web attributes of two streams: Implications for food-web science[END_REF][START_REF] Thompson | Is resolution the solution?: the effect of taxonomic resolution on the calculated properties of three stream food webs[END_REF][START_REF] Thompson | Allocation of effort in stream food-web studies: the best compromise?[END_REF][START_REF] Thompson | Impacts on stream food webs of native and exotic forest: an intercontinental comparison[END_REF][START_REF] Thompson | Energy availability, spatial heterogeneity and ecosystem size predict food-web structure in streams[END_REF], were obtained from the Interaction Web Database (available online at http://www.ecologia.ib.usp.br/iwdb/). The data acquired corresponded to 26 adjacency matrices representing food web networks observed in multiple locations and seasons of streams of New Zealand, Maine (USA), and North Carolina (USA). The 26 adjacency matrices were then combined into a single hypothetical food web network containing 173 species that preserves the original observed trophic interactions. The species taxon were analyzed using the NCBI Taxonomy Database [START_REF] Schoch | NCBI taxonomy: a comprehensive update on curation, resources and tools[END_REF]. The combined food web network will be used in the second case example developed in this work.

Two types of models are usually applied to characterize species disposed on a food web: those considering topological measurements of the species on a food web network [START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF], and those based on trophic levels (TL) [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF]. These two types of models are not independent, as TLs can be determined by analyzing the topological features of the species on the food web. In general, adopting these models to characterize species facilitates the comprehension of the roles of organisms in different ecosystems [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF][START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF][START_REF] Christian | Organizing and understanding a winter's seagrass foodweb network through effective trophic levels[END_REF]. Furthermore, these models can assist the study of energetic flows, bioaccumulation, and biomagnification in the environment [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF][START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF][START_REF] Christian | Organizing and understanding a winter's seagrass foodweb network through effective trophic levels[END_REF][START_REF] Won | Importance of accurate trophic level determination by nitrogen isotope of amino acids for trophic magnification studies: A review[END_REF].

There are many models based on topological measurements. A commonly used measure to study food webs is the node degree, which is calculated for each available species within a food web network [START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF]. A node in a food web network represents a species, and the node degree is defined as the number of other nodes connected to it [START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF]. The in-degree of a species in a food network corresponds to the total number of species it preys on [START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF]. The out-degree of a species in a food network is the number of species that predates on it [START_REF] Jordán | Topological keystone species: measures of positional importance in food webs[END_REF]. We used the node degree to assemble two datasets composed of species with an in-degree equal to 1 (species with only one prey) and an out-degree equal to 0 (species without predators).

One of the simplest models based on the TL characterizes species into integer TLs based on their topological distance from a primary producer species (a species that do not predate any other organism) [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF]. There are multiple ways to calculate the TL of a species based on network distances, for example, by considering the shortest distance between a species and a primary producer, the mean distances from any primary producer, or even the longest observed distance [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF]. Particularly, the longest TL is defined as one plus the longest distance between the species and a primary producer [START_REF] Williams | Limits to trophic levels and omnivory in complex food webs: theory and data[END_REF]. According to this definition, species that do not predate other organisms belong to TL 1; species that predate only those in TL 1 belong to TL 2; species that predate organisms in TL 1 and 2 belong to TL 3; and so on. Applying the longest TL to analyze the food web of the Thompson and Townsend data [START_REF] Jaarsma | Characterising food-webs in two new zealand streams[END_REF][START_REF] Townsend | Disturbance, resource supply, and food-web architecture in streams[END_REF][START_REF] Thompson | The effect of seasonal variation on the community structure and food-web attributes of two streams: Implications for food-web science[END_REF][START_REF] Thompson | Is resolution the solution?: the effect of taxonomic resolution on the calculated properties of three stream food webs[END_REF][START_REF] Thompson | Allocation of effort in stream food-web studies: the best compromise?[END_REF][START_REF] Thompson | Impacts on stream food webs of native and exotic forest: an intercontinental comparison[END_REF][START_REF] Thompson | Energy availability, spatial heterogeneity and ecosystem size predict food-web structure in streams[END_REF], we assembled four more distinct datasets corresponding to the collection of species whose longest TL equals 1, 2, 3, and 4 or more (4+).

The two datasets assembled using the node degree measures and the four datasets defined using the longer TLs were integrated into a single environment domain E. The models corresponding to each dataset in E were united into a single modeling framework M . This mixture of models was conceived in order to evaluate how these different approaches to studying a food web complement and overlap one another in the analysis of the species available in a food web.

In summary, the six datasets assembled using the species of the food web constructed using the Thompson and Townsend data [START_REF] Jaarsma | Characterising food-webs in two new zealand streams[END_REF][START_REF] Townsend | Disturbance, resource supply, and food-web architecture in streams[END_REF][START_REF] Thompson | The effect of seasonal variation on the community structure and food-web attributes of two streams: Implications for food-web science[END_REF][START_REF] Thompson | Is resolution the solution?: the effect of taxonomic resolution on the calculated properties of three stream food webs[END_REF][START_REF] Thompson | Allocation of effort in stream food-web studies: the best compromise?[END_REF][START_REF] Thompson | Impacts on stream food webs of native and exotic forest: an intercontinental comparison[END_REF][START_REF] Thompson | Energy availability, spatial heterogeneity and ecosystem size predict food-web structure in streams[END_REF] had respective models defined by the rules presented in Table 3.

Table 3: The six rules defining the datasets and respective modeling framework of the species of the food web constructed using the Thompson and Townsend data.

Dataset Rule and Model 1. Node out-degree equals 0 2. Node in-degree equals 1 3. TL 1 4. TL 2 5. TL 3 6. TL 4+

Results and Discussion

In this section, we report and discuss the evaluation of specificity metrics on various simulated metamodels and on two real-case examples. First, based on artificial data, we evaluate the characteristics of unspecific and redundant models within modeling frameworks and identify conditions for constructing new models through the logical composition of alternative models. Then, using these results, we develop and analyze the metamodels of two case-examples: the modeling of symmetries in 3-by-3 pixel patterns and the modeling of species trophic classification in a food web.

Characterization of Modeling Frameworks

In this section, we characterize the modeling frameworks of 180 simulated metamodels from the perspective of four measurements: mean element multiplicity, mean connections specificity, total connections diversity and mean dataset cardinality. The three first measurements address the quantification of inter-specificity between datasets and models through different perspectives -respectively, the mean number of element repetition among datasets, the mean fraction of connections received by a model that comes from its respective dataset, and the equivalent number of disjoint datasets that could connect to the models with the same level of uncertainty. The last measure, the mean dataset cardinality, was used to further characterize how the datasets were assembled in each metamodel. Two examples of the simulated metamodels are present in Figure 7.

These four measures were analysed together through generalized pair plots [START_REF] Emerson | The generalized pairs plot[END_REF], shown in Figures 9 to 11. In each pairs plot, one different parameter of the metamodel simulations was highlighted with colors: the number of elements N distributed into their 6 datasets, the probability distribution Probability A of assigning the levels of multiplicity 1 to 3 to each element, the probability distribution Probability B of assigning specific elements to the datasets 1 to 6, and the probability distribution Probability C of assigning unspecific elements to the datasets 1 to 6. In the upper right squares of the pair plots (Fig. 9), the Pearson correlation coefficient between the respective two variables, considering all 180 simulations, is shown in black. The correlation between mean dataset cardinality and multiplicity of elements is trivial (Eq. 8 and 11), as for a specific number of elements observed in a model, a low average cardinality indicates less repetition of elements and, therefore, smaller multiplicity.

For similar reasons, the mean cardinality of datasets is strongly correlated with the other specificity metrics. As the percentage of the total number of elements present in each dataset increases, the repetition of elements between datasets is inevitable, especially when this percentage is higher than 100/n% (the total number of elements divided by the total number of datasets). Thus, simulations with higher mean dataset cardinality will present elements repeated in different datasets and models that receive connections from both their respective dataset and alternative datasets. Therefore, this will decrease the metamodel mean connections specificity (due to the accumulation of unspecific elements in the datasets) and increase the total connections diversity of the models (due to the greater variety of unspecific connections between datasets and models).

In order to visualize how these correlations emerge, we can compare the two metamodel examples in Figure 7, both simulated considering N = 500 elements. The metamodel (a) has a mean dataset cardinality of 96.67 (18.53% of N = 500), mean connections specificity of 0.82, total connections diversity of 12.74 and mean element multiplicity of 1.112. The metamodel (b) have a mean dataset cardinality of 154 (30.80% of N = 500), mean connections specificity of 0.51, total connections diversity of 23.97 and mean element multiplicity of 1.848. To have a higher mean dataset cardinality in comparison to the metamodel (a), the metamodel (b) had to repeat elements between different datasets, which increased the weights of its alternative connections (thicker diagonal arrows in Fig. 7 (b)), thus decreasing the overall connections specificity of the models and increasing the connections diversity.

The observed tendencies are strictly related to one parameter simulation: P robA, the distribution of probabilities of each multiplicity value to be drawn for each element. In Figure 8, the points with the same colors represent metamodels with the same probabilities P robA. Along the main diagonal of the pairs plot, it is shown the distribution of the measurements corresponding to each group of simulations with the same P rob A. Because P rob A determines the distribution of the elements multiplicity in each simulation, the observed distributions of the mean element multiplicity for each group are markedly segregated. This group segregation persists in the distributions of the mean connections specificity and the total connections diversity. The last line of plots in Figure 8 combines these metrics in scatter plots that show how the P rob A parameter creates layers of values of mean element multiplicity that can be further analyzed through the other specificity metrics.

The range of the measurements distributions for the most unspecific group of simulations, group P rob A 5, gives different ideas of how the unspecificity of metamodels can arise:

• Mean element multiplicity: distribution ranges from 2 to 2.25, hence unspecific metamodels can arise from having a considerable amount of elements with a minimal degree of multiplicity (duplication).

• Mean connections specificity: values are lower than 0.5, thus the number unspecific elements available over the different datasets is, in total, at least equal to the number of specific elements.

• Total connections diversity: distribution ranges from 24 to 29, indicating that its models receive connections with enough diversity to uniformly satisfy more than four times the available number of datasets.

Therefore, it is the accumulation of unspecific elements in the datasets and the wide distribution of unspecific connections between datasets and models that can lower the overall specificity of a modeling framework.

Interestingly, the only simulation parameter that has a high impact on the specificity of metamodels is P rob A. In Figure 9, 10 and 11, the simulations are grouped according to the parameters P rob B and P rob C, respectively. In these three cases, the formed groups of simulations have distributions of specificity values that are almost identical to each other.

However, it is worth noticing that the P rob C parameter creates subgroups of higher (P rob C 1), medium (P rob C 2), and lower (P rob C 3) total connections diversity within the layers created by the groups of simulations with the same P rob A. After the P rob A determines the number of unspecific elements available, the metamodels that uniformly distribute those elements in their datasets following P rob C 1 will be less specific than those that accumulate them in just some datasets (P rob C 2 and P rob C 3). This indicates that, given a number of unspecific elements in the environment domain, for the modeling framework to be specific, it is better to have a few highly unspecific models than several models with the same mid-level unspecificity.

Composing Models within Modeling Frameworks

After evaluating the specificity of modeling frameworks, we aim at describing the conditions in which a modeling framework is redundant. Redundancy occurs in a modeling framework when one of its models is equivalent to some other available model, or some logical composition The metamodel simulations are divided into three groups according to the probability distribution P rob C used to distribute the unspecific data elements in their datasets: 1, uniform distribution of elements between the six available datasets; 2, linear distribution of elements between the six available datasets; 3, irregular distribution of elements between the six available datasets. Each group has a corresponding color in the graph. The plots on the lower left triangle are scatter plots, with each point corresponding to a metamodel simulation. The points are colored according to its P rob B group. The graphs on the main diagonal are the distributions of the respective metrics for each group of metamodels. The upper right triangle shows the Pearson correlation coefficient between the correspondent pairs of metrics. The correlation considering all 180 simulations is in black, and the correlations calculated for each group of metamodels are in the respective colors. Refer to section 3.1 for further information on the probability distributions.

of other models.

The maximum level of redundancy of a modeling framework can be measured by evaluating the maximum coincidence value achieved between the datasets of any available model and any possible logical composition of alternative models. This measurement, which we called Top Coincidence (TC) value, is defined in Equation 18. To calculate the TC value, we compared any ω k dataset of a metamodel with any set combinations ω X generated by the union of two parts (Eq. 13):

i̸ =k (ω i ) C , a fixed part that represents all exclusive elements of ω k ; and ω O , a variable part that is equal to any set combination of alternative datasets ω i̸ =k .

In order to study the above type of redundancy in modeling frameworks, we simulated and analyzed 90 metamodels, following the method in section 3.2. The results obtained are shown in Figure 12, which is organized in box plots grouped according to different simulation parameters of the 90 studied metamodels. Figure 12(a) shows the distributions of the TC value of the 90 metamodels, grouped according to the parameter of simulation P rob A. In Figure 12(b), the TC values of the metamodels are shown in box plots grouped according to the parameter of simulation P rob B, while Figure 12(c) has box plots grouped according to the parameter simulation P rob C.

We expect that highly redundant modeling frameworks will be characterized by higher TC values, and we assume this redundancy emerges from how the data elements are distributed among the explained datasets. First, based on the previous analysis of specificity, we expect that the probability P rob A should have a major impact on the overall redundancy of modeling frameworks. The metamodels simulated with a high probability of elements being assigned to one dataset, such as P robA1 and P robA2, should be more redundant. This should occur because these simulations create modeling frameworks whose models are more easily replaced by the logical negation of the others. Alternatively, metamodels simulated with extreme probabilities of elements being assigned to multiple datasets, such as P rob A 9 and P rob A 10, should also be more redundant, as their models explain datasets whose elements are also explained by others.

Based on the analysis of all the results in Figure 12, some simulation parameters can be noticed to have a visible impact on the overall redundancy, i.e. TC values, of the metamodels: (i) the P rob A parameters, the set of three probabilities of the elements being assigned to 1, 2 or 3 different datasets; (ii) and P rob C, the set of probabilities of the unspecific elements to be assigned to the dataset 1, 2, 3, 4, 5 or 6. These parameters also interfere in the overall specificity metrics of the metamodels (Fig. 8 and11).

In Figure 12(a), the P rob A categories on the x-axis are arranged, from left to right, in increasing value of the third probabilities. Since the total sum of the three values must add to 100%, there are changes in the first and second probabilities. These probabilities are related to the overall unspecificity of the elements in the metamodel. The first probability, P rob A 1, gathers the most specific simulations, with a 90% probability of elements being assigned to a single dataset, 8% of being assigned to two datasets, and 2% of being assigned to three datasets. The last probability, P rob A 10, groups the most unspecific simulations, with a 0% probability of elements being assigned to a single dataset, 30% of being assigned to two datasets, and 70% of being assigned to three datasets.

The median of the box plots of the TC values in Figure 12(a) follow two distinct tendencies: first, there is a decrease in the TC values observed from P rob A 1 to P rob A 4; then, there is an increase from P rob A 5 to P rob A 10. This distinction also occurs respectively to the number of alternative models used in the logical composition that achieved the TC values: from P rob A 1 to P rob A 4, numerous TC values are reached for logical compositions of 2, 3 and 4 alternative models; from P rob A 5 to P rob A 10, the TC values corresponds mostly to combinations of 4 alternative models.

The progression from P robA1 to P robA4 in Figure 12(a) occurs because the simulations that have lower percentages of unspecific elements (elements that fit into multiple datasets) involve almost mutually exclusive models. In these simulations, at least one model m k is sufficiently described as the model that explains all elements that are not explained by the other models. In this case, the dataset respective to this m k model would be ω k ∼ i̸ =k (ω i ) C . One example of a metamodel simulated using P rob A 1 is depicted in Figure 13. In the diagram, the dataset colored in gray, ω 5 , corresponds to the most redundant dataset in the metamodel, yielding the maximum TC value. The dot representing the respective model of ω 5 , m 5 , is omitted to indicate that this model is being compared to the logical compositions of the other available models.

The TC value of the metamodel of Figure 13 is calculated using Equation (i) of the diagram. This equation, obtained in an exhaustive comparison between any available dataset and any set combination of alternative datasets, yields the maximum TC when we compare the dataset ω 5 with a set combination of the other datasets available, called ω X,T C for short. The set combination of alternative datasets, ω X,T C , is expressed in Equation (ii) as a combination of the datasets ω 1 , ω 2 , ω 3 , ω 4 and ω 6 . In that way, ω X,T C turned out to correspond to a dataset Each dot corresponds to one modeling framework simulated using the parameter indicated on the x-axis in (a) and y-axis in (b) and (c). The P rob A represents the probabilities of each element in the simulation to be assigned to one, two, or three different datasets -e.g. in P rob A 1, elements had 90% probability of being assigned to one dataset, 8% of being assigned to two datasets, and 2% of being assigned to three datasets, organized as a triple of values between vertical bars 90|8|2. The P rob B and P rob C represent the simulations' probability distributions to assign specific and unspecific elements, respectively, into datasets 1 to 6. The six weights of the distribution, corresponding to each dataset, are indicated between vertical bars (e.g. 1|1|1|1|1|1). The TC value of a metamodel is the max coincidence calculated between any model m k and any logical composition mX of the alternative models m j̸ =k available in the simulated modeling framework. The combination mX respective to the TC value of a modeling framework is a combination of 2, 3, or 4 alternative models, indicated by the dot colors.

Figure 13: Weighted bipartite representation of a metamodel simulated using P robA1, P robB1 and P robC 1. The representation is modified to highlight the dataset ω5, which yields the TC value of the represented metamodel. The dataset ω5 is colored in gray on the left-hand side of the diagram, while its respective model m5 is omitted on the right-hand side. The TC value of the represented metamodel is ≈ 0.96. As shown in equation (i), the TC value, in this case, is the coincidence index between the dataset respective to model m5, ω5, and the dataset respective to a logical composition of models, ωX,T C . The set operation that defines the ωX,T C dataset is indicated in equation (ii). In this metamodel, multiple ωO,T C can be used to achieve the same TC value. Three examples of ωO,T C are provided in equations (iii), (iv), and (v), each one involving datasets respective to a different number of alternative models: 2 models in (iii), 3 models in (iv), and 4 models in (v).

obtained from a logical composition of the alternative models available in modeling framework, m 1 , m 2 , m 3 , m 4 and m 6 .

In this example, ω 5 is sufficiently explained by the logical exclusion of all alternative models -as can be verified by the fact that the weights of its diagonal connections with alternative models (weights equal to 1, 1, and 2) are small compared to the size of dataset ω 5 (equal to 18 elements). Thus, the model m 5 can be regarded as the model that explains almost all data elements that are not explained by any other models, which can be mathematically expressed as ω 5 ∼ i̸ =5 (ω i ) C . The particular analysis of the example in Figure 13 helps to clarify why there is a decreasing tendency of TC values between P robA1 to P robA4 in Figure 12(a). The metamodels simulated with these probabilities have most of their elements being a part of only one dataset. Therefore, the model m k that yields the TC value of the metamodel have respective dataset ω k similar to i̸ =k (ω i ) C . However, as the percentage of elements exclusive to one dataset decreases, the ω k datasets becomes more different from i̸ =k (ω i ) C , thus making the ω O part of the set combination more important as a means to yield high TC values.

The decreasing tendency of TC values between P rob A 1 and P rob A 4 changes between P rob A 5 and P rob A 10, where TC values tend to increases. This occurs because these metamodels simulations have high percentages of unspecific elements. Since the datasets have fewer exclusive elements, the TC value calculated using any ω k and ω X depends almost completely on the ω O part. In that case, the higher the probability of the elements being part of multiple datasets, the easier it is to find a set combination ω O that coincides with ω k . This explains why the metamodels simulated with P rob A 10, with 70% probability of elements being on three datasets, have higher TC values than the metamodels simulated with P rob A 5, with a probability of only 18%.

Anyhow, for the tendencies between P rob A 1 and P rob A 4 and between P rob A 5 and P rob A 10, higher values of TC should be achieved when the distribution of unspecific element is not uniform. This is the behaviour observed in Figure 12(c), where the simulations following P rob C 2 and P rob C 3 have higher TC values than those following the uniform distribution P rob C 1. For the simulations following P rob A 1 to P rob A 4, one single model having fewer unspecific elements will produce a higher TC value. For simulations following P rob A 5 to P rob A 10, a model with more unspecific elements will be more easily explained by the logical composition of the other models, thus reaching a higher value of TC. Therefore, non-uniform distributions of unspecific elements will yield higher TC values for all cases. In addition, the more extreme the non-uniformity is, the higher the TC values will be. These results provide the conditions under which a model could be successfully described in terms of the alternative models available within the same modeling framework. This can be used to access the redundancy of the modeling framework and to indicate models (the model respective to the TC value) that could be decomposed and simplified in the framework in order to increase its simplicity.

Furthermore, this examination also helps to define the conditions to successfully build new models through logical compositions. As discussed previously (section 2.1.4), within a metamodel, it is possible to create a new model to explain a new dataset solely using logical compositions of models already established within a modeling framework. The results from the TC analysis specify the modeling framework requirements to assemble a model through logical compositions depending on the characteristics of its respective dataset: either if it is specific (shares few elements with the other available datasets) or unspecific (shares many elements with the other available datasets). In order to explain a new specific dataset, a new model can be created using a metamodel with a non-uniform distribution of few unspecific elements among the available datasets. In order to explain a new unspecific dataset, a new model can be created using a metamodel with a non-uniform distribution of many unspecific elements among the available datasets.

Real-World Case Example 1: Symmetries in 3-by-3 Pixel Patterns

In this section, we analyze a M * metamodel concerning the symmetries that appears on 3-by-3 pixel patterns. The metamodel, constructed as described in section 3.3, encompasses the relationship between six datasets of 3-by-3 patterns of black and white pixels and six models that respectively explains these datasets. The six models analyzed in the metamodel, enumerated as follows, describe patterns that have:

1. Rotational symmetry around their central cell; 6. More than half of their cells with black pixels.

These models can be further developed conceptually and mathematically. For example, the model for the 3-by-3 pixel patterns that have reflection symmetry respectively to their main diagonal (model 2) can be expressed using the matrix equation P t = P , where P is a 3x3 matrix of 0s and 1s that represents the 3-by-3 patterns. Anyhow, the data-driven M * metamodel depends solely on the relationship between the available models and datasets and not their detailed structures.

Figure 14 depicts the metamodel constructed using the six models presented above. The specificity metrics calculated over the whole modeling framework -namely the mean element multiplicity, mean connections specificity and total connections diversity -, as well as other descriptive metrics of the metamodel, are presented in Table 4. The metamodel in Figure 14 reveals a highly unspecific modeling framework. Only 24 of the 512 data elements distributed among the datasets are specific, i.e. belongs to only one dataset. On average, each data element belongs to more than two different datasets, thus being explained by two distinct models. According to Table 4, four times as much diversity can be observed than would be necessary for a specific mapping between datasets and models, which emphasizes the high level of the interrelationship between the different models. We can associate this definition for the m Extra model with the information available for the m X (m 5 ) in order to create an exact logical composition of models that is equivalent to m 5 . This is achieved by combining the m X (m 5 ) and the negation of m Extra as expressed below:

ω 1 / m 1 0.500 i̸ =1 ω c i ∪ (ω 2 ∩ ω 3 ) ω 2 / m 2 0.250 i̸ =2 ω c i ∪ (ω 1 ∩ ω 3 ) ω 3 / m 3 0.250 i̸ =3 ω c i ∪ (ω 1 ∩ ω 2 ) ω 4 / m 4 0.939 i̸ =4 ω c i ∪ (((ω 6 ∪ ω c 1 ) ∪ ω c 2 ) ∪ ω c 5 ) ω 5 / m 5 0.947 i̸ =5 ω c i ∪ (((ω 6 ∪ ω c 1 ) ∪ ω c 2 ) ∪ ω c 4 ) ω 6 / m 6 0.561 i̸ =6 ω c i ∪ (((ω 1 ∩ ω 2 ) ∪ ω 4 ) ∩ ω 5 ) Dataset (ω k ) and Model (m k ) Maximum Coincidence (C) Respective Logical Composition of Alternative Models (m X ) ω 1 / m 1 0.500 i̸ =1 ¬m i ∨ (m 2 ∧ m 3 ) ω 2 / m 2 0.250 i̸ =2 ¬m i ∨ (m 1 ∧ m 3 ) ω 3 / m 3 0.250 i̸ =3 ¬m i ∨ (m 1 ∧ m 2 ) ω 4 / m 4 0.939 i̸ =4 ¬m i ∨ (((m 6 ∨ ¬m 1 ) ∨ ¬m 2 ) ∨ ¬m 5 ) ω 5 / m 5 0.947 i̸ =5 ¬m i ∨ (((m 6 ∨ ¬m 1 ) ∨ ¬m 2 ) ∨ ¬m 4 ) ω 6 / m 6 0.561 i̸ =6 ¬m i ∨ (((m 1 ∧ m 2 ) ∨ m 4 ) ∧ m 5 )
m 5 =     i̸ =5 ¬m i   ∨ (((m 6 ∨ ¬m 1 ) ∨ ¬m 2 ) ∨ ¬m 4 )   ∧ ¬m Extra (19)
Although this logical composition and its interpretation are much more complex than the straightforward explanation of the model m 5 (model that describes patterns with a black pixel in at least one of the north, south, west, and east cells), in case this explanation is unavailable, this exact logical composition can be useful as a general model for the 480 patterns in the ω 5 dataset. This result illustrates how the data-driven metamodel approach described in this work can be effectively used to construct a model through logical compositions within a defined modeling framework.

Real-World Case Example 2: Species in a Food Web

In this section, we analyze a M * metamodel concerning the classification of species in a food web. The metamodel is constructed as described in section 3.3, and it encompasses the relationship between six datasets and six respective models of species described in the Thompson and Townsend food web data [START_REF] Jaarsma | Characterising food-webs in two new zealand streams[END_REF][START_REF] Townsend | Disturbance, resource supply, and food-web architecture in streams[END_REF][START_REF] Thompson | The effect of seasonal variation on the community structure and food-web attributes of two streams: Implications for food-web science[END_REF][START_REF] Thompson | Is resolution the solution?: the effect of taxonomic resolution on the calculated properties of three stream food webs[END_REF][START_REF] Thompson | Allocation of effort in stream food-web studies: the best compromise?[END_REF][START_REF] Thompson | Impacts on stream food webs of native and exotic forest: an intercontinental comparison[END_REF][START_REF] Thompson | Energy availability, spatial heterogeneity and ecosystem size predict food-web structure in streams[END_REF]. The six models composing the modeling framework of this metamodel, enumerated as follows, describe species that:

1. Have node out-degree equal to 0 in the food web; 2. Have node in-degree equal to 1 in the food web; 3. Belong to TL 1 (defined as the longest TL possible); 4. Belong to TL 2; 5. Belong to TL 3; 6. Belong to TL 4 or above.

These models can be developed conceptually and biologically. For example, model 3 above, of species belonging to TL 1, classifies all algal species available in the Thompson and Townsend food web data, specifically the species belonging to the phyla Bacillariophyta (diatoms) and Rhodophyta (red algae). Species from these phyla obtain chemical energy through photosynthesis, thus being autotrophs and primary producers of food webs [START_REF] Raven | Inorganic carbon acquisition by marine autotrophs[END_REF]. Although insightful, this detailed taxonomic or ecological analysis of the species is not required to analyze the modeling framework, as the M * metamodel depends only on evaluating the relationship between the models and datasets.

Figure 16 depicts the metamodel constructed using the six models presented above. The specificity metrics calculated over the whole modeling framework and other descriptive metrics of the metamodel are presented in Table 7. The characterization of each individual model is provided in Table 8.

The weighted bipartite representation of the metamodel, available in Figure 16, reveals a highly specific modeling framework, with more than three quarters of the data elements (133 out of 173) being specific to their respective datasets.

However, according to Table 7, there is almost three times more diversity in the distribution of connections than necessary for a specific mapping between datasets and models. This diversity mainly originates from models 1 and 2, whose respective datasets are entirely composed of unspecific elements. In fact, each one of these two models has connections with five datasets: 1, 2, 4, 5, and 6.

The connections between models 4, 5, and 6 with the datasets of models 1 and 2 induced their higher connection diversity values. It is worth noticing that models 3, 4, 5, and 6, defined by analyzing trophic levels, are mutually exclusive, having no data elements shared between their respective datasets. Thus, the inclusion of topological models 1 and 2 into the modeling framework accounts for this observed unspecificity. Particularly, model 3 is the only model that does not have unspecific connections, as its dataset does not share data elements with the datasets of models 1 and 2. to the following rules: the pair ω1/m1 comprises species with node out-degree equal to 0; ω2/m2 comprises species with node in-degree equal to 1; ω3/m3 comprises species that belong to TL 1 (defined as the longest TL possible); ω4/m4 comprises species that belong to TL 2; ω5/m5 comprises species that belong to TL 3; and ω6/m6 comprises species that belong to TL 4 or above. The points on the left side of the scheme represent datasets, and the numbers below them represent the cardinality of those datasets. The points on the right-hand side of the representation are associated with the models respective to each available dataset. The connection between each dataset and each alternative model is represented by weighted arrows, whose weights represent the number of data elements in a dataset that can be explained by the targeted model. This analysis of specificity metrics can be combined with the knowledge available on the models to reveal further characteristics of the analyzed food web data.

Model 1 describes species with node out-degree equal to 0, i.e., species without predators.

These species are at the top of their food chains (a linear stretch of a food web), thus being called apex or top predators. Disturbances in the apex predators populations are often associated with the occurrence of the trophic cascade phenomenon: a cascade of population disruptions of multiple species participating in a food chain due to the ecological disturbances of species in higher trophic levels [START_REF] Pace | Trophic cascades revealed in diverse ecosystems[END_REF]. Possibly having a critical role in maintaining the ecological equilibrium of a community, apex predators are often regarded as keystone species [START_REF] Sergio | Top predators as conservation tools: Ecological rationale, assumptions, and efficacy[END_REF]. Apex predator species are necessarily included at a trophic level, thus they must also be explained by models 3 to 6. In the metamodel analysis, this means that the connections diversity of model 1 will always be higher than unity, and the higher it is, the most spread across all trophic levels the apex predators are.

Figure 16 shows that around 40% of each dataset 4, 5, and 6, respective to species on TL2, TL3, and TL4+, are indirectly connected to model 1. This highlights two aspects of the studied food web. First, the number of top predator species in each trophic level is closely proportional to the number of species available in the trophic level. Second, the analyzed food web data are composed of food chains with various lengths, ending in any TL from 2 to 4+. This could suggest incompleteness of the food web, with a consistent lack of data on predators from higher trophic levels, meaning that some observed apex predators are actually intermediate predators. Furthermore, this might indicate a tendency of apex predators in the same food web to feed on distinct trophic levels, a strategy observed in estuary ecosystems [START_REF] Franco-Trecu | Individual trophic specialisation and niche segregation explain the contrasting population trends of two sympatric otariids[END_REF][START_REF] Franco-Trecu | Trophic relationships in apex predators in an estuary system: A multiple-method approximation[END_REF].

Model 2 describes species that have a node in-degree equal to 1, i.e., species that prey on only one other species. These species, also classified as specialist predators, are highly vulnerable, as changes in the habitat and population of only one species, its prey, can directly influence its survival [START_REF] Ryall | Response of predators to loss and fragmentation of prey habitat: a review of theory[END_REF]. Figure 16 and Table 8 show that only 10 of the 173 species in the food web are explained by model 2, from which 8 are also related to model 1, thus being specialist and apex predators. This indicates that most of the predators available in the food web prey on multiple species, and the number of vulnerable specialist species across all trophic levels is low.

Table 8 also points out that the metamodel has a non-uniform distribution of few unspecific elements among the available datasets. Following the results of section 4.2, it is expected that modeling frameworks with these characteristics can be used to build a logical composition of models that explains highly specific datasets.

We performed the search for the maximum coincidence value between each available model and each composition of alternative models, as described in Equations 12, 13, and 18. The maximum coincidence index value achieved for each model, the respective logical composition of alternative models used to obtain that coincidence value, and the respective set combination of datasets are presented in Table 9.

The Top Coincidence value of the metamodel is 1. This coincidence value is achieved in the analysis of models m 3 , m 4 , m 5 , and m 6 . The high specificity and the mutually exclusive nature of these four models help to find perfectly equivalent models m X simply through the logical composition of the alternative models available.

The model m 3 is equal to a model m X (m 3 ) created through the logical composition of the negation of all alternative models ( i̸ =3 ¬m i ). Conceptually, this is coherent, as model 3 describes autotrophs, while the other models biologically define predators, i.e., heterotrophs.

The model m 4 is equal to a model m X (m 4 ) obtained from the logical composition of the negation of all alternative models ( i̸ =4 ¬m i ) and a logical composition of models m 3 , m 5 and m 6 ((¬m 3 ∧ ¬m 5 ) ∧ ¬m 6 ). The first term is satisfied within the second term, and the second term is the expression of the complementarity of models m 3 , m 4 , m 5 and m 6 in explaining the entire data environment. The same rationale can be used for defining the m X that coincides with models m 5 and m 6 .

These results illustrate how the data-driven metamodel approach can be employed to detect a set of mutually exclusive models within a modeling framework, and how they relate to non-exclusive models. Furthermore, we show how the metamodel could be used to extract information about the data environment and the modeling framework.

Concluding Remarks

In order to integrate data and models, with special attention to the first, Models can be constructed without a data structure a tool to integrate data to model to improve the evaluation and construction of new models.

The present work describes the development of a new metamodeling approach to scientific data, the M * metamodel. The metamodel constructed provides a means to evaluate the specificity and redundancy of the relationship between collections of scientific data and its respective scientific models. Due to its explicit dependence on data availability, the M * metamodel is characterized as data-driven and it is designed to be used for the evaluation and improvement of modeling frameworks applied in classification and pattern recognition problems.

The M * metamodel can be developed by dividing modeling approaches into three main domains and formalizing mapping operations between them. The first domain is the universe Ω, which harbors all available data that can be collected through scientific methods. Each data element in Ω maps into a organized dataset in the data environment domain E, and each organized dataset on E maps into respective models available in the modeling framework domain M . These mappings can be effectively appreciated from the domain mapping representation of the M * metamodel.

A third mapping operation can also be described, now between each data element of each dataset on E and each model available on M based on the capability of the models to explain each individual data element available on the different datasets. This operation can be used to create a weighted bipartite representation of the metamodel that can be extensively used to measure the specificity of the modeling framework M to the collected datasets on E. Based on this representation, three specificity metrics were developed: the mean element multiplicity, the mean connections specificity and the total connections diversity.

The analysis of the application of the specificity metrics on simulated metamodels has singled out multiple conditions where modeling frameworks can be unspecific: when multiple models explains multiple elements repeated among the collected datasets; when a model can explain multiple data elements assigned to alternative datasets; or when models have high uncertainty (entropy) of explanation of the elements distributed into the different collected datasets. Particularly, modeling frameworks with multiple pairs or trios of models explaining the same elements repeated into multiple datasets were observed to be, as expected, the most unspecific configuration of models. Still, the overall unspecificity of the framework could be mitigated when the unspecificity of few models are maximized in advantage of increasing the specificity of other minor models.

Furthermore, with the construction of the M * metamodel, it was possible to elaborate a paired algebra of set operations between datasets and logical operation between models. This relationship enabled the creation of new models based on the existing data environment and the established modeling framework. The created models could be used to substitute redundant or unspecific models within a modeling framework, thus providing a strategy to improve datadriven modeling approaches. Specially, with the help of simulated metamodels, we identify two scenarios where this strategy could be applied: (i) metamodels with non-uniform distribution of few unspecific elements in their datasets have modeling frameworks that could be used to build specific models; and (ii) metamodels with non-uniform distribution of many unspecific elements in their datasets have modeling frameworks that could be applied to construct unspecific models.

The metamodeling procedure was applied for the construction of metamodels for two realworld cases: the modeling of symmetries in 3-by-3 pixel patterns and the modeling of species in a food web. In both cases, we briefly analyzed sources of unspecificity and created logical compositions of models that could be used to recover an alternative model. The metamodeling of symmetries in 3-by-3 pixel patterns showed a great unspecificity of six models constructed to explain different forms of symmetries. This unspecificity was used to replace a model used to explain patterns with a black pixel in at least one of north, south, west, and east cells with a logical composition of the other five models available, plus one minor additional model. The metamodeling of species in a food web showed how a set of mutually exclusive models behave with the assistance of unspecific models to explain the same data environment. In this last case, we could combine the specificity metrics with further knowledge on the models to extract additional information of the trophic levels of the source food web data.

We expect that the described methodology for the construction of data-driven metamodels can be used to assist the characterization and improvement of database and knowledgebase systems in multiple scientific fields. For instance, in computational sciences, the metamodel could be used to access the specificity of pattern recognition and fuzzy clustering algorithms. In biology and chemistry, the reported approach could be applied to analyze databases of protein families and chemical compound classes, helping to identify stronger and weaker definitions of groups. In physical and material sciences, the metamodel approach could be used to improve the classification of different materials. Additionally, future works on the M * metamodel approach can explore outlier detection in datasets.

Figure 1 :

 1 Figure1:The main components of the data-driven modeling approach considered in this work. In order to model a phenomenon of interest, one starts by observing the phenomenon, collecting datasets, and creating respective models. The set Ω corresponds to the respective universe, containing all the accessible data related to the studied phenomenon. Particular data of interest are collected from Ω and organized into datasets that become available to the data environment E. In this way models can be constructed and combined into a single modeling framework, M , which can be used to explain the studied phenomenon.

Figure 2 :

 2 Figure 2: The cutlery metamodel. (a) Domain mapping representation. (b) Weighted bipartite representation.The universe domain Ω contains all the available data elements in the real world, for example, all the cutlery available in a kitchen drawer. Data elements are drawn from Ω to form datasets, which are available in the environment domain E. Each dataset is modeled by a respective model available in the modeling framework domain M . The domain mapping representation (a) depicts the metamodel with two mapping operations: first, each data element in Ω maps into each dataset in E that contains it; second, each dataset in E maps into its respective model in M through a bijective association (E, M ). The mapping between Ω and E is non-injective, as the two elements in orange belong to both datasets in E. The weighted bipartite representation (b) replaces the mapping (E, M ) with weighted connections, whose weights highlight the number of elements within each dataset that are indirectly connected to alternative models. In (b), the connections with dashed lines show that two elements, the orange elements, can be indirectly associated with the alternative model, as they also make part of the alternative dataset.

Figure 3 :

 3 Figure 3: The polygon metamodel representations. (a) Domain mapping representation. (b) Weighted bipartite representation.Although all the elements on the squares dataset are explained by both the rectangles and the squares models, the squares dataset is mapped only to the second. This follows the rule that, in order to ensure the bijective mapping between E and M , a model is bijectively associated with the largest dataset that it can entirely explain. Therefore, being the smallest of the two datasets that satisfy the rectangles model, the squares dataset is mapped only to the squares model.

Figure 4 :

 4 Figure 4: The incorporation of new datasets into the metamodel. (a) The new dataset (green) is not explained by any available model, which prompts the building of a new label model, the "Knife" label. (b) The new dataset (dashed green lines) is fully explained by an available model that already has a respective dataset (dashed black lines), prompting the union of both datasets into a single one. (c) The new dataset (green) is a set combination of the two available datasets in E, and it can be explained by a respective logical composition of the two models available in M .

  (b). In Figure5(c), the model with two weighted

Figure 5 :

 5 Figure 5: The diversity of connections D of a model m indicates how many datasets can be connected, with the same observed entropy, to an equivalent model m with a uniform distribution of weights p. The model ma in (a) only connects with its respective dataset, ωa. It has a distribution of weights with null entropy, H(ma) = 0, thus having a connections diversity D(ma) equal to 1. This model has entropy equivalent to that of the arbitrary model ma, which connects to only one dataset with weight p, represented in (b). The model m b in (c) connects to three datasets, including its respective dataset ω b , with a weight distribution with entropy H(m b ) ≈ 0.69. The connections diversity of m b is D(m b ) ≈ 2, which is the approximated number of datasets uniformly connected to the arbitrary model mb that have the same entropy of connections, represented in (d).

•

  P rob A 6 : (0, 70, 30)/100; • P rob A 7 : (0, 60, 40)/100; • P rob A 8 : (0, 50, 50)/100; • P rob A 9 : (0, 40, 60)/100; • P rob A 10 : (0, 30, 70)/100;

Figure 7 :

 7 Figure 7: Weighted bipartite representation of two simulated metamodels. Both metamodels were simulated considering m = 6 models and N = 500 elements. The metamodel in (a) follows the probability distributions P rob A 1 (very low chance of elements being a part of two or three datasets), P rob B 1 (uniform distribution of specific elements), and P rob C 1 (uniform distribution of unspecific elements). The metamodel in (b) follows the probability distributions P rob A 4 (high chance of elements being a part of two or three datasets), P rob B 3 (irregular distribution of specific elements), and P rob C 3 (irregular distribution of unspecific elements).

Figure 8 :

 8 Figure 8: Pairs plot of specificity metrics for 180 metamodel simulations using multiple simulation parameters.The metamodel simulations are divided into five groups according to the probability distribution P rob A used to simulate the multiplicity of data elements in their datasets: 1, very low chance of elements being a part of two or three datasets; 2, low chance of elements being a part of two or three datasets; 3, medium chance of elements being a part of two or three datasets; 4, high chance of elements being a part of two or three datasets; 5, very high chance of elements being a part of two or three datasets. Each group has a corresponding color in the graph. The plots on the lower left triangle are scatter plots, with each point corresponding to a metamodel simulation. The points are colored according to its P rob A group. The graphs on the main diagonal are the distributions of the respective metrics for each group of metamodels. The upper right triangle shows the Pearson correlation coefficient between the correspondent pairs of metrics. The correlation considering all 180 simulations is in black, and the correlations calculated for each group of metamodels are in the respective colors. Refer to section 3.1 for further information on the probability distributions.

Figure 9 :

 9 Figure 9: Pairs plot of specificity metrics for 180 metamodel simulations using multiple simulation parameters.The metamodel simulations are divided into four groups according to the number of elements N distributed in their datasets. Each group has a corresponding color in the graph. The plots on the lower left triangle are scatter plots, with each point corresponding to a metamodel simulation. The points are colored according to its group. The graphs on the main diagonal are the distributions of the respective metrics for each group of metamodels. The upper right triangle shows the Pearson correlation coefficient between the correspondent pairs of metrics. The correlation considering all 180 simulations is in black, and the correlations calculated for each group of metamodels are in the respective colors. Refer to section 3.1 for further information on the probability distributions.

Figure 10 :

 10 Figure 10: Pairs plot of specificity metrics for 180 metamodel simulations using multiple simulation parameters.The metamodel simulations are divided into three groups according to the probability distribution P rob B used to distribute the specific data elements in their datasets: 1, uniform distribution of elements between the six available datasets; 2, linear distribution of elements between the six available datasets; 3, irregular distribution of elements between the six available datasets. Each group has a corresponding color in the graph. The plots on the lower left triangle are scatter plots, with each point corresponding to a metamodel simulation. The points are colored according to its P rob B group. The graphs on the main diagonal are the distributions of the respective metrics for each group of metamodels. The upper right triangle shows the Pearson correlation coefficient between the correspondent pairs of metrics. The correlation considering all 180 simulations is in black, and the correlations calculated for each group of metamodels are in the respective colors. Refer to section 3.1 for further information on the probability distributions.

Figure 11 :

 11 Figure 11: Pairs plot of specificity metrics for 180 metamodel simulations using multiple simulation parameters.The metamodel simulations are divided into three groups according to the probability distribution P rob C used to distribute the unspecific data elements in their datasets: 1, uniform distribution of elements between the six available datasets; 2, linear distribution of elements between the six available datasets; 3, irregular distribution of elements between the six available datasets. Each group has a corresponding color in the graph. The plots on the lower left triangle are scatter plots, with each point corresponding to a metamodel simulation. The points are colored according to its P rob B group. The graphs on the main diagonal are the distributions of the respective metrics for each group of metamodels. The upper right triangle shows the Pearson correlation coefficient between the correspondent pairs of metrics. The correlation considering all 180 simulations is in black, and the correlations calculated for each group of metamodels are in the respective colors. Refer to section 3.1 for further information on the probability distributions.

Figure 12 :

 12 Figure 12: Distribution of TC from 90 modeling frameworks grouped following the simulation parameter P rob A(a), P rob B(b) and P rob C(c).Each dot corresponds to one modeling framework simulated using the parameter indicated on the x-axis in (a) and y-axis in (b) and (c). The P rob A represents the probabilities of each element in the simulation to be assigned to one, two, or three different datasets -e.g. in P rob A 1, elements had 90% probability of being assigned to one dataset, 8% of being assigned to two datasets, and 2% of being assigned to three datasets, organized as a triple of values between vertical bars 90|8|2. The P rob B and P rob C represent the simulations' probability distributions to assign specific and unspecific elements, respectively, into datasets 1 to 6. The six weights of the distribution, corresponding to each dataset, are indicated between vertical bars (e.g. 1|1|1|1|1|1). The TC value of a metamodel is the max coincidence calculated between any model m k and any logical composition mX of the alternative models m j̸ =k available in the simulated modeling framework. The combination mX respective to the TC value of a modeling framework is a combination of 2, 3, or 4 alternative models, indicated by the dot colors.

2 .; 3 .

 23 Reflection symmetry respectively to their main diagonal (lower left corner to upper right corner)Reflection symmetry respectively to their antidiagonal (upper left corner to lower right corner); 4. A black pixel in at least one corner; 5. A black pixel in at least one of north, south, west, and east cells;

Figure 14 :

 14 Figure 14: Metamodel of symmetries in 3-by-3 pixel patterns. (a) Domain mapping representation, showing some examples of patterns present in each dataset. (b) Weighted bipartite representation. The datasets ωi in the environment domain E are groups of distinct 3-by-3 pixel patterns that can be collectively explained by a respective model mi available in the modeling framework M . Each pair of dataset and model comprises different pixel patterns according to the following rules: the pair ω1/m1 comprises patterns with rotational symmetry around their central cell; ω2/m2 comprises patterns with reflection symmetry respectively to their main diagonal (lower left corner to upper right corner); ω3/m3 comprises patterns with reflection symmetry respectively to their antidiagonal (upper left corner to lower right corner); ω4/m4 comprises patterns with a black pixel in at least one corner; ω5/m5 comprises patterns with a black pixel in at least one of north, south, west, and east cells; and ω6/m6 comprises patterns with more than half of their cells with black pixels. The points on the left side of the scheme represent datasets, and the numbers below them represent the respective cardinalities. The points on the right-hand side of the representation are associated with the models respective to each available dataset. The connection between each dataset and each alternative model is represented by weighted arrows, whose weights represent the number of data elements in a dataset that can be explained by the targeted model.

Figure 15 :

 15 Figure 15: The 27 3-by-3 pixel patterns explained by the logical composition of models mX (m5) = i̸ =6 ¬mi ∨ (((m1 ∧ m2) ∨ m4) ∧ m5) that are not explained by model m5. They include patterns that do not have black pixels in the north, south, west, and east cells and satisfy at least one of the following statements: (i) do not have any rotational or reflection symmetry, have white pixels in the corners, and have less than half of their cells with black pixels; (ii) have more than half of their cells with black pixels; (iii) do not have rotational symmetry around their central cell; (iv) do not have reflection symmetry respectively to their main diagonal; (v) do not have black pixels in the corners.

Figure 16 :

 16 Figure 16: Metamodel of species in a food web. (a) Domain mapping representation, showing some examples of species present in each dataset. (b) Weighted bipartite representation. The datasets ωi in the environment domain E correspond to groups of different species that can be collectively explained by a respective model mi available in the modeling framework M . Each pair of dataset and model comprises different species accordingto the following rules: the pair ω1/m1 comprises species with node out-degree equal to 0; ω2/m2 comprises species with node in-degree equal to 1; ω3/m3 comprises species that belong to TL 1 (defined as the longest TL possible); ω4/m4 comprises species that belong to TL 2; ω5/m5 comprises species that belong to TL 3; and ω6/m6 comprises species that belong to TL 4 or above. The points on the left side of the scheme represent datasets, and the numbers below them represent the cardinality of those datasets. The points on the right-hand side of the representation are associated with the models respective to each available dataset. The connection between each dataset and each alternative model is represented by weighted arrows, whose weights represent the number of data elements in a dataset that can be explained by the targeted model.

Table 4 :

 4 Specificity metrics of the metamodel of symmetries in 3-by-3 pixel patterns.

	Number of elements in E	512
	Number of specific elements in E	24
	Mean element multiplicity ( Ū )	2.69
	Mean connections specificity ( S)	0.31
	Total connections diversity (D T ) 28.41
	Mean cardinality ( C)	229.33

Table 6 :

 6 Maximum coincidence values found comparing set combination of datasets and logical composition of models with each pair of dataset and model available in the metamodel of symmetries in 3-by-3 pixel patterns.

	Dataset (ω k ) and	Maximum	Respective Set Combination
	Model (m k )	Coincidence (C)	of Alternative Datasets (ω X )

Table 7 :

 7 Specificity metrics of the metamodel of species in a food web.

	Number of elements in E	173
	Number of specific elements in E	133
	Mean element multiplicity ( Ū )	1.28
	Mean connections specificity ( S) 0.63
	Total connections diversity (D T ) 16.26
	Mean cardinality ( C)	36.83

Table 8 :

 8 Specificity metrics for each pair of dataset and model available in the metamodel of species in a food web.

	Dataset (ω k ) and Model (m k )	Cardinality	Unspecific Elements	Connections Specificity (S)	Connections Diversity (D)
	ω 1 / m 1	38	38	0.452	4.179
	ω 2 / m 2	10	10	0.357	4.311
	ω 3 / m 3	80	0	1.000	1.000
	ω 4 / m 4	37	18	0.661	2.200
	ω 5 / m 5	25	10	0.625	2.460
	ω 6 / m 6	31	12	0.689	2.111

Table 9 :

 9 Maximum coincidence values found when comparing set combination of datasets and logical composition of models with each pair of dataset and model available in the species metamodel. / m 1 0.409i̸ =1 ¬m i ∨ (¬m 3 ) ω 2 / m 2 0.196 i̸ =2 ¬m i ∨ (m 1 ∧ ¬m 4 ) ω 3 / m 3 1.000 i̸ =3 ¬m i ω 4 / m 4 1.000 i̸ =4 ¬m i ∨ ((¬m 3 ∧ ¬m 5 ) ∧ ¬m 6 ) ω 5 / m 5 1.000 i̸ =5 ¬m i ∨ ((¬m 3 ∧ ¬m 4 ) ∧ ¬m 6 ) ω 6 / m 61.000 i̸ =6 ¬m i ∨ ((¬m 3 ∧ ¬m 4 ) ∧ ¬m 5 )

	Dataset (ω k ) and	Maximum	Respective Set Combination
	Model (m k )	Coincidence (C)	of Alternative Datasets (ω X )
	ω 1 / m 1	0.409	i̸ =1 ω c i ∪ (ω c 3 )
	ω 2 / m 2	0.196	i̸ =2 ω c i ∪ (ω 1 ∩ ω c 4 )
	ω 3 / m 3	1.000	i̸ =3 ω c i
	ω 4 / m 4	1.000	i̸ =4 ω c i ∪ ((ω c 3 ∩ ω c 5 ) ∩ ω c 6 )
	ω 5 / m 5	1.000	i̸ =5 ω c i ∪ ((ω 1 ∩ ω c 4 ) ∩ ω c 6 )
	ω 6 / m 6	1.000	i̸ =6 ω c i ∪ ((ω 1 ∩ ω c 4 ) ∩ ω c 5 )
	Dataset (ω k ) and	Maximum	Respective Logical Composition
	Model (m k )	Coincidence (C)	of Alternative Models (m X )
	ω 1

Acknowledgements

L. da F. Costa thanks CNPq (307085/2018-0) and FAPESP (2015/22308-2) for support. R. dos Reis thanks CAPES (88887.637015/2021-00) for financial support. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -Brasil (CAPES) -Finance Code 001.

The specificity indices of each model are presented in Table 5. The six models have low connections specificity and high connections diversity, with less than 40% of their connections coming from their respective datasets and more than three times the expected diversity for a specific mapping between datasets and models. Models 1, 2, 3, and 6 are totally composed of elements that also belong to other datasets, thus being the least specific models.

Interestingly, models 2 and 3 have the same values for each of the considered characterization metrics. This can be explained using knowledge about the structure of these models, i.e. model 2 explains 3-by-3 patterns that have reflection symmetry respectively to their main diagonal, and model 3 describes patterns with reflection symmetry respectively to their antidiagonal. Therefore, for each data element explained by model 2, there is a mirror image of that element described by model 3, and vice-versa. This creates a symmetry between these two datasets, as well as in their characterization metrics.

Furthermore, the metamodel presents a non-uniform distribution of many unspecific elements among the available datasets. As pointed out in section 4.2, it is expected that modeling frameworks with these characteristics have the potential for constructing a model composition that explains unspecific datasets.

We performed the analysis of redundancy in the modeling framework by searching for the maximum coincidence value between each available model and each composition of alternative models, as described in Equations 12, 13, and 18. The maximum coincidence index value achieved for each model, the respective logical composition of alternative models used to obtain that coincidence value, and the respective set combination of datasets are presented in Table 6.

The Top Coincidence value of the metamodel is 0.947. This coincidence value is achieved when comparing the dataset respective to the model m 5 and a model m X (m 5 ) created via the logical composition of four alternative models (m 1 , m 2 , m 4 and m 6 ) united with the negation of all alternative models ( i̸ =5 ¬m i ). Therefore, the model m 5 can be seen as redundant, as the other models can be used to explain its dataset. Given that the model m 5 is the most redundant among the six models, we will henceforth focus our attention on its respective discussion.

Using the previous definitions for each model listed at the beginning of the section, m X (m 5 ) is the model that explains: (i) patterns that do not have any rotational or reflection symmetry, have white pixels in the corners, and have less than half of their cells with black pixels; or (ii) patterns with more than half of their cells with black pixels; or (iii) patterns that do not have rotational symmetry around their central cell; or (iv) patterns that do not have reflection symmetry respectively to their main diagonal; or (v) patterns that do not have black pixels in the corners. This broad model explains 507 patterns, including all the 480 patterns explained by the model m 5 plus 27 more patterns, depicted in Figure 15.

The 27 extra patterns explained by m X (m 5 ) can form a new dataset ω Extra with respective model m Extra , which can be easily defined by direct inspection of its data elements. The ω Extra dataset has patterns with black pixels restricted to its diagonals, with either zero, one, two, three, or four consecutive corners occupied by black pixels.