Summary statistics are fundamental to data science, and are the buidling blocks of statistical reasoning. Most of the data and statistics made available on government web sites are aggregate, however, until now, we have not had a suitable linked data representation available. We propose a way to express summary statistics across aggregate groups as linked data using Web Ontology Language (OWL) Class based sets, where members of the set contribute to the overall aggregate value. Additionally, many clinical studies in the biomedical eld rely on demographic summaries of their study cohorts and the patients assigned to each arm. While most data query languages, including SPARQL, allow for computation of summary statistics, they do not provide a way to integrate those values back into the RDF graphs they were computed from. We represent this knowledge, that would otherwise be lost, through the use of OWL 2 punning semantics, the expression of aggregate grouping criteria as OWL classes with variables, and constructs from the Semanticscience Integrated Ontology (SIO), and the World Wide Web Consortium's provenance ontology, PROV-O, providing interoperable representations that are well supported across the web of Linked Data. We evaluate these semantics using a Resource Description Framework (RDF) representation of patient case information from the Genomic Data Commons, a data portal from the National Cancer Institute.
One of the most common forms of data analysis involves the use of basic statistics over groups of things. Sums, counts, and averages provide the foundation
for understanding data, and because of this, these aggregation functions form
the basis of statistics and statistical analysis that underpins much of modern
science. We provide a means to express aggregate facts about classes of entities
in a way that avoids issues with the open world assumption by asserting those
facts, relative to speci c graphs, and then closing those graphs using
cryptographic graph hash identities. This method can be used to automatically create
classes through user interaction with semantically-aware, aggregation-based data
exploration and analysis tools based on OnLine Analytical Processing (OLAP)
[
This paper provides background on existing work to formalize aggregation
criteria as OWL classes with URIs derived from those aggretation criteria using
the RDF graph digest algorithm RGDA1 [
The expression of summary statistics requires a vocabulary that includes support for attributes, objects, and their interrelationships. As an integrated ontology, the Semanticscience Integrated Ontology (SIO) ontology includes terminology for rei ed roles, attributes (including quantities and qualities), time, and processes. SIO is an integrated ontology - it provides both an upper level framework for semantics and detailed semantics for general science. SIO is often extended for particular domains, and has been used e ectively, for example, for modeling scienti c data using relationships between entities and attributes. The top-level class of sio:entity 4 is subclassed by sio:object (continuents), sio:attribute (characteristics of any entity), and sio:process (occurrents). All kinds of entities can have attributes. The subtype of an attribute determines which attribute is being characterized, and sio:`has value' relates literal forms of the value of the attribute to it. Entities are related to their attributes using sio:`has attribute'.
necessary.
We use case data for our examples from the Genomic Data Commons (GDC)
[
and sio: in relation to some sio:investigation)
This can be expressed as this SPARQL query: select ?GDC_Subject WHERE { ?GDC_Subject a sio:SIO_000485; # human sio:SIO_000228 [ # has role a sio:SIO_000883; # study subject sio:SIO_000668 [ # in relation to
a sio:SIO_000747 # investigation
]
When this class is applied to the sample data in Supplemental Materials it results in 33,549 matches, one for each human:
GDC Subject 8 case:d4f90900-3b81-4015-8e11-4b4525345063 9 < =
case:d52a195d-7d63-4eb6-81c2-3c473ba57979 : : : : ; An aggregate query in Calvanese et al. is expressed as:
q (x; (y)) where x is a sequence of grouping variables, (y) is the aggregation term, and is the query condition expressed in rst order logic. We translate this into manchester notation through the following template: Class: x SubClassOf: 5 The following pre xes are used in all SPARQL, Manchester notation, and Turtle examples: rdf: http://www.w3.org/1999/02/22-rdf-syntax-ns# rdfs: http://www.w3.org/2000/01/rdf-schema# sio: http://semanticscience.org/resource/ prov: http://www.w3.org/ns/prov# case: http://example.com/gdc/case/ project: http://example.com/gdc/project/ We will introduce y in Section 5, as Calvanese et al. do not provide a way to represent y in a knowledge graph. In order to more explicitly treat their uses of variables, we de ne a function G (g1; : : : ; gn) = x: Class: G ( g1; : : : ; gn) SubClassOf: An aggregate query of study subjects by investigation can now be expressed as: Class: G(?x) SubClassOf: sio:human and sio: has role some (sio: subject role
and sio: in relation to value ?x) The selection SPARQL query would look like this: select ?GDC_Subject ?x WHERE { ?GDC_Subject a sio:SIO_000485; # human sio:SIO_000228 [ # has role a sio:SIO_000883; # study subject sio:SIO_000668 ?x # in relation to ].
?x a sio:SIO_000747 # investigation A class is de ned for every matched value in the knowledge base: Class: G(case:FM-AD) EquivalentTo: sio:human and sio: has role some (sio: subject role
and sio: in relation to value case:FM-AD) Class: G(case:TARGET-NBL) EquivalentTo: sio:human and sio: has role some (sio: subject role
and sio: in relation to value case:TARGET-NBL)
The members of each grouping criterion G (g1; : : : ; gn) are therefore members of the generated class G(), and the RDF graph that describes these summary statistics can provide rdf:type statements for each member as provenance. There are 45 di erent investigations in GDC total, above we show the three with the most subjects. These variables can replace classes, properties, and individuals, and can be mixed in with non-variable criteria, as was shown in the above example. Calvanese et al. discuss the computation of the aggregate operations MIN, MAX, COUNT DISTINCT, SUM, and AVG (mean), but do not specify how the values relate to the computed classes. The following sections relate the work done by Calvanese et al. to a complete representation of both the grouping criteria and aggregate statistics in RDF. This includes methods for computable URIs for the grouping criteria classes G(), how to provide aggregate statistics on all G() using sio:`has attribute', and how to link those aggregate statistics to their source graphs. 4
We use the following method for computing URIs for G, which allows for alignment of grouping criteria independent of the source graph or individual naming schemes. Take the concise bounded description (CBD, or C()), minus annotations, of G in a separate RDF graph C(G), where G itself is represented as a blank node. A CBD is the direct connections of a resource in the graph along with the transitive direct connections of any blank nodes in the description. Compute the graph digest of C(G) and use the digest value to rewrite the URI for G in the original graph. The CBD means that the URI should be computable in any case. The URI will be di erent if the inferred closure of statements on G is computed instead of the minimal grouping critieria, so users will need to maintain consistency there. 5
We use OWL 2 punning [
Once a grouping operation has been performed over a set of data to produce owl:Classes, it is now possible to compute aggregate functions over the members of the class. The aggregate functions available in SIO include mean, median, standard deviation, count, mode, minimal and maximal values, and can be extended with new statistical concepts using similar representation patterns. This is accomplished by using the following steps, with predicates mapped to OWL properties as shown in Table 1: 1. De ne an OWL class that is a subclass of the aggregation criteria, as shown in Section 3. 2. Pun the owl:Class to an owl:Individual in order to make assertions about it.
In RDF nothing needs to be explicitly done to do this. 8S; P; O
8S; P; O 3. If the predicate is not a subproperty of sio:`is related to', reify the predicate of the values being aggregated using the following rules:
P (S; O) ^ P 6 rel^ P 2 DatatypeProperty ) 9A A 2 P ^ attr(S; A) ^ val(A; O)
P (S; O) ^ P 6 rel^ P 2 ObjectProperty ) 9A A 2 P ^ role(S; A) ^ to(A; O) (1) Note that this operation is simply mapping a non-SIO property into the SIO framework. Punning is not needed for native SIO attributes.
We can now determine a way to represent the aggregation term expressed in RDF where y is an attribute of G(g1; : : : ; gn), and each attribute of type : (y). This is (y) is the 8G; (y)9A 2 ; Y 2 yattr (G; Y ) ^ attr (Y; A) ^ val (A; (y)) Following the data in the Supplemental Materials, when G(case:TCGA-BRCA) is de ned as an aggregate as expressed above: Class: G(case:TCGA-BRCA) SubClassOf: sio:human and sio: has role some (sio: subject role
and sio: in relation to value case:TCGA-BRCA) the aggregate facts can then be asserted about G(case:TCGA-BRCA) in this way: G(case:TCGA-BRCA) sio:has-attribute [ a sio:count; sio: has value 1098 ], [ a sio:age; sio: has attribute [ a sio:mean; sio: has value 21582 ], [ a sio: maximal value ; sio: has value [ a sio: minimal value ; sio: has value ].
By providing formal representations for aggregations, it becomes possible to formally de ne them using grouping criteria. Additional facts can be provided about each set through aggregate functions, which can be extended with more sophisticated statistical functions. Since each aggregation becomes a de ned and denoted thing, it becomes possible to provide the provenance of those de nitions, which would include the members of the class and aggregate query used to de ne it. These classes and facts about these classes can now be de ned automatically using grouping functions and instances. Because of this, OLAP-like tools can use and generate assertions about the aggregate sets that they produce through user interaction, since OLAP relies on GROUP BY, ltering criteria, and aggregation functions. These classes can then be subjected to statistical analysis
rel()
Property sio:'is related to' attr()
sio:'has attribute' val() role() to() sio:'has value' sio:'has role' sio:'in relation to' der() prov:wasDerivedFrom
attribute where an attribute is an intrinsic characteristic such as a quality, capability, disposition, function, or is an externally derived attribute determined from some descriptor
actual value (numeric, date, text, etc).
and the de nitions can be re-applied to further datasets for hypothesis testing or published as nanopublications. 6
Aggregation techniques face a number of challenges when dealing with the open
world assumption. First, aggregation functions assume that the available data
is complete and whole. For instance, asserting that a class has ten instances is
implicitly means that ten known instances have been counted, but that number
could be higher (because of unknown instances). Additionally, because of the
non-unique naming assumption, if those instances are actually identical but not
known to be, then the class may actually have fewer than ten instances.
Expressing aggregate values is useful, but the open world assumption prevents any
nal conclusions about aggregate statistics, because there can always be more
facts to be discovered. We need to close the graph to additional statements. A
number of approaches have been proposed to compute the content digest of an
RDF graph [
; Y 2 y (attr (G; Y ) ^ attr (Y; A) ^ val(A; (y)) ^ der(A; N ))
This allows for the reuse of the classes G across aggregations, while providing evidence for computation of A; Y in speci c graphs N . The grouping criteria on G can be re-applied to other datasets for further validation. For instance, a new graph can either be merged with the original, or computed separately for independent validation. Members of each G() can be classi ed using OWL inference (due to the equivalence restrictions), and aggregation across those sets can be computed within the new members. Each aggregate value would, because of its derivation statement, be traceable to a speci c RDF graph, and changes in the input can be validated simply by comparing URIs of the derivations to determine the reproducibility and/or repeatiblility of the aggregate value. 7
Aggregation semantics in logic programming is well researched, [
In most publications, aggregate statistics are expressed in human-readable tables or in gures. While human readers can conceptually make the connection back to the input data that has been aggregated, since no internal representation exists, the connection between source and aggregate are not maintained and cannot be used. Using our knowledge representation, we can query and persist those statistics as linked data that integrates with the original data. Using the supplementary materials script create summaries.ipynb, we were able to compute
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Carcinoma Squamous Cell Carcinoma
Ductal Breast Carcinoma Endometrioid Adenocarcinoma
Glioblastoma s i s o n g a i D
Serous Cystadenocarcinoma Gastric Papillary Adenocarcinoma
Melanoma
Non-Small Cell Carcinoma Diffuse Large B-Cell Lymphoma
Acinar Cell Carcinoma Neuroendocrine Carcinoma
Small Cell Carcinoma
Papillary Carcinoma Mucinous Adenocarcinoma
Thymoma Adult Cholangiocarcinoma
Cervical Adenocarcinoma Acute Myeloid Leukemia Not Otherwis… 0 1,000 2,000 3,000 4,0005,000 # of cases count statistics and age mean, min, and max by diagnosis6. The source data is contained in gdc cases.nt. The class URIs for these aggregates are recomputable from their de nitions, and we are able to construct summary visualizations using the graphs. Figure 2 contains summary counts for the top 20 diagnoses in GDC, retrieved from a summary semantics graph. This gure was generated from the RDF graph in the supplementary materials le age by diagnosis.ttl. 8.1
Description Logic Complexity The Description Logic (DL) complexity of G(g1; : : : ; gn) is not impacted by the way we express these classes, since they have no direct semantics in DL,
and the DL-based de nitions of each class are used to compute a URI for each G(). The generated G(g1; : : : ; gn) classes are themselves in ALE . SIO is in SRIQ(D). Additionally, were able to use Pellet to perform a full inference of the age by diagnosis.ttl with SIO without any inconsistencies or errors. 8.2
Overhead The storage overhead of our knowledge representation is fairly limited. For instance, in our GDC dataset, expressing age by diagnosis for the entire dataset required 4,992 statements using 304 classes, requiring about 16 statements (with RDFS labels mixed in) per class. When using multiple grouping criteria, the number of expressed classes will expand geometrically based on the number of combined class criteria. The underlying data provided contained 8,048,537 statements, resulting in a signi cant reduction in statements. 9
The approach of OWL representations for grouping criteria plus SIO-based attributes for the summary statistics is a natural extension of both representations, and makes interoperability within Linked Data much easier. Assertions of aggregate information about groups of entities can now be formally expressed in ways that are traceable to their source. In some cases, these assertions can even be computed based on the available summary statistics of the data at hand. This improves the ability for researchers to build facts and hypotheses about their entities of interest from their data. This method builds on existing ontologies in provenance, such as Prov-O, and eScience, (e.g. SIO) and results in assertions with justi able explanations. These assertions and their explanations can be published as nanopublications. The proposed knowledge representation is extensible across any aggregate functions that can be applied to de ned sets of entities. This paper also provides a way of computing URIs for classes based on their necessary and su cient conditions using graph digests of those OWL restrictions, making access of those parameterized classes stable across datasets. Finally, use of these aggregate semantics enables the use of existing analytical tools to generate explainable and exportable assertions about the data being analyzed and produce grouping criteria that can be re-applied to other datasets for further validation. 10
Thank you to James Michaelis and John Erickson for feedback and examples. This work is supported by IBM Research AI through the AI Horizons Network.