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https://ceur-ws.org/Vol-2549/article-04.pdf
A Linked Data Representation for Summary
Statistics and Grouping Criteria
James P. McCusker1[0000−0003−1085−6059] , Michel
2[0000−0003−4727−9435]
Dumontier , Shruthi Chari1[0000−0003−2946−7870] , Joanne S.
3[0000−0002−1753−2885]
Luciano , and Deborah L.
McGuinness1[0000−0001−7037−4567]
1
Department of Computer Science, Rensselaer Polytechnic Institute. 110 8th Street,
Troy, NY, USA {mccusj2,charis}@rpi.edu, dlm@cs.rpi.edu
2
Maastricht University. Minderbroedersberg 4-6, 6211 LK Maastricht, Netherlands
3
University of the Virgin Islands. Charlotte Amalie, St. Thomas, USVI, USA
Abstract. Summary statistics are fundamental to data science, and are
the buidling blocks of statistical reasoning. Most of the data and statis-
tics 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. Addi-
tionally, many clinical studies in the biomedical field rely on demographic
summaries of their study cohorts and the patients assigned to each arm.
While most data query languages, including SPARQL, allow for compu-
tation 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 In-
tegrated Ontology (SIO), and the World Wide Web Consortium’s prove-
nance ontology, PROV-O, providing interoperable representations that
are well supported across the web of Linked Data. We evaluate these se-
mantics using a Resource Description Framework (RDF) representation
of patient case information from the Genomic Data Commons, a data
portal from the National Cancer Institute.
Keywords: Knowledge Representation · Linked Data · Provenance · Summary
Statistics · Data Science · Transparency · Interoperability · Data Exploration.
1 Introduction
One of the most common forms of data analysis involves the use of basic statis-
tics over groups of things. Sums, counts, and averages provide the foundation
Copyright 2019 for this paper by its authors. Use permitted under Creative Com-
mons License Attribution 4.0 International (CC BY 4.0).
2 J. McCusker et al.
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 specific graphs, and then closing those graphs using crypto-
graphic 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)
[4] and statistical languages such as R [12].
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 [17]. We extend it with a method for
asserting aggregate facts about those OWL classes using the Semanticscience
Integrated Ontology (SIO) [7] in order to provide support for realist representa-
tions of scientific data and knowledge [8]. These aggregate facts are treated as
attributes of the class, grouping both different aggregate measures (like count,
mean, and standard deviation) to specific kinds of attributes (like age and sur-
vival). Identifiers for the source RDF graph using the graph digest algorithm
RGDA1 [17] can, if used in the provenance of those aggregate facts, will provide
a closed set of assertions that the aggregate facts were computed over. We there-
fore provide a formal method to express the semantics of aggregate functions
over well-defined grouping criteria. This allows us to create summary graphs of
data that can be used in multiple contexts and introspected using reusable soft-
ware. These mappings are illustrated using patient case information from the
Genomic Data Commons (GDC) [11], a National Cancer Institute data portal
for reusable cancer data.
2 Example Data Using Semanticscience Integrated
Ontology
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 reified roles, attributes (including quantities and qualities), time, and pro-
cesses. 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 effectively, for example, for modeling
scientific data using relationships between entities and attributes. The top-level
class of sio:entity 4 is subclassed by sio:object (continuents), sio:attribute (char-
acteristics of any entity), and sio:process (occurrents). All kinds of entities can
have attributes. The subtype of an attribute determines which attribute is be-
ing 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’.
4
SIO properties and classes are expressed here using their labels, and quoted when
necessary.
J. McCusker et al. 3
We use case data for our examples from the Genomic Data Commons (GDC)
[11], that includes patient demographics, diagnosis, and survival. This kind of
data is representative of biomedical information and includes information at-
tributes with and without temporality, roles between entities, and some pro-
cesses. GDC is particularly interesting because much of the exploration of data
within the GDC is based on displaying aggregate statistics, even though there is
no way to use those statistics outside of the user interfaces they provide. Our ex-
ample can set us on a path to pre-computing aggregate statistics and providing
not just visualizations of them, but also provide generalized API access to that
information in new ways. While GDC does not provide examples of all possible
RDF-based data, it is sufficient to illustrate our summary statistics approach.
We also discuss below how to apply these approaches to all possible RDF by
providing reification rules for both datatype and object properties. The data is
provided as supplementary material as gdc cases.ttl [16], and contains case in-
formation about 33,549 patients that are currently available in the portal as of
March, 2019. The RDF data was converted from the cases’ API endpoint. The
structure of the study information follows major classes and properties from
SIO, as shown in Figure 1. We resolve data values to classes in NCI Thesaurus
[6] using the BioPortal Annotator [13]. The resulting RDF is loaded into a triple
store, along with the summary statistics, which is also represented in RDF.
Fig. 1. A conceptual map of the GDC patient case information. We incorporate de-
mographics such as race, ethnicity, gender, age at diagnosis, life (vital) status, disease
type, and survival duration, and a link to the specific GDC investigation. This re-uses
predicates and classes from SIO and is extended using classes from the NCI Thesaurus,
to which the GDC data has been aligned.
3 Grouping Criteria as Classes
In order to create aggregate function semantics, we first have to formally define
the grouping criteria. Fortunately, the grouping criteria used in OWL restric-
tions covers most cases. We therefore define an aggregate value as an attribute
4 J. McCusker et al.
of an OWL Class. The aggregation semantics from Calvanese et al. [2] effec-
tively introduce variables into OWL class definitions. A conventional OWL class
contains references to classes, properties, individuals, and literals:5
Class: GDC_Subject
EquivalentTo: sio:human
and sio:'has role' some (sio:'subject role'
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:
case:d4f90900-3b81-4015-8e11-4b4525345063
GDC Subject ⊇ case:d52a195d-7d63-4eb6-81c2-3c473ba57979
...
An aggregate query in Calvanese et al. is expressed as:
q (x̄, α (ȳ)) ← φ
where x̄ is a sequence of grouping variables, α (ȳ) is the aggregation term, and
φ is the query condition expressed in first order logic. We translate this into
manchester notation through the following template:
Class: x̄
SubClassOf: φ
5
The following prefixes 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/
J. McCusker et al. 5
We will introduce ȳ in Section 5, as Calvanese et al. do not provide a way to
represent ȳ in a knowledge graph. In order to more explicitly treat their uses of
variables, we define a function G (g1 , . . . , gn ) = x̄:
Class: G ( g1 , . . . , gn )
SubClassOf: φ
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 defined 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 different 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
6 J. McCusker et al.
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 Computable URIs for Grouping Criteria
We use the following method for computing URIs for G, which allows for align-
ment of grouping criteria independent of the source graph or individual naming
schemes. Take the concise bounded description (CBD, or C()), minus annota-
tions, 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 different 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 Generating Facts About Classes using Aggregate
Attributes
We use OWL 2 punning [23, 10] to provide natural metamodeling of aggregate
attributes of classes and to reify non-SIO-based triples into a SIO-compatible
format. “Punning” here refers to the ability to separate OWL Classes, Properties,
and Individuals, even if they share the same URI. For instance, giving OWL class
a meta-type in addition to owl:Class would place that ontology in OWL-Full, as
would any facts (as opposed to annotations).
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 ex-
tended 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. Define 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.
J. McCusker et al. 7
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∧
∀S, P, O ⇒ ∃A A ∈ P ∧ attr(S, A) ∧ val(A, O)
P ∈ DatatypeProperty
P (S, O) ∧ P 6⊂ rel∧
∀S, P, O ⇒ ∃A A ∈ P ∧ role(S, A) ∧ to(A, O)
P ∈ ObjectProperty
(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 α (ȳ). This is
expressed in RDF where ȳ is an attribute of G(g1 , . . . , gn ), and each α (ȳ) is the
attribute of type α:
∀G, α(ȳ)∃A ∈ α, Y ∈ ȳattr (G, Y ) ∧ attr (Y, A) ∧ val (A, α(ȳ))
Following the data in the Supplemental Materials, when G(case:TCGA-BRCA)
is defined 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' 2009 ];
[ a sio:'minimal value'; sio:'has value' 32872 ],
].
By providing formal representations for aggregations, it becomes possible to
formally define 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 defined and
denoted thing, it becomes possible to provide the provenance of those definitions,
which would include the members of the class and aggregate query used to define
it. These classes and facts about these classes can now be defined 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, filtering criteria, and ag-
gregation functions. These classes can then be subjected to statistical analysis
8 J. McCusker et al.
Predicate Property Definition
rel() sio:’is related to’ A is related to B iff there is some relation between
A and B..
attr() sio:’has attribute’ A relation that associates an entity with an
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
val() sio:’has value’ A relation between an informational entity and its
actual value (numeric, date, text, etc).
role() sio:’has role’ A relation between an entity and a role that it
bears.
to() sio:’in relation to’ A comparative relation to indicate that the
instance of the class holding the relation exists in
relation to another entity.
der() prov:wasDerivedFrom A derivation is a transformation of an entity into
another, an update of an entity resulting in a new
one, or the construction of a new entity based on a
pre-existing entity.
Table 1. First Order Logic predicates mapped to SIO and PROV properties, with
definitions. These predicates may be mapped to other equivalent properties as needed.
and the definitions can be re-applied to further datasets for hypothesis testing
or published as nanopublications.
6 Closing Graphs Over Aggregates
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. Ex-
pressing aggregate values is useful, but the open world assumption prevents any
final 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 [20, 3, 17, 15], and an implementation of [17] has been published as
part of RDFlib [18]. The algorithm in [17] has the additional benefit of efficiently
computing reproducible identifiers for blank nodes within the graph, producing
stable identifiers for all RDF graphs. The approach in [15] uses nanopublications
to provide a mechanism for referencing RDF graph content by URI similar to
the approach in [17], but its approach to blank node skolemization (by providing
J. McCusker et al. 9
a UUID for each blank node) means that the graph identifiers are not consistent.
By creating different graph identifiers for the same graph, it becomes impossible
to verify that identical graphs from different sources are potentially the same.
By encoding the graph digest as part of a URI scheme, the aggregate attributes
can be encoded with a prov:wasDerivedFrom link to the digest-identified graph:
∀G, α(ȳ), N ∃A ∈ α, Y ∈ ȳ (attr (G, Y ) ∧ attr (Y, A) ∧ val(A, α(ȳ)) ∧ der(A, N ))
This allows for the reuse of the classes G across aggregations, while providing
evidence for computation of A, Y in specific 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 classified 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 specific 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 Related Work
Aggregation semantics in logic programming is well researched, [22, 21, 1] but in
many ways, these works do not address how to integrate the output of aggregate
functions into a knowledge representation that integrates with the original data.
The HiFun query language provides the means to express analytic queries di-
rectly through a relational algebra, but does not provide a formalized knowledge
representation in the process. The RDF Data Cube Vocabulary, or QB, expresses
entities-in-context as information artifacts, instead of as attributes of entitites
[5]. It has been used in a number of cases to create specialized representations
of summary statistics [14, 19, 9]. The formalisms of aggregation aren’t clear with
this approach, as the aggregate values are not expressed in RDF. Additionally,
RDF Data Cube Vocabulary treats statistical data as information artifacts, and
does not support for realist representations of scientific data and knowledge [8].
8 Evaluation: Exploring the Genomic Data Commons
In most publications, aggregate statistics are expressed in human-readable tables
or in figures. 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
10 J. McCusker et al.
Adenocarcinoma
Carcinoma
Squamous Cell Carcinoma
Ductal Breast Carcinoma
Endometrioid Adenocarcinoma
Glioblastoma
Serous Cystadenocarcinoma
Gastric Papillary Adenocarcinoma
Melanoma
Diagnosis
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
Fig. 2. The top 20 diagnoses for cancer in the GDC, retrieved from summary semantics.
This figure was generated from the aggregate statistics encoded in age by diagnosis.ttl,
and is reusable for viewing similar aggregations.
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 definitions, 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 figure was generated from the
RDF graph in the supplementary materials file 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,
6
age by diagnosis.ttl
J. McCusker et al. 11
and the DL-based definitions 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 in-
stance, 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 state-
ments, resulting in a significant reduction in statements.
9 Conclusions
The approach of OWL representations for grouping criteria plus SIO-based at-
tributes for the summary statistics is a natural extension of both representations,
and makes interoperability within Linked Data much easier. Assertions of ag-
gregate 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 on-
tologies in provenance, such as Prov-O, and eScience, (e.g. SIO) and results in
assertions with justifiable 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 defined sets
of entities. This paper also provides a way of computing URIs for classes based
on their necessary and sufficient 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 Acknowledgements
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.
12 J. McCusker et al.
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