=Paper=
{{Paper
|id=None
|storemode=property
|title=Order Theoretical Semantic Recommendation
|pdfUrl=https://ceur-ws.org/Vol-1059/ordring2013-paper2.pdf
|volume=Vol-1059
|dblpUrl=https://dblp.org/rec/conf/semweb/JoslynHPPST13
}}
==Order Theoretical Semantic Recommendation==
https://ceur-ws.org/Vol-1059/ordring2013-paper2.pdf
Order Theoretical Semantic Recommendation
Cliff Joslyn1 , Emilie Hogan1 , Patrick Paulson1 ,
Elena Peterson1 , Eric Stephan1 , and Dennis Thomas1
Pacific Northwest National Laboratory, firstname.lastname@pnnl.gov
Abstract. Mathematical concepts of order and ordering relations play
multiple roles in semantic technologies. Discrete totally ordered data
characterize both input streams and top-k rank-ordered recommenda-
tions and query output, while temporal attributes establish numerical
total orders, either over time points or in the more complex case of start-
end temporal intervals. But also of note are the fully partially ordered
data, including both lattices and non-lattices, which actually dominate
the semantic strcuture of ontological systems. Scalar semantic similari-
ties over partially-ordered semantic data are traditionally used to return
rank-ordered recommendations, but these require complementation with
true metrics available over partially ordered sets. In this paper we report
on our work in the foundations of order measurement in ontologies, with
application to top-k semantic recommendation in workflows. We con-
clude that true ordered set metrics are strongly preferable to traditional
semantic similarities.
1 Introduction
Order and sequence occur frequently in the semantic web and semantic tech-
nologies, but in different contexts and manners. First, data themselves can be
ordered, for example in a time sequence or order of arrival to a data stream.
When time stamps are available, a mathematical “total” (or “linear”) order
arises. If events can occur simultaneously, then a “weak order” is needed so as
to allow “ties”. And if event occurences can extend over time, then “interval
orders” are needed to reflect the interactions between start and stop times of
possibly overlapping events with different durations.
But data can be ordered from other ways and sources. In fact, the heart of se-
mantic technology may be considered to be class structures, which sit at the
cores of ontologies as taxonomic semantic categorization hierarchies. Such struc-
tures are mathematically “partial orders”, that is tree-like hierarchies supporting
multiple inheritance, including lattices. Note that this is the most general case,
as all total orders are weak orders, which are all interval orders, which are all
partial orders. Also, all lattices are partial orders.
Whether input data or semantic categories are totally, weakly, intervally, or
partially ordered, in any semantic selection task there is a need to rank-order the
resulting calculations, in order to recommend particular choices to the user. And
the algorithmic methods used to make such recommendations must measure the
data in a way which acccommodates and respects its underlying mathematically
ordered nature, perhaps up to and including being fully partially ordered.
2
Our interest is 1) the proper use of metrics in partially-ordered data in order to
2) make top-k rank-ordered recommendations in semantic systems. We report
on the performance of order metrics in a test analytic semantic recommendation
task within multi-domain scientific workflows as part of the Signature Discov-
ery Initiative1 (SDI) [1] hosted by the Pacific Northwest National Laboratory
(PNNL). Comparing true order metrics against traditional, non-metric seman-
tic similarities, we conclude that within our test knowledgebase, the rank-order
recommendations provided by partial order metrics are strongly preferable.
2 Workflows Using the Sequence Analysis Knowledgebase
We now introduce our test knowledgebase and use case within which we make
recommendations. Our use case concerns identifying signatures within generic
sequential data objects from multiple domains. While alignment of biosequences
has been in use for decades, it has only been recently recognized as a significant
method for analyzing non-biological sequences [8]. The overall workflow is shown
in Fig. 1. In the initial “alphabet construction” phase, members of an arbitrary
data set “input corpus” are translated into a linear sequence of exchange prim-
itives here called “symbols”. Each primitive is then assigned a unique character
called a “token” drawn from a limited alphabet using a “map file”, a data struc-
ture mapping symbols to tokens via a lossy encoding. This encoding of data to a
limited character set allows linear structure to be maintained for analysis while
reducing the vocabulary of characters to a computationally manageable number.
Sequences encoded as these token strings are represented in the FASTA file for-
mat2 as used in bio-informatics. We call encoding outputs “pseudo-proteins”, as
they are represented in FASTA files, similarly to real proteins, even if they are
representing any arbitrary input. Once encoded, pseudo-proteins are BLASTed
against themselves, then hierarchically clustered. Additional stages involve mul-
tiply aligning pseudo-proteins to generate sequence features for downstream de-
tection, but these do not concern the current effort being reported on here.
While this workflow targets any sequential data object which can be translated
and encoded into a pseudo-protein, the obvious cases are from computational
biology. Here inputs can be either gene sequences, with a map file implementing
the genetic code of codons to amino acids; or protein sequences directly (with a
null map file). But as shown in Table 1, we also include cyber objects, in par-
ticular disassembled executable files with actual processor opcodes as symbols;
and system log files, with log events (e.g. “login” or “logoff”) as symbols.
The goal for recommendation is to rank order available choices of data objects
and component connections in terms of semantic appropriateness, e.g. not rec-
ommending a cyber objet for a genetics task. We developed a Sequence Analysis
Ontology (SAO) to model our use case, including the types of data objects passed
in to and out of the workflow components, and the operations performed by the
workflow components. The SAO, and our methodology, uses object relations to
1
http://signatures.pnnl.gov
2
http://zhanglab.ccmb.med.umich.edu/FASTA/
3
Fig. 1. Sequence analysis workflow.
represent semantic relations of interest. For analytical and testing purposes, we
augment the SAO with OWL individuals to create a Sequence Analysis Knowl-
edgeBase (SA-KB) as a static representation of one time state of the overall
workflow environment. Each OWL individual thus represents a specific data ob-
ject or component port which may be available at any one time.
Domain Subdomain Symbols
Genomics Codons
Biology
Proteomics Amino acids
Executables Opcodes
Cyber
Log files Log events
Table 1. Domains and symbols used in sample workflows.
SA-KB statistics are shown in Table 2. Note that we regard the class hierarchy,
the core of the ontology, as a (partially) ordered data object of 157 classes
organized into nine levels. But it’s not a tree, with an average of 1.68 parents
per class. Thus methods in order theory are necessary to properly measure it.
Fig. 2 show an illustrative portion of the SA-KB around the individual FASTAs_Linux,
which is a KB entry for a particular FASTA file produced from encoding a
Linux executable file. FASTAs_Linux is a member of the class FASTA_File_Set
(classes shown in ovals, dashed lines show class membership). Below we will de-
note C(x) for the classes (potentially multiple) of an individual x, so here we
have C(FASTAs Linux) = {FASTA File Set}.
FASTAs_Linux is connected to other individuals (shown in boxes), represent-
ing data objects on which it depends semantically, by object properties. For
4
Property Value
# Classes (asserted) 157
# Classes (inferred) 168
Average # children/class 2.02
Average # parents/class 1.68
Class hierarchy height 9
# Object properties 31
# Individuals 116
Table 2. SA-KB statistics.
Fig. 2. A portion of the SA-KB around the FASTAs Linux individual. Individuals are
in boxes, classes are in ovals. Dashed lines indicate class membership, all others are
object properties or class hierarchy.
5
example, FASTAs_Linux was encoded with a particular mapfile represented by
MapFile_Executables, of class Map_File, and is thus linked to it by Depends_On.
MapFile_Executables effects a mapping from opcode symbols to tokens, and
so is itself related by File_Constructed_Using to two individuals: Opcodes, of
class Symbol_Set, representing the symbols of opcodes; and Cyber_Alphabet of
class Alphabet, representing the token set used by the cyber domain.
Recommendation proceeds when, given a fixed target individual (e.g. the FASTA
file FASTAs_Linux) at a particular point in the workflow (e.g. as input to the
BLASTer), calling for an object of a given type (e.g. scoring matrix), which of
many candidates are recommended? We wish to see recommendations rank or-
dered by being most semantically similar, for example the matrix SM_Executable
used for executable files being highly ranked, while SM_Genetics used for genetic
sequences loweest ranked.
3 Ontology Metric Annotation Methodology
We now describe our semantic annotation metric methodology. Given a class
hierarchy, we have a number of metrics to measure pairwise distances between
classes. We collect the sets of classes associated with individuals, and can then
measure the overall extent of those sets of classes within the ontology, and using
a Hausdorff set distance can measure the overall metric similarity between them.
This supports the rank-ordered recommendation of the most closely related ob-
jects of the same data type otherwise.
3.1 Distances in Ordered Semantic Structures
Real-world semantic hierarchies such as subsumption and composition are usu-
ally cast as Directed Acyclic Graphs (DAGs) on a finite set of classes P . They
are typically bounded above with a top element > ∈ P , with ∀a ∈ P, a ≤ >;
can have (a moderate amount of) multiple inheritance; and branch downward
very strongly. These are best represented mathematically as partially ordered
sets (posets) P = hP, ≤i [3], where ≤ ⊆ P 2 is a reflexive, anti-symmetric, and
transitive binary relation. Each DAG as described above uniquely determines a
bounded poset P by taking its transitive closure and including a bottom bound
⊥ ∈ P such that ∀a ∈ P, ⊥ ≤ a.
The first thought towards a metric in a graphical structure such as a semantic
hierarchy is as the length of the minimum undirected path
dM P (a, b) = min |p|.
p∈undirected paths(a↔b)
But such a distance ignores both the directional nature and the level structure of
the semantic hierarchy. Instead, the knowledge systems literature has focused on
semantic similarities [2, 10] to perform a similar function. These are available
when
P P is equipped with a probabilistic weighting function p : P → [0, 1], with
a∈P p(a) = 1. While p can be derived, for example, from the frequency with
which terms appear in documents [4], or which genes are annotated to bio-
ontology nodes [5], the simplest approach is to uniformaly weight each a ∈ P
proportionally, so that ∀a ∈ P, p(a) ≡ π, where π = |P1 | .
6
This then allows the definition of the information content of a node as
X
IC(a) = − log2 π p(b) = − log2 (π| ↓ a|) ,
b≤a
where ↓ a : = {b ≤ a} is the downset of a, then set of its descendants, so that
| ↓ a| is the number of such descendants. Then the information content of the
most informative common ancestor of a, b ∈ P is
M (a, b) : = max IC(c).
c∈↑ a∩↑ b
where ↑ a is defined analogously to ↓ a as the upset of a.
This then allows us to define the first of three standard semantic similarity-based
(SS-based) distances, the Jiang-Conrath distance [10], as
dJC (a, b) = IC(a) + IC(b) − 2M (a, b).
The other two follow from the Resnik and Lin similarity measures [10], after
normalization, to dervie corresponding distances as:
M (a, b) 2M (a, b)
dR (x, y) = 1 + , dL (x, y) = 1 − .
log2 (π) IC(a) + IC(b)
While all of these, including dM P , are somewhat standard metrics, our purpose
is more general, since we may not have such a weighting function available,
and semantic similarities are not required to be metrics satisfying the triangle
inequality. In seeking out the proper mathematical grounding, we turn to or-
der metrics [6, 9] which can use, but do not require, a quantitative weighting.
For details about order metrics built from isotone and antitone lower and up-
per semimodular functions on ordered sets, see [9]. In this work, we use the
cardinality-based distances. First we have the upper and lower distances:
du (a, b) = | ↑ a| + | ↑ b| − 2| ↑ a ∩ ↑ b|, dl (a, b) = | ↓ a| + | ↓ b| − 2| ↓ a ∩ ↓ b|,
Having two distances, upper and lower, may appear arbitrary or unfortunate, but
they behave differently.3 It may at first appear to be more natural to use upper
distance, since we’re then “looking upwards” towards the top bound > ∈ P . But
when P is top-bounded, strongly down-branching, and with multiple inheritance
(as in our cases), then it might be argued that it is preferable to use lower
distance, since up-sets are typically very small and narrow, frequently single
chains; where down-sets are large, branching structures. Additionally, this allows
siblings deep in the hierarchy to be closer together than siblings high in the
3
Note that it can be the case that the upper distance du (a, b) is the same as dM P (a, b),
but this is only required to be true when P is an upper-bounded tree: path length
and all these other metrics are generally unrelated.
7
hierarchy (this will be demonstrated below). This is considered valuable, for
example, where e.g. “mammal” and “reptile” are considered farther apart than
“horse” and “goat”. In practice, these can be combined. Defining the hourglass
of a node Ξa = ↑ a ∪ ↓ a, we then have the hourglass-based measure:
dT H (a, b) = |Ξa| + |Ξb| − 2|Ξa ∩ Ξb|
dT H is a direct analog of dl and du , but it is not a true metric.
We need to normalize distance to the size of the structure, so that we are mea-
suring the relative proportion of the overall structure two nodes are apart, or
in other words, what proportion of their potential maximum distance. These
normalized forms are (all ∈ [0, 1]):
du (a, b) dl (a, b) dT H (a, b)
d¯u (a, b) : = , d¯l (a, b) : = , d¯T H (a, b) = .
|P | − 1 |P | − 1 |P | − 2
3.2 Annotation Collection and Measurement
The metrics above measure distances between classes in an ontology, and in
particular, for any two individuals x, y in our KB, it can measure the distance
between its classes C(x), C(y). But in order to compare the entire semantic
context of an individual x, we need to identify not just its classes C(x), but
additionally the classes C(z), C(w) of any individuals z, w which x is related to,
that is, to which x is connected via object properties. We may not wish to invoke
all object properties in such connections, but perhaps particular ones. And these
should be transitive, so that through transitive closure we can find both directly
and indirectly related individuals.
The SAO has two transitive object relations , sao:depends on and sao:has part,
to play this role. But we don’t wish to restrict ourselves to just sao:depends on
and sao:has part proper, but rather all of their sub-relations as well, for exam-
ple sao:has corpus, which inherits from sao:depends on. So, for an individual
x, let A(x) be the class annotations of x, defined as: A(x) is the set of all classes
which either x or some other individual y, to which x is either directly or indi-
rectly linked by a particular, transitive object property, or some sub-relation of
one, are members of. From our example from Fig. 2, we have:
A(FASTAs Linux) = { sao:alphabet, sao:corpus, sao:cyber sequence,
sao:executable file, sao:FASTA file set, sao:linux executable file,
sao:map file, sao:opcode, sao:symbol set }
(1)
Consider an individual x with classes C(x) and full set of class annotations A(x).
We wish to have measures of A(x) for various x, and to compare A(x) and A(y)
for different individuals x, y. We first wish to capture the “spread” of A(x) within
the ontology. Given a distance d, we use
P P ¯
C1 ∈A(x) C2 ∈A(x) d(C1 , C2)
Ed (x) = |A(x)|
∈ [0, 1],
2
8
as the extent of x, which is normalized by the number of pairs of classes drawn
from A(x). Next, given two individuals x, y with classes C(x), C(y) and anno-
tations A(x), A(y), then we are interested in comparing the aggregate distance
d(A(x), A(y)) between those sets of annotations. To do so, we use a Hausdorff
distance [7] as a standard method. Given a distance d, then we have
¯ ¯
Hd (x, y) = max max min d(C, D), max min d(C, D) ∈ [0, 1].
C∈A(x) D∈A(y) D∈A(y) C∈A(x)
If d is a true distance function on P (positive definite, symmetric, satisfies tri-
angle inequality) then Hd is also a distance function.
4 Metrics Behavior
The left side of Fig. 3 shows distributions of the 169
2 = 14, 196 distances
d(C1 , C2) between the 168+1 (for the bottom node) classes in the SA-KB.
The right side shows the distributions of the 116 2
= 6, 670 Hausdorff distances
Hd(x, y) between the individuals. The left side of Fig. 4 shows the distribution of
the extents E(x) of the individuals, while the right side shows the distribution of
|A(x)|, the number of classes annotating each individual. We note the following:
Fig. 3. (Left) Distances d(C1 , C2 ) between classes C1 , C2 . (Right) Hausdorff distance
Hd(x, y) between annotation sets of individuals A(x), A(y).
– In Figs. 3 and 4, the distributions shown are not mutually correlated, rather
each is sorted separately in descending order, and then overlaid in the figure.
– The SS-based distances are convex, and of the SS-based distances, JC is
likely the best in the sense of the most discriminative, although it has large
regions in the high distances where it fails to distinguish class pairs.
– For the poset distances, lower dominates upper (as discussed above), and is
concave. When combined with lower, the resulting hourglass distance dT H
is less concave and more linear, providing a wide range of discriminatory
values, although less in the lower values. Below we will show that SS-based
distances provide poor rank orderings for recommendations, wo while we will
return to this issue, in the sequel dT H will be considered preferentially.
9
Fig. 4. (Left) Extents E(x) of the individuals. (Right) The number of classes |A(x)|
annotating each individual.
Fig. 5. A portion of the inferred SAO class hierarchy showing class annotations for
just FASTAs Linux (marked with ‘FL’ in blue), just SM Executable (marked with ’SM’
in red), and both (marked with both).
10
– The Hausdorff distances are a “selection” of the values appearing in the
class distances, in virtue of its max-min composition, effectively acting as a
coarsened representation.
– Extents Ed (x) and counts |A(x)| are shown separately, and our normal-
ization factor is effective, with generally small correlations coefficients be-
tween Ed and |A|, especially for the poset distances, with Pearson correlation
ρ(ET H (x), |A(x)|) = 0.23, while for the SS-based distances ρ > 0.5.
5 A Detailed Recommendation Example
In our example in Fig. 2, from Equ. 1 we have that FASTAs_Linux is annotated
by |A(x)| = 9 classes and, which are marked by ‘FL’ in blue in Fig. 5, and has
extent ET H (x) = .0381. This is relatively low extent, but a relatively high count,
as can be seen in Fig. 4.
Referring back to Fig. 1, consider that we are at the portion of the workflow
where a BLASTer component has been identified, and connected to its input
FASTA file using the output of the upstream encoder component. In order to
complete the instantiation of the BLASTer component, we need to identify its
one lacking input, namely a scoring matrix, and we need one which is as seman-
tically compatible as possible. The five available scoring matrices are shown in
Table 3, each of which has its own set of annotations, as shown in Table 4, which
is arranged with gaps to show where annotations sets overlap and differ.
Hd(FASTAs Linux, y) Minpath TH Resnik Lin J+C
sao:SM Executable 0.012 0.041 0.749 0.676 0.607
sao:SM Server Logfiles 0.012 0.053 0.749 0.676 0.607
sao:SM Logfiles 0.012 0.059 0.749 0.676 0.607
sao:SM Genetics 0.012 0.148 0.749 0.676 0.607
Table 3. Hausdorff distances between the annotating classes from FASTAs Linux and
all the scoring matrices in SA-KB, by different distances d. Sorted up by TH.
Consider the one example scoring matrix SM_Executable. On the one hand,
it differs from FASTAs_Linux in that it is annotated to sao:scoring_matrix,
since it is a scoring matric, but to neither sao:FASTA_file_set, since it is not
a FASTA file, nor to sao:linux_executable_file, since it can be used with
any executable, not being specific to any particular operating system. The right
side of Fig. 5 shows this annotation set in the dual ‘FL SM’ markings.
In measuring the difference in the annotation sets A(FASTAs Linux) and
A(SM Executable), the first thought is to cast them as sets X, Y and measure
their symmetric difference |X∆Y | = |(X \ Y ) ∪ (Y \ X)|, the number of different
elements. We have |A(FASTAs Linux)∆A(SM Executable)| = 3 for these three
differing classes. Table 4 shows ∆ for all the scoring matrices, along with HT H ,
the Hausdorff for the hourglass measure. But we actually observe that HT H is
strongly preferred to ∆, in that it not only gets the correct rank ordering, but
can also distinguish SM_Server_Logfiles as being preferable to SM_Logfile,
in virtue of the fact that it doesn’t just count the differences, it can weight
them by semantic distance, in this case that server_log_file is more specific
11
x =FASTAs Linux y =SM Executable y=SM Server Logfiles y=SM Logfiles y =SM Genetics
HT H (x, y) 0.041 0.053 0.059 0.148
A(x)∆A(y) 3 8 8 12
E(y) 0.042 0.038 0.041 0.076
|A(y)| 8 8 8 9
edam:data 2976
sao:alphabet sao:alphabet sao:alphabet sao:alphabet sao:alphabet
sao:codon
sao:corpus sao:corpus sao:corpus sao:corpus sao:corpus
sao:cyber sequence sao:cyber sequence sao:cyber sequence sao:cyber sequence
sao:executable file sao:executable file
sao:FASTA file set
sao:file
sao:gene sequence
sao:linux executable file
sao:log event sao:log event
sao:log file
sao:map file sao:map file sao:map file sao:map file sao:map file
sao:opcode sao:opcode
sao:scoring matrix sao:scoring matrix sao:scoring matrix sao:scoring matrix
sao:server log file
sao:symbol set sao:symbol set sao:symbol set sao:symbol set sao:symbol set
Table 4. Annotation sets for FASTAs Linux vs. all scoring matrices.
than log_file. Finally SM_Genetics is the least recommended scoring matrix,
coming from another domain entirely, it is most divergent in both ∆ and HT H .
Considering the other metrics available, and referring back to Table 3, we can see
the Hausdorffs for other distances, and that HT H is preferable to any of the SS-
based measures, in that it can distinguish all of these examples, and rank-orders
them correctly, where the SS-based distances cannot.
6 Overall Comparison
Broadening from our one example, we can consider an evaluation of the perfor-
mance of order metrics relative to SS-based distances. To accomplish this, we
considered all possible recommendations of a candidate individual against a fixed
source individual in the four cases implied by the upper parts of our workflow
in Fig. 1, specifically: 1) recommend symbol sets (candidates) against a mapfile
(target); 2) recommend a map file against a fixed symbol set; 3) a scoring matrix
against a FASTA file; and finally 4) a FASTA file against a scoring matrix.
There were 33 targets across all four groups, and for each we independently es-
tablished a “ground truth” for the ordering of the candidates. For example, in
Table 3, ground truth for candidate scoring matrices against the FASTAs Linux
target is the rank ordering shown, but with Server_Logfiles and Logfiles
reversed. Then for each recommendation, we measured the Spearman rank cor-
relation rs (averaging ranks over ties) for ground truth using each of the metrics.
Results are shown in Fig. 6. Note that rs = 1 indicates complete agreement on
rank order, while rs = −1 is a complete reversal of that order. Also, rs does not
exist when all scores are the same, as exemplified by the SS-based metrics in
Table 3. This was always true for minpath dM P , which was thereby eliminated,
and was frequently true for the SS-based distances, as can be seen in Fig. 3.
12
Fig. 6. (Left) Spearman’s rank correlation rs for each of the 33 recommendations;
(Right) Average rs for each group of recommendations.
While our results only hold within our test use case and KB, these were con-
structed independently of our evaluation. Poset metrics, especially lower and
hourglas, provided strong performance for top-k semantic ranking, with high
rank correlations above 80%. They were almost always preferable to SS-based
metrics, and sometimes remarkably so, including negative rs values for some
SS-based metrics.
Acknowledgments
This research is part of the Signature Discovery Initiative (http://signatures.pnnl.gov)
at Pacific Northwest National Laboratory. It was conducted under the Labora-
tory Directed Research and Development Program at PNNL, a multi-program
national laboratory operated by Battelle for the U.S. Department of Energy.
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