We argue that ontology-structured knowledge graphs can play a crucial role in generating predictions about future events. By leveraging the semantic framework provided by Basic Formal Ontology (BFO) and Common Core Ontologies (CCO), we demonstrate how data-such as the movements of a fishing vessel-can be organized in and retrieved from a knowledge graph. These query results are then used to create Markov chain models, allowing us to predict future states based on the vessel's history. To fully support this process, we introduce the term 'spatiotemporal instant' to complete the necessary structural semantics. Additionally, we critique the prevailing ontological model of probability, according to which probabilities are about the future. We propose an alternative view, where at least some probabilities are treated as being about actual process profiles, which better captures the dynamics of real-world phenomena. Finally, we demonstrate how our Markov chain-based probability calculations can be seamlessly integrated back into the knowledge graph, enabling further analysis and decision-making.
Often people wonder what the probability is that some event might occur in the near future. We ask, for example, what the chance is that it will rain tomorrow, and we ask about the likelihood of an imminent economic recession. In order to answer these questions, we take into account the conditions that precede rainy days and recessions. The goal of the calculations is to plan an eficient use of resources: if there is a 90% chance of rain tomorrow, we bring an umbrella; if there is a high probability of a recession, we shore up our investments.
Prior to the use of knowledge graphs, e.g. relational databases structured by general data models, these probabilities would be the result of a calculation whose input data was collected, or structured, for the sole purpose of determining the chance of rain or a recession. Knowledge graphs facilitate the collection and organization of information, and the querying of that information for any analysis of the data that can benefit from the logical structure of the knowledge graph. This database structure ofers a clear advantage: information needs only to be collected and structured once, but can be used to answer any number of questions without significant restructuring. Such uses include answering questions about the probabilities of possible events.
Ontologies - logically structured vocabularies - give structure and meaning to the information in the knowledge graphs. Ontologies therefore allow for the integration of data collected from disparate databases and diferent schemas. Imagine these organizational and computational issues when stored information about rain and economic conditions is contained in disparate databases organized according to diferent schemas. A knowledge graph structured by an ontology allows for integration and analyses of the probabilities of these events regardless of the structure of the source databases.
This paper provides a general way to use knowledge graphs and Markov analyses to support queries that return the probabilities of future events. Section II explores the methods used to acquire and store the data for input into the Markov analyses. Section III demonstrates a first-order Markov analysis.
CEUR Workshop
ISSN1613-0073
Section IV demonstrates a second-order Markov analysis. Section V discusses how the results of the Markov analyses can be integrated back into the knowledge graph for additional uses. Section VI presents future directions of research. Section VII ofers conclusions.
Basic Formal Ontology (BFO) is a top-level ontology, which means it is a domain neutral representation
of reality at its most general [
The Common Core Ontologies (CCO) are a suite of mid-level ontologies that extend BFO toward domains of interest [2]. The Information Entity Ontology, for example, extends BFO toward domains of interest that include information, and the Artifact Ontology extends BFO toward domains of interest that include artifacts.
Both BFO and CCO are realist ontologies, which means that they are models of reality according to subject matter experts [3]. In many cases, subject matter experts are scientists, but in other cases, they are the data stakeholders who have a privileged understanding of the domain they are interested in.
In this paper, we shall use BFO and CCO to model the following hypothetical case: There is a fishing vessel of interest to fisheries ecosystem management analysts that is at some location. After observing the fishing vessel for 100 days and collecting information about the ifshing vessel’s movements, analysts correctly conclude that each day the fishing vessel either stays at its location, or travels to one of two other fixed locations.
Based only on previous behavior of the fishing vessel, what is the probability that the fishing vessel will travel to locationX the day after it travels to locationY? ‘LocationX’ and ‘locationY’ in this question are each intended to be interchangeable with ‘location1,’ ‘location2,’ and ‘location3.’ As such, ‘travel to’ is intended to be read as accounting for cases where the vessel goes to a diferent location, and cases where the vessel remains at the same location.
In order to model this case, and ultimately write a SPARQL query for our question, we focus on the
classes in Table I, and properties in Table II, all of which are from in BFO [
Since this is a hypothetical case, we created a set of dummy data that includes the randomly generated locations of a fishing vessel of interest over one hundred days. The fishing vessel can be located at location1, location2, or location3, at any day, and, once per day, the vessel either stays where it is or moves to a diferent location. Table III shows the first three days, and the last two days of data.
To create our knowledge model, we used an ontology development tool with a plug-in that ingests spreadsheet data. Additional data was added to the spreadsheet in order to more eficiently use the plug-in to ingest our data into a BFO conformant knowledge graph. This includes columns for instances of fishing trip, spatiotemporal region, spatial region, and temporal region, as well as a column for a single instance of fishing vessel. In the interest of space, we do not show the full spreadsheet here. A spatiotemporal region is an occurrent that is an occurrent part of spacetime.
A spatial region is a continuant entity that is a continuant part of the spatial projection of a portion of spacetime at a given time.
A temporal region is an occurrent over which processes can unfold.
A temporal instant is a zero-dimensional temporal region that has no proper temporal part. is a process means is an occurrent that has some temporal proper part and for some time , has some material entity as participant.
A temporal part of a process that has no proper temporal parts.
A history is a process that is the sum of the totality of processes taking place in the spatiotemporal region occupied by the material part of a material entity.
An occurrent that is an occurrent part of some process by virtue of the rate, pattern, or amplitude of change in an attribute of one or more participants of said process.
A vehicle that is designed to convey passengers, cargo, or equipment from one location to another by water travel.
A measurement information content entity that is a measurement of the likelihood that a process or process aggregate occurs.
An object track point that is where a vehicle is or was located during some motion.
A disposition is a realizable entity such that if ceases to exist then its bearer is physically changed; ’s realization occurs when and because this bearer is in some special physical circumstances, and this realization occurs in virtue of the bearer’s physical make-up.
We refer to the fishing vessel of interest as “fishingVessel.” FishingVessel is ingested as an instance of Watercraft and ingested as participating in a single fishing trip, which is a Process that we refer to as “fishingTrip.”
FishingTrip is identified as occupying some Spatiotemporal Region, which temporally projects only onto Days 1-100. FishingTrip is codified as having one hundred occurrent parts that signify the activities of each day of the trip. All one hundred parts of fishingTrip are ingested as instances of Process. Each part of fishingTrip bfo:precedes and bfo:is_preceeded_by some other part of fishingTrip, except for the temporally first and last parts of fishingTrip, which, respectively, only bfo:precede or only bfo:is_preceeded_by some part of fishingTrip. Each part of fishingTrip is ingested with a single occurrent part which is the occurrent during which fishingVessel undergoes observation. The occurrent parts of the occurrent parts of fishingTrip are ingested as instances of Process Boundary. Precedes is a relation between occurrents and ′ such that if is the temporal extent of and ′ is the temporal extent of ′, then either the last instant of is before the first instant of ′, or the last instant of is the first instant of ′, and neither nor ′ are temporal instants. is_a_measurement_of if is an instance of Information Content Entity and is an instance of Entity, such that describes some attribute of relative to some scale or classification scheme.
No definition in CCO.
A measurement value of an instance of a quality, realizable entity, or process profile.
Spatially projects onto is a relation between a spatiotemporal region and a spatial region such that at some time , is the spatial extent of at .
Temporally projects onto is a relation between a spatiotemporal region and a temporal region which is the temporal extent of .
Participates in holds between some (either a specifically dependent continuant, generically dependent continuant, or independent continuant that is not a spatial region) and some process , such that participates in in some way. inheres in if is a specifically dependent continuant and is an independent continuant that is not a spatial region, and specifically depends on .
Realizes is a relation between a process and a realizable entity such that inheres in some , and for all , if has participant then exists, and the type instantiated by is correlated with the type instantiated by . occupies spatial region if is an independent continuant that is not a spatial region and is a spatial region, and there is some time such that every continuant part of occupies some continuant part of at , and no continuant part of occupies any spatial region that is not a continuant part of at .
Occupies spatiotemporal region is a relation between a process or process boundary and the spatiotemporal region which is its spatiotemporal extent. spatial part of if , , , and are instances of immaterial entity, such that for any connected with , is also connected with , and is connected with but not connected with .
Occurrent part of is a relation between occurrents and when is part of . temporal part of if is an occurrent part of and either (i) and are temporal regions, or (ii) and are spatiotemporal regions and temporally projects onto an occurrent part of the temporal region that temporally projects onto, or (iii) and are processes or process boundaries and occupies a temporal region that is an occurrent part of the temporal region that occupies.
History of is a relation between history and material entity such that is the unique history of .
Process boundaries lack proper temporal parts, so each process boundary instance occupies an instance of Spatiotemporal Instant. A spatiotemporal instant is a spatiotemporal region that spatially projects onto a zero-dimensional spatial region and temporally projects onto a temporal instant at the same moment in time. In other words, it is a spatiotemporal region without some spatiotemporal region as a proper part. We introduce the term ‘spatiotemporal instant’ in order to complete the structural semantics necessary for modelling our use case.
We are ultimately interested in fishingVessel’s location at specific points in time, so we created one-hundred instances of Temporal Instant. Each temporal instant is the temporal projection of a spatiotemporal instant. We ingested each time in the spreadsheet as a datetime value of some temporal instant.
We also created a single instance of Vehicle Track Point, which we refer to as “vesselTrackPoint.” VesselTrackPoint is the zero-dimensional spatial region that fishingVessel01 always occupies at some discrete time. Ultimately we can find out fishingVessel’s location at a time by determining what spatial region has vesselTrackPoint as a part at that time.
Instances of location are ingested into the graph as instances of spatial region.1
Now that we have specified the structure of our application ontology and modeling conventions, Fig. 1 is the resulting graph for randomly chosen day_62. 1Since locations can be absolutely measured in relation to the center of the Earth’s geoid, it is appropriate to use bfo:spatial_region here.
We want to use the structure of the graph in Fig. I to allow us to calculate the probability that the ship of interest will travel to locationX after locationY. It’s important to note that Fig. 1 is only a snapshot of reality - at one particular temporal instant. A full graph would connect fishing trips at diferent temporal instants using the bfo:precedes object property, as noted earlier.
One way to return such results is to write a query that leverages the datetime values to return locations in order.
SELECT ?datetime ?location WHERE { fishingVessel bfo:occupies_spatial_region ?fishingVesselTrackPoint . ?fishingVesselTrackPoint cco:spatial_part_of ?location . ?spatiotemporalInstant Bfo:spatially_projects_onto ?fishingVesselTrackPoint . ?spatiotemporalInstant Bfo:temporally_projects_onto ?temporalInstant .
?temporalInstant cco:has_datetime_value ?datetime . } ORDER BY ?datetime
This query gives us results that may look like Table IV. In this case, the query looks at the locations of fishingVessel at every time period in question.
In this case, the query looks at the locations of fishingVessel at every time in question. Another sort of query returns results across times. It does so by leveraging ‘bfo:precedes’ to return a list of locations as well as the prior locations that fishingVessel occupied.
SELECT ?startLocationOfFishingVessel ?endLocationOfFishingVessel WHERE { ?fishingTripPart1 bfo:precedes ?fishingTripPart2 . ?fishingTripPart1 bfo:has_occurrent_part ?beingObserved1 . ?beingObserved1 bfo:occupies_spatiotemporal_region ?spatiotemporalInstant1 . ?spatiotemporalInstant1 bfo:spatially_projects_onto ?fishingVesselTrackPoint1 . ?fishingVesselTrackPoint1 bfo:spatial_part_of ?startLocationOfFishingVessel . ?fishingTripPart2 bfo:has_occurrent_part ?beingObserved2 . ?beingObserved2 bfo:occupies_spatiotemporal_region ?spatiotemporalInstant2 . ?spatiotemporalInstant2 bfo:spatially_projects_onto ?fishingVesselTrackPoint2 . ?fishingVesselTrackPoint2 bfo:spatial_part_of ?endLocationOfFishingVessel . }
A Markov chain is a discrete-time sequence of random variables 0, 1, 2, … with values in a finite set co,infdititfioolnloalwlys itnhdeeMpeanrdkeonvtporfotpheertpya.sTthe0,M…a,r ko−v1 pgriovpeenrtthyesptarteessetnhtastt,aatte an y[4t]im.Ien a,tthi meen-ehxot mstoagtee n e+o1usi2s Markov chain, the transition probabilities do not depend on the time parameter , so the transition matrix remains constant at each step. In this context, each step represents one day.
The state of the sequence at time is denoted by a random variable , which takes values in . FishingVessel01 has three possible locations (states). Thus, our state space may be defined as = { location1, location2, location3}.
Moving from one state to another is called a transition. This includes transitions to the same state (often called self-loops). In this way, transition probabilities may be understood as the probabilities of transitioning from one state to another in a single step. We refer to the resultant transition matrix as . It is important to note that transition matrices are × matrices when the chain has possible states. The entry represents the probability of transitioning from a state of state-type to a state of state-type .
Note that in the present context, the relevant state-types (in the parlance of BFO) are occupation-oflocation1, occupation-of-location2, and occupation-of-location3. These types represent the particular states of occupying location1, location2, or location3 that are individuals on particular days. A point about usage: in what follows, for the sake of brevity, we will often use location1 to refer not only to the spatial location that is ’location1’ but also to particular states of location1-occupation and to the state-type occupation-of-location1; similarly for ’location2’ and ’location3’.
While some may argue that the simplicity of Markov chains limit their applicability, we leverage their simplicity when integrating results into the knowledge graph. The straightforward structure of Markov chains facilitates clear interpretation and updating of data in the model. Markov chains play a foundational role in more complex models, providing us with a robust foundation for more comprehensive analyses where we can incorporate more features and uncertainty.
We turn our attention to the SPARQL query, which returns a list of ’previousLocation’ and ’currentLocation’, representing the transitions that FishingVessel01 makes each day. To populate the transition matrix, we sum each unique transition from state ’previousLocation’ to state ’currentLocation’. We divide that number by the total number of transitions that originated from state ’previousLocation’. For example, there are 9 transitions from location1 to location2, and there are 32 transitions originating from location1. The estimated Markov probability3 of moving to location2, given the present location 2Time-homogeneity is assumed here for simplicity and practicality. An example of time-inhomogeneity is explained in Section VII 3”Markov probability” refers to the probability of moving between states, whilst conforming to the Markov property as stated above. being location1, as
12 = 392 = 0.281.
This process may be automated using SPARQL queries or Python scripts to eficiently compute transition probabilities.
The resultant matrix in Table 6 is populated by rows indicating the present location and columns indicating the next location at the subsequent time step.
The transition matrix in Table 6 allows stakeholders to answer questions about future locations of FishingVessel01. For example, given that the vessel of interest is presently at location3 on day 100 (row), we conclude that there is a 29.0% chance that this vessel will be at location2 on day 101 (column).
Now that we have constructed a first-order matrix, we may make predictions about the vessel’s location beyond only the next day.
Recall that the (, ) entry of the transition matrix represents the probability that the Markov chain, starting in a state of state-type , will be in a state of state-type after steps. Table 7 shows matrix 5, estimating the probability of the vessel’s location after = 5 days.
What if the vessel’s movement is more dependent on previously made consecutive steps? Using a higher-order model allows us to capture more complex patterns in the movement.
Second-Order Markov chains function similarly to First-Order Markov chains but with a key diference in how the transitions are determined. In a First-Order Markov chain, the probability of transitioning to the next state depends solely on the present state. However, in a Second-Order Markov chain, the probability of transitioning to the next state depends on both the present state and the previous state. To count as nevertheless adhering to the Markov property, we look at transitions from a state pair ( −1 , ). refers to the ’present state’, of which the present state is an individual of type . −1 refers to the ’immediate past state, of which the immediate past state was an individual of type −1 . The probability that the entity in question will transition to a state of a given type +1 is given as follows:
( +1 ∣ −1 , ) .
Including more history when determining future probabilities allows the model to capture more patterns. If a vessel’s current movement is influenced by more of its past behavior, a Second-Order Markov chain will capture this with greater accuracy.
To determine the transition probabilities for a Second-Order Markov chain, we examine each possible state pair ( −1 , ). Similar to the process used for the First-Order Markov chain, we examine transitions in the historical data from each possible state pair to the subsequent state. For FishingVessel01, there are three locations that may be visited. There are 9 possible state pairs.
We can use SPARQL to retrieve data where each row represents a transition from a specific state pair to a subsequent state. In this format, we can calculate the transition probabilities. What we considered the “present state” in the First-Order Markov chain is now treated as a state pair in the Second-Order model. Because of this, the resulting transition matrix will be larger, reflecting the increased complexity.
The Second-Order matrix in Table 8 captures more historical context. This allows us to ask questions such as: Given the vessel was at location1 and is presently in location2, what is the probability that the vessel will move to location3? This can be directly answered from our matrix. In this case, we expect a 22.2% chance of this movement.
The Second-Order Markov chain provides additional context, allowing for more accurate explanations of processes that rely heavily on patterns.
The primary concern with higher-order models is not the computational complexity, but the requirement of more suficient historical data. In cases where data is sparse, higher-order Markov models may struggle to capture these sequential dependencies.
The resulting transition matrix will be larger and more complex, increasing the amount of information required to be represented back into the graph. The richer historical context allows us to capture more nuanced relationships and movement patterns, more accurately predicting these state changes in a real-world scenario.
This section takes steps toward updating knowledge graphs with probabilities. First, this section presents and assesses the model of probability in CCO. Second, desiderata for a satisfactory model of probability is extracted from the assessment of CCO’s model. Third, a new model is developed according to which probabilities are about process profiles.
In CCO, probability is an information content entity. A “Probability Measurement Information Content Entity,” as probability is labeled, is a “Measurement Information Content Entity that is a measurement of the likelihood that a Process or Process Aggregate occurs.” [2] The process or process aggregate that PMICEs are about can either be past processes or future processes. For example, we can ask what the probability was that a particular asteroid would hit Earth after it safely passes by. Such a measurement is about the past because it is about some process whose time to occur is over. But we can also ask what the probability is that a particular asteroid will hit Earth as it approaches. This is about the future because it is about some process whose time to occur has not begun.
In this paper, interest is in probabilities that inform us about the future, so we focus on the second case where the processes time has not yet begun. The way that the Common Core Ontologies models information that is about future entities is through modal relations. For our purposes, these relations are forward looking, whereas the non-modal versions are backward looking.
Every Probability Measurement Information Content Entity in CCO is “made in a particular context given certain background assumptions.” [2] This guides us toward defining kinds of Probability Measurement Information Content Entity. For example, a Markov probability, in CCO terms, can be defined as a Probability Measurement Information Content Entity that assumes the Markov Property holds for the entity or entities that it measures. The Markov Property is the property that makes it such that probabilities can be calculated only considering system’s previous state.
Using this model, we can produce the following graph of the probability that fishingVessel at location_01 either goes to location_02, goes to location_03, or stays at location_01.
In this graph, there are three instances of Markov Probability Measurement Information Content Entity, which correspond to each possible transition in location. Each instance of Markov PMICE is modally about fishingTripPart_101, which is a future process. What the graph tells us is that: there is a 0.375 probability that fishingVessel remains at location01 during fishingTripPart_101; there is a 0.281 probability that fishingVessel travels from location01 to location02 during fishingTripPart_101; and there is a 0.344 probability that fishingVessel travels from location01 to location03 during fishingTripPart_101.
The CCO model faces the following issues.4 First, probabilities are not about future processes. Instead, probabilities are about past or present entities that provide us with the ability to make predictions about the future. Second, even if probabilities could be about future processes, the CCO model is silent on what aspects of the future processes the probabilities are about. This becomes a larger issue if we accept that probabilities are not about the future because we then need to know what aspect of past processes probabilities need to be about to allow people to make predictions using them. This subsection spells these issues out in more detail. 5.2.1. Probabilities are not about future processes This subsection provides motivation to think that probabilities are not about future processes. A full argument for this point will have to wait for a future project. But it is important to provide motivation for the view here because once the connection between probabilities and the future is plausibly severed, we are left with the task of explaining the forward-looking nature of probabilities not about the future. A view that provides such an explanation is a main contribution of this paper.
The view is motivated by arguing (1) statements about the future are not derivable from a set of statements about the past; and (2) probabilities are quotients X/Y such that X and Y are both sets of only statements about the past.
We now propose that a statements about the future – predictive statements – cannot be derived from a set of statements about the past – descriptive statements. The issue here is one of logical validity, which is the property of arguments that makes it such that if all premises are true, the conclusion must be true. The problem is that for any set of descriptive statements with a predictive conclusion, it is possible to deny the conclusion without denying any of the descriptive statements. For example, given the following argument: (P1) # of times, did after -ing. (P2) # of times, did anything after -ing.
(C1) / of the time, will after -ing.
One can deny (C1) and accept (P1) and (P2) without doing anything contradictory. This is because, as far as the argument is concerned, it may be that facts about the past are not the kinds of things that can inform predictions about the future. On the other hand, accepting (P1) and (P2) while denying (C2) /
of the time, did after -ing. is contradictory. To accept (P1) and (P2) is just to accept (C2).
We shall now show that Markov calculations have the structure of the (P1), (P2), (C2) argument. We will do this through examination of the probability calculations done earlier in this paper. An example Markov probability calculation was described earlier as follows.
To populate the transition matrix, we sum up each unique transition from state ‘previousLocation’ to state ‘currentLocation.’ For example, there [is a sum of] 9 transitions from location1 to location2.
This sum is therefore a numerical representation of the nine past state transitions from location1 to location2. But it can be propositionally represented as (P1*) 9 times, fishingVessel travelled from location1 to location2. Next,
We divide that number by the total sum of transitions that originated from state ‘previousLocation’. [For example,] there are 32 transitions originating from location1. 4In addition to the more substantive issues presented in this subsection, there are legitimate questions about labelling and defining probability-related entities in CCO.
This sum is therefore a numerical representation of all the past state transitions that began at location1. But it can be propositionally represented as (P2*) 32 times, fishingVessel travelled from location1 to any location. Next,
The estimated Markov probability of moving to location2, given the present location being location1, as (P1*) 9 times, fishingVessel travelled from location1 to location2. (P2*) 32 times, fishingVessel travelled from location1 to any location.
(C2*) 9/32 of the time, fishingVessel travelled from location1 to location2.
So, probabilities that are calculated by finding the quotient of two sets of descriptive statements are not about the future. They are about the past. For this paper, we only wish to show that the Markov probabilities of interest have this structure. However, we do think that this argument extends to other kinds of probabilities, including other Markov probabilities, Bayesian probabilities, and so on. This is because, we think, fundamentally all probabilities are quotients of two sets of descriptive statements. If so, no probability is about the future.
However, predictions can be made by naturally assuming that the future will mirror the past as the probability calculations quantify it. The point is that this is a seperate assumption that allows us to make predictions using the probability calculation. It is not, strictly speaking, the probability (i.e., the output of the probability calculation) alone that is a prediction about the future. This is because that assumption does not feature in the probability calculation, or the propositional representation of the probability calculation. As will be shown starting in section 5.2.3, the insights of the current section matter to the ontology modelling of probability. 5.2.2. Probabilities are about certain aspects of processes that remain unidentified Even if probabilities are about future processes, CCO is efectively silent on the aspect of processes that probability measures. CCO says that probability measures the “likelihood that a process occurs,” but CCO also says that an alternative label of Probability Measurement Information Content Entity is ‘Likelihood Measurement.’ Given this, one can only conclude that ‘likelihood’ and ‘probability’ are synonymous in CCO.5 Thus, according to CCO, probability measurements are measurements of probability. This provides no additional information on what probabilities are about. 5.2.3. We need to know what probabilities are about If it is true that probability is not about future processes, then we need to know what it is about. This is for three reasons. One is that because any complete model of probability will have as parts both the information content aspect of probability and an informative model of what that information content measures. Currently, CCO lacks the second part, as shown. The second reason is that we need to know what it is we are basing predictions on. Just saying “probability” is not good enough since it raises the question: “what is the probability based on?” An answer to this question will be explored in the next section. The third, related, reason is that we want to know that we are justified in making predictions based on probabilities. To be justified, the entities that probabilities are about must have some bearing on the future even though they exist in the present. 5There is a fine distinction between ‘likelihood’ and ‘probability’ in probability theory, but there is not in common sense parlance.
In this section, we improve upon the CCO model of probability. We do so by addressing the issues just mentioned. We show that probabilities are intimately connected to realizable entities that inhere in the participants of processes that we want to make predictions about. This allows us to do the following things: (i) model probability in a way more consistent with what probability calculations are about; (ii) model probability in a way that assists in making predictions about future processes; (iii) understand what probabilities are based on; (iv) understand why predictions based on probability can be justified. 5.3.1. Are probabilities about single realizable entities? Probabilities are not about future processes, but they may be about parts of past processes that have potential characteristics. A potential characteristic is a characteristic that some continuant has but which can be fulfilled under some set of circumstances. Salt has the potential to dissolve but will not dissolve unless it is placed in the right set of circumstances, like a glass of water. I have the potential to finish writing this paper but will not until I have the correct mindset to fulfill this potential. In BFO, what I have called potential characteristics are called realizable entities. Realizable entities can be realized in processes of a certain type, like being in water, or being focused on finishing writing a paper.
One possibility, then, is that probabilities are about realizable entities. For example, fishingVessel bears three relevant realizables: the realizable to travel from location_01 to location_02; the realizable to travel from location_01 to location_03; and the realizable to stay at location_01. Each instance of Markov PMICE would then be about the correlated realizable entity instance.
This solution is insuficient. Probabilities are not just about single realizable entities. They are about realizable entities as they compare to other realizable entities. They are about, for example, the realizables one is most interested in as compared to all relevant realizables. 5.3.2. Are all kinds of probabilities about aggregates of realizable entities? If any probability is about aggregates of realizable entities, it is the probability that a fair six-sided die comes up on one. We intuitively know that there is a 1/6 probability that such a die comes up on one. But why is this the case? The explanation consistent with the view that probabilities are about aggregates of realizable entities is that the die bears six relevant realizables, none of which is in circumstances to increase the chance that it is realized over another relevant realizable. Knowing that the die can only realize one realizable at a time and that there is an aggregate of six realizables that inhere in the die, we get a 1/6 probability. But this explanation does not work for the Markov probabilities that fishingVessel travels to locations 01, 02, or 03. The reason for this is that a Markov probability of a system depends on its previous state and the overall pattern of states of the system over time. Conversely, the probability of a die coming up on one only depends on the aggregate of realizables at a single point in time. So, there are at least some probabilities that are not about aggregates of realizables. 5.3.3. Are some kinds of probabilities about process profiles? A process profile is “an occurrent that is an occurrent part of some process by virtue of the rate, or pattern, or amplitude of change in an attribute of one or more participants of said process.” [5] Some process profiles are magnitudes of changes in attributes of continuants. Call these “magnitude process profiles.” 6 Examples, are changes of mass, changes in temperature, changes in amounts.
There are also process profiles that are abstractions of magnitude process profiles over time. In particular, a rate process profile is an occurrent that is an occurrent part of some process by virtue of the rate of change in an attribute of one or more participants of said process. Examples are heartbeat (e.g., beats per minute), speed (e.g., miles per hour) and baseball pitch count average (e.g., pitches per inning). 6[5] calls these “quality process profiles,” but change in some realizable entities, like strength or solubility, can be measured and plotted in a graph in just the same way as change in mass or temperature. Pattern process profiles An unexplored kind of process profile is the pattern process profile . A pattern process profile is an occurrent that is an occurrent part of some process by virtue of the pattern of change in an attribute of one or more participants of said process. For us, patterns are observable regularities in the world. So, a pattern process profile is an occurrent that is an occurrent part of some process by virtue of an observable regularity of change in an attribute of one or more participants in said process. Pattern of life A pattern of life is an occurrent that is an occurrent part of some process by virtue of the pattern change in realizables that are realized by one or more participants of said process. Thus, patterns of life are pattern process profiles. Examples are an individual’s pattern of online activity, an individual’s morning routine, and an individual’s excersize regimen. Patterns of life need not be restricted to parts of processes that a single individual is an agent in. Indeed, some important patterns of life are parts of processes that groups of people are agents in. The travel pattern of a partiular convoy, and the pattern of a guard patrol, are examples.
From pattern of life to probability Patterns of life are often used to determine the probability that some agent will take a future action. If some individual takes route x to work at around 8:30 am, and then takes route x (in reverse) home at 5pm, every workday for a year, then there is a very good chance that they will do the same thing on the next workday. There a couple reasons why an analysis of pattern of life as a pattern process profile allows us to do this. First, patterns of life profile realizable entities. In other words, the attributes that patterns of life exist in virtue of are realizable entities – in particular, realizable entities that have been realized in the past. In contrast with the view that probabilities are about aggregates of realizable entities, the process profile view allows us to consider patterns of realized realizables over time. Second, the view that patterns of life profile realized realizables over time, allows us to explain what probabilities are about, and why we are justified in using them to make predictions. The explanation is that probabilities are measurements of the potentials of realizables. That is, they are measurements of the chance that some realizable will be triggered. However, these measurements are taken by measuring proxies, since potentials are not directly measurable. These proxies are patterns of life.
Demonstrating this view with the fishingVessel case In our case, the pattern of change we are interested in is the change in the pattern of realizables that are realized in transitions between states. In particular, we are interested in the pattern of change in realizables realized in changes in location of ifshingVessel. This pattern has already been discussed and used to calculate probabilities earlier in the paper. So, we can straightforwardly use this new construct to model the fishingVessel case.
Recall that the fishingVessel is currently, on day 100, at location 01 and next, on day 101, may either stay at location 01, move to location 02, or move to location 03. Each option requires the fishingVessel to realize a disposition: either the disposition of being at location 01 and remaining at location 01; the disposition of being at location 01 and moving to location 02; or the disposition of being at location 01 and moving to location 03.
Since the fishingTrip is a process that has thus far occurred over 100 days, with the fishingVessel either staying in place, or moving to one of two other fixed locations, a probabilistic pattern of change in realized realizables has been established. This is fishingVessel’s pattern of life during fishingTrip (herein we just call this fishingVessel’s pattern of life). In the graph this is called “fishingVessel_PoL.”
The pattern of life of fishingVessel – fishingVessel_PoL – has parts that we care about more than others. Since the vessel is at location 01, these are the parts relevant to the pattern where the last realization moved fishingVessel to location 01. We can pick these out in the graph by specifying that there occurrent parts of fishingVessel_PoL, namely, 1to1_PoL_Part, 1to2_PoL_Part, and 1to3_PoL_Part. Each part, respectively, is the part of the pattern of change that profiles the realizations of the disposition of staying at location 01, the disposition of moving from location 01 to location 02, and the disposition of moving from location 01 to location 03. See Fig. 3 for a visual representation.
Next, we want to move from these parts of fishingVessel’s pattern of life to the probability values themselves. The first step toward doing this is to count the indivdual times that each disposition is realized in each part of fishingVessel’s pattern of life. After, those counts need to be summed in order to get the total number of times each disposition in question was realized. This is shown visually in Fig. 4. In Fig. 4: 1to1TransitionCount is the count of times the 1to1_Disposition was realized; 1to2TransitionCount is the count of times the 1to2_Disposition was realized; the 1to3TransitionCount is the count of times that the 1to3_Disposition was realized; and total1toXTransitions is the sum of the three other counts.
Finally, the probability values are reached by putting the count of interest into the numerator of a fraction, and the total count into the denominator of a fraction. These fractions are the Markov Probability Information Content Entities. They are ultimately about fishingVessel’s pattern of life, since they are the result of dividing a count that is about the pattern of life.
In this paper, we have used Discrete-Time Markov Chains (DTMCs), where transitions between states occur at fixed, discrete-time intervals. Recall that the fishing vessel’s location was recorded once per day over a period of 100 days. This approach has advantages, particularly in the stability of transition probabilities over time. Once the transition probabilities are calculated, they do not change over time. Thus, they can be easily stored in the graph, allowing for straightforward predictions regarding the vessel’s future movements.
However, this discrete-time model does not capture the randomness of real-world movements. In reality, the movement of a fishing vessel is unlikely to occur at fixed time intervals. Various external factors (e.g., weather conditions, fishing regulations, or equipment functionality) could force the vessel to move at any continuous point in time. To model this more realistic behavior, we propose exploring Continuous-Time Markov Chains (CTMCs).
CTMCs allow transitions between states to occur at any continuous point in time. The time between state changes is modeled using an exponentially distributed random variable and represented in a rate matrix . The memoryless nature of the exponential distribution allows CTMCs to adhere to the Markov property. By incorporating the time spent at each location into the model, CTMCs provide a more accurate representation of the vessel’s movements. For example, if the fishing vessel spends five hours in one location before weather conditions force it to relocate, CTMCs can capture this behavior.
Despite these advantages, CTMCs introduce additional challenges. Transition probabilities are no longer fixed over time. Instead, the transition-rate matrix describes the instantaneous rate at which the chain transitions between states. From this rate matrix , we can generate a collection of transition matrices
() = .
Because the transition matrix now depends on for any future movement of interest, a new matrix must be calculated for each time point. To store these future probabilities in the graph, the time at which the future event occurs must be known so that the corresponding probabilities can be computed. For any time , a new matrix is required. This greatly increases the complexity of representing such probabilities within the graph, as it becomes necessary to predict when probabilities should be stored for specific time intervals.
In this work, transitions between states were calculated without accounting for external factors. One way to model a more realistic scenario while still using a DTMC is to allow transition matrices to vary depending on contextual factors. For example, a vessel’s movement may difer significantly between weekdays and weekends. To address this, we could construct distinct transition matrices, such as one representing weekday transitions and another representing weekend transitions. Depending on the day of the week, the appropriate matrix could then be applied to predict future movement.
This same principle could be applied to other factors, such as fishing regulations, weather seasons, or operational hours. Incorporating time-inhomogeneity would make the model more dynamic and better aligned with real-world variations in vessel movement.
An assumption in this example relies on the locations or states to be somewhat general. In real-world scenarios, when observing a vessel, locations may not be recorded in a discrete manner. We therefore would likely see some geo-coordinates that represent the vessel’s location at some time. This would fundamentally increase our state space to be somewhat immeasurable and the resultant Markov model would produce somewhat meaningless results.
To deal with geo coordinates, while still implementing a Markov chain, we cluster observations as a pre-processing step. Instead of treating each geo coordinate as its own state, we may group observations together if they fall within some area on a map. Let’s say, for example, within one mile of some known landmark. If we create n-number of these boxes to group observations into, we have reduced our state space to be discrete. [6] deals with this problem in a similar manner. We may then apply the same analytical methods detailed in this work to this discretized state space.
This paper presented an ontological model for Markov probabilities, which we view as generalizable to other types of probabilities derived from observations of past processes. Future work will explore how the model developed here can be extended to represent Bayesian probabilities, for example.
Future research will also examine how named graphs should be used to represent future-directed knowledge based on Markov and Bayesian probabilities. This work will include recommendations on how probabilities should be related to future-directed representations, such as expectations, as well as how to model the processes that are expected to occur.
Finally, future work will address how probabilities can be linked to dispositions in BFO-conformant OWL ontologies. Some work in this direction already exists in the context of a first-order logic version of BFO that recognizes non-actual instances [7]. The oficial OWL version of BFO does not recognize non-actual instances. Because our work is expressed in OWL and recognizes only actual instances, we are well positioned to make the insights of [7] implementable within BFO-OWL ontologies.
In this paper we showed how knowledge graphs structured according to an ontology can be directly accessed to calculate predictions about future processes. We provided two ways to query a knowledge graph for information that can be used to measure the Markov probabilities for fishingVessel to realize dispositions to travel to locations of interest. We then provided the ontology that can be used to structure the information about probabilities, and integrate it back into the knowledge graph. This methodology can be scaled to many similar or dissimilar objects exhibiting the same patterns of behavior. Importantly, the standardized representation of the knowledge in the graph allows us to align our knowledge of the domain with a machine-understandable representation of data so that we can layer additional advanced analytics on top of the knowledge in the graph in a way that will provide an audit trail for techniques that are otherwise considered “blackbox,” rote learning algorithms. This feature of predictive analysis using ontology-based knowledge graphs is important when decisions must be supported by auditable analytics and data that is stored in a standardized, logical construct.
[2] CUBRC, Inc., Common core ontologies, https://github.com/CommonCoreOntology/
CommonCoreOntologies/, 2022. Accessed: Dec. 19, 2022. [3] B. Smith, W. Ceusters, Ontological realism: A methodology for coordinated evolution of scientific ontologies, Applied Ontology 5 (2010) 139–188. doi:10.3233/AO- 2010- 0079. [4] R. Serfozo, Basics of Applied Stochastic Processes, Springer Science & Business Media, 2009. [5] B. Smith, Classifying processes: An essay in applied ontology, Ratio 25 (2012) 463–488. doi:10.
1111/j.1467- 9329.2012.00557.x. [6] I. Nizetic, K. Fertalj, D. Kalpić, A prototype for the short-term prediction of moving object’s movement using markov chains, in: Proceedings of the 31st International Conference on Information Technology Interfaces (ITI 2009), IEEE, Cavtat, Croatia, 2009, pp. 559–564. doi:10.1109/ITI.2009. 5196147. [7] A. Barton, A. Burgun, R. Duvauferrier, Probability assignments to dispositions in ontologies, in: Formal Ontology in Information Systems, 2012.