Spatial interaction models : developments and future directions

(1)

Spatial interaction models:

developments and future directions

Luuc van der Zee

Sociale Geografie & Planologie (afstudeerrichting planologie) 10853251

19-06-2017

Begeleider: Prof. Dr. Pieter Tordoir Tweede lezer: Ori Rubin

(2)

Spatial interaction can be defined as a movement or communication between two objects in space. Spatial interaction modeling is used to explain or predict the intensity of this interaction. There are many factors that might affect the intensity of interaction between two objects. A spatial interaction model (SIM) is, in essence, a function that takes these factors as input, and gives the intensity of interaction as output. Tobler’s first law of geography states that “everything is related to everything else, but near things are more related than distant things” (Tobler, 1970: 234). SIMs apply this idea

by (usually) taking into account some measure of separation, such as Euclidean distance, travel time or transport cost, which decreases the intensity of interaction as it gets larger. In addition, most SIMs contain a ‘generator’ and/or ‘attractor’ term, such as population size, both of which will increase the intensity of interaction and thus have the opposite effect of distance.

This thesis will summarize and explicate the general development of SIMs since their emergence, provide an overview of innovations in spatial interaction modeling up to this day, and finally suggest some directions which spatial interaction modelers could explore in the future. This will be undertaken by reviewing the body of literature produced on the topic in the past ±60 years. Because of the large body of work produced in this period, this analysis will be limited to the most

influential works and occasionally some less influential works.

This thesis consists of five sections. The remainder of this section will be dedicated to some of the broader developments in the field of geography, of which SIMs are part. The changing

epistemological attitudes that have dominated throughout the years, and how these have influenced the development of quantitative geography and SIMs, will be discussed in paragraph 1.1. Paragraph 1.2 will briefly discuss some current developments in planning and other disciplines, building the argument that SIMs are now more relevant than ever. The second section will be concerned with the general development of spatial interaction models, subdivided over four ‘eras’. Each era grows out of theoretical deficiencies of the previous era, which are usually addressed by importing concepts from other fields such as physics or economics. In the third section, some general developments that do not fit the classification of the second section, but are nevertheless considered important, will be examined. The fourth section discusses synergies between spatial interaction modeling and other modeling techniques, and identifies some possible fruitful directions that spatial interaction modelers might explore in the future. The fifth and last section will provide some concluding remarks.

1.1 Quantitative geography

During the 1950s the discipline of geography went through a transformation, often referred to as the quantitative revolution (Burton, 1963). Similar transitions have taken place in other scientific disciplines, but often earlier. In the physical sciences, the mathematical approach had been the norm for centuries (according to Newton: “God created everything by number, weight and measure”), and other social sciences embraced quantitative methods years before geography did as well. Psychologists, for example, discovered the experimental design in the second half of the 19th

(4)

century, and American psychologists in particular started stripping the philosophical and

metaphysical heritage from the discipline in exchange for a positivist and empiricist framework (Davies, 2015). Economists took a different approach to understanding human behaviour and envisioned humans as rational agents that try to maximize utility- an assumption which easily lend itself to quantification- around the same time. Perhaps the quantitative revolution in geography was delayed because experiments are hard to incorporate in geographic theory due to ecological issues, in addition to the subject matter of geography being hard to measure, especially before the

computers, GIS and big data. Another theory is that quantification was (and, sometimes, still is) associated with determinism and the search for ‘universal laws’, which geographers were reluctant to engage in after the environmental determinism of the late 19th_{/early 20}th_{century, e.g. Ratzel and}

the idea of lebensraum (Burton, 1963). Nevertheless, the revolution did take place, and geographers moved towards more sophisticated statistical methods.

Quantitative geography flourished and became the dominant paradigm in geography, developing an increasingly strong theoretical base (Harvey, 1969) until it experienced a sudden downturn in popularity in the early 1980s/ mid-1990s. There is no easy explanation for this downturn, but Fotheringham et al (2000) suggest a combination of causes:

• The rise of new schools of thought in human geography like Marxism and postmodernism. After becoming the dominant paradigm in human geography, the ontological and

epistemological underpinnings of quantitative geography came under attack.

• Geographers jumping on the bandwagon of these new schools of thought after growing tired of the established paradigm.

• The rise of geographic information systems (GIS) and its associated field of study,

geographic information science (GISc). In some places, the new school of geographers saw GISc as either equivalent to quantitative geography (which is not true, although they certainly complement each other) or as a backdoor to sneak quantitative methods back into geography (Johnston, 1997; Taylor & Johnston, 1995). GISc became separated from geography and, as a result, quantitatively oriented geographers have been drawn to GISc programs rather than old-fashioned geography programs.

• The rapid development of quantitative geography and its increasingly complex mathematical methods. Geography students often lack a strong mathematical base or mindset, and if these are not sufficiently developed during in the undergraduate years (as is currently the case (Johnston, 2008)) the threshold to learn quantitative geography is quite high.

The last two points are related to the first two, and have now turned into a positive feedback mechanism: the less often quantitatively-oriented geographers encounter quantitative methods in their curriculum, the more will switch to either GISc or qualitative analysis, slowly draining geography departments of people with a quantitative mindset, resulting in even less quantitative theory in the curriculum.

(5)

that quantitative geographers have not ignored these critiques.

Firstly, there is the epistemological divide between the positivist base of quantitative geography and the constructionist base of postmodern methods. Generally, adopting a positivist view works well for researching concrete, measurable phenomena like traffic, while constructivism makes more sense in abstract domain where phenomena are less tangible, like nationalism. The subject matter geographers are interested in occupies the middle ground on this spectrum. Understanding this middle ground requires the geographer to think both positivist and constructivist- replacing one with the other seems senseless (Wilson, 2010). Most geographers (the radical positivists or

constructionists excepted) would largely agree with this view. The two can (and should) coexist. It is this dualistic way of thinking about the world with which, combined with GIS, geography could take the lead in tackling the current complexity paradigm (Wilson, 2010). This challenge is

currently undertaken by a variety of disciplines like ‘geospatial science’(Berry et al, 2008), computer and/or data science, and complex systems science. Geography and its unique dualistic approach has a lot to contribute here.

Secondly, there are the critiques on the theoretical base of concepts in quantitative geography. These are not necessarily postmodern, since they are not attacking positivist methods in general, but are nonetheless part of the same assault on quantitative geography (Fotheringham et al, 2000). SIMs, for example, were (and still are) criticized for stealing Newton’s law of universal gravitation and assuming people will behave the same way as planets, completely ignoring the cognitive and social processes that determine their behaviour (Fotheringham et al, 2000). Although this criticism was definitely valid for the earliest models, the theoretical base for SIMs has grown greatly,

incorporating ideas from statistical mechanics, economics, and psychology, among others. Still, some theoretical issues remain, and these will be under scrutiny in the second section of this thesis. Furthermore, although the theoretical base for SIMs might be somewhat shaky compared to models in e.g. physics, this does not mean that SIMs do not describe the real world at all. Humans are sentient, autonomous beings that make rational and irrational choices and are thus fundamentally different from dead physical particles. Nonetheless, their behaviour is captured very well by SIMs. This empirical success is probably the largest reason for the long-lasting popularity of SIMs. The success of Newtonian physics was not in explaining exactly why objects attract each other, but in predicting phenomena like the return of Halley’s comet. That is to say: it is highly unlikely that a theory that consistently makes good predictions does not correspond to reality in the slightest. And if the mathematics are the same, does it matter whether we tell ourselves that we are dealing with sentient beings or dead particles? Arguing whether a SIM is dealing with ‘interaction fluxes’ or ‘the aggregate of individual decisions’ while using the same mathematics might be seen as “some kind of metaphysics or maybe even a play on words” (Sokal, 2008: 238). Of course, having a strong theoretical base is better than having a weak theoretical base, and unifying different theories in science is extremely important. But a model that is empirically strong and theoretically weak is definitely superior to one that is theoretically strong but empirically weak.

(6)

1.2 Spatial interaction models: relevance

SIMs are among the oldest tricks up the sleeve of the quantitative geographer (e.g. Carey, 1858). SIMs can (with minor tweaks) be applied to a plethora of different topics, ranging from traffic and international trade to communication and migration. The models are often simple, and yet perform well empirically. For these reasons, spatial interaction modeling has been an active topic of research throughout the 2nd_{half of the 20}th_{century, although development stagnated slightly around the turn}

of the millennium, for reasons discussion above. In the last years, however, due to an abundance of computational power and data, a renewed interest in SIMs emerged.

Another reason for a renewed interest in SIMs can be found in urban and transport planning, where a transition from planning for mobility to planning for accessibility has taken place (Handy, 2002; Rodrique et al, 2006). To clarify: mobility-enhancing strategies are focused on improving the functioning of transport systems themselves and improving access to these systems. For example, when a road suffers from congestion, the capacity of the road might be increased by adding new lanes, or a new road might be built altogether. Accessibility-enhancing strategies, on the other hand, focus on whether people are able to reach the places they need to reach with ease. The difference between the two is that in a mobility-based framework, this goal is attained through the transport system alone (e.g. decreasing congestion by adding lanes) while in an accessibility-based

framework the problem might also be solved by land-use planning (e.g. moving activities closer to the people that need to participate in them, reducing the need to travel altogether).

The superiority of accessibility-based strategies might seem obvious when spelled out, but the recognition has fundamental implications for planning- it has even been named a paradigm shift by some (Todman, 2017). Accessibility-based planning has led to popular strategies and buzzwords like compact cities and transit-oriented development (Handy, 2002). SIMs play a key role in accessibility-based planning, as measures of accessibility are commonly derived from SIMs (Hansen, 1959). For some applications of SIMs in the context of accessibility, see Geurts et al (2012) and Condeço-Melhorado et al (2014).

2. Theoretical base of spatial interaction models

This section will be concerned with the development of the theoretical base of SIMs. Fotheringham et al (2000) have argued that the development of SIMs can be divided into four paradigms (Figure 1):

1. Spatial interaction as ‘social physics’ and the dominance of the gravity-analogy 2. Spatial interaction as statistical mechanics

3. Spatial interaction as ‘aspatial information processing’ at the micro-level 4. Spatial interaction as ‘spatial information processing’ at the micro-level

This structure will be used as a guideline in this section, too. Note that this is not a chronological account of all developments in spatial interaction modeling; the models mentioned in the following

(7)

any of these paradigms will be discussed in section 3. Section 4 will be concerned with some more recent innovations in spatial interaction modeling.

Besides being a straightforward way to describe how SIMs have developed over time, the more or less chronological structure used here serves another purpose. The paradigms represent large leaps in understanding of the theoretical base of SIMs, and usually counter some of the criticism

associated with the models of the previous paradigm. Paragraph 2.1, describing the first paradigm, will explain how the first SIMs were largely based on Newton’s law of universal gravitation. This analogy lacked a proper theoretical base, but the models performed empirically well. The paragraph ends with a discussion of the theoretical and empirical weaknesses associated with the models presented. The next paragraph, then, will introduce the next paradigm, and how a leap in

understanding counters some of the criticism that was associated with the previous paragraph. Put differently: a new paradigm is ‘born’ out of the criticism of the previous one. This pattern will be repeated throughout this section.

2.1 Gravity and social physics

Social physics can be defined as “the application of the concepts of physics in the social sciences” (Wilson, 1969: 159). The idea that social phenomena might be described by laws similar to those found in physics dates back to 1803, when Henri de Saint-Simon explained it in his first book (Iggers, 1959). Auguste Comte, a student of de Saint-Simon (Iggers, 1959) provided the first definition of social physics and developed the required epistemological framework- for this, he is often credited as the founder of positivism and (positivist) sociology.

Figure 1: The increasingly strong theoretical base of SIMs. Source: Fotheringham et al, 2000: 215

(8)

2.1.1 Development of ‘gravity models’

It might be argued that the first SIM was Carey’s application of Newton’s law of universal gravitation to inter-city migration flows (1858). Newton’s original law states that every particle attracts every other particle; the strength of attraction between two particles is proportional to the product of the masses of both particles and inversely proportional to the distance between the two particles squared. In the SIM used by Carey, the particles are replaced by cities, the masses by population and the attraction by migration. The use of this SIM extended well into the 20th_century.

Stewart, an astrophysicist who liked to meddle in social sciences, was still using the same concept as Carey as late as 1948 to calculate what he called the ‘demographic force’ ( ) between two cities (Stewart, 1948):

(1) where and are populations of respectively city and city j, is the distance between city and and is used for scaling.

This so-called ‘gravity model’ went through some significant changes in the subsequent years. Since the model was used to model various phenomena, it would not make sense for all these phenomena to be affected by distance in the same way. People doing their daily groceries, for example, will experience a stronger effect of distance than people who are shopping for a piano, ceteris paribus; people will buy a piano maybe once every 30 years, and will likely be willing to travel a little further to buy one of superior quality than they would for their dailygroceries. For this reason, the inverse square of the distance was replaced by a parameter which could vary for

different phenomena, places and scales. Furthermore, the ‘masses’ of the cities (or whatever objects were modeled) were also given parameters and/or weights. This makes sense too, as in Newton’s formula, two particles would always attract each other with the same strength, but this is not necessarily true for social phenomena. The parameters and/or weights are to be estimated from empirical data. These developments gave the ‘gravity model’ increased empirical strength, and the contributions to spatial interaction modeling in this period can be summarized by the following model (Isard, 1960: 510):

(2) where is the expected interaction flow between and j, , and are parameters and and are weights to be estimated for empirical data.

In the 1960s, spatial interaction modeling developed rapidly in various directions (Hua & Porell, 1979). First, the model in (2) was being applied to various phenomena. Just using populations of cities and did not cut it anymore; when modeling, for example, shopping trips, one might want to use something like shop floorspace rather than population. Second, while Newton’s formula and the model in (2) use only one variable to describe the ‘mass’ of an object, the spatial interaction modeler might want to use many. Shopping trips to shopping center might depend, besides

floorspace, on the quality of the service of the employees, or whether the shopping mall has free parking space. In essence, all attributes of that might increase outgoing traffic from , and all

(9)

makes sense to come up with a general notation that captures this idea:

(3) (4) where represents a function of all attributes of through and their complementary

parameters through , from now on called propulsiveness, and represents a function of all attributes of through and their complementary parameters through , from now on called attractiveness (Fotheringham & O’Kelly, 1989; Hua & Porell, 1979). Note that the attributes might increase or decrease propulsiveness or attractiveness, and that parameters may therefore be positive or negative; more floorspace will increase attractiveness, but a higher price will decrease attractiveness, for example. (3) and (4) attempt to give the most general definition of and . In reality, and are almost always assumed to have the following form (Fotheringham & O’Kelly, 1989):

(5) (6) Third, euclidean distance often got replaced by a different variable like travel time, travel cost or other measures of separation, since humans by and large do not travel as the crow flies. The

requirement that the distance decay effect has the functional form of a power law was also relaxed (Hua & Porell, 1979). Instead of , one may write

(7) where is ‘cost’ or any other measure of separation between and j. Fotheringham and O’Kelly call this the separation function (1989: 10), although (7) is frequently called a cost function too. This function can take a variety of forms (see paragraph 3.1), but the form most commonly used in this period was the power function:

(8) In Fotheringham and O’Kelly’s general definition of a SIM (1989), and most other literature in fact, there is assumed to be just one ‘cost’ variable and one associated parameter. Since ‘cost’ implies a negative effect on the intensity of interaction, is assumed to be negative. Hua and Porell (1979) are more generous, and define

(9) where is a set of variables related to the route from to j. is a function of and may be called connectivity. The variables that make up can, in contrast to the cost function (7), exercise a positive or negative effect on the intensity of interaction . When the effect is positive, it might be called facility; when it is negative, it might be called friction (Hua & Porell, 1979).

(10)

One might even abandon this notion of connectivity altogether, and replace it with whatever can be defined as an attribute of the pair that will increase or decrease the intensity of interaction . This idea is applied in the intervening opportunities model (Stouffer, 1960):

(10) where and represent respectively the intervening opportunities and the competition between (or in a circle around with a radius of the distance between) and j, and and are parameters to be estimated. Stouffer’s approach will not be used here, though, because competition can be included in our model in other ways, which require a measure of connectivity like . Hua and Porell’s (1979) is attractive, and could even be extended to include anything that can be defined as an attribute at the level of the pair ; when modeling international trade, for example, a dummy variable taking a value of 1 when two countries speak the same language.

Fourth, and finally, SIMs were empirically strengthened by the use of constraints. A SIM can be constrained at the origin (i), the destination (j) or both. An origin-constrained SIM means a model where the outgoing flow from every origin ( ) is known (from, for example, empirical data), and used in the model instead of . The model will be extended with a balancing factor to ensure that the estimated outgoing flows from an origin does, in fact, add up to :

(11) A destination-constrained model, equivalently, takes the incoming flows for a destination ( ) as given, and the balancing factor in the model will ensure the estimated incoming flows will add up to

:

(12) Whether using constraints is appropriate depends on the phenomenon which is being modeled. When modeling labour markets, it can be reasonable to constrain both the origin and the

destination, as a job can only be done by one person and one person can only have one full-time job. When modeling phone calls between cities, however, it is not, since there is no fixed amount of phone calls at the origin waiting to distributed over various destinations. An origin-constrained model can be written as

(13) (14) where is the estimated intensity of interaction between and j, and is a balancing factor, summing the attractiveness times connectivity over all competing locations (including j) to ensure (11) is met. This model is widely known as ‘Huff’s probabilistic retail model’ (Huff, 1963). The destination-constrained model, similarly, can be defined as

(11)

(16) where is the balancing factor to ensure that (12) is met. Finally, the

origin-destination-constrained model can be expressed as

(17) where (18) and (19)

2.1.2 Criticism

The unconstrained (2) model, the single-constrained ((13) through (16)) models and the double-constrained ((17), (18), (19)) model are all, essentially, based on the ideas of gravity and social physics. However, where the unconstrained model still resembles Newton’s original law, in the constrained model the law of universal gravitation is hard to recognize. The

double-constrained model did, at the time, not resemble any other mathematical model, and the validity of its form was strongly questioned (Fotheringham et al, 2000). Although some attempts where made to explain the gravity models from an economic perspective (e.g. Niedercorn & Bechdolt, 1969), these were not convincing enough and required making many assumptions about human behaviour- an affair that geographers, contrary to economists, are generally not keen to participate in.

So, the gravity models ended up between two stools. For the radical positivist, the gravity models and their inconsistent use of variables and parameters disqualified them from being universally applicable in the way that laws of physics are universally applicable. Geography, unlike physics or perhaps economics, is the study of difference rather than similarity, so for a geographer this is not a big issue; in fact, parameters varying across places can provide interesting information. The

criticism on the other end of the spectrum is that it is unclear what the model is studying. The notion of spatial interaction as introduced at the start of this thesis is vague and abstract, and can apply to people as well as money. The movement of people can be seen as the aggregate of human

behaviour. If the same model is applied for money, does this mean money has a will of its own? Empirically, the gravity models introduced in this paragraph worked just fine, but how or why exactly was not always clear.

2.2 Spatial interaction and statistical mechanics

The gravity models introduced above, though performing empirically well, were subject to the criticism of being theoretically unjustified. As discussed in the first section, this does not necessarily

(12)

mean that they do not correspond to reality at all. Still, having a deeper understanding of the underlying mechanics is a worthy goal an sich, and can lead to new formulations that improve theories empirically as well. A big step in improving the theoretical base of SIMs was undertaken by Wilson (1967; 1970) after realizing that SIMs could also be derived from principles based in statistical mechanics, rather than Newtonian mechanics. The general idea will be described here based on Wilson (1970), followed by some alternative frameworks.

2.2.1 Wilson’s entropy-maximizing framework

First, it must be clarified that this explanation is aimed at the origin-destination-constrained model and is purely concerned with the distribution of trips, not the generation of trips (although the unconstrained and single-constrained models can also be derived from a statistical mechanics-perspective (Wilson, 1971)). In other words, the trips departing from all places and arriving at all places are known, so any distribution given by the model must satisfy the constraints (11) and (12). Second, Wilson’s model is a model of ‘trip distribution’ often used in traffic modeling, where it often makes sense to use a double-constrained model. Hence the sudden switch in jargon from the more abstract ‘intensity of interaction’ to ‘trips’. A small modification will be made to the model introduced in (17):

(20) where is replaced by cost function , which will make explaining the

entropy-maximizing method introduced below easier.

Every pair can be seen as a ‘box’ ( ) containing a number of trips. The number of boxes will be the number of origins times the number of destinations j. Next, there is a total number of trips, for now called population:

(21) which is distributed in some way over our boxes. The amount of trips that go in one box will be called the occupation number of said box. Many of such distributions are possible. For example, the entire population could be put in the first box , giving it a occupation number equal to . Alternatively, the entire population could be put in the first box except for one person who will be in box 2, giving and . If every individual making up is unique, there is a certain number of ways a distribution could be configured. Every such configuration of a

distribution will be called a state. The distribution where everyone is in the first box ( ) could only arise in one way: namely, the entire population being in the first box. Put differently: only one state gives rise to this distribution. The distribution and , on the other hand, can arise in a lot of different ways; there just has to be one person in the second box, and this could be anyone in the population as long as the rest of the population is in the first box. The number of states that could give rise to this distribution is, therefore, . Probability theory tells us that the number of states that can give rise to any distribution is

(13)

In statistical mechanics it is (initially) assumed that every state is equally likely to take place. From this assumption, it can be concluded that the most likely distribution is the one that could arise from the largest number of states. Hence, the distribution that maximizes will be the most likely distribution of population (trips). Note that, in this approach, the predicted trip distribution is not deterministic but should be seen as the most likely result- the prediction it will likely be a bit off, and this is acknowledged. From (22) it can be deduced that will be largest when is divided equally over every box through , meaning every box gets a population of where k is the number of boxes. This is, of course, unrealistic and does not satisfy our constraints (11) and (12). Besides, it completely ignores the differences in cost ( ) between boxes. In the

entropy-maximizing framework, this is taken into account by adding the total expenditure on travel as a constraint, too:

(23) which has an equivalent role to the constraint of the total energy in a physical system, the usual type of system studied through statistical mechanics. It should be noted that this approach is known as the ‘entropy-maximizing’ framework. Maximizing entropy is not the same as maximizing . It can, however, be shown that maximizing and will yield the same result- see Appendix 1. To maximize (22) subject to constraints (11), (12) and (23), the technique of Lagrangian multipliers is used. (21) is, in fact, another constraint that must be satisfied- but this already happens as a side-effect of (11) and (12). The Lagrangian that has to be maximized is:

(24) where , and are the Lagrangian multipliers to be solved for. For an overview of all mathematical steps required to accomplish this, see Wilson, 1967. In the end, the steps will result in (20), with:

(25) (26) (27) This entropy-maximizing SIM has a few peculiar properties. First, can be calculated numerically if the total ‘expenditure on travel’ is known. This makes sense for physical systems, when would represent the total amount of energy in the system, which can be derived from a number of other macro-variables. In the case of spatial interaction, can only be exactly known if every value of

is known. If that is the case, can also be estimated using statistical methods (more on parameter estimation in the paragraph 3.1). If this is not the case, however, and could in some

(14)

way be accurately estimated, this property would be quite useful. Second, when origin-constrained model is derived from entropy-maximizing principles, the cost function is found to be an

exponential function with ‘travel cost’ as its sole variable (27). Whether the exponential form is always the best choice has been theoretically and empirically challenged- but this discussion will be postponed until paragraph 3.1. Third, it is commonly desirable to add more information to the model than just the univariate cost function (27) and the constraints (11) and (12), like the influence of different modes and their corresponding advantages and disadvantages. Rather than defining some composite measure of connectivity , Wilson (1970) introduces a new method that takes into consideration the availability of modes for different groups of people. This method is now shortly explained.

2.2.2 Entropy-maximization and mode choice

Following Wilson’s (1970) example, two person types ( ) will be defined: car owners and non car owners. To keep the example simple, only two mode types ( ) will be defined: car and transit (which included all types of transit as well as trips made by foot or bicycle). Let be the set of modes that person type has access to- for example, would be since car owners can choose both, as opposed to to non car owners who can only choose . A mode in such a set is written as . Let be the number of trips between and by person type using mode ; let be the number of trips generated in by people of type ; and let be the cost of travel between and by mode .

It is assumed that car owners compete for the same activities as non car owners. Our constraints can consequently be defined as

(28) (29) (30) The model resulting from these constraints is

(31) where

(32) (33) From this model, the modal split by person type can be derived as follows:

(15)

where is the fraction of traveling by mode .

Wilson treats this method in greater detail, addressing issues like: • aggregation over person types or modes

• the role of in deciding mode choice and/or trip length • the use of composite costs like both price and travel time for

• different assumptions about distance decay, e.g. that it is unique for mode type rather than person type

• different available routes for the same mode

Readers interested in these topics are encouraged to consult Wilson, 1970.

2.2.3 Choukroun’s model

Choukroun (1975) offers an approach that, like Wilson’s model, can be derived from principles grounded in statistical mechanics. Choukroun’s model differs from Wilson’s in that, instead of assuming and to be known, it returns to the notions of propulsiveness and attractiveness

. Choukroun shows that the model can also be derived from a micro-economic perspective, but this derivation makes a large amount of unreasonable assumptions, like the assumption that an individual does not know his or her own place of residence (Choukroun, 1975) and will therefore be ignored here. Like Wilson’s model, Choukroun’s model was developed primarily for trip

distribution. A simplified account of statistical mechanics derivation will now follow.

Let and be the fractions of respectively the total amount of propulsiveness and attractiveness: (35) and can be seen as the probabilities for a trip to respectively start or end in or . From this it follows that the probability for 2 trips to start in is , and for trips the probability is . If there exists some matrix containing a distribution of trips from all ’s to all ’s, and for every we can calculate the outflow using (11), and for every we can calculate the inflow using (12), the probability of any given or is

(36) The probability of having any distribution of trips is then

(16)

which can be interpreted intuitively as follows: if is high when is high, and is high when is high, and the same when all are low, this particular distribution of trips has a relatively high probability of occurring. In subparagraph 2.2.1 it was shown that the number of states that can give rise to a certain distribution can be computed using (22). Combining (22) with (37) gives

(38) where represents the amount of states that can give rise to a distribution of trips, weighted by the probability of such a distribution occurring given the probability defined in (37). To find the distribution that has the likeliest chance of occurring, is then maximized, constrained by the total amount of trips (21) and total cost (23). This is process is similar to Wilson’s method, and is described in detail in Choukroun (1975). The resulting model is:

(39) where

(40) which resembles the unconstrained model.

2.2.4 Alonso’s framework and quasi-constrained models

Alonso (1973) proposed the system of equations (41) – (45) to model inter-city migration. After realizing these equations could function as an overarching framework to represent different spatial interaction models and measures of accessibility, this system of equations was rechristened the ‘theory of movements’ (Alonso, 1978):

(41) (42) (43) (44) (45) where in (41) and (42) summations over are the predicted outflow and inflow of respectively places and from their respective propulsiveness and attractiveness, and are parameters that can be set to 0 or 1 to generate different models, and are ‘endogenous’ factors summarizing the the external conditions of places and (equivalent to Hansen’s (1959) notion of accessibility and the balancing factors in the other constrained models) and (45) is the unifying model.

(17)

to see this system of equations as a systematic way of writing down different models and measures of accessibility, showing immediately their connections and serving as a tool of understanding or a lingua franca rather than a model that has strong practical applications (Hua, 1980). The model, for example, clearly communicates the relations between the expected outflow ( ) of an origin ( ) and the location’s propulsiveness ( ), as well as its position relative to all other origins and their relation to all destinations ( ). Of course, these relations were already embedded in other SIMs, but Alonso’s theory of movements does a good job at communicating them.

Wilson (1980) draws a similar conclusion and illustrates the similarities between Alonso’s framework and his own family of SIMs (1971). Setting and to 1 results in the unconstrained model:

(46) Setting to 0 and to 1 and assuming to be known as results in the origin-constrained model (13); setting to 1 and to and assuming to be known as results in the

destination-constrained model (15); and, finally, setting and to 0 and assuming and to be and results in the origin-destination-constrained model (17) (Wilson, 1980).

Fotheringham and Dignan (1984) suggest a different interpretation of the parameters and . (41) and (42) show how the expected out- and inflows and depend on a combination of

attributes related to places or , called site-related variables, and the positioning of places or in relation to all other places or , called situation-related variables. As shown by Wilson (1980), if the total out- and inflows are known, the situation-related variables become obsolete and the parameters are set to 0. It is, however, conceivable that the out- and inflows are not exactly known, or represented by a different variable or combination of variables like or that do not exactly represent or , but still represent it to some accuracy. In this case, using a double-constrained model may yield better empirical results, but since it does not make sense to assume the out- and inflow to be known with complete certainty, a small bit of flexibility is desirable. This can be achieved by setting and to a value between 0 and 1, depending on how strongly and resemble and . Models using this technique are called quasi-constrained models (Fotheringham & Dignan, 1984) and can be seen as a hybrid between constrained and unconstrained.

2.2.5 Criticism

The most obvious way of critiquing the statistical mechanics framework is that it “...replaces one physical analogy … with another” (Fotheringham et al, 2000: 221). While this seems like a logical conclusion to draw, it is not entirely valid. First, rather than ‘replacing’, the statistical mechanics derivation can also be seen as strengthening to the original gravity analogy. In the natural sciences, if two theories are formulated from very different standpoints or on different scales, it is regarded as a large gain in credibility if the theories are eventually ‘united’. In fact, one of the largest

occupations of physicists nowadays is to find a ‘theory of everything’ that unifies the theory of general relativity and quantum field theory. Not to say that showing how gravity models can also be

(18)

derived from an entropy-maximizing perspective is an accomplishment of this magnitude, but the fact that it is possible definitely makes the theory of spatial interaction more plausible.

Second, although statistical mechanics is most commonly applied to studying systems in physics, it is not just a physical theory like gravity. Statistical mechanics should rather be seen as a

mathematical concept that has myriad applications, like (regular) statistics and probability theory. Wilson (2010) describes statistical mechanics and entropy as ‘super concepts’ that cut through different fields- something that has definitely taken place after the ground-breaking work by Wilson, finding applications in traffic modeling, economics, sociology and more. Besides countering the point that it is merely another analogy, this also increases the justification of applying SIMs to different topics.

However, one issue that was already mentioned under the criticism of gravity models remains unresolved: the explanation of macro-level structures is there, but the micro-level processes giving rise to these structures are still hardly accounted for. Wilson (1970) has taken some steps in the right direction by using a ‘cost function’, which makes sense from an economical perspective in that cost is an example of dis-utility that would make is less attractive for individuals to travel to places with high costs. Moreover, in subparagraph 2.2.2 it is shown how differences between people, e.g. car ownership, can influence their preferences or available options. Later in his book, Wilson (1970) makes some suggestions how to relate entropy-maximizing to utility-maximizing theory. To summarize, the first attempts to descent to the micro-level were there, albeit limited in numbers. Theory on micro-level processes in spatial interaction, however, remained very limited until the end of the 70s, when a new approach was introduced- this approach will be discussed now.

2.3 The aspatial information processing framework

Fotheringham and O’Kelly (1989) define spatial interaction as “… movement or communication over space that results from a decision process”. In the introduction of this thesis, a similar

definition is provided, though deliberately without inclusion of the part about the decision process. This choice of definition was made on the grounds that not all spatial interaction is the result of a (conscious) decision. This point is will be explored in more depth near the end of this paragraph, as part of the criticism on this framework. There are, nonetheless, plenty of situations conceivable where it is reasonable to assume that the interaction between two places is the aggregate result of numerous individual decisions. To better understand this decision-making process at the micro level, discrete choice models can be deployed. The basics of discrete choice modeling will now be discussed.

2.3.1 Discrete choice modeling

The purpose of discrete choice modeling is to describe, explain and predict choices made by individuals among a finite set of mutually exclusive and collectively exhaustive alternatives (Ben-Akiva & Lerman, 1985). Each alternative has a set of properties, and each individual has a set of preferences. The choice is determined by a combination of properties inherent to the available alternative in relation to preferences of the individual making the choice. This nature of this relation

(19)

interested in the behaviour or larger groups of people. For this reason, the models constructed must have a higher level of abstraction and generalizability than models found in, for example,

psychology. This inevitably leads to a trade-off. No person has the exact same preferences, which influences the (perceived) utility of alternatives and thus the choices made. The different

preferences of individuals are assumed to be a result of socio-economic properties that describe an individual; this means that all individuals that have the same socio-economic properties have, as far as discrete choice theory is concerned, exactly the same preferences. The understanding of all cognitive processes involved in making a choice and how these are influenced by socio-economic attributes would require extremely detailed data on the profiles of the individuals studied, as well as a complete list of attributes of the alternatives that might play a role in the decision process. Clearly, this is both practically and theoretically impossible. On the other hand, in many cases certain properties of alternatives will lead to similar effects for a large group of people- the maximum amount of time people are willing to spend on traveling to the grocery store, for example, can already explain a large portion of grocery store choice. The discrete choice modeler has to find the balance between the level of detail that the model can explain and the practical and theoretical limits of how complicated a model can get.

The first step in discrete choice modeling is to list all alternatives, and to figure out which

alternatives are available to which individuals. The set of all possible alternatives might be called the universal set ; the subset of this set which is available to the individual in question can be called the choice set: (Ben-Akiva & Lerman, 1985). A choice set consists of

alternatives:

(47) An alternative is either chosen or not, and only one alternative can be chosen:

(48) Every alternative has a vector of attributes , seen from the perspective of individual (for example, distance will vary depending on where an individual lives) called :

(49) An alternative will be chosen by an individual based on the attributes of the alternatives (Lancaster, 1966). This happens according to a decision rule. Ben-Akiva & Lerman (1985) distinguish four types of decision rules:

1. Dominance: an alternative is chosen where every attribute has a superior value to the other alternatives.

2. Satisfaction: an attribute has a certain threshold. If this threshold is not met by an alternative, this alternative is excluded from the possibilities.

(20)

3. Lexicographic rules: attributes are ranked by ‘importance’. The highest value (or range of values) on the most important attribute determines the chosen alternative. If there is a tie, the tied alternatives will be judged on the next most important attribute, and so forth.

4. Utility: attribute levels correspond to a certain amount of utility for the individual making the choice. The total utility of an alternative is calculated with an utility function. The alternative with the highest utility is chosen.

Method 1 cannot deal with situations where alternatives score higher on different attributes, method 2 will not always lead to a single winner and method 3 performs theoretically and empirically poor on its own in most situations (Fotheringham & O’Kelly, 1989). Decision rules can be chains or combinations of these four rules, which can resolve some of these issues. However, the most common approach in discrete choice modeling is to use the concept of utility, since it does not have the issues of the other three in addition to being easier to estimate from empirical data. For this reason, the maximum utility rule will be explored in more depth.

The utility associated with an alternative from the perspective of an individual is a function of and a vector of socio-economic variables that determine the individual’s preferences:

(50) Discrete choice models can never fully explain or predict the choices made by individuals. Given a certain decision rule, two individuals might have the exact same socio-economic properties and choice set, and still make a different choice. Discrete choice theory offers two different explanations for this. The first is called constant utility and assumes that there is some inherently probabilistic aspect to human behaviour. This method relaxes the assumption that an individual always chooses the alternative with the highest utility by stating that an individual will most likely choose the alternative with the highest utility. Formulated differently, every alternative has an associated probability ( or ) which is a function of its utility relative to the utilities of the other alternatives. This function has, in most cases, the following form:

(51) The second explanation is deterministic and implies that an individual will always choose

(consciously or unconsciously) the alternative with the highest utility, but that the unexplained observations are due to a lack of knowledge on behalf of the researcher. This approach is called random utility- a slightly confusion term, since it essentially states that there is no true randomness, only perceived randomness as a result of observational deficiency. This ‘perceived randomness’ will be represented by an error term . The ‘random utility’ including can then be defined as

(52) where is the observable component of attributes of an alternative. Since the individual always chooses the option with the highest utility, the probability that an individual chooses an alternative is

(21)

utilities of all other alternatives in the same choice set. Alternatively:

(54)

2.3.2 Discrete spatial choice and spatial interaction models

Discrete choice models’ sudden rise to fame can be explained in twofold. First, there is the

simplicity and strength of the theoretical underpinnings, which is obviously an important factor. But the most important factor, which also explains why this theory was not introduced earlier (if it is so simple), was the development of the required statistical methods like logit and probit (Wrigley, 1985). Where the parameters expressing preference for continuous quantities in utility-maximizing theory can be calculated with calculus, for discrete choice modeling this was not possible (Ben-Akiva & Lerman, 1985). But the advent of revolutionary new regression techniques finally made discrete choice modeling possible.

Discrete choice models, as the name implies, can be applied to everything decision process

involving discrete options. These options can be aspatial, like the choice between brands, or spatial, like the choice between shops. In both situations, the underlying theory is the same. Spatial choices, initially the turf of single-constrained SIMs, can using discrete choice theory be approached from another point of view. But the similarities between SIMs and discrete choice models are not limited to their application: in fact, it can be shown that in many cases, the models are one and the same. The easiest demonstration of this structural similarity is to compare origin-constrained SIMs (13) and (14) and the constant utility approach in discrete choice modeling (51). In the previous subparagraph, it was shown that utility is a function of attributes of the alternative from the individual’s point of view and socio-economic variables related to the person: . For now, the socio-economic variables are assumed to be equal for everyone and the choice is assumed to be determined solely by attribution of the alternative from the individual’s point of view. can then be divided into two groups: a set of attributes that are inherent to the alternative , namely , and a set of attributes that are related to the individual ’s point of view, namely . If it is then assumed that every individual has the same choice set ( ), the probability of a person picking an alternative becomes

(55) If all individuals living in the same place are assumed to have the same socio-economic

characteristics and point of view towards alternative , and an alternative is a spatial object like a store, and might as well be replaced by and , yielding

(56) which is the same to the origin-constrained model (13) and (14), given that

(22)

This formulation is better known as Huff’s probabilistic retail model (Huff, 1963). To allow for easy explanation, some weak assumptions have been made on part of the discrete choice model, like the assumptions regarding the choice set and socio-economic attributes. The single-constrained SIMs could be greatly improved by taking into account these concepts from discrete choice theory. Fotheringham and O’Kelly (1989), with greater mathematical rigour, prove that the

origin-constrained models are, in fact, structurally similar to random utility models (54), too. Anas (1983), furthermore, demonstrates that the unconstrained models and double-constrained models can be deduced from discrete choice principles as well, and in the same paper highlights the similarities between the entropy-maximizing and utility-maximizing approach.

2.3.3 Criticism

While the derivation of spatial interaction models from discrete choice theory can be seen as a huge improvement, especially regarding the explanatory power of micro-level processes, it has a

downside; by importing a framework grounded in economics, one imports a point of view from a predominantly aspatial discipline containing many peculiar assumptions about human behaviour as well.

The first issue that arises from is that discrete choice theory and economics in general does not explain spatial processes very well. In fact, the only way space is included in the equation is through some measure of connectivity between the individual and the alternative. More complex interactions between the alternatives, like clustering-effects, spatial autocorrelation or increased competition between alternatives in proximity, are not captured.

Second, discrete choice theory makes some serious assumptions about human information

processing capacity and willingness. When choosing between a few candy bar brands, the human mind is likely to come up a good comparison and pick a bar that compliments the taste of the individual. With spatial choice, however, there can be thousands of possibilities with each numerous attributes. If an individual could somehow calculate the utility of every one, it is unlikely that the individual will do this every time he or she has to make a spatial choice. Also, in many situations choices are not made by listing literally every option and choosing one- often, alternatives are subdivided in (mental) categories, and the decision maker will apply a multi-level strategy by first choosing a category and then an individual alternative. Such processes can be modeled within a discrete choice framework, although for ‘spatial categories’ this process is more difficult.

Third, discrete choice modeling takes some classic assumptions about human behaviour typically for economics, which can take different levels of plausibility in different situations. The name ‘discrete choice’ implies that there is a choice being made. This assumption is grounded in logic of free-market economics, where individuals are free to choose between alternatives. The alternatives will compete for the individual, who chooses the best alternative based on its attributes. But, as anyone who has attempted to find a job lately can confirm, this rosy view does not correspond to reality very well in the situation of labour markets. Finding a job is for many people extremely hard, but still extremely necessary if they desire food on the table. An individual without income will

(23)

the job that provides ‘the maximum utility’ and risking starving to death.

In the situation sketched above, it might be argued that the individual still makes a choice, albeit one with very limited options. There are, however, situations conceivable where the is no choice involved at all. Take, for example, a police station and a neighbourhood , where the interaction between the two is the number of arrests made in that neighbourhood by police officers from . This interaction could be modeled by some spatial interaction model, taking into account the size of the police station, some variables that explain the crime rate in the neighbourhood and the distance between the two. Clearly, the criminals in the neighbourhood are not ‘deciding’ to be caught, nor are the police officers in the police station trying to ‘maximize utility’ by choosing to arrest people from a neighbourhood. Still, the number of arrests made could well be modeled by SIMs.

2.4 The spatial information processing framework

The spatial information processing framework (Fotheringham et al, 2000) is largely an extension of the discrete choice theory-based framework set out in the previous paragraph and can be described as a collection of methods that try to integrate the different types of criticism made in subparagraph 2.3.3. Although there is still no perfect method that counters all issues, some interesting

improvements have been made. This paragraph will discuss three.

2.4.1 Utility trees and clusters

The criticism made in subparagraph 2.3.3 can partly be addressed by using a utility tree (Strotz, 1957). A utility tree requires some a priori categorization of alternatives. For aspatial alternatives, a categorization of dishes might be Italian cuisine, French Cuisine and Korean Cuisine, which will be called the ‘branches’ or ‘clusters’. The alternatives in every cluster would then be the dishes

belonging to that cuisine. The individual making the decision is assumed to first decide on a national cuisine (cluster), and after that decide on a dish (alternative) that belongs to that cuisine. A cluster is chosen on its (perceived) utility, and the alternative from that cluster is then chosen on its (perceived) utility. This may be written as

(58) where is the number of branches or clusters, the number of alternatives per cluster and

determines the separability between clusters.

In spatial choice, this method can be applied in two ways, depending on whether the clusters themselves are aspatial or spatial. Both methods can be illustrated by an example. Consider an individual who has to make a decision where to spend his or her Friday night. If the clusters are assumed to be aspatial and the alternatives spatial, the clusters might be a category of entertainment, like music, cinema or theatre the spatial alternatives belonging to each cluster are then respectively music venues, cinemas or theatres. If, alternatively, the clusters are spatial and the alternatives aspatial, the clusters might be cinemas and the alternatives the movies that are currently playing in each cinema. Lastly, both the clusters and the alternatives might be spatial- for example, the clusters might be close-by cities, and the alternatives the music venues, cinemas or theatres in each city.

(24)

The spatial clusters described in the example above are discrete. In reality, this is not always the case- spatial clusters can better be described as ‘fuzzy’ (Zadeh, 1965). Choosing a place to get a drink, for example, might work cognitively by first selecting a spatial cluster that contains a lot of nice bars, and on arrival picking the exact bar. But where the cluster starts and ends might be unclear. Mathematically, if represents whether two alternatives and belong to the same cluster, for discrete clusters this relationship may be written as

(59) while in a fuzzy, spatial cluster, the relationship is better described as

(60) where is some measure of connectivity between and . In order words: two alternatives are likely to be perceived in the same cluster if they are close or well connected (Fotheringham & O’Kelly, 1989). To explain how this works on a micro-economic level using the concepts of discrete choice and fuzzy clusters, it makes sense to first explain the macro-level equivalent: the competing destinations model.

2.4.2 Competing destinations and agglomeration effects

The competing destinations model (Fotheringham, 1983) was proposed as an extension of existing SIMs to better account for the spatial structure of the area which is being modeled. Traditionally, SIMs deal with spatial structure by measuring the connectivity between and origin and a

destination , which can be seen as a summary of various aspects of spatial structure like distance, the structure of transport systems or even mountains and rivers. However, the relations between different destinations themselves are not included. The role of that these relations might have on spatial interaction can be illustrated by an example. Consider a neighbourhood and three identical stores at equal distance from the neighbourhood. In a traditional SIM, all three stores would receive an equal amount of customers from the neighbourhood. But if two stores are directly next to each other, while the third is located for a away from the other two, it might be expected that the third one receives a larger portion of customers since the two other stores experience more competition due to their mutual proximity.

To include these competition effects, Fotheringham (1983) includes a competition term and a parameter to indicate the strength of the competition. The competition term is computer as follows:

(61) where is the destination in question and represents another destination which is not . This measure of competition is similar to Hansen’s (1959) measure of accessibility. Consequently, an origin-constrained competing destinations model would look as follows:

(25)

are in proximity of each other, will have a negative value. But in some cases, will in fact be positive. If this is the case, proximity to other destinations will have a positive effect on the

expected interaction from , which indicates the presence of agglomeration effects (Fotheringham, 1985). Agglomeration effects might be observed when e.g., instead of similar stores, nightlife venues are modeled.

The process of including the effect of competing destinations in discrete choice-based models is described in Fotheringham, 1988. As mentioned before, the continuity of space does not allow for discrete categories. Another issue that comes up when applying discrete choice models to spatial choices is that of independence from irrelevant alternatives (IIA) (McFadden, 1974). IIA means that when a destination has a higher chance of being chosen than another destination, the superiority of the first destination over the second destination will remain unchanged when another alternative is introduced. That this is not always a reasonable assumption in spatial choice problems can be demonstrated by returning to the example of competing stores. If people in a neighbourhood have two choices, store A and B, store A might for some reason be a more popular choice than B. However, if another store C opens right next to store A, store A might lose a lot of customers to store C, while store B is less affected by the opening of store C. As a result, B might become more popular than A, and thus the principle of IIA is violated.

If is a cluster which is a subset of ( ), the probability of a person choosing an alternative from the cluster would be:

(63) Since the clusters are not discrete, some uncertainty on whether alternatives and are in the same cluster must be included. This can be achieved by replacing the two-step process where first a cluster is chosen, and then an alternative, by the following:

(64) where is the probability that individual perceives alternative to be in cluster . The definition of depends on the phenomenon that is being modeled and how the individual would group this phenomenon in clusters. A common approach in aspatial discrete choice modeling is to include attributes of the alternatives or correlation between these to indicate that alternatives are in some sense similar (Fotheringham, 1988). In spatial choice modeling, unsurprisingly, is often assumed to depend on spatial characteristics, such as the proximity/connectivity to other alternatives (Borgers & Timmermans, 1987), reflecting the idea of (60):

(65) or a combination of proximity/connectivity weighed by their attractiveness, which can be achieved by including the competition term (61) (Fotheringham, 1988):

(26)

(66) where is the number of destinations and a parameter reflecting the effect of the competing destinations. For a more extensive explanation, see Fotheringham, 1988.

2.4.3 Time geography

Hägerstrand’s time geography (1970) is a sobering approach to human behaviour, developed largely as a response to the methods of regional scientists and quantitative geographers at the time. SIMs, for example, can be used to model commuter behaviour by taking as starting point the residential location of individuals and the locations of jobs, and then proceed to calculate some probabilistic aggregate behaviour making assumptions about how people decide on jobs. Time geography, in contrast, sees the individual as a continually existing entity that, during the course of his or her life, moves through space and time. This ‘life path’ is shaped partly by some preferences of the

individual, but mostly by a combination of capability, coupling and authority constraints.

Capability constraints can be biological, like the need to eat and sleep, or can refer to the maximum speed a person can travel by foot or in a vehicle. Coupling constraints describe situations where a group of individuals has to meet up, to attend a lecture at university or to exchange physical goods. Authority constraints, lastly, are set by the formal and informal laws of the society where a person is part of, like not being allowed to wander into other people’s houses.

Although time geography has many practical limitations (if taken literally, all humans should from their moment of birth constantly be tracked by a GPS device), some of the ideas developed by Hägerstrand could definitely improve how SIMs correspond to reality. Instead of assuming single-purpose trips from home to work, and from home to the store, models could incorporate the idea of the individual moving through space and time by taking into account the path from home to work to the store. This would imply that the attractiveness of a store relates to its proximity to a person’s job, rather than his or her home. This would, however, mean that multiple phenomena have to be included in one model, which in the context of spatial interaction modeling is rather difficult. It has, however, been attempted- for an example, see the multipurpose travel study in Fotheringham and O’Kelly (1989). More research on how other ideas of Hägerstrand’s can be incorporated in SIMs could prove fruitful.

3. Other contributions to spatial interaction modeling

3.1 Decay functions and connectivity

In the earliest gravity models, e.g. Carey (1858) (1), it was assumed that the interaction of ‘force’ between two objects would be inversely proportional to the square of Euclidean distance , similar to the effect of distance in Newton’s original formula. This proportionality is in geography called the distance decay. In the remaining part of paragraph 2.1, the inverse square was replaced by parameter (2), and then again replaced by a more general term which allowed for multiple

(27)

gravity model was derived from entropy-maximizing principles, followed by Wilson’s (1970) method of disaggregation by e.g. modal split which made multiple measures of and possible. After Wilson’s contributions, finally, was reintroduced. The reader might justifiably wonder what is going on here. Essentially, three related debates are taking place simultaneously: the first is concerned with which the choice of variable to represent the connectivity between two places; the second with the way distance decays; and, the third is on how to include multiple measures of connectivity. All three will be discussed shortly.

3.1.1 Choice of connectivity measure

In Carey’s formula, the ‘counterweight’ the to ‘masses’ of the populations was the Euclidean distance between them. Later, alternative measures like travel time or travel cost starting being used. Of course, instead of using a variable that decreases the intensity of interaction (causing friction), it should also be possible to use variables that have the opposite effect and increase the intensity of interaction (causing facility). As argued in paragraph 2.1, any other property of the pair

that does not necessarily relate to separation, but will still influence the intensity of

interaction, might be included. The definition of connectivity was proposed to avoid having to list all these options every time a new spatial interaction model was introduced. Which measure is most appropriate will, of course, depend on the phenomenon that is modeled and the research question.

3.1.2 ‘Distance’ decay functions

After the previous subparagraph, the term distance decay seems like a misnomer; many other (often superior) measures of connectivity exist. An alternative is the cost function, where distance and travel time can be seen as different measures of cost for the individual, but this still fails to capture the broader definition of connectivity. For unknown reasons, the concept of connectivity is not often used in spatial interaction modeling, and authors prefer to fall back on the old notions of distance and cost. For the sake of completeness, a ‘connectivity function’ could serve as an umbrella term for all types of distance decay and cost functions, but this would not be in accordance with the

interpretation of connectivity as a variable that can be composed of different other functions and variables. The best solution to this (perhaps tedious) discussion regarding nomenclature is to just speak of decay functions.

The decay functions that dominate the literature are the power function (67) and the exponential function (68):

(67) (68) where represents some measure of connectivity. The power function dominated in the social physics paradigm, until the exponential function emerged as a result of the entropy-maximizing derivation. A few issues must be considered when choosing one of these two functions. The first

Spatial interaction models : developments and future directions