Spatial Interaction
From Brett J. Lucas
Spatial interaction is the flow of products, people, services, or information among places, in response to localized supply and demand.It is a transportation supply and demand relationship that is often expressed over a geographical space. Spatial interactions usually include a variety of movements such as travel, migration, transmission of information, journeys to work or shopping, retailing activities, or freight distribution.
Edward Ullman, perhaps the leading transportation geographer of the twentieth century, more formally addressed interaction as complementarity (a deficit of a good or product in one place and a surplus in another), transferability (possibility of transport of the good or product at a cost that the market will bear), and lack of intervening opportunities (where a similar good or product that is not available at a closer distance).
Complementarity
The first factor necessary for interaction to take pace is complementarity.In order for trade to take place, there has to be a surplus of a desired product in one area and a shortage or demand for that same product in another area.
The greater the distance, between trip origin and trip destination, the less likelihood of a trip occurring and the lower the frequency of trips. An example of complementarity would be that you live in San Francisco, California and want to go to Disneyland for vacation, which is located in Anaheim near Los Angeles, California. In this example, the product is Disneyland, a destination theme park, where San Francisco has two regional theme parks, but no destination theme park.
Transferability
The second factor necessary for interaction to take pace is transferability. In some cases, it is simply not feasible to transport certain goods (or people) a great distance because the transportation costs are too high in comparison to the price of the product.
In all other cases where the transportation costs are not out of line with price, we say that the product is transferable or that transferability exists.
Using our Disneyland trip example, we need to know how many people are going, and the amount of time we have to do the trip (both travel time and time at the destination). If only one person is traveling to Disneyland and they need to travel in the same day, then flying may be the most realistic option of transferability at approximately $250 round-trip; however, it is the most expensive option on a per person basis.
If a small number of people are traveling, and three days are available for the trip (two days for travel and one day at the park), then driving down in a personal car, a rental car or taking the train may be a realistic option. A car rental would be approximately $100 for a three day rental (with for to six people in the car) not including fuel, or approximately $120 round-trip per person taking the train (i.e., either Amtrak's Coast Starlight or the San Joaquin routes). If one is traveling with a large group of people (assuming 50 people or so), then it may make sense to charter a bus, which would cost approximately $2,500 or about $50 per person.
As one can see, transferability can be accomplished by one of several different modes of transportation depending on the number of people, distance, the average cost to transport each person, and the time available for travel.
source : http://geography.about.com/od/culturalgeography/a/ucspatialint.htm?1234
Spatial Interactions
One methodology of particular importance to transport geography relates to how to estimate flows between locations, since these flows, known as spatial interactions, enable to evaluate the demand (existing or potential) for transport services.
A spatial interaction is a realized movement of people, freight or information between an origin and a destination. It is a transport demand / supply relationship expressed over a geographical space. Spatial interactions cover a wide variety of movements such as journeys to work, migrations, tourism, the usage of public facilities, the transmission of information or capital, the market areas of retailing activities, international trade and freight distribution.
Economic activities are generating (supply) and attracting (demand) flows. The simple fact that a movement occurs between an origin and a destination underlines that the costs incurred by a spatial interaction are lower than the benefits derived from such an interaction. As such, a commuter is willing to drive one hour because this interaction is linked to an income, while international trade concepts, such as comparative advantages, underline the benefits of specialization and the ensuing generation of trade flows between distant locations. Three interdependent conditions are necessary for a spatial interaction to occur [Ullman, 1956]:
- Complementarity. There must be a supply and a demand between the interacting locations. A residential zone is complementary to an industrial zone because the first is supplying workers while the second is supplying jobs. The same can be said concerning the complementarity between a store and its customers and between an industry and its suppliers (movements of freight).
- Intervening opportunity. There must not be another location that may offer a better alternative as a point of origin or as a point of destination. For instance, in order to have an interaction of a customer to a store, there must not be a closer store that offers a similar array of goods.
- Transferability. Freight, persons or information being transferred must be supported by transport infrastructures, implying that the origin and the destination must be linked. Costs to overcome distance must not be higher than the benefits of related interaction, even if there is complementarity and no alternative opportunity.
Spatial interaction models seek explain spatial flows. As such it is possible to measure flows and predict the consequences of changes in the conditions generating them. When such attributes are known, it is possible for example to better allocate transport resources such as highways, buses, airplanes or ships since they would reflect the transport demand more closely.
2. Origin / Destination Matrices
Each spatial interaction, as an analogy for a set of movements, is composed of an origin / destination pair. Each pair can itself be represented as a cell in a matrix where rows are related to the locations (centroids) of origin, while columns are related to locations (centroids) of destination. Such a matrix is commonly known as an origin / destination matrix, or a spatial interaction matrix.
| O/D Pair | Destinations | ||||
| A | B | C | Total | ||
| Origins | A | Ti | |||
| B | |||||
| C | |||||
| Total | Tj | T | |||
In the O/D matrix the sum of a row (Ti) represents the total outputs of a location (flows originating from), while the sum of a column (Tj) represents the total inputs (flows bound to) of a location. The summation of inputs is always equals to the summation of outputs. Otherwise, there are movements that are coming from or going to outside the considered system. The sum of inputs or outputs gives the total flows taking place within the system (T). It is also possible to have O/D matrices according to the age group, income, gender, etc. Under such circumstances they are labeled sub-matrices since they account for only a share of the total flows.
In many cases where spatial interactions are relied on for planning and allocation purposes, origin / destination matrices are not available or are incomplete, requiring surveys. With economic development, the addition of new activities and transport infrastructures, spatial interactions have a tendency to change very rapidly as flows adapt to a new spatial structure. The problem is that an origin / destination survey is very expensive in terms of efforts, time and costs. In a complex spatial system such as a region, O/D matrices tend to be quite large. For instance, the consideration of 100 origins and 100 destinations would imply 10,000 separate O/D pairs. In addition, the data gathered by spatial interaction surveys is likely to become obsolete quickly as economic and spatial conditions change. It is therefore important to find a way to estimate as precisely as possible spatial interactions, particularly when empirical data is lacking or is incomplete. A possible solution leans on the use of a spatial interaction model to complement and even supplant empirical observations.
3. the Spatial Interaction Model
The basic assumption concerning many spatial interaction models is that flows are a function of the attributes of the locations of origin, the attributes of the locations of destination and the friction of distance between the concerned origins and the destinations. The general formulation of the spatial interaction model is as follows:
- Tij : Interaction between location i (origin) and location j (destination). Its units of measurement are varied and can involve people, tons of freight, traffic volume, etc. It also relates to a time period such as interactions by the hour, day, month, or year.
- Vi : Attributes of the location of origin i. Variables often used to express these attributes are socio-economic in nature, such as population, number of jobs available, industrial output or gross domestic product.
- Wj : Attributes of the location of destination j. It uses similar socio-economic variables than the previous attribute.
- Sij : Attributes of separation between the location of origin i and the location of destination j. Also known as transport friction. Variables often used to express these attributes are distance, transport costs, or travel time.
The attributes of V and W tend to be paired to express complementarity in the best possible way. For instance, measuring commuting flows (work-related movements) between different locations would likely consider a variable such as working age population as V and total employment as W. From this general formulation, three basic types of interaction models can be constructed:
- Gravity model. Measures interactions between all the possible location pairs. The gravity model is covered in more details here.
- Potential model. Measures interactions between one location and every other location.
- Retail model. Measure the boundary of the market areas between two locations competing over the same market.
Spatial Interaction Models
There is a large body of literature concerning gravity and spatial interaction models. They are largely concerned with description and sometimes prediction of interaction (flows) between defined regions. They work on the idea of describing interaction between regions as
 | (1) |
where Tij is the interaction (trips)between regions i and j, 
is a property of region i or j (analogous to mass or gravity), and dij is the ``distance'' (spatial or cost-wise) between regions i and j.
These equations are descriptive, similar to general linear models in regression statistics. They are a way to fit observed data to a concise mathematical model with potential predictive capabilities. They are a standard tool for geographical study; several works give excellent descriptions of their formulations and histories (Golledge and Stimson, 1997; Haynes and Fotheringham, 1984; Lowe and Moryadis, 1975; Wilson and Bennett, 1985). Typical use includes descriptions or analysis of travel linkages between regions (Ivy, 1995) or labor migrations (Fik et al., 1992). They have also been used for parameterization of traffic simulations (Cascetta and Cantarella, 1991) and definition of functional regions based on possible interaction (Noronha and Goodchild, 1992).
One of the major criticisms of gravity models has been what many consider to be a too literal translation of a Newtonian physics model to social science (Haynes and Fotheringham, 1984, page 17). Wilson and Bennett (1985) alleviated part of this doubt by deriving some of the parameters independently through entropy maximization. However, whatever the analytic justification for the parameters, it can still be inappropriate for a spatial representation of a system. They are an inherently static representation of spatial patterns, though many of the processes that it is used to model are quite dynamic (Fik, 1997, page 399). When one is fitting the model to data, one may not know whether the data are long-term averages, a snapshot in time, or a transition between states. This limitation is not always acknowledged by the people using it.![[*]](http://www.gis.usu.edu/%7Esanduku/public_html/dissertation/outline/foot_motif.gif)
Dendrinos and Sonis (1990) gave a rigid mathematical treatment to general spatial interaction models, and showed that in equations describing even the simplest cases (one population, or stock interacting in two regions) there are many cases where no equilibrium exists. The implications are that many kinds of spatial interaction are capable of chaotic, complex, or unpredictable behavior, even when described in terms of assumed homogeneity that the gravity model implies. This should serve as an important caveat for any attempts to model dynamic spatial processes as static or equilibrium phenomena.
Gravity models and others similar ones have shown themselves to be valuable for fitting data and parameterizing conceptual relationships, but are useful only to the extent that a sufficiently large body of macroscopic system data is available in a form that the modeler can confidently use for extrapolation.
