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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

Author : Dr. Jean-Paul Rodrigue

1. Overview

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 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:

• T_ij : 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.
• V_i : 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.
• W_j : Attributes of the location of destination j. It uses similar socio-economic variables than the previous attribute.
• S_ij : 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.

source : http://people.hofstra.edu/geotrans/eng/ch5en/meth5en/ch5m1en.html

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 T_ij is the interaction (trips)between regions i and j, is a property of region i or j (analogous to mass or gravity), and d_ij 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.

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.

source : http://www.gis.usu.edu/~sanduku/public_html/dissertation/outline/node24.html

Data - Production of Crops

Production of CottonCrop

Productionof Sugarcane

Production of BarleyCrop

Production of Guava

Production of MungCrop

Districts of Pakistan

Districts of Pakistan

Subdivision	Number of Districts
Balochistan Province	30
North-West Frontier Province	24
Punjab Province	35
Sindh Province	23
Islamabad Capital Territory	1
Federally Administered Tribal Area	7 Tribal Agencies plus 6 Frontier Regions
Azad Kashmir	8
Northern Areas	6
Pakistan	124 Districts plus 7 Tribal Agencies

Islamabad Capital Territory

Districts	Area (km²)	Population (1998)	Density (people/km²)
Islamabad	906	805,235	889

Federally Administered Tribal Areas

Agency	Area (km²)	Population (1998)	Density (people/km²)
Bajaur	1,290	595,227
Khyber	2,576	546,730
Kurram	3,380	448,310
Mohmand	2,296	334,453
North Waziristan	4,707	361,246
Orakzai	1,538	225,441
South Waziristan	6,620	429,841
Six Frontier Regions combined	4,813	235,083
F.R. Bannu District	745	19,550
F.R. D.I.Khan District	2.008	39,373
F.R. Kohat District	446	90,806
F.R. Lakki Marwat District	132	6,955
F.R. Peshawar District	261	53,902
F.R. Tank District	1,221	27,339
FATA	27,220	3,176,331	117

Districts of Balochistan

There are 27 districts in Balochistan province.

District	Area (km²)	Population (1998)	Density (people/km²)
Awaran	29,510	118,173	4
Barkhan	3,514	103,545	29
Bolan	7,499	288,056	38
Chagai[3]	50,545	202,564	4
Dera Bugti	10,160	181,310	18
Gwadar	12,637	185,498	15
Jafarabad	2,445	432,817	177
Jhal Magsi	3,615	109,941	30
Kalat	6,622	237,834	36
Kech (Turbat)	22,539	413,204	18
Kharan	48,051	206,909	4
Khuzdar	35,380	417,466	12
Kohlu	7,610	99,846	13
Lasbela	15,153	312,695	21
Loralai	9,830	295,555	30
Mastung	5,896	179,784	30
Musakhel	5,728	134,056	23
Nasirabad	3,387	245,894	73
Nushki[3]
Panjgur	16,891	234,051	14
Pishin	7,819	367,183	47
Qilla Abdullah	3,293	370,269	112
Qilla Saifullah	6,831	193,553	28
Quetta	2,653	744,802	281
Sibi	7,796	180,398	23
Zhob	20,297	275,142	14
Ziarat	1,489	33,340	22
Balochistan Province	347,190	6,563,885	19

Districts of the North-West Frontier Province

There are 24 districts in the North-West Frontier Province.

District	Area (km²)	Population (1998)	Density (people/km²)
Abbottabad	1,967	880,666	448
Bannu	1,227	675,667	551
Batagram	1,301	307,278	236
Buner	1,865	506,048	271
Charsadda	996	1,022,364	1,026
Chitral	14,850	318,689	21
Dera Ismail Khan	7,326	852,995	116
Hangu	1,097	314,529	287
Haripur	1,725	692,228	401
Karak	3,372	430,796	128
Kohat	2,545	562,644	221
Kohistan	7,492	472,570	63
Lakki Marwat	3,164	490,025	155
Lower Dir	1,582	717,649	454
Malakand	952	452,291	475
Mansehra	4,579	1,152,839	252
Mardan	1,632	1,460,100	895
Nowshera	1,748	874,373	500
Peshawar	1,257	2,019,118	1,606
Shangla	1,586	434,563	274
Swabi	1,543	1,026,804	665
Swat	5,337	1,257,602	236
Tank	1,679	238,216	142
Upper Dir	3,699	575,858	156
North-West Frontier Province	74,521	17,735,912	238

Districts of Northern Areas

There are 6 districts in Northern Areas administrative region.

Agency	District	Area (km²)	Pop.(1998)	Headquarter
Baltistan	Ghanche	6,400	88,366	Khaplu
	Skardu	15,000	214,848	Skardu
Diamer	Astore	8,657	71,666	Gorikot
	Diamer	10,936	131,925	Chilas
Gilgit	Ghizer	9,635	120,218	Gahkuch
	Gilgit	26,300	243,324	Gilgit
Northern Areas	6 districts	69,971	970,347	-

Districts of Azad Kashmir

There are 8 districts in Azad Kashmir

Division	District	Area (km²)	Pop.(1998)	Headquarter
Mirpur	Bhimber	1,516	301,633	Bhimber
	Kotli	1,862	563,094	Kotli
	Mirpur	1,010	333,482	Mirpur
Muzaffarabad	Bagh	1,368	393,415	Bagh
	Muzaffarabad	2,496	638,973	Muzaffarabad
	Neelum[4] [5]	3,621	106,778	?
	Poonch	855	411,035	Rawalakot
	Sudhnati	569	334,091	Pallandari
Azad Kashmir	8 districts	13,297	2,972,501	Muzaffarabad

Districts of Punjab

There are 35 districts in Punjab province.

District	Area (km²)	Population (1998)	Density (people/km²)
Attock	6,857	1,274,935	186
Bahawalnagar	8,878	2,061,447	232
Bahawalpur	24,830	2,433,091	98
Bhakkar	8,153	1,051,456	129
Chakwal	6,524	1,083,725	166
Dera Ghazi Khan	11,922	1,643,118	138
Faisalabad	5,856	5,429,547	927
Gujranwala	3,622	3,400,940	939
Gujrat	3,192	2,048,008	642
Hafizabad	2,367	832,980	352
Jhang	8,809	2,834,545	322
Jhelum	3,587	936,957	261
Kasur	3,995	2,375,875	595
Khanewal	4,349	2,068,490	476
Khushab	6,511	905,711	139
Lahore	1,772	6,318,745	3,566
Layyah	6,291	1,120,951	178
Lodhran	2,778	1,171,800	422
Mandi Bahauddin	2,673	1,160,552	434
Mianwali	5,840	1,056,620	181
Multan	3,720	3,116,851	838
Muzaffargarh	8,249	2,635,903	320
Narowal	2,337	1,265,097	541
Nankana Sahib[2]
Okara	4,377	2,232,992	510
Pakpattan	2,724	1,286,680	472
Rahim Yar Khan	11,880	3,141,053	264
Rajanpur	12,319	1,103,618	90
Rawalpindi	5,286	3,363,911	636
Sahiwal	3,201	1,843,194	576
Sargodha	5,854	2,665,979	455
Sheikhupura	5,960	3,321,029	557
Sialkot	3,016	2,723,481	903
Toba Tek Singh	3,252	1,621,593	499
Vehari	4,364	2,090,416	479
Punjab Province	205,345	73,621,290	359

Districts of Sindh

There are 23 districts in Sindh province.

District Area (km²) Population (1998) Density (people/km²) Badin 6,726 1,136,044 169 Dadu 19,070 1,688,811 89 Ghotki 6,083 970,549 160 Hyderabad 5,519 2,891,488 524 Jacobabad 5,278 1,425,572 270 Jamshoro[1] Karachi 3,527 9,856,318 2,795 Kashmore[1] Khairpur 15,910 1,546,587 97 Larkana 7,423 1,927,066 260 Matiari Mirpurkhas 2,925 1,569,030 536 Naushahro Feroze 2,945 1,087,571 369 Nawabshah 4,502 1,071,533 238 Kamber and Shahdad Kot Sanghar 10,728 1,453,028 135 Shikarpur 2,512 880,438 350 Sukkur 5,165 908,373 176 Tando Allahyar Tando Muhammad Khan Tharparkar 19,638 914,291 47 Thatta 17,355 1,113,194 64 Umerkot[1] Sindh Province 135,306 30,439,893 225 source : http://en.wikipedia.org/wiki/Districts_of_Pakistan