Location and Space-Economy (Cambridge, M assa­chusetts : The M .LT. Press, 1956). pp. 207-220; Bela Ba lassa, " An Empiri cal Demonstration of Classica l Comparative Cost theory," Review of Economics and Stat ist ics, Vo l. 45 (August, 1963), pp. 213-221.

(14) L. A. Brown, " Diffusion o f Innovation : A Macro­View," Economic Development and Cultural Change, Vol. 17 (1969), pp. 189-211 .

THE SEARCH FOR SPATIAL THEORIES: ON THE ROLE OF CONCEPTUAL AND COMPUTATIONAL MODELS IN GEOGRAPHIC RESEARCH

Daniel A. Griffith

Mr. Griffith is a doctoral candidate in Geography at the Universit y of Toronto, Canada. The paper was pre­sented initially at the annual meeting of the East Lakes Division, Association of American Geographers at the Universi ty of Pittsburgh, October 18-20, 1973. He was a member of Gamma Theta Upsilon at Indiana Univer­sity of Pennsylvania .

(15) See for example: Edward J. Taaffe, ed ., Geogra­phy, The Behaviora l and Socia l Sciences Survey, (Englewood Cliffs, N.J.: Prentice-Hall , Inc., 1970) pp. 132-135.

At present academic inquiry is guided by various research paradigms, the most popular ones being those affiliated with metaphysics, phenom­enology and positivism, while the literature to date lacks an explicit structural typology identifying and in­terrelating these formats . As a result, numerous peculiar classification schemes have been created . For ex­ample, Kerlinger indicates that re­esearch in general is comprised of three different modes, namely behav­ioral, historical and methodological~ Harvey, following ' Hempel, distin­guishes between deductive-nomologi­ca l and inductive-systematization, viewing the hypothetico-deductive system as a special case of this former type 2 Olsson talks about formal as opposed to substantive research~ And Relph , exemplifying a fourth possible outlook, posits the dichotomy of phe­nomenology versus positivism 4 Ob­viously these authors' views do render distinct, mutually exclusive group­ings. The ensuing discussion pertains to that body of literature associated with what Relph calls the positivistic approach to research.

Embedded in this positivistic atti­tude toward research is a procedure commonly known as the " scientific method ." Accordingly, the disciples of positivism are in constant search of universalities, whether absolute or probabilistic, such that a priori ex­pectations about the real world may

7

be established. Such foresights re­volve around the notions of theory and model. According to Hempel a theory is based on a system of empiri­cal uniformities exhibited by some phenomena, and seeks to afford a deeper and more accurate under­standing of these phenomena, ulti­mately construing them as manifesta­tions of certain underlying entities and processes.5 Meanwhile, Rudner describes a model as both a form of scientific shorthand and an heuristic device for gaining insights into the nature of phenomena.6 In simple terms, models may be thought of as the uniformity inputs to theory, merely describing a facet of reality whilst theory supplies a rational for this facet's emergency and existence. Intuitively speaking, then, it seems that a discipline's model-oriented re­search should be examined before its theory-oriented research can be ade­quatelyexplicated.

Consequently, this exposition is primarily concerned with models in­stead of theories. Again, a goodly number of classifications, stemming from independent attempts to sustain some degree of order among these items, have appeared in the literature. For instance, a model may be labeled static or dynamic, descriptive or pre­scriptive, deterministic or probabilis­tic , linear or curvilinear, conceptual or computational? The purpose of this paper is to illuminate the differ­ences between the components of this last categorization, especially in a geographic context. As such the idea is not a novel one, for many writers have mentioned the relationship be­tween these two model types . But, its explication appears to be a worth­while endeavor since the geographic literature reveals that many geograph­ers display an ignorance about the nature of the corresponding relation­ship.

8

Quantitative Geography: A Reconnaissance

During the 1950's geography under­went a radical transformation of both spirit and purpose, since termed the "quantitative revolution," which re­sulted in a mathematization of much of the discipline. This so-called revo­lution started suddenly, and was exerting Significant influences as early as 1954. It seemingly gained its mo­mentum from several sources, most notably the advent of modern high speed computers, the increased avail­ability of published numerical data, the popularization of the "new math" and statistical techniques, the pur­ported revival of a mathematical tra­dition, and particularly the belief that its accompanying methodology was capable of generating a more theoret­ical geography. Several authors have already traced the major temporal developments of this event, and thus a recapitulation of its history would be superfluous.~ Nevertheless, one outcome, namely a confusion of the roles of quantifier and statistical ana­lyst, merits closer examination here because of its relevance to the theme of this essay.

The search for generality in geogra­phy has frequentty resulted in a con­fusion of the roles of theory-seeking geographers and spatial statisticians.9

These theory-seeking academicians are preoccupied with the application of normal process of analytical rea­soning to geographical problems, investigating some predetermined set of relationships existing in a portion of reality.lO Meanwhile, the crux of the matter for the spatial statisticians is Berry's "geographic matrix," educ­ing a tradition of collecting data, ap­plying computational models, and achieving sophisticated statistical description of a portion of reality.l1 For their statistical results to be

meaningful, though, a working re­lationship is needed between these two practices, with theory as the leader.12 Unfortunately, this has not always been the case in geography. Many published articles have claimed theory where only statistical descrip­tion has been achieved. Rectifying this situation demands an under­standing of the differences between conceptual and computational spatial models.

The Role of Conceptual Models in Geographic Research

The theoretical framework is deemed an integral element of any positivistic research procedure even by such general directives ac Camp­bell's Form and Style in Thesis Writ­ing.13 However, the utility of this notion is predicated upon the existing body of theory articulated by a dis­cipline, since it refers to that set of relevant theories coupled with their interlinkages which both provides a structure suggesting how research for a given problem might best be pur­sued and logically relates the research at hand to subject matter of the dis­cipline in question . But, its utility in geography is almost nil, since the existing body of spatial theories is relatively meager. Examples of those in human geography include von ThUnen , Weber, Fetter and the Christaller-Losch central place theo­ries, whereas some for physical geog­raphy are the wave theory of cyclones and the hydrologic model.14

Although this field suffers from a paucity of spatial theories, it does possess a fruitful cluster of concepts generically exempl ified by the ideas of spatial distribution, spatial struc­ture, locational process, region, and so on. Furthermore, by transcending its formal scholastic boundaries, con­cepts developed by other academic communities have been added to this

assemblage. For instance, the gravity model of human interaction was bor­rowed from Newtonian physics. In light of this status, a number of authors have recently suggested that geography could improve its theoreti­cal basis by temporarily substituting a conceptual framework for the much desired theoretical one.15 Such a structure actually constitutes a con­ceptual model, providing economy of thought as well as insights into the re­search problem.I 6

In summary, it is not justifiable merely to restate accepted ideas in numerical form. Some model of men as organizers of space is necessary to guide one's work . For now, concep­tual models furnish geographic re­search with this very item. These interim frameworks enable germane spatial relationships and spatial con­structs to be identified, an economy of thought to be expressed, and an in­crease in the likelihood of geographic theory development. They also incor­porate analytical thinking into the re­search, yielding a higher degree of clarity and conviction for more rogor­ously obtained conclusions . In terms of, say, the hypothetico-<leductive system, they supply the research en­deavor with a set of universal postu­lates .

The Role of Computational Models in Geographic Research

Regardless of how elegant scholars conceive a given conceptual model to be, it embraces little substantive meaning unless it is amenable to em­pirical testing. In other words, although a conceptual model yields certain expectations about a given set of phenomena, it remains sterile with­out a method of comparing these re­sults and their real world counter­parts?7 The computational model furnishes a sufficient vehicle for ac­complishing this task?8 Moreover, it

9

may be thought of as a mathematical version of some conceptual model, its component variables being based on those constructs which are, at worst, acceptable surrogates for the con­cepts embedded in this latter model.

For the sake of comparability and the maintenance of a uniform degree of credence in outcomes all of science needs to adhere to a common group of computational models, which in fact it does. Of these, those commonly employed by geographers today are statistical in nature, usually the regression and/or factor analytic models.

Consider the application of the linear regression model in geographic research. According to the literature this model has often been used to predict the spatial distribution of some phenomenon based on various spatial associations, or to obtain a descriptor of the functional relation­ship between two or more spatial variates . A thorough discussion of its different geographic contexts has been provided by Griffith 's recent literature review, and anything more at this time would be redundant .19 In either of the aforementioned cases, this model produces an array of ex­pected values for which techniques have been devised to check how closely this array aligns with the origi­nal phenomenon's distribution . Con­sequently, the linear regression model seems to be somewhat of a prototype computational model , a contention to which its popularity is probably at­tributable.

To summarize then, a computation­al model is a mathematical emulator of reality whose fundamental ob­jectives include estimating parame­ters of some conceptual model, eluci­dating structures among the variables for this conceptual model, and man­ipulating numbers such that expected values are generated by, or derivable

10

from this conceptual model. In so doing it can be viewed as a piece of equipment required to provide those data necessary for evaluating how close some theoretical principle ap­proximates reality.

A Review of Selected Articles : Tom Swift Rides Again?20

One highly successful conceptuali­zation of a spatial phenomen is the gravity model of human interaction . In its simplest form this conceptual model posits that migration, for ex­ample, is some function of the population of origins and destina­tions, plus the distance separating these places. More succinctly,

where lij

Pi, Pj

the interaction be­tween area i and area j; the population of areas i and j, respectively; the distance sepa­rating area i from area j;

K a constant of pro­portionabi lity; and,

a, b, c = constants.

Garrison has shown how this model can be related to the real world via regression?l Consequently, the re­gression or computational model yields estimates of the constants K, a, b, and c. Unfortunately, too many subsequent studies have focused solely on calculating values of these parameters for different settings.

For instance, Black and Larson re­cently evaluated four different dis­tance functions in terms of twenty­four distinctive commodity flow groupings, failing to embed their undertaking within a conceptual framework that would sanction such extensions.22 Meanwhile Curry has earnestly pointed out that only when

tl n ti g

\

t

o +

intervening obstacles + + o

origin destination

FIGURS l:A diagram depicting the structure of Lee's conceptual model.

SOURCE: E. Lee, "A Theory of Higra tion," Demography, Vol. 3 (1966), pp. 47-57.

the unlikely event of zero autocor­relation occurs does such a calibra­tion of distance exponents for the gravity model become meaningful.23

Another feasible conceptual model which portrays the process of migra­tion has been devised by Lee, and de­picts this particular phenomenon as a response to contripetal , centrifugal, and neutral forces located at both source areas and destinations, the response being constrained by inter­vening obstacles .24 Meanwhile, Rid­dell has proposed that migration can be represented by a factor analytic regression model.25 By scrutinizing these two designs it appears as though Riddell has suggested a rea­sonable computational model to ac­commodate Lee's conceptual model. This last idea has been further ex­plicated elsewhere by this writer.26

As a final exemplar, consider Logan 's analysis of spatial economic development.27 Although the litera­ture review in his study serves to identify sixteen indicators of econom­ic development, Logan hypothesizes that those indices stemming from a principal components analysis of

these variables are functionally re­lated to various measures of distance

'and the occupational structure of the workforce . However, these last two notions, i .e. the influence of distances and an occupational structure, are not founded upon a sound concep­tual framework . Consequently, one should seriously wonder why Logan expects such relationships to prevail.

Concluding Comments In conclusion, two of the kinds of

models harbored by positivistic re­search have been the topic of this paper The theoretical or conceptual model, depending upon a discipline's developmental stage, should act as a guide to research , while the computa­tional model should function as a means of assessing the adequacy of this former model. Such a perspective is not only consistent with, but also isomorphic to Torgerson's ideas con­cerning the structure of a science.28

Furthermore, the computational model is some mathematical formula, whereas the conceptual model may be expressed either as a mathematical equation or a structural schema, a

11

point explicitly illustrated in the pre­ceding section. This feature alludes to the fact that a computational model is mostly used for prediction. On the other hand, a conceptual model is usually used for rationalizing the existence of some phenomena, es­pecially as the output from specific relationships .

Finally, many geographers need to heed these differences, along with the many more that are not mentioned here, for the literature to date a­bounds with computational models mistaken to be conceptual models .

FOOTNOTES

(1) F. Kerlinger, Foundations of Behavioral Research (New York: Holt, Rinehart and Winston, Inc ., 1964), especial ly pp. 690-702.

(2) D. Harvey, Expla nation in Geograph y (New York : St. Martin 's Press, 1969) , p. 36; and C. Hempel, Philosophy of Natural Science (Englewood Cli ffs, N.J.: Prentice-Hall , Inc., 1966), pp. 48-49.

(3) G. Olsson, "Trends in Spatia l Model Building," Geographical Analysis, Vol. 1 (1969), pp. 219-224.

(4) E. Relph, "An Inqui ry in to the Relations Between Phenomenology and Geography," The Canadian Geographer. Vol. 14 (1970), pp. 193-201.

(5) Hempel , op. cit., footnote 2, p. 70.

(6) R. Rudner, Philosophy of Social Science (Engle­wood Cliffs, N .J.: Prentice-Hall , Inc., 1966), pp. 25-26.

(7) L. Lowenstein , "On the Natu re of Analytical Models," Urban Studies, Vol. 3 (1966), pp. 112-119.

(8) See inter alios B. Berry, "The Quantitative Bogey­man," Economic Geography, Vol. 36 (1960), fa cing p. 283; O . Spate, "Quanti ty and Quali ty in Geography," Annals , Assoc iation of Ameri can Geographers, Vol. 50 (1960) , pp. 377-394; I. Bu rton, "The Quantitative Revo lu t ion and Theo­retical Geography," The Canadian Geographer, Vol. 7 (1963) , pp. 151-162; L. Curry , "Quantitative Geography, 1967," The Canadian Geographer, Vol. 11 (1967), pp. 265-279; P. LaValle, H. McCon­nell , and R. Brown , "Certain Aspects of the Expans ion of Quantitative Methodology in Ameri ­can Geography," Anna ls, Association o f Ameri can Geographers, Vol. 57 (1967), pp. 432-436; A . Wilson , "The Use of Analog ies in Geography," Geographical Ana lysis , Vol. 1 (1969), pp. 225-233; and S. Gregory, "The Quantitat ive Approach in Geography," Quantitative and Qualitative Geog­raphy : La Necess ite d 'un Dialogue, compiled and edited by H. French and J. Rac ine (Ottawa : Uni­versity of Ottawa Press, 1971), pp. 25-33.

(9) Curry, op. cit ., footnote 8, p. 265; A. Pred, Behav­ior and Location: Foundations for a Geographic

12

and Dynamic Locat ion Theory-I (Lund, Sweden : The Roya l University of Lund, 1967), p. 17; and L. Guelke , " Problems of Scientif ic Explanation in Geography," The Canadian Geographer, Vol. 15 (1971), p. 40.

(10) Curry, op. cit ., footnote 8, p. 265; and Gregory, op. cit., footnote 8, p. 27.

(11 ) Cu rry , op. cit. , footnote 8, p. 265; and Guelke, op. cit. , footnote 9, p. 40.

(12) L. Curry , " A Note on Spat ial Associat ion ," The Profess ional Geographer , Vol. 18 (1966), pp. 97-99; J. Hudson , " A Location Theory for Rural Settlement," Annals, Association of American Geographers, Vo l. 59 (1969), p. 366; and L. King, "The Ana lysis of Spat ial Form and its Re lation to Geographic Theory," Anna ls, Association of Ameri can Geographers, Vol. 59 (1969) , pp. 594-595 .

(13) W . Campbell , Form and Style in Thesis Writing (Boston : Houghton M ifflin Company, 1969).

(14) A . Strahler, Physica l Geography (New York : John Wiley and Sins, Inc. , 1969); and P. Lloyd and P. Dicken, Location in Space: A Theoretical Ap­proach to Economic Geography (New York : Harper & Row, Publishers, 1972) .

(15) See for example D . Janelle, "Spatia l Reorganiza­t ion : A Model and Concept," Annals , Associati on of Ameri can Geographers, Vo l. 59 (1969), pp. 348-349; W ilson, op. c it. , footnote 8, pp. 230-231; L. Wo lf , "The Metropoli tan Tida l Wave in O hio," Economic Geography, Vol. 45 (1969), pp. 133-136; L. Brown and D. Longbrake, "Migration Flows in In t ra-Urban Space : Place Utility Considerations, " Annals , Associat ion of Ameri can Geographers, Vol. 60 (1970), p. 369; S. Cohen and L. Rosenthal, "A Geographica l Model for Politi ca l Systems Analysis," Geographical Review, Vol. 61 (1971), pp. 5-6; T. Harts horne, " Inner City Residential Structure and Decline," Annals , Assoc iation of American Geographers, Vol. 61 (1971), p. 72; and G. Steed, " Plant Adaptation, Firm Env ironments and Location Analysis ," The Professional Geogra­pher , Vol. 23 (1971), pp. 324-326.

(16) C. Alexander, No tes on the Synthesis of Form (Cambridge, M ass. : Harvard University Press, 1970), p. 62; and M . Marx, "Theory Construction and Eva luation," Learning: Theories, compiled and edited by M . Marx (New York : The MacMillan Company, 1970), pp. 3-45 .

(17) J. Kuhn , The Structure of Scientific Revolutions (Chicago : University of Chicago Press, 1964), pp.31-32.

(18) Rudner, op. cit. , footnote 6, pp. 23-28; and Curry, op. cit. , footnote 8, p. 265.

(19) D. Griffith , "The Use of Regression Models in Geographic Research : A Classroom Method­o logy," The Pennsylvania Geographer, Vol. 10, No. 2 (ju ly, 1972), pp. 7-14.

(20) Tom Swift is a mythi cal character whose intellec­tua l venture was fabricated by Professor Arm­strong to illustrate that factor analys is, by itself, may be misleadi ng as far as the development of theory is concerned .

(21) W . Garrison, "Estimates of the Parameters of Spatial In teract ion," Papers and Proceedings, Regional Science Association, Vol. 2 (1956), pp. 280-288.

(2:

(2~

T L

L

[

t

n: L. in 15

y ,

e,

e ) .

11 n

a ,f

g

n

1-

(22) W. Black and R. Larson , " A Comparative Evalua­tion of Altern ative Friction Factors in the Gravity Model," The Pro fessional Geographer, Vol. 24 (1972), pp. 335-337.

(23) l. Curry, " A Spatial Analysis of Gravity Flows," Regional Studies , Vol. 6 (1972), p. 132.

(24) E. Lee, " A Theory of Migration ," Demography, Vol. 3 (1966), pp. 47-57.

(25) J. Riddell , "On Structuring a Migration Model, " Geographica l Analysis , Vol. 2 (1970), pp. 403-409.

THE CITY AT THE BOTTOM OF LAKE MICHIGAN

Lawrence C. Wolf

Dr. Wolf is Associate Professor of Geography at the University of Cincinnati.

(26) D. Griffith , " Student Enrollment at the University of Toronto, 1971," unpublished paper, Depart­ment of Geography, University of Toronto, 1973.

(27) M . Logan , "The Spatial Dimensions of Economic Development : The Case of the Upper Midwest," Regional Studies , Vol. 4 (1970), pp. 117-125 .

(27) W . Torgerson , Theory and Methods of Scaling (New York : John Wiley and Sons, Inc., 1958), pp . 2-8 .

The New York Times, Sunday, Feb­ruary 17, 1974, reported a subter­ranean heat source extending from Mexico to above the Salton Sea. If subterranean geothermal sources can be gotten out of the air over the Salton Sea, what next? A week later a radio announcer was heard saying a balloonist was lost over the ocean below Morocco. If the ocean is now beneath Morocco, no wonder he was lost!

Journalists are not the only ones living in an unreal world . The United Nations and the Population Reference Bureau end the North American con­tinent at the Rio Grande! The conti­nents of North and South America are joined by the isthmus of Panama, are they not? Although geologists tell us the continents are drifting, to my knowledge Panama has not slid be­tween Mexico and our Southwest. The two cultural regions of the Americas, Anglo-Merica and Latin America, are hot coterminous with the two Ameri­ca continents . Language is so impre­cisely used of late, that such con­fusions are becoming the norm .

Even geographers are at home in this fictitious and confused world . Many of the textbooks and scholarly journals refer to Africa south of the Sahara as Sub-Saharan Africa. Now "sub" in my dictionary (Random House, 1966) means " under, below, beneath ." There's nothing under, be­low, or beneath the Sahara but rock

13