Geographia Polonica (1973) vol. 25

Zbyszko Chojnicki, Adam Mickiewicz University, Poznan
Teresa Czyż [tczyz@amu.edu.pl], Institute of Socio-Economic Geography and Space Economy, Adam Mickiewicz University, Fredry 10, 61-701 Poznań, Poland

Abstract

It is the purpose of this paper to describe and illustrate a number of interpretivedevices which have recently emerged and which have not before beenutilized in the geographic literature. These devices are the computation of thecanonical factor structure matrix, the variances extracted from each measurementdomain by the canonical factors, the redundancy of the canonical factorsof one set given those of the other, and the canonical factor scores. The canonicalfactor structure matrix provides the correlations of the variables (or measurementdomain) with the canonical factors and takes the place of the raw canonicalvectors in which the variances are uncontrolled. The variances extractedfrom a measurement domain by a canonical factor may shrink to insignificanceif its canonical correlation with the corresponding factor for the othermeasurement domain is low. A better measure of the interrelationships betweenthe two measurement domains being analyzed is the redundancy measure,which is the product of the variance extracted and the variance shared foreach pair of canonical factors. This paper also introduces the notion of canonicalfactor scores which correspond to the scores computed in principal componentsanalysis and which provide a mapping of the observation units into thecanonical factor space. The computation of these indexes is described in the mathematicalsection which follows. Two research examples are then providedto illustrate the application and interpretation of the technique.

D. Michael Ray, State University of New York, Buffalo
Paul R. Lohnes, State University of New York, Buffalo

Abstract

In recent years the geographer has been exposed to a bewildering variety ofquantitative techniques the usefulness of many of which have yet to be demonstrated.Furthermore, discussions of the lesser known techniques tend tobe very technical or, alternatively, are limited in scope. In consequence therelative values of these procedures are difficult to assess by the majority ofgeographers and are quickly dismissed. One such technique is spectral analysiswhich is often wrongly classified as being too complicated or being applicableonly to periodic data sets of which there are few. This paper attempts toreview briefly at a relatively non technical level the scope of the technique,or better, group of techniques, which may be labelled "spectral" or "Fourier",and to describe in detail the simple though long calculations involved. Centralto these is the Fourier transformation which is nothing more than a particularform of curve fitting by least squares. At the outset it should be noted thatthese techniques usually apply to data which have equally spaced coordinatesin space and/or time. Other arrangements of data are possible but they willnot be included in the present discussion.

John N. Rayner, The Ohio State University, Columbus

Abstract

Several authors have recently suggested that urban growth can be representedas a wave-like diffusion process (Blumenfeld 1954, Boyce 1966, Morrill1968 and 1970, Korcelli 1969, 1970 and 1972). It may be assumed that this lineof research will also expand in the future. Its relation to other approaches, aswell as some insights it gives into the nature of the spread of urbanization, arediscussed elswhere (Korcelli 1972). This form of analysis, however, beside certainadvantages it offers, brings also some dangers, which can not be overemphasized.Two particular problems may be noted:

The improvement of the concepts has not been supported by an extensivebody of sound, empirical evidence. This, in part, is a consequence of scarcity ofadequate data, especially when large spatial and temporal series should be employed.If it persists, such a gap may eventually prevent further developmentof the theory.

The second problem relates to the methodology itself. While it is usuallytempting to classify a phenomenon under investigation as a part of a broadersystem, one may loose, by doing so, some of its rather essential properties. Thefollowing citation from Beckmann (1970, p. 116) well illustrates the point:"Although it is interesting that the same mathematical equation appears toapply to a particle, heat diffusion, and to human migration, this conclusionshould not be accepted uncritically. After all, we do not seek to reproduce thewell-known equations of mathematical physics but to develop models that bestreflect economic conditions".

The objections of this paper are, therefore, twofold. First, we attempt tofind some statistical evidence, however limited, for the aforementioned conceptsof urban growth. The method applied is believed to be consistent with the theorytested. Second, we want to trace, on the basis of the data employed, some of thespecific features of the urban growth process, as opposed to other spatial diffusionprocesses.

Piotr Korcelli [korcelli@twarda.pan.pl], Institute of Geography and Spatial Organization Polish Academy of Sciences, Krakowskie Przedmieście 30, 00-927 Warszawa, Poland
Beniamin Kostrubiec, Wrocław University

Waldo R. Tobler, University of Michigan, Ann Arbor

Nurudeen Alao, Department of Geography, University of Lagos, Lagos, Nigeria