On scale and size

I recently attended a workshop organised by the SPatial Analysis Research Center at Arizona State University which was focused on a discussion around a whole series of questions pertaining to scale but where the emphasis was largely on inductive models for extracting meaningful scale from areal data. The workshop threw up many questions involving what we mean by scale, the ambiguities posed by its definition, and how the objects and processes forming the phenomena in question were transformed with respect to scale when they changed in size. I will not repeat the way the workshop was organised – you can read about it on the relevant website https://sgsup.asu.edu/sparc/workshops – but suffice it to say that as soon as the workshop began, the range of contributions revealed (to me) a considerably wider portfolio of ideas, both methodological and substantive, with respect to the concepts of scale in question than anything that I had anticipated in reflecting on the meeting beforehand.

In fact, in the prologue to the meeting, the organisers made clear the breadth of possible issues saying that scale ‘… is one of a small number of quintessential geographic topics that defines geography as a discipline’. Although this focus began by restricting itself to two-dimensional space measured largely by area, it became apparent that as the meeting progressed, several participants blew open Pandora’s box with respect to many geographical objects, patterns and processes that could not be cast into the conventional logic of scale pertaining to area. Even points and lines in Euclidean space pose differences and difficulties in terms of measuring scale from those of areal definition and once the domain is extended to networks and processes such as diffusion which operate across space and time, the confusion in our notions about scale in spatial analysis becomes readily apparent. And this is not simply a matter of semantics.

To me, one of the major issues is grounding the concept of scale in related conceptions of space, geometry and particularly size. I tend to think of scale as pertaining to size, with size being a considerably more general concept than scale, with my own definition of ‘scale in essence being an approach to size’. When we look at the size of an object, we define a scale which then enables us to measure various properties of the object. In some senses, size which we usually characterise on a continuum from small to big or large, does not mean scale although as size changes, scale is defined as the level or levels at which the size of the object can be measured. The question of whether you can define a scale without a size in mind or even discuss size without mentioning scale is ill-posed. To an extent, size is not intrinsically anything that is related to space or time per se or for that matter to any particular set of characteristics pertaining to the object in question. But it becomes significant in terms of scale when we invoke its measurement, thus defining different scales which usually reflect an implicit if not explicit ordering across some continuum or range. In two-dimensional geographic space, scale which is defined pertaining to some size, implicitly defines an area or perimeter in some geometry; or if one is dealing with phenomena which has a temporal dimension as well, then scale defines the interval of time over which size is measured.

If one adopts the idea that scale must always be associated with size, then scale must change as size changes. In this sense, we can also speak of scale in terms of size but the semantics are tricky. Large scale can often be thought of as fine scale which implies we are dealing with small sizes. On the other hand, small scale and coarse scale usually means large sizes. So in terms of two-dimensional city geographies, for example, fine scale and large scale deal say with streets and buildings, whereas small scale and coarse scale deal with much more aggregate areas pertaining to how we represent land use, demographic and economic activities at metropolitan and regional levels. The term ‘level’ is also pertinent to this discussion and to some extent this is invoked as a proxy for changes in size at different scales. Moreover, this introduces the concept of hierarchy and pushes us hard towards the notion of what happens as size changes systematically. In fact, there is nothing intrinsic in scale change with respect to hierarchy but as soon as one begins to think of such transformations of size, then the notion of what different levels or scales mean becomes significant.

A rather different perspective of our concern for scale emerges when we move the lens to changes in size without worrying too much about scale or area. This is where our concern shifts not to extracting the influence of scale at different levels but to searching for invariance across scale. This is referred to in physics as ‘scaling’. The best example is through fractal geometry which articulates the system of interest as a set of scales over which some transformation takes place that is invariant to size but not to scale itself. The clearest signature of this is in the notion of self-similarity which within the development of fractals is usually reflected in some process of change that operates in a similar way at different levels of scale. In fact, very often the process kicks in at the lowest scale and as it continues to operate in building structure by growing any object, the same principle of similarity is reflected in the growth process at every subsequent stage as the object gets bigger. We can characterise this kind of transformation as a process which we can scale (up or down); that is, some function of the scale x we call F x which we scale by a growth rate λ . If the resultant size is simply a linear growth of the original function, then the object is said to scale, that is λ F x ∼ F λ x . Now the most familiar example of this is a power law. If F x ∼ x - α , then it is easy to show that F x ∼ λ - α x - α = ( λ x ) - α , and as you can see this has little to do with scale and everything to do with size.

The best example of this is a tree whose branches simply scale proportionately in some way as the branches composing the tree grow and get bigger and as new branches are continually generated at the finest scale. In fact in many systems, particularly those in the human domain, such strict self-similarity is rare. The similarity might be statistical, it might be affine but in certain cases, the idea that there may be many fractal processes operating on the same structure has been invoked. This is the idea of multifractals and the fact that this is usually associated with growth processes also brings in the temporal dimension which opens the debate to a whole new domain of spatiotemporal models (Murcio et al., 2015).

Much of the workshop was aimed at spatial models and analyses that revealed the meaning of scale, that is, the meaning of patterns and correlations that could be extracted using a variety of linear models such as geographically weighted regression and multi-level variants of the same class of models in traditional inductive fashion. In this sense, prior theories of the system of interest might be used to select the independent variables that drive such model explanations. There is however less significance given to such a prior set of plausible input variables that need to be included in any acceptable explanatory model than in the case of the contrasting deductive approach to exploiting the impact of scale where the initial explanatory variables are strong determinants of a basic theory of how the world is spatially structured. For example, central place theory, of which the hierarchy of neighbourhoods is a part, is predicated on the basis of strong interactional forces and economic logics that almost dictate what variables are required for good explanation. To an extent of course, there is always a to-ing and fro-ing between induction and deduction in spatial modelling but unless the latter more deductive approach is invoked, ideas about structure and hierarchy rarely appear in conventional spatial analysis. This is a limitation but at the same time, the explanatory power of such linear models is often much greater than those built deductively on a priori theory of how the world works. All too often the deductive models do not perform with the explanatory power that the more ad hoc models discussed in this workshop are able to realise.

My definition of scale as the way one looks at size needs further qualification for geographical systems that reflect area. The notion of area is highly volatile in that unless there is strict control over the area associated with the phenomenon, biases that are difficult to disentangle creep in and sometimes become serious to the point where any conclusions associated with correlations and relationships are flawed. The modifiable areal unit problem originally posed by Openshaw (1983) relates to this, while the ecological fallacy represents another spin on the same (Gehlke and Biehl, 1934). To control for area, strictly and possibly always, one should normalise and work with densities. In fact, it is remarkable that so little discussion of counts versus densities has occurred in spatial analysis, even in statistics generally (although uniform intervals are usually assumed in mainstream statistics). If areal counts are turned into densities, modifiable areal unit effects still occur but these are very much due to the nature of the data being an approximation to a true density and they reduce dramatically as the fineness of the spatial lattice or grid or set of polygons – whatever the areal unit is – get smaller. To an extent, this is a generic issue in all scaling and spatial analysis in that any variable which has an extensive base such as population should be normalised with respect to the unit of analysis; for example, variables such as income which should be normalised as income per head and so on.

My last concern with respect to scale and size was voiced at the meeting but as a point for further thought and reflection. Besides telling us how to think hard about the way we define space and scale and size which is a topic under continual discussion, the notion that these ideas might help us in thinking about the best scale and the most appropriate space and the way these might nest into each through ideas about hierarchy, were raised. Many years ago, in fact nearly 100 years ago, the geneticist, JBS Haldane (1926), wrote a wonderful essay entitled ‘On being the right size’. In it he showed that as plants and animals changed in size, their proportions did not stay the same but simply to function, they needed dramatic changes to their body shape, their morphology. In short, he showed that for any animal to function, major qualitative changes were required with respect to its dimensions when it changes in size. Scale, he argued, should thus reflect this and indeed it does in that if size dictates scale, then the morphology of the object determines the obvious and correct way to examine and measure it. ‘Haldane’s Principle’ as some have called it, thus shows that as objects scale, they change qualitatively and this is particularly true of cities. His essay of course represents the foundations of allometry which is the study of this kind of morphogenesis but it serves as a wakeup call to all of us using models which rely on basic ideas about size and scale (West, 2017). Nor is it confined to the physical and natural sciences. In a series of TV programmes for the Canadian Broadcasting Corporation, Jane Jacobs (1979) invoked Haldane’s principle in thinking about what the best size of a nation should be, arguing that there was no guarantee that big nations would produce bigger and better economic prosperity. Far from it. Her argument about bigness and smallness, about scale of course, is well worth reading.

The challenge amongst many that came from discussion in our workshop is this: how do we inject ideas about size into models that attempt to identify the most appropriate scales which enable them to capture the most significant spatial variations. Size is something that we have quite substantial control over in cities and regions especially where we can see the effects of different sizes quite clearly. Our argument is that we need to figure out the best sizes and by definition the most appropriate scales for selecting the best variables to drive our explanations; and these need to be related much more closely to what our theories are telling us about with respect to what we intuitively know about how the world works. And inevitably in Catch-22 like fashion, we need much better theories about how our world (of cities here) works to achieve better inductive explanations and predictions.