The correlation coefficient is high (Adj.85). This method also provides the m parameter, that.829 (p.10-4) and can be interpreted as the average fractal dimension of the road networks for the 43 cities analyzed:.171. Robustness analysis It is crucial though to test the robustness of such a model. This will be done by carrying out a multivariate analysis integrating econometric parameters. This concern was at the center of the criticisms that have emerged after Newman and Kenworthys paper. This is why it is crucial to test if there is any omission bias due to any hidden econometric explanatory variable. The multivariate analysis will aim at regressing the average energy consumption for private transport versus a set of explanatory variables. These explanatory variables have to be chosen carefully to cover the spectrum of parameters that could explain the consumption associated with private transport.

## Modifiable areal unit problem - wikipedia

Fractal dimension of sad the road network is a good proxy for accessibility, in analogy with natural systems such as lungs. In the case of road networks, the closer the fractal dimension to two, the more accessible the city. That is why it is the difference 2-D that matters. The term 2-D has a significant impact on urban transport phenomena. Baba (2000) corroborates this hypothesis by proving a significant correlation (R0.658) between 2-d, density and average journey length for uk cities, using an inverse power law model. The constant average travel time budget hypothesis allows to extend this result to the average speed. This reasoning leads to the following power law, linking average speed, population density and fractal dimension of the network, with a a constant parameter: (7 it is crucial though to note that diminishing the fractal dimension of the network would decrease the accessibility, and. Finally, the average private transport energy consumption can be expressed as a function of urban density and fractal dimension of the network: (8) is the average private transport energy consumption d is the population density d is the fractal dimension of the road network. Before using an ols method, it is indeed necessary to transform this relationship into a linear one, by applying the natural logarithm on both sides of the equation: (9) Concerning the existence of a spurious correlation (Brindle, 1994) due to the hidden per capita term. This transformation insures that there is no spurious correlation (Naess, 1993). The analysis is based on 1990 data from a sample of 43 cities, gathered and updated by barter (1999).

Intuitively speaking, the denser the city, the more contracted the road network. A high population density means a high building density, letting fewer space for the road network. The correlation between average speed and density is thus expected to be an inverse one, looking for instance like: (5 but it is possible to refine this equation using fractal theory, reviews which allows describing objects distributions with an invariance of scale. As explained here above, trees for instance are fractals: the size r of the branches is linked to the number n(r) of branches of size r by the following power law, m being the fractal dimension of the structure, a a constant: (6 the fractal. Density contracts the network, letting less space to roads and paths within the city. But for a given density, and a given area dedicated to the road network, fractal networks are the most efficient ones. It is the result of a constrain optimization.

To the first order, average energy consumption for private transport is directly related to the average distance traveled per capita per year. These two parameters are linked by the average efficiency parameter of the car fleet in J/km: (2 average distance traveled per year per capita can be split into a product of two terms: the average speed and the average time spent traveling per year: (3. The average annual time spent traveling can thus be considered as being 365 hours p). This leads to the following equation, where the average annual consumption for private transport per capita is directly proportional to the average speed of private transports in the city. (4 this reasoning leads to an outstanding result: urban consumption for transport is proportional to the average speed of private transport. This is not without listing going against certain common beliefs, defending that faster and more fluid networks have to be fostered for a better efficiency. Based on the 1990 data gathered by barter (1999) for 43 cities, the correlation between average private transport energy consumption (including both gasoline and diesel) and average traffic speed is high (Adj.73 which provides a sound argument for this assumption. The key point is then to link the average speed of private transports in cities with urban density.

Table 1 displays values of road networks fractal dimensions for uk cities, ranging from 1. Table 1: Fractal dimension of road networks in 10 uk cities. Adapted from Baba (2000). City Fractal dimension of the road network. A power law model for private transport energy consumption. The main issue of this paper is to propose a simple model to link private transport energy consumption and urban density. Authors present hereunder a naïve and simple model that could explain the emergence of such a power law linking private transport consumption per capita and urban density. The following reasoning is based on average values.

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According to heitor-reiss (2006) and Salat bourdic (2011b based on thermodynamics and energy arguments, this type of structure is the most efficient one to distribute a flow (the sap in a tree for instance) into a 2 or 3 dimension space. A two dimension tree-like structure is much more efficient than a grid structure, where all the veins are the same size. This is the intrinsic reason why so many network based structures are fractals: lung, river basins, road networks, internet, electricity networks, etc. The fractal dimension of such a structure is a fundamental parameter, that is called non-euclidian. In Euclidian theory (the classic one a point has a topological dimension of zero, a line 1, a plane 2 and a volume. Fractal theory allows this value not to be an integer: fractal structures have a non integer topologic dimension. A simple square grid made of lines has a fractal dimension that can be considered as being one.

A two dimension fractal tree on the contrary is somewhere between a line and a plane and its fractal dimension is between one and two. The better efficiency of such networks is understandable: most of the flow transits via the big veins, where friction is lower, making the energy losses smaller. As one goes along the network, there is fewer flow to be distributed, and the diameter decreases, in order to feed every little part of the surface. This will be referred to as accessibility further in this paper. The analogy with urban networks is outstanding: wide highways and boulevards to roughly and quickly irrigate the city, and narrower and narrower streets essay to irrigate all the places within the city: fractal means accessible.

In particular, nobody seems to have realized that the hunt for, and the interpretation of, invariants of this type might lay the foundations for an entirely novel type of theory. Schumpeter (1949 on the subject of power laws and Pareto distributions. Power laws have a tremendous importance in many natural and man-made phenomena. It is a relation of the type: (1 with y and x are variables, α a constant exponent and k a constant. This type of law is also known as a scaling law, as it structures the relationship between size (or scale) of elements in a set and their multiplicity. This type of regularity is omnipresent in both: natural phenomena: river networks (Maaritan., 1996 biological networks and size of animals (Brown., 2000; West, 1999 trees, lungs, coast lines (Mandelbrot, 1982 etc.

socio economic phenomena: city size (Zipf, 1949 income distribution and wealth, quantitative linguistics (Montemurro, 2001 website occurrence in google (Adamic., 2000 citations in scientific literature (Bradford, 1985 etc. Power laws present a remarkable number of regularities, and allow to describe a wide range of distributions with analogue properties: many small objects and few large objects, many small events, and few large events. The omnipresence of this type of heavy tail distribution has a lot to do with the presence of flows and networks in the background of all these phenomena: blood system, river networks, human networks, financial flows, etc. Salat and bourdic (2011a) have shown that this type of distribution within networks can be explained on the basis of thermodynamics arguments: for certain types of flows, this type of structure is the most efficient one. Systems and networks thus naturally evolve towards this type of distribution. The omnipresence of this type of structure is directly related to networks efficiency.

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Ewing and Cervero (2001) eventually explored the impact of the built environment on fuller transportation variables and proved that high urban density reduces the average journey distance and thus encourages soft and public transports. Few if any literature on the topic though proposes a model that could physically explain the link between urban density and private transport energy consumption. Density for sure recovers a wide range of parameters, and the results obtained by newman and Kenworthy are a radical shortcut, as based on aggregated variables. However, despite the lack of sharpness of this analysis, this relationship is worth being investigated, as it has been fueling the debate around urban form and energy consumption for the last 20 years. Using power laws and fractal theory, authors aim at providing new insights, by proposing a physical interpretation of this curve. Power book laws and transport efficiency. Few if any economists seem to have realized the possibilities that such invariants hold for the future of our science.

He argued that plotting two per capita parameters was a guarantee to find a hyperbolic relationship between the two parameters. The log-log multivariate analysis that will be carried out in this paper will avoid the trap of this spurious correlation. On the statistical standpoint however, evill (1995) discredited Brindles arguments, by showing that the correlation found by newman and Kenworthy had nothing to do admission with spurious per capita correlations. According to rickwood (2009) yet, despite all the criticisms, there is too little evidence to refute newman and Kenworthys thesis, as it remains one of the most comprehensive international study analyzing the effect of urban form on energy consumption figures. Numerous authors since have confirmed these results. Naess (1993) confirmed them analyzing energy consumption figures for 22 Scandinavian cities, avoiding at the same time the cultural bias Newman and Kenworthy could be criticized for. Ecotec report (1993) also got to the same result, showing an inverse relationship between modal transfers and density for uk cities.

increasing car tenure and transport expenses, they are supposed to be responsible for the increase in the average distance traveled, and. These authors cast doubt on the causal link established by newman and Kenworthy between density and gasoline consumption, the better living standards causing both the drop in urban density and the rise in automobile mobility (Allaire, 2007). This concern about causality though boils down to the chicken-or-egg dilemma. The improvement of living standards is certainly partly responsible for the increased use of cars and the associated energy consumption, fostering at the same time urban sprawl and low density urban tissues. But turning the question round in the way illich (1973) did reverses this causal link. In the current state, there is no choice for people not to use their car if they live in a non dense city. Lengthening urban distances, the car has excluded all other transportation means, literally following the definition of a radical monopoly. Another controversy was raised by Brindle (1994 who criticized the statistical theory behind Newman and Kenworthys results.

A robustness analysis is then carried out to discuss the validity of such a fractal model. Keywords: Transport, Energy, fractals, Urban Efficiency, power Laws. Introduction, the debate around the link between private transport energy consumption (for gasoline first, and then as an aggregated diesel-gasoline figure) and urban density has emerged after Newman and Kenworthys seminal work (1989). It has raised since a huge amount of criticisms. Gordon and Richardson (1989) wrote the first violent reply to newman and Kenworthys paper. Based on the. Experience, they defended the liberalization of land market, as opposed to any kind of planning based on arguments such as density. The main criticism raised by gordon and Richardson emerged from the lack of complete multivariate analysis, criticism which has been further developed by gomez-ibanez (1991). Further criticisms stressed the fact that socio-economic factors should be the main drivers for write mobility behaviors.

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Salat 16th International Conference of dates Hong Kong Society for Transportation Studies. 17-20 December 2011, hong Kong. Abstract, cities are dense and complex structures. Fractal theory provides a theoretical framework to analyze complex systems that is worth being used for urban systems analysis. Fractal dimension is one of the useful tools to quantify urban complexity. This paper investigates through the prism of fractal theory the relationships between urban density and private transport energy consumption. A simple model based on a power law function is proposed to provide a physical interpretation to the relationship first coined by newman and Kenworthy. It notably interprets the exponent of the curve as the fractal dimension of the road network.

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