Table Of ContentGeographical Models with Mathematica
Geographical Models
with Mathematica
André Dauphiné
First published 2017 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd
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Introduction
This book shows how to construct geographical knowledge by linking together
theories, models and techniques. This book is not a geography course, but rather a
course which provides a means to develop geographical knowledge. We will
endeavor to articulate a comprehensive geographical approach.
I.1. The scientific practice of the geographer
When spearheading all scientific activity, a hypothesis always arises.
The geographer never departs from this rule. Within experimental sciences, an
experiment is thus projected, but in geography, experimentation is hardly possible,
except in rare cases. Statistical processing or a simulation model substitutes
experimentation. Whatever the reasoning used, a proven assumption takes the form
of a scientific law. However, the majority of such laws stemming from the statistical
approach are only valid for the sphere, which is covered by the data processed.
Statistical laws are therefore of a local nature. On the other hand, laws deduced by a
simulation often tend to be of a more general nature.
A scientific theory is a body of laws, which are linked by internal principles. The
theories in question are therefore abstract constructs. However, through the
intermediary of its external principles, a theory orders deductive reasoning and
supplies an explanation of the world. As with other scientists, in order to develop a
theory, the geographer has three strategies available to him. He may borrow a theory
from a related discipline and transpose it into the geographical sphere. Thus,
É. Levasseur showed that the pull of urban areas is directly proportional to their
respective population totals. Levasseur resorted to gravitational theory in order to
explain a geographical phenomenon. There is also a second strategy, which involves
generalizing an existing theory. Very unassumingly, we have generalized the theory
of Norwegian perturbation by proposing a theory of fractal perturbation. Finally, it is
xii Geographical Models with Mathematica
possible to gather together dispersed laws to establish a single theoretical group. In
the sphere of physics, Maxwell worked out the electromagnetic theory by collating
the laws of Coulomb and Faraday, and Ampère’s law.
Within an implicit or explicit theoretical framework, the geographer constructs
models. These are supposed to represent either reality or a law or theory. Classical
geographers were, in fact, not proceeding any differently by drawing magnificent
block diagrams of the reliefs they were studying at the time. Modeling became the
principal activity of all scientists. However, these models were representations and
not at all reflective of reality. A model eliminates a number of very real
characteristics, which are considered, a priori, to be of no effect. Conversely, a
model sometimes creates new false characteristics. All geographers, consulting a
topographical map, may observe its contour lines. These contour lines are highly
visible on the map, but no geographer going down four levels has tripped on a
contour line.
To build a model, the geographer proceeds using a process of abstraction. This
abstraction may take different pathways. However, in science, mathematical
language is at the center of formalization. The refusal for measurement in social
sciences rests upon several misunderstandings, in particular that which assimilates
both quantitative and mathematical aspects. Certainly, not everything is quantifiable
in human sciences. However, on the other hand, everything is able to be
mathematically measurable. Indeed, it is necessary to avoid confusing quantification
and mathematical measuring. For more than 5,000 years, there has been qualitative
mathematics, in the form of geometry. Moreover, the qualitative approach to
differential equations is more efficient and useful. However, although the world is
entirely mathematically measurable, the construction of models with a “common”
language remains a form of abstraction, which is both necessary and useful. Darwin
never wrote a single equation when setting out the theory of evolution. However,
this formalization suffers from three flaws. First, researchers deprive themselves of
formal simple proof. Moreover, any theory, which is not formalized into
mathematical language, has no formal predictive value. Nobody is in a position to
predict the disappearance or appearance of a given species using Darwin’s theory.
Finally, despite its flexibility, literary language is linear, whether it involves words
or written texts. Moreover, nonlinear mechanisms with interactive cycles are the rule
for the majority of geographical phenomena.
I.2. The three forms of geography projects
No science is defined by its purposes. Cities are subject to investigation
by economists, sociologists, urban planners, ecologists and experts in many
other disciplines. A given science is defined by a particular project, that is to
Introduction xiii
say questions relating to given purposes, an urban area or a mountain. Moreover, the
geographer considers three groups of questions. The first project makes geography a
science of the relationships between human beings in society and the natural
environment. The image of the geographer, as an individual synthesizing natural
sciences and social sciences, has a long history. This history starts with the works of
George P. Marsh, and the writings of the first German geographers. Friedrich Ratzel,
for example, makes the distinction between peoples in their natural environment,
subject to natural conditions, and so-called “culture peoples” who are freed from
natural constraints. Ratzel’s form of geography, which was translated belatedly into
French, has certain similarities with human ecology. The school of thought in
classical French geography accepts this definition of the geographer.
Classical and contemporary geographers adhere to a second project; this being
understanding and explaining the spatial location of objects and places, for example,
the sinkholes in a karstic landscape, or coastal towns along a coastline. Generally, it
revolves around the distribution of individuals on the earth’s surface. The
geographer can thus test theories nearly always with a view to responding to the
questions Where and Why there? The geographer relies upon all forms and logic of
localization. Although the absolute location of phenomena, which was so significant
during the Renaissance era of major discoveries, is now resolved thanks to new
technologies, issues born out of the study of relative location are countless. It is a
question not only of explaining the location of given activities, such as industry or
tourism, but also of examining the interactions occurring within such activities. An
urban area or a region is a location-based clustering of agricultural, industrial or
service activities, which either exhibit pull or push factors. Moreover, physical and
biological constraints increase potential combinations. More recently, the terms of
relocation, resettlement and geolocation have been put forward within the context of
globalization and technological innovation, in order to respond to these questions.
Finally, contemporary geographers strive towards a third project; this is to
understand and explain structures and spatial dynamics. Etymologically, geography
is a narrative of the earth. Over time, it has become the science for the organization
of onshore areas. The geographer must identify the differences, disparities and the
various categories of physical and human discontinuities which separate predictable
temporal phases or spaces with a greater or lesser degree of homogeneity. The
geographer, therefore, analyzes morphologies. However, these disparities and
discontinuities are not static. The geographer then studies the dynamics of forms –
morphology. These morphogenic aspects characterize physical phenomena, for
example, the emergence of a mountain or the incision of river networks, as much as
human forms such as urban segregation or the densification of the Internet network.
xiv Geographical Models with Mathematica
I.3. Plan of the work
This work is made up of three parts. The first part approaches classical
geography and the science of relationships between man and his natural
environment. The second part deals with the scientific geography of locations.
Finally, the third part studies issues relating to spatial structures and their
morphology, and considers the territorial dynamics of given lands. Each part is
constructed following the same approach. In the first chapter, we present central
concepts and theories put forward by classical or contemporary geographers. We
then devote a number of chapters to modeling, taking account of the questions which
the geographer poses, and by distinguishing empirical models or more theoretical
simulation models. These elementary models are formalized using the Mathematica
language, which has recently been renamed Wolfram language in honor of the
individual who instigated it. Of educational value, these models may however be
used by students, teachers and researchers to process available data and respond to
the questions they ask. After giving brief conclusions, there is a question set out
which takes the form of a case study and a wider exercise.
I.4. How should this work be read?
The reader, discovering the Mathematica language, must first immerse
themselves in the Annex Section 1. This annex is a simple initiation into this
modeling language, but it is constructed by using a concrete example. The reader
will then engage in the part which best corresponds to his lines of questioning and to
his own concept of geography.
I.5. Appendix 1: a general modeling language Mathematica
To build models and analyze data, scientists use three general types of software:
R, MATLAB and Mathematica. Contrary to what its name leads you to believe,
Mathematica is a complete formalization language including graphs and
cartographic representations, and it has several advantages. First, it is a highly
comprehensive language. It has over 5,000 functions available, which allows an
approach to all modeling spheres that a geographer frequents. These include
statistics, probabilities, time series analysis and stochastic processes, macro-
simulation through ordinary or partial differential equations, micro-simulation
through cell automatons and multi-agent systems, graph theory, processing images,
and graphical and cartographic representations. The geographer may draw from an
endless stream of resources to construct his models. Moreover, various aids provide
virtually infinite wealth. Each function is set out with the help of a plethora of
examples. It is not unusual for the definition of a single function to stretch over
Introduction xv
several pages, with numerous case studies, which may be subject to a simple copy
and paste function. Moreover, Internet communities exist which allow us to ask
questions and to obtain very rapid responses. For more complicated problems, the
Wolfram Demonstrations Project provides genuine ready-to-use programs. For
these, it is often sufficient to introduce your own data to immediately obtain results.
Another source of wealth is outward orientation. Options within the Import
function[] ensure a direct link with numerous files. Moreover, Mathematica has its
own databases, of which some are of direct interest to the geographer (CityData,
CountryData, WeatherData and other databases). Indeed, the geographer may
import any piece of data (Annex 2). Moreover, this opening is not limited to data.
Mathematica may communicate with other programing tools, for example, R or
NetLogo.
For a brief initiation into Mathematica, let us start with a specific example, i.e. a
series of data, monthly temperatures, the population of European states or a series of
our own creation. We wish to calculate various parameters and produce graphic
representations. Having launched the program Mathematica, in the File menu
choose New, then Notebook. A blank page appears. In this notebook, we can enter
instructions and results are listed. To create a series of 20 datasets, we place the
cursor in the notebook, we click and we write:
data=RandomInteger[10,20]
Then, we tap Shift+Enter. By way of output, we obtain a list of 20 whole
numbers varying between 0 and 10. The reader may remark that the result follows
an “out” form. Every function or instruction starts with a capital letter, and to avoid
any error, functions written by the user should thus start with a lowercase letter.
Often, the function is described by two or three words, each of which corresponds to
a capital letter in terms of function. For example, ListLinePlot plots a graph in the
form of a continuous line. If a semicolon is placed at the end of the instruction, the
operation is completed but the result is not listed. It is practical to use when
processing large volumes of data or complex charts that the computer posts with a
time delay. All functions are followed by square brackets within which various data,
data to process, options and sometimes even other instructions are supplied. Within
the above example, the instruction RandomInteger has two options: the number 10,
which signifies that the random numbers may vary between 0 and 10, and the
number 20, which imposes a sequence of 20 numbers.
The list of numbers, as with any other list of words, maps or images, is in
brackets. The lists are an essential component of Mathematica. Numerous functions
xvi Geographical Models with Mathematica
allow us to construct and work upon these lists. For example, it is easy to remove the
first two pieces of data from a list using the instruction:
Drop[data,2]
However, many other instructions are currently used, in particular when
translating a table, which is a list of lists, into one single list. Many such instructions
are used to execute the converse; translation from a single list to a list of lists. To
partition 20 numbers into two lists of 10 numbers, we should write:
Partition[data,2]
By way of output, this gives a table of 10 lines and two columns. Conversely, to
move from this 10-line and 2-column to a single list, we write:
Flatten[tab],,,,
In addition to square brackets and braces, Mathematica uses parentheses to set
the order of calculation, and double square brackets. These double square brackets
serve to designate the position of a single element in a list. We are then able, for a
given series of data, to calculate the parameters of central tendency and dispersion as
we wish. For example, to obtain the average, all we have to do is write:
Mean[data]//N
However, it is possible to program the output for several results by using the
instruction Manipulate[], that is to say the following instruction line:
Manipulate[ moments[data], {moments, {Mean, Median, StandardDeviation, Skewness,
Kurtosis}}, SaveDefinitions -> True]
The moments function, applied to data, may correspond to the instruction Mean[]
or one of the other instructions that is included within the list. The result will be
listed in the form of a dynamic image. By clicking on the chosen instruction, we
obtain the result. We must be careful when a line includes several instructions, as
square brackets and braces come in pairs, to open and close an instruction or insert a
list.
Having calculated various parameters, we may illustrate the series by using a
graph. As data takes the form of a list, it is recommended to use the instruction
ListPlot, which includes a large number of options. The instruction below:
ListPlot[data, Filling -> Axis, FillingStyle -> {Red}]
Introduction xvii
allows us to plot a graph featuring the points, linking the abscissae by a line colored
red. However, there are many other options available. For all instructions, these
options are set out in the Help facilities. To obtain data relative to a given
instruction, the most rapid solution consists of typing out the instruction, selecting it
and then in the Help menu choosing Find Selected Function. Each Help screen takes
the same format; presentation of the function, its details and its Help screens,
examples, Scope, generalizations and extensions, the description of each option,
case studies, a number of properties and the list of similar functions. All of the
examples dealt with in these Help screens are directly usable through a simple
copy-paste function. Another instruction, Histogram[], represents the histogram of
data. Moreover, as with the previous instruction, this instruction offers numerous
options. We thus get the instruction:
Histogram[data, Automatic, “Probability”]
which gives the histogram of relative frequencies or probabilities in a series.
