Table Of ContentNew Methods of
Geostatistical Analysis and
Graphical Presentation
Distributions of Populations
over Territories
N e w Methods of
Geostatistical Analysis and
Graphical Presentation
Distributions of Populations
over Territories
Roberto Bachi
Late of The Hebrew University of Jerusalem
Jerusalem, Israel
Published in cooperation with
The Israel Academy of Sciences and Humanities
Kluwer Academic / Plenum Publishers
New York, Boston, Dordrecht, London, Moscow
Library of Congress Cataloging in Publication Data
Bachi, Roberto, 1909-1995.
New methods of geostatistical analysis and graphical presentation: distributions of popula-
tions over territories / Roberto Bachi.
p. cm.
"Published in cooperation with The Israel Academy of Sciences and Humanities."
Includes bibliographical references and index.
ISBN 0-306-45544-7
.1 Spatial analysis (Statistics) 2. Population geography--Statistical methods. 3. Geology--
Statistical methods. 4. Statistics----Graphic methods. I. Akademyah hale 'umit ha-Yisre'elit
ie-mada ira. II. Title.
HA30.6.B33 1999 99-17060
304.6'01'5192---dc21 CIP
ISBN 0-306-45544-7
© 1999 Kluwer Academic / Plenum Publishers
233 Spring Street, New York, N.Y. 10013
10987654321
A C.I.P. record for this book is available from the Library of Congress.
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form
or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise,
without written permission from the Publisher
Printed in the United States of America
Foreword
The chief aim of this book is to present the reader with an integrated system of
methods for dealing with geographical statistic(sg eostatistics) and their applications.
It sums up the developments based on the vast experience accumulated by Professor
Roberto Bachi over several decades of research.
Interest in the quantitative Iocational aspects of geography and the common
ground of geography and statistics has grown rapidly, involving an ever-increasing
spectrum of scientific disciplines. This expansion has gained momentum particularly
since the advent of user-friendly statistical software facilitating research in these
areas. The geostatistical methods described in this book make it possible to
overcome many of the difficulties encountered in graphically presenting, processing,
analyzing, and comparing spatial data over different territorial units. It is our hope
that the present volume will fill a genuine need- as a textbook, as a reference work,
and as a tool for geographers, applied statisticians, demographers, ecologists,
regional planners, economists, professional staff of statistical agencies, and others.
The book's central theme si the presentation, treatment, and analysis of large
collections of data subdivided into territorial units of irregular shape. Specifically,
the methods presented enable one to: (a) graphically display the information
supplied by the raw data in a way that permits a quick grasp of the geographical
characteristics of the territorial distributions being studied; (b) produce accurate
computerized maps comparing their features; )c( present geostatistical parameters
summarizing the main characteristics of position, dispersion, and shape of a
distribution under study and compare them even for different territories and at
different points in time; (d) measure the geographical discrepancy between any two
distributions.
The title of the book relates to distribution of populations over territories.
However, the term population should be understood in its widest possible sense, to
include events such as births, deaths, and accidents; establishments and institutions
of different kinds; and various services and economic activities--in fact, any kind of
phenomenon that can be described in spatial-geographical terms. More formally,
the units under investigation here are geographical population sets or geosets, i.e.,
sets of triplets. Each triplet consists of the weight (or frequency) of a particular item
of population in a territorial entity and the Cartesian plane coordinates of its
location. These, then, are the basic building blocks and raw material with which this
book deals, comparing them with each other and with the geographical features of
the territory itself.
vii
viii Foreword
The impact of the methods described herein on applied geographical statistics
constitutes a milestone in geostatistics. Professor Bachi was able to try out these
methods in a series of extensive applications. Foremost among these were the studies
carried out at the U.S. Bureau of the Census during the late 1970s, and the creation
of the Statistical Atlas of Italy, a two-volume work of vast proportions produced by
a team of specialists from the Italian National Institute of Statistics and coordinated
by Professor Bachi himself during his frequent visits to Italy. More recently, he and
his colleagues, supported by a grant from the US-Israel National Science Founda-
tion applied the geostatistical methods described here to a study of the distribution
of Jews in many cities of the Diaspora all over the world. One can therefore affirm
with confidence that the methods outlined in the present volume have undergone
extensive and thorough testing under a wide range of circumstances and conditions.
A few words should also be said about the beginning of Roberto Bachi's interest
in, and involvement with, geostatistical research. His earliest papers in geostatistics
were published over 60 years ago. The first, on a measure of distance between
geographical distributions, was published in Italy in 1934. On the basis of this
preliminary step he developed, chiefly during the 1950s and 1960s, a complete
methodology for quantitatively analyzing and comparing geostatistical distributions
by measuring the spatial position of the points constituting them, together with their
thematic content, and then computing the shape, extension, and direction of the
entire distribution, expressing them with the aid of a small number of both
dimensioned and dimensionless parameters. These make it possible to draw quanti-
tative as well as qualitative comparisons between different populations. Somewhat
later, and then in parallel with the former, he devised a technique to facilitate the
simple graphical presentation of geostatistical data, the graphical rational patterns
(GRP) method, which allows geographical data over large territorial units to be
displayed and grasped in a less area-generalized manner than they are in, for
example, choropleth maps. The integration of these two fields of interest-- the GRP
and the earlier geostatistical method--led to the series of investigations that
eventually resulted in the present volume.
As his granddaughter Clila notes in the Preface, Professor Bachi's book was
almost finished at the time of his death in November A 1995. team of his colleagues
undertook to assist her in completing the editing of the text. We all considered it an
honor and a privilege to do this, feeling that we were fulfilling a debt of gratitude to
the author, whose intellectual integrity and stature, unfailing commitment to this
profession, respect for his colleagues, kindness to his students, devotion to his family,
and gentleness toward all we learned to admire and esteem.
E. Peritz
G. Nathan
Jerusalem N. Kadmon
Preface
The death of Professor Roberto Bachi on November 26, 1995, brought to a close an
unusually prolific career in the fields of demography and statistics. It also cut short
his work on the present volume, the culmination of his research in geostatistics, to
which he devoted his postretirement years. As his granddaughter, I had the pleasure
of assisting him in his work on this book during the last two years of his life. This
unique opportunity was a special privilege because he was truly a remarkable man,
with outstanding scientific achievements to his credit and an innate graciousness and
nobility.
It was my grandfather's dearest wish to see this book published. On his
deathbed he told the family that it was almost finished and asked me to carry out
the task of seeing it through to completion. Accordingly, I assumed responsibility
for its publication, guided by my understanding of his intentions. For the most part,
the text as he wrote it was left unchanged. In some cases, such as the last two sections
of Chapter ,01 the text was reconstructed from a draft; while in other instances,
incomplete material had to be omitted. The maps and illustrations comprised the
most problematic area by far: at the time of his death, many remained undone, and
I had to rely on the figures as they were referred to in the text as my guide in their
preparation.
I hope and trust that in fulfilling my grandfather's wish, I have made an
important scientific document available, which will also constitute a step toward the
adoption of his methods for theoretical and field studies.
This volume could not have been completed without the assistance of several
people. I would like to thank Dr. Shai Ever-Hadani for his invaluable help in the
preparation of the illustrations. Dr. Ever-Hadani' drew upon his long years of
collaboration with my grandfather. He dedicated both time and effort unstintingly
to solving the many problems with which I was confronted.
I am also greatly indebted to three of my grandfather's colleagues for their
assistance in preparing the text for publication. I am especially grateful to Prof. Gad
Nathan for shouldering the largest part of the burden. He reviewed Chapters 8
through 61 and was particularly helpful in the reconstruction of Chapter 10 and
Appendixes E and H. I would also like to express my appreciation to the late Prof.
Eric Peritz for his work on Chapters 1 through 4 and for his pertinent comments.
Special thanks must also be extended to Prof. Naftali Kadmon for his profound and
insightful observations on Chapters 5 through .7
xi
x Preface
I would like to thank Evelyn Grossberg for her excellent and very professional
editorial work on the text as well as for her valuable comments and suggestions.
Finally, I wish to express my deep gratitude to my grandmother Vera, who, after
standing by my grandfather's side for over 60 years as an unfailing source of
encouragement in his academic pursuits, did her utmost to see this work published.
Words cannot express my deeply felt appreciation for her assistance and support.
Clila Gerassi Tishby
Jerusalem
Contents
List of Plates . . . . . . . . . . . . . . . . . . . . . . . . . xxv
Part I: Basic Concepts
Chapter 1
Introducing Geostatistical Methods
1.1. Primary Aims of this Volume . . . . . . . . . . . . . . . . . . 3
1.2. Introducing Rational Maps and Comparisons among Them ..... 3
1.2.1. The Distribution of Population over the Territory of Italy . . . 3
1.2.2. Comparing Distributions of Populations to a Uniform
Distribution . . . . . . . . . . . . . . . . . . . . . . . 5
1.3. Formalizing Population Distribution as Geosets . . . . . . . . . . 10
1.3.1. Primary Geosets (gP, g U) and Geosets Derived from Comparing
Them . . . . . . . . . . . . . . . . . . . . . . . . . . I0
1.3.2. An Interpretation of the Methods above for Comparing
gP and gU . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4. Two or More Geosets over the Same Territory . . . . . . . . . . . 12
1.4.1. Further Examples of Geosets for Italy . . . . . . . . . . . . 12
1.4.2. Geosets Derived from the Comparison of Two Primary
Population Geosets . . . . . . . . . . . . . . . . . . . . 17
1.5. Examples of Geostatistical Parameters . . . . . . . . . . . . . . 22
1.5.1. Order of Presentation of the Parameters . . . . . . . . . . . 22
1.5.2. Interpretation of the Crosses Representing Geosets
Superimposed on the Maps . . . . . . . . . . . . . . . . 24
1.5.3. The Mean Center . . . . . . . . . . . . . . . . . . . . . 25
1.5.4. Measuring Territorial Dispersion and Oblongity . . . . . . . 27
1.5.5. Measuring Territorial Coverage and Net Density . . . . . . . 29
1.5.6. Comparing Dispersion to Area Covered and Measuring
Looseness . . . . . . . . . . . . . . . . . . . . . . . . 30
1.5.7. Concise Measure of the Discrepancy between Two Geosets . . . 32
xi
iix stnetnoC
Chapter 2
Selected Applications of Geostatistical Methods
2.1. Generalizing the Approaches in Chapter 1 to Other Fields ...... 35
2.2. Applying Geostatistical Methods in Different Fields . . . . . . . . 36
2.3. Scope of this Volume . . . . . . . . . . . . . . . . . . . . . . 38
2.4. A Brief Survey of the Development of Geostatistical Methods .... 40
2.4.1. Limitations of this Review . . . . . . . . . . . . . . . . . 40
2.4.2. Centrography, Social Physics, and Geostatistics . . . . . . . . 40
Chapter 3
Geosets of Populations Distributed over a Predetermined Territory
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1. Why Geosets? . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2. Sets and Set Operations . . . . . . . . . . . . . . . . . . 44
3.1.3. Introducing Population Geosets . . . . . . . . . . . . . . 45
3.2. Comparing the Actual Population Distribution to the Uniform
Distribution over the Territory . . . . . . . . . . . . . . . . . 48
3.2.1. The Uniform Distribution over the Territory . . . . . . . . . 48
3.2.2. Comparing the Actual and Uniform Distributions Using the
Cell Network ~ . . . . . . . . . . . . . . . . . . . . . 52
3.2.3. Distinguishing between Inhabited and Noninhabited Cells . . 55
3.2.4. Transforming gP into gU by Means of Operations over Geosets . 56
3.2.5. Splitting the Habitat into Territories of Matched Elements
and Surpluses . . . . . . . . . . . . . . . . . . . . . . 59
3.3. Comparing Two or More Geosets Distributed over the Same
Territory . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.3.1. The Importance of this Topic . . . . . . . . . . . . . . . 62
3.3.2. Introducing the Comparison between gP and Qg . . . . . . . 63
3.3.3. Population Geosets Derived from Comparing "Pg and "Qg . . . 64
3.3.4. Territorial Geosets Derived from Comparing 9p. and gQ".... 65
3.4. Comparisons between Geosets Representing Large Populations . . . 69
3.4.1. The Problems Dealt with in this Section . . . . . . . . . . . 69
3.4.2. Reduction of the Number of Cells in the Geostatistical Analysis
of Large Populations . . . . . . . . . . . . . . . . . . . 69
3.4.3. Dealing with Input Formed by Aggregated Data . . . . . . . 70
3.4.4. Determining the Reduced Number of Elements to Be Used
in the Analysis of Large Populations . . . . . . . . . . . . 71
3.5. Operations on Geosets versus Operations on Sets . . . . . . . . . 72
3.5.1. Similarities and Differences . . . . . . . . . . . . . . . . 72
3.5.2. Match of Geosets versus Intersection of Sets . . . . . . . . . 73
.6.3 A Compound Geoset Formed by Two Component Subsets ..... 73
3.7. General Scheme of the Geosets Considered in this Chapter and
the Symbols Used in this Volume . . . . . . . . . . . . . . . . 77
Contents xiii
.8.3 Extending the Approach Suggested in Section 3.6 to Other Settings . . 80
3.8.1. Selected Extensions . . . . . . . . . . . . . . . . . . . . 80
3.8.2. Geosets of a Compound Population Formed by Many
Subpopulations . . . . . . . . . . . . . . . . . . . . . 82
3.8.3. Geosets of a Compound Population Formed by Positive and
Negative Subpopulations . . . . . . . . . . . . . . . . . 84
3.8.4. Averaging or Compounding Two Geosets . . . . . . . . . . 84
3.8.5. Weighted Average of Two Geosets . . . . . . . . . . . . . 84
3.8.6. Average of Many Geosets . . . . . . . . . . . . . . . . . 85
3.8.7. Union of Geosets versus Average of Geosets . . . . . . . . . 85
3.8.8. Intermediate Geosets . . . . . . . . . . . . . . . . . . . 86
3.8.9. Decomposition of a Geoset . . . . . . . . . . . . . . . . 87
3.9. Location-Free Sets of Relative Cell Weights . . . . . . . . . . . . 87
3.10. Expressing Geosets of Surpluses Through Rates or Ratios ...... 88
Chapter 4
Further Considerations on Geosets of Territories
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.2. Defining the Territorial Gross Range for Unbounded Population
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1. The Problem . . . . . . . . . . . . . . . . . . . . . . 90
4.2.2. Criteria for Solutions . . . . . . . . . . . . . . . . . . . 92
4.2.3. The Convex Hull as Gross Territorial Range . . . . . . . . . 93
4.2.4. Excluding Outliers from the Convex Hull . . . . . . . . . . 94
4.3. Tracts of Territory Covered by Individual Population Elements . . 96
4.4. Ellipses of Equivalent Scatter . . . . . . . . . . . . . . . . . . 98
4.4.1. An Ellipse as a Substitute for Gross Territorial Range ..... 98
4.4.2. The Ellipse of Equivalent Scatter as a Substitute for
Habitat . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.5. The Geoset of a Simply Connected Territory Defined without
Reference to a Population . . . . . . . . . . . . . . . . . . . . 105
4.5.1. A Geoset Formed by an Infinite Number of Infinitesimal
Territorial Elements . . . . . . . . . . . . . . . . . . . 105
4.5.2. Finite Territorial Geoset . . . . . . . . . . . . . . . . . . 106
4.6. Geosets of Compound Territories . . . . . . . . . . . . . . . . 107
4.6.1. A Territory Formed by Separate Fragments . . . . . . . . . 107
4.6.2. Geosets of Subterritories . . . . . . . . . . . . . . . . . 107
4.6.3. Territories with Internal Gaps . . . . . . . . . . . . . . . 108
4.6.4. Territories with External Concavities . . . . . . . . . . . . 108
4.7. Operations on Territorial Geosets . . . . . . . . . . . . . . . . 108
4.8. Types of Scales Used to Determine Geosets of Populations and
Territories . . . . . . . . . . . . . . . . . . . . . . . . . . 111
