INTRODUCTION

Contents

Page

A. General ........................................................................................................................ 2

1. The Concept of Compactness and Its General Aspects ............................................. 2

2. Applications of the Index.......................................................................................... 3

3. Previous Studies on the Subject ............................................................................... 4

B. Geographical Basis and Variables.............................................................................. 5

1. General Definitions................................................................................................... 5

2. Object of Measurement – Urban Fabric..................................................................... 7

C. Calculation of Index Components............................................................................... 9

1. Selection of Variables............................................................................................... 10

2. List of Variables, Definitions and Properties............................................................. 11

D. Calculation of the Overall Index.................................................................................

.......................................................................................................................................

....................................................................................................................................... 15

1. Factor Analysis......................................................................................................... 15

2. Cluster Analysis........................................................................................................ 17

3. Summary Tables....................................................................................................... 20

E. References..................................................................................................................... 21

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A. General

The Index of Compactness presented in this study characterizes and classifies municipalities and local councils in Israel by spatial patterns of urban built-up areas. The study was commissioned by the Ministry of the Interior, and introduces a tool that helps distinguish between compact localities, whose spatial urban development is concentrated within a boundary of a regular shape, and non-compact localities, whose spatial development is dispersed and fragmented.

The new index supplements other indices developed by the Central Bureau of Statistics, such as the Socio-Economic Index, which has been calculated since the 1990s, and the Peripherality Index, which was developed in 2008. These indices allow for quantitative and objective evaluation of complex aspects that may influence the functioning of local authorities, especially the provision of municipal services to the population of the local authority.

1. The Concept of Compactness and Its General Aspects

The term compactness has been used in international research to characterize patterns of urban and rural development, and refers to a pattern of filling an area through different land uses. The term is intuitively understood in terms of its extreme manifestations: compact at one end, and dispersed at the other end.

Spatial development of land uses, especially urban development, has multiple economic, social, and ecological implications. In order to examine the impact of spatial urban development on these areas, it is essential to derive tools for measuring that development. The issue of measuring the compactness of localities has drawn considerable attention in international research for over 20 years, but it is a relatively new topic in Israel (Frankel & Ashkenazi, 2005). Although numerous measures have been proposed and implemented, only a few researchers have provided definitions of the term, and none of the definitions that have been provided are more than a list of different aspects of development measured on a continuum from compact to non-compact. There is a general consensus in the literature that the compactness of a locality is a multi-dimensional abstraction, which can be quantified by groups of measures, where each group describes a specific dimension of urban spread that is not necessarily related to other dimensions (Ewing, 2002). The choice of measures depends on the aims of a specific project as well as on the availability of data for calculating the measures.

Although there is no exact definition of the term, compactness is intuitively understood as non-dispersion or concentration. According to the geometric definition, a figure is referred to as compact if it takes up a relatively small space or if its parts are close together. Thus, a circle is considered the most compact shape because it maximizes the area within a given perimeter. For an object consisting of separate elements that are at some distance from one another, this perspective of compactness may be expressed and measured in terms of closeness of the elements to each other or to some central point. In Israel, the Ministry of the Interior has been using a measure of compactness of regional councils, which consist of a number of small, separate localities. This measure is calculated as the average of the distances by the shortest road from the localities in the council to its functional center. Measuring the compactness of a municipality or a local council is more complicated, because each of them consists of a single locality, and cannot be considered as a composite of separate elements.

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This publication introduces an Index of Compactness in Israeli municipalities and local councils, based on measurement methods applied world-wide that have been adapted to the nature of spatial development of local authorities in Israel and to the availability of geographic infrastructure data.

The index is constructed using the factor analysis method, as a combination of measures that characterize different spatial dimensions of urban development. These measures can be ascribed to two main aspects which are not necessarily interconnected:

Configuration, which refers to such geometric dimensions as size, shape, and spread of urban built-up areas;

Concentration and internal continuity of urban built-up areas.

A locality is considered more compact when its urban built-up areas are concentrated in a smaller space, when there are fewer outlying areas, when the outlying areas are closer to the central area, and when there is more internal continuity in the built-up areas.

The study was conducted in five main stages:

1. A review of international literature on measuring the compactness of localities (see Section 3 of this Chapter and Appendix).

2. Establishing the criteria to define the object of measurement of spatial urban development – the urban fabric, and identifying the urban fabric for each municipality and local council. This process is described in Chapter B, Section 2.

3. Selection of relevant variables to characterize the level of compactness of municipalities and local councils. The variables were selected so that the index would reflect most of the spatial aspects for all the localities. The selection process is described in detail in Chapter C, Section 1.

4. Calculating the variables for each locality, see Chapter C, Section 2.5. Statistical processing of the data. Processing was based on factor analysis, an accepted

statistical method for combining the values of several variables into one quantitative scale, which is the index. The statistical analysis was conducted for all the municipalities and local councils in the various sectors (Jewish and Arab), and yielded a common index for all of them. The cluster analysis method was used to classify the localities into clusters that were as homogeneous as possible with respect to the values of the index. The statistical methods are described in Chapter D.

Stages 2, 3, and 4 included tests, which were performed on pilot localities. Localities representing different types of spatial urban development (in terms of size, configuration, land-use combinations, etc.) were chosen as pilot localities. In that way, we aimed to establish criteria and methods of calculation that would be best suited to all of the localities included in the study.

2. Applications of the Index

The study on the Index of Compactness was commissioned by the Ministry of the Interior, as part of a project that aimed to characterize the local authorities on the basis of various dimensions that may influence their functioning. The index of compactness can contribute to formulation of policies of various ministries and other major governmental agencies, including policies for allocation of resources to local authorities.

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Applications at the Ministry of the Interior

Of all government ministries, the Ministry of the Interior is the main one that deals with local authorities, because it is responsible for matters such as regular budgets, development budgets, personnel in local authorities, areas of jurisdiction, municipal-owned enterprises, organizational development, and physical planning. In each of these areas, the local authority's level of compactness needs to be considered. The manner and extent to which this is considered are relevant to the topic at hand.

Additional Applications

(1) Other ministries dealing with issues of urban development, such as the Ministry of Construction and Housing, can use the index for various purposes related to budget allocation (e.g., the neighbourhood rehabilitation and renewal project).

(2) Local authorities can use the index in their ongoing activities.

(3) The index can be used by academic and research institutions in Israel as a basis for further research on different municipal issues.

Limitations of Use

The Index of Compactness was developed as a tool for measuring spatial patterns of urban development. The Index does not relate to the circumstances underlying any given spatial development, such as the geographical characteristics of the area, the history behind the establishment of the localities, the legality of construction, and other issues. We hope this tool will be used wisely in order to facilitate formulation of policies and decision-making, while taking other relevant issues into account if necessary.

3. Previous Studies on the Subject

During the past decade, theoretical and applied research throughout the world has devoted considerable attention to the subject of measuring the compactness of urban development. Appendix presents numerous approaches and methods for measuring various dimensions of spatial urban spread.

In Israel, a comprehensive study on urban sprawl was conducted by Frenkel and Ashkenazi (2005). The applied part of the study was based on a sample of 78 localities, including 71 Jewish localities and 7 mixed localities. The urban sprawl measures were calculated for the urban fabric of each locality (see the definition in Chapter B, Section 2), and identification of the urban fabric was performed manually. The set of variables used to calculate the Index of Sprawl included area-perimeter measures calculated for the central node alone, measures of gross and net population density, a measure of built-up area density, gross and net percentage of outlying areas, and weights of different land uses.

B. Geographical Basis and Variables

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The Index of Compactness presented in this publication was constructed for all of the municipalities and local councils in Israel in 2006, and included 197 localities, according to their municipal status at the end of that year.1 Of these, 77 are Arab localities, and 120 are Jewish localities (including 8 mixed localities). Based on the size of their population, six of the localities are defined as rural, and 191 are defined as urban. The population size of the localities ranges from 1,291 residents in the smallest locality to 733,329 residents in the largest locality, and their built-up area ranges from less than 1 square km to more than 80 square km.

1. General Definitions

Locality - a permanently inhabited place that meets the following criteria:

a. It is usually inhabited by 40 or more adult residents.b. It has an independent administration.c. It is not within the municipal boundaries of another locality.d. Its establishment was authorized by planning institutions.

Every year, changes occur in the list of localities due to several reasons: establishment of new localities, merging of a number of small localities into one locality, merging of one or more small localities with a large locality, and splitting of localities. These changes result from decisions approved by the Minister of the Interior.

Localities are defined as urban or rural, based on the population size: urban localities have 2,000 or more residents, and rural localities have less than 2,000 residents.

Localities are classified as Jewish or non-Jewish according to the majority population in the locality. There are eight urban localities in Israel defined as "mixed", with the large majority of Jews, but with a considerable minority of Arabs: Jerusalem, Tel Aviv-Yafo, Haifa, Akko, Ramla, Lod, Ma'alot-Tarshiha, and Nazerat Illit.

Municipal status of localities – in accordance with legislative and administrative regulations, local authorities are divided into three types:

a. Municipality – a local authority of one locality only, which has received the status of a municipality.

b. Local council – a local authority of one locality only, which has not received the status of a municipality.

c. Regional council – generally includes a number of rural localities. Sometimes urban localities are also included, e.g., Qesaryya (included in the regional council Hof HaKarmel), Kefar Habad (included in the regional council Emek Lod). Some of these urban localities are later granted the status of a local council. Included in regional councils are localities with a representative on the council, as well as localities that are within the municipal jurisdiction of the council but are not represented on it.

In addition, there are localities with no municipal status, i.e., localities that are situated in an area that does not belong to any local authority.

The municipal status of a locality may change over the years. A local council may receive the 1 See: File of Localities, their Population and Codes 2006, the website of the Central Bureau of Statistics, http://www.cbs.gov.il/ishuvim/ishuv2006/bycode.xls

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status of a municipality, a locality within a regional council may receive the status of a local council, and it is even possible for a locality to transfer from one regional council to another.

Land use refers to the way in which land is exploited for different human purposes or economic activities such as residential use, industrial use, agriculture, forestry, etc. Land use refers to the current situation, as opposed to land designation, which refers to future use.

The spatial database on land use was constructed in 2004 by the GIS Sector of the Central Bureau of Statistics in collaboration with the Ministry of the Interior.2 The land use information is nationwide and continuous, covering the entire area of the country except for military bases and security areas. As such, the database can be used to produce data for different geographical units (local authorities, natural areas, etc.).

In the 2004 database, land uses are divided into two main categories: built-up areas and open areas.

A. Built-up areas are divided into sub-categories according to the different functions:

Residence: Area used for residential purposes, as well as built-up area for which no other purposes have been defined.

Schooling and education: Area used for schooling and educational purposes, e.g., kindergartens, schools, universities, community centers, and yeshivot.

Health and welfare: Area used for health and welfare purposes, e.g., public clinics, hospitals, and day centers for the elderly.

Public services: Emergency and rescue services, public administration services, and religious services, including cemeteries.

Culture, leisure, and sports: Area used for cultural purposes, e.g., theatres, cinemas, museums, public libraries, zoological gardens, and archeological sites; tourist and leisure areas, e.g., hotels, hostels, restaurants, amusement parks, etc.; sports areas, e.g., stadiums, swimming pools, etc.

Commerce: Area used for commercial purposes, e.g., shopping and commercial centers.

Industry and infrastructure: Industrial areas, waste treatment centers, mining and quarrying areas, and infrastructure facilities, e.g., airports, sea ports, sewage treatment plants, water reservoirs, etc.

Transportation: Parking lots, gas stations, railway stations, taxi and central bus stations.

Agricultural structures: Structures used for agricultural purposes, e.g., hothouses, fish ponds, etc.

B. Open areas are divided to the following sub-categories:

Public open area: Area used for specific municipal purposes, e.g., municipal parks and authorized beaches.

Forest and brushwood: Area covered by planted forest or natural brushwood.

2 See: Land-Uses in Israel 2002, the website of the Central Bureau of Statistics, http://gis.cbs.gov.il/website/landuse_2002/mavo/main.html (Hebrew only).

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Groves, orchards and olives: Area covered by groves, orchards, or olives.

Cultivated fields: Area of cultivated fields.

Other open area: Unclassified area, which is not defined by any purposes. In local authorities that border the sea and whose area of jurisdiction includes sea areas, the sea areas are a part of the other open area.

2. Object of Measurement – Urban Fabric

The aim of the present study was to characterize localities that have the municipal status of a municipality or local council, by spatial spread of built-up areas. Accordingly, the localities were ranked on a continuum from "compact" to "non-compact". As in the study conducted by Frenkel and Ashkenazi (2005), the urban fabric was chosen as the object of measurement.

The urban fabric was defined as an area within the municipal boundary of the locality, which includes built-up areas as well as open areas surrounded by the built-up areas. In localities characterized by discontinuity of built-up areas, a distinction was made between the central area and outlying (leapfrog) areas.

For example, Figure 1 presents the urban fabric of Rosh HaAyin, which includes a central area (number 1) and two outlying areas (numbers 2 and 3).

Figure 1. Urban Fabric of Rosh HaAyin

Legend

Municipal boundary

Nodal boundary and number

Circumscribing polygon

Residential built area

Non-residential urban built area

Geometric center of the central node

The urban fabric areas were delineated by the GIS Sector of the Central Bureau of Statistics through a computerized process, based on uniform criteria established for all of the local authorities. The criteria were defined in the course of the study as described below.

Land-use data

The land-use data for 197 municipalities and local councils in 2006 were provided by the GIS Sector of the Central Bureau of Statistics. The data were based on the 2004 Land-Use Project (see Chapter B, Section 1), and updated for new construction up to 2006. The update

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was made specifically for the purpose of delineating the urban fabric areas in the present study.

The basic layer used to derive the urban fabric for each locality, which was called an urban built-up area, included all of the categories of built-up land use in the locality, excluding agricultural structures. Open land use and agricultural structures were included in the urban fabric only when they were surrounded by the urban built-up area.

Delineation of urban fabric areas

The aim of the process described here was to identify the nodes that constitute the urban fabric of each locality. Each node was defined as the continuous area within the municipal jurisdiction boundary of the locality, which includes urban built-up areas of the locality, as well as inner areas surrounded by the built-up area. In localities where there is discontinuity in the urban built-up areas (based on a distance threshold of 100 m between adjacent nodes, as defined during the study), a distinction was made between the central node and outlying (leapfrog) nodes, which are distant from the central node but municipally connected to it (e.g., residential districts, industrial zones, institutions, etc.). In the course of the study, a minimal nodal area of 60 dunams (60,000 square m) was defined, in order to avoid dealing with dispersed areas that are too small on the one hand, and in order to make sure that large built-up areas are not missed on the other hand.

The delineation process included the following steps:

a. All basic layer polygons were surrounded by 50 m wide buffers to ensure that polygons with a distance of 100 m or less between them would be assigned to the same node.

b. Nodal areas beyond the jurisdiction boundaries of the locality were excluded from the urban fabric.

c. Nodes with an area of less than 60 dunams were excluded from the urban fabric.

d. Nodes with no clear municipal linkage (for example, nodes comprised solely of cemeteries or infrastructures such as mining and quarrying areas, sewage treatment plants, gas and fuel plants, or water reservoirs) were excluded from the urban fabric.

e. Of the remaining nodes, the one with the largest area was defined as the central node. Others were defined as outlying nodes.

Urban fabric maps

The urban fabric maps were produced for each of the 197 municipalities and local councils are presented in. The following are several important notes for understanding the maps.

a. The headings of each map include the name of the locality followed by the code of the locality in parentheses.

b. All of the maps were produced in the same scale (1:20,000), in order to allow for comparison between localities of different size. The legend indicates which paper size (A2 or A3) retains the scale when the maps are printed. The maps of a few localities with a large urban fabric were divided into several sheets for printing.

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c. The layer of urban fabric areas was produced at the nationwide scale, and the division by local authorities was performed for purpose of the presentation. Therefore, the maps of the urban fabric of most localities also show some parts of the urban fabric of adjacent municipalities or local councils.

d. The municipal boundaries of the local authorities are marked with a thick red line. The names of local authorities are also marked in red. In the computation of the Index of Compactness, the municipal area of a locality is irrelevant. Therefore, in many maps only part of the municipal area is shown. The maps were made so as to include the entire urban fabric of each locality (except for a few localities, whose maps were divided), but do not necessarily include the entire jurisdiction area.

e. The boundaries of nodes that constitute the urban fabric of each locality are marked with a black line. The nodes within the locality are numbered according to their area size: the central node (with the largest area) is numbered 1, the next largest node is numbered 2, and so on.

f. The boundary of the smallest polygon circumscribing the urban fabric of the locality is marked with a blue line. This polygon was defined by the GIS Sector for each locality through a computerized and uniform process. The circumscribing polygon is only a geometric indicator, and its boundary may extend beyond the jurisdiction boundary of the locality. The area of the circumscribing polygon serves to calculate one of the components of the Index of Compactness (see Chapter C, Section 2).

g. In the computation of the Index of Compactness presented here, the land-use categories are irrelevant. Therefore, the maps do not show the land-use categories of the urban built-up area, except for the distinction between the residential built-up area (in yellow) and the non-residential urban built-up area (in green). This was done for the presentation purposes only, and other areas are not highlighted by colour.

h. The geometric center of the central node is marked by a brown point. This point serves to calculate one of the components of the Index of Compactness (see Chapter C, Section 2).

C. Calculation of Index Components

The Index of Compactness is constructed using factor analysis, as a combination of components (spatial characteristics of the urban fabric), which can be ascribed to the following major aspects of spatial development that are not necessarily interrelated:

Configuration, which refers to geometric dimensions such as size, shape, and spread of the urban fabric;

Concentration and internal continuity of the urban fabric.

A locality is considered more compact when its urban fabric is more circular, when there are fewer outlying areas, when the outlying areas are closer to the central area, and when there is more internal continuity in the urban built-up areas.

Each spatial dimension can be described in a number of ways. During the course of the study, seven variables (components) were selected from a wide set of possible measures. The

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variables were evaluated on a continuum from "compact" to "non-compact". Each variable can be ascribed to more than one dimension of compactness, but no variable alone can reflect the entire multidimensional phenomenon. The overall index value was calculated by the factor analysis method as a weighted average of the seven selected measures.

1. Selection of Variables

The selection of variables for calculation of the Index was carried out in an iterative process, where the following aspects were taken into account: properties of the variables as described in the professional literature, availability of data for all of the investigated units, and the statistical requirements of the factor analysis model.

Statistical Analysis

Statistical analysis was carried out for the original list of variables (a large number of measures). In the analysis, the distribution of each variable was examined separately, from several perspectives: parameters of location; parameters of dispersion; symmetry of distribution; and evaluation of extreme values. In addition, the correlations between each pair of variables were calculated. All these steps were taken in order to reduce the number of variables included in the calculation, and to prevent the inclusion of variables that had too much of an effect or variables that correlated strongly with each other. When the Pearson's correlation between two variables exceeded 0.8, the possibility of not including one of the variables in the calculation of the Index was considered. Variables characterizing different spatial dimensions were included in the calculations even if correlated strongly with each other. When several variables characterized the same spatial dimension, preference was given to those with a symmetric distribution, high variance (i.e., considerable differences between the units under investigation), and a smaller correlation with other variables relating to that dimension.

In addition, variables were eliminated according to Kaiser's Measure of Sampling Adequacy, which was calculated for the entire set of variables as well as for each variable separately. This measure has two uses. On the one hand, it is used to examine whether the variables belong to the same content area. On the other hand, it is used to examine the contribution of a single variable to the group in which it is included. Regarding each group of experimental variables, the measure was always greater than 0.5 (on a scale ranging from 0 to 1). That is, the entire set of variables belongs to the same content area. Regarding each separate variable, an attempt was made to include those variables for which the value of the measure was greater than 0.5. The final decision was based on the extent to which the variable contributed to explaining the overall variance in the factor analysis, as well as on the extent to which the association of the factors with the other variables would be affected if that variable were not included.

The following is an example of this kind of analysis: the choice between the "measure of shape" and the "fractal dimension of district boundary" (two perimeter-to-area measures, which are multi-correlated and relate to the same spatial dimension, see Appendix). The correlation coefficient between these variables (when they are calculated as a weighted sum of the values obtained for each node, see Section 2 further) is 0.78. As a result of the statistical analysis, the variable "weighted measure of shape" was chosen due to its relatively large contribution to explaining the general variance in the factor analysis model (0.79 as opposed to 0.37 of the "fractal dimension" variable). In addition, the overall explained

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variance is larger in the factor analysis model, which includes this variable (75%, as opposed to 68% in the model that includes the "fractal dimension").

Main considerations for selection of variables

The considerations guiding the final choice of the set of variables were as follows:

a. Balanced coverage of the characteristics of the aspects comprising the Index of Compactness.

b. Degree of adequacy of the variables for factor analysis. The inclusion of variables with high intercorrelations may artificially inflate the variance, and may influence or even change the relative weights of the variables. However, the correlations between the variables should not be too low, so that they should belong to the same content area.

c. A small number of factors must account for a substantial amount of the variance in the spatial measures in order to maximize the distinction between the units under investigation. The larger the amount of variance explained by the factors, the greater the distinction between the units under investigation.

Changes and improvements in original formulas

In the course of the study, some changes and improvements were introduced in the approaches to measuring the compactness that have been proposed in the literature. These changes and improvements were made in order to adjust the formulas to the spatial development of the local authorities in Israel, and to characterize the spread of the built-up areas in the best possible way. The following are examples of these changes:

a. The measure of shape for each urban fabric was calculated as the weighted sum of shape measures of each node (see Section 2, variable C3).

b. In the calculation of remoteness of outlying nodes, the distances were weighted by the size of outlying nodes (see Section 2, variable C5).

c. In the calculation of measure of mutual proximity, all of the raster cells in the urban fabric were taken into account, and not just the cells with built areas (see Section 2, variable C7).

2. List of Variables, Definitions and Properties

The calculation of components included in the following formulas was performed by the GIS Sector of the Central Bureau of Statistics through unique spatial processing, and according to the definitions formulated in the process of the study.

Configuration Measures

C1 Urban fabric area – calculated as a sum of the areas of the central node and the outlying nodes. The values for the municipalities and local councils range from 0.21 square km (Ghajar) to 81.57 square km (Jerusalem).

C2 Coefficient of Variation (CV) of radials (Siegel, 1996) – calculated as the Coefficient of Variation of the lengths of radials that pass from the geometric center of the central node to the most remote point on the boundary of the urban fabric at equivalent angles

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of one degree. The values of the variable range from 13.86 (Arrabe) to 171.30 (Qazir-Harish). Lower values mean that the localities have a more circular configuration, and are more compact in this dimension.

The formula for calculating the variable is:

where ; ; ;

are the lengths of the radials that pass at equivalent angles of one degree from the

polygon center of the central node to the most remote point of the urban fabric (see

Figure 2).

C2 equals zero in the hypothetical case where the urban fabric forms a circle, and large values are obtained for elongated serpent-shaped areas.

C3 Weighted measure of shape – based on the measure of shape (McGarigal and Marks, 1995), and calculated as the sum of ratios between the perimeter of each node and the perimeter of the circle with an equal area, weighted by the relative size of each node. The values of the variable range from 1.29 (Mazkeret Batya) to 5.18 (Jerusalem). Lower values mean that the perimeter of each node is closer to the perimeter of the circle, and the locality is more compact in this dimension.

The formula for calculating the variable is:

where is the total number of nodes comprising the urban fabric; is the perimeter of node ; is the area of node ; is the total area of the urban fabric.

C3 equals one in the hypothetical case where each node is a circle, and obtains larger values when the nodal shapes are far from the circle.

Measures of Concentration and Internal Continuity

C4 Percentage of outlying area within the urban fabric area (Hanson and Freihage, 2001). The variable values range from zero (for localities without outlying nodes) to 69.39 (Ka'abiyye-Tabbash-Hajajre). Higher values mean that the outlying area is larger, and the locality is less compact in this dimension.

The formula for calculating the variable is:

where is the total number of nodes comprising the urban fabric; the numbers of outlying nodes start from 2; is the area of node ; and is the total area of the urban fabric.

C5 Weighted remoteness of outlying nodes – calculated as the sum of the minimal aerial distances from the boundary of the central node to the boundary of each outlying node, weighted by the percentage of the outlying areas within the urban fabric. The values of

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the variable range from zero (for localities without outlying nodes) to 322.09 (Qazir-Harish). Higher values mean that there are more outlying areas and they are more remote, and the locality is less compact in this dimension.

The formula for calculating the variable is:

where is the total number of nodes comprising the urban fabric; the numbers of outlying nodes start from 2; is the area of node ; is the total area of the urban fabric; is the minimal aerial distance, measured in kilometers, between the boundary of node and the boundary of the central node (for example, the purple line in Figure 2 indicates ).

C6 Percentage of urban fabric area within the area of the smallest circumscribing polygon. The values of the variable range from 9.45 (Basma) to 94.76 (Lehavim). High values mean that the urban fabric fills up most of the area of the circumscribing polygon, and the locality is more compact in this dimension.

The formula for calculating the variable is:

where is the total area of the urban fabric; is the area of node ; is the area of the smallest polygon circumscribing the urban fabric (in the urban fabric maps, the smallest circumscribing polygon is marked by a blue line, for example, see Figure 2).

C7 Measure of mutual proximity – based on Thinh et al. (2001), and calculated on a network of 100 x 100 m raster cells as the average reciprocal power of attraction between each pair of cells (see the illustration in Figure 3). The reciprocal power of attraction between two raster cells is calculated by analogy with the law of gravitation, as the product of the built area size in these cells divided by the square of the aerial distance between the centers of the cells. The values of the variable range from 0.07 (Jerusalem) to 6.83 (Mazra'a). Higher values mean that the built-up cells are more concentrated, and locality is more compact in this dimension.

The formula for calculating the variable is:

where is the reciprocal power of attraction between two cells calculated by

analogy with the law of gravitation; is the built area of cell , measured in square meters; is the aerial distance between the centers of cells and , measured in meters; is the number of cells with at least 50% of the area belonging to the urban fabric; and is a constant which turns into a non-dimensional value ( equals 100 square m).

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Figure 2 illustrates the transfer of radials and the calculation of minimal distances between the nodes. Figure 3 illustrates the network of raster cells and the distances between the centers of cells. The illustrations were produced for the urban fabric of Rosh HaAyin.

Figure 2. Illustration of the transfer of radials andcalculation of the distances between nodes

Figure 3. Illustration of the network of raster cells and calculation of the distances between cell pairs

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D. Calculation of the Overall Index

The statistical technique used to calculate the Index of Compactness was factor analysis. After obtaining the Index values, the municipalities and local councils were allocated to homogeneous groups by means of cluster analysis. The following is a review of these techniques, together with a description of the way they were applied.

1. Factor Analysis

Factor analysis is a group of statistical techniques aimed to express a large number of variables on the basis of a smaller number of factors and thus to characterize the units under study (in our case, municipalities and local councils) in a synthesized way that can be conveniently used (Morrison, 1967).

Factors are new variables, calculated as linear combinations (weighted averages) of the original standardized variables (i.e., each variable has a mean of 0 and a variance of 1). The need to standardize variables stems from the differences in the measuring units: the value of a variable can be expressed as a number, quotient or percent, and can be measured for example, by meters or by square meters. Standardization makes it possible to convert the variables into a uniform scale ("standard score" in Tables 1 and 2), and to further combine them into one synthetic score. The weights of the original standardized variables are determined mathematically so as to attain maximum distinction between the units under study, subject to some normalization restrictions. For p variables, there exist p factors that can explain all of the variance of these variables. Because the variables are standardized, the total variance of the original variables is equal to their number.

The factors are determined sequentially, so that the first factor is the linear combination that accounts for a maximum amount of the variance of the variables. Hence, the first factor has a maximum power of discrimination between the units under study. The second factor accounts for a maximum amount of the variance not accounted for by the first one, etc. The next step is to find the minimal number of factors that explain a considerable amount of the variance. The optimal number of factors is determined by statistical testing that examines the amount of information added by a factor versus increasing their number. The addition of an extra factor, beyond the optimal number, increases the dimension that the Index is based on, while its contribution to explaining the variance is negligible.

The factors described above define an orthogonal set of axes in the multidimensional variable space (because each factor is a linear combination of the original variables, and the factors are orthogonal). This type of factor analysis is called principal component analysis.

In the study and interpretation of the derived factors, an important concept is that of factor loadings. These are the correlation coefficients between the original variables and the factor. Their size is a measure of the relative importance of each variable in differentiating between the geographical units. In particular, if a variable has a low loading on all factors, this is an indication that it may be removed from the analysis. It should be mentioned that for the sake of convenience, some of the original variables were multiplied by (-1) in order to obtain positive correlation coefficients, so that a higher standardized value would signify a higher level of compactness (see notes in Tables 1 and 2).

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Various options are available in factor analysis, including a rotation of axes (factors) with the aim of strengthening the relationship between each variable and one factor only, while weakening the relationship between the same variable and the rest of the factors. In that way, it is often possible to reach a situation where each factor is significantly associated with a well defined set of variables that belong to a specific domain, such as configuration measures or internal continuity measures. In the attempt to interpret the meaning of the different factors, it is important to bear in mind, that specific interpretation is one of many possible explanations that may be obtained by another rotation.

After the rotation, the first factor is no longer the linear combination with the maximum variance. Moreover, in a non-orthogonal rotation the overall variance explained by the variables is reduced. In the present study, the orthogonal rotation was used (the same total amount of variance explained as before the rotation), which may cause some loss in the explanatory power of the first factor.

The Index value, which expresses the level of compactness of the unit under study, was calculated as a weighted average of the factors, where the weighting was based on the percent of variance explained by each factor. The Index values were converted into standard scores, so that the most compact locality obtained the smallest (negative) value, and the least compact locality obtained the largest (positive) value. The local council Mazra'a obtained the smallest Index value (-2.207), and the local council Qazir-Harish obtained the largest value (4.189). All of the municipalities and local councils were ranked according to their Index values, from 1 (most compact) to 197 (least compact), see Table 2.

In the process of factor analysis, two factors were determined. These two factors explained about 75% of all of the information contained in the original set of variables. This amount (total variance) is equal to the number of variables, i.e., 7. The first factor accounted for about 45% of the variance, and this percent of the explained variance decreased as a result of rotation (it was more than 46% before the rotation). Table A presents the variance, the percent of variance, and the cumulative percent of variance explained by each of the factors.

Table A.- Variance and Percent of Variance Explained by the First Two Factors in the Model of the Index of Compactness

Factor Explained variance Percent of explained variance

1 3.13 44.76

2 2.10 29.93

Total 5.23 74.69

Table B presents the correlation coefficients between the factors and the variables included in the model of the Index of Compactness. The variables are arranged according to the size of their correlations with each of the factors, so that the first set of variables (4 variables) has the highest correlation with the first factor, and the second set (3 variables) has the highest correlation with the second factor. The variables in the first set have correlation coefficients

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higher than 0.81. This group comprises configuration measures and characteristics of outlying areas. The variables "coefficient of variation of radials" and "percentage of urban fabric area within the area of the smallest circumscribing polygon" have the highest correlation (0.91 and 0.88) with the first factor. The set of variables that correlate strongly with the second factor (0.73-0.89) comprises measures of size, configuration, and internal continuity.

The last column in Table B presents the final communality estimates for the variables used in the model of the Index of Compactness. These estimates reflect the correlation between the variable and the calculated index. Notably, the sum of the communality estimates is equal to the total variance explained by the factors, as presented in Table A. The variables included in the model have final communality estimates over 0.67. The communality estimate of the variable "percentage of urban fabric area within the area of the smallest circumscribing polygon" is the highest (0.85).

Table B.- Correlation Coefficients between Variables and Factors, and Final Communality Estimates of Variables in the Model of the Index of Compactness

Variable Factor 1 Factor 2Final

Communality Estimate

Coefficient of variation of radials 0.91 0.00 0.83

Percentage of urban fabric area within the area of the smallest circumscribing polygon 0.88 0.27 0.85

Percentage of outlying area within the urban fabric area 0.85 0.06 0.73

Weighted remoteness of outlying nodes 0.81 -0.11 0.67

Weighted measure of shape -0.07 0.89 0.79

Urban fabric area -0.06 0.83 0.69

Measure of mutual proximity 0.36 0.73 0.67

Note: Loadings greater than 0.5 are shaded in grey.

2. Cluster Analysis

Cluster analysis is a technique for allocating the units under study to groups or clusters that are as homogeneous as possible with respect to a set of variables. Allocation is based on a measure of distance (similarity) between clusters. The current analysis used only one variable, the Index of Compactness value, and the distance between clusters was calculated on the basis of Ward's distance. For a given number of clusters, the variance of Index values within clusters is minimized, and the variance of Index values between clusters is maximized, i.e., two locations belonging to the same cluster are similar to each other, and two locations belonging to different clusters are different from each other.

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The localities were classified into 10 clusters based on the value of the Index: from cluster 1, which included the most compact localities, to cluster 10, which included the least compact localities. The clusters are not equal in size. Table C presents the number and ranking range of localities in each cluster.

Table C.- Distribution of Municipalities and Local Councils by Cluster of Compactness

Cluster of Compactness Number of Localities

Ranking Range

1 17 1 - 17

2 17 18 - 34

3 44 35 - 78

4 22 79 - 100

5 21 101 - 121

6 15 122 - 136

7 27 137 - 163

8 19 164 - 182

9 12 183 - 194

10 3 195 - 197

Figure 4 presents the urban fabric maps (in equivalent scale) of the localities that were ranked approximately in the middle of each of the 10 clusters. The localities representing clusters 1, 2, and 3 show a high level of compactness. The shape of their urban fabric is relatively simple, there is internal continuity in their built-up areas, and they don't have outlying areas. The localities representing clusters 4, 5, and 6 show a moderate level of compactness. The shape of their urban fabric is irregular, and there is less internal continuity in their built-up areas. The localities representing clusters 7, 8, 9, and 10 show a low level of compactness. They are characterized by fragmentation of built-up areas, and they have numerous and distant outlying areas.

The analysis presented here is subject to the basic limitations inherent in any attempt to reduce a vast set of multidimensional data on the spatial spread of built-up areas to a one-dimensional ranking. Notwithstanding these limitations, the comparison between the urban fabric maps of the local authorities allocated to different clusters of compactness demonstrates that the index developed in the present study reasonably reflects the types of spatial development of the municipalities and local councils.

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Figure 4. The Urban Fabrics of LocalitiesRanked in the Middle of Each Cluster of Compactness

Mazkeret BatyaRank 7

Cluster 1

DimonaRank 90Cluster 4

ShelomiRank 53Cluster 3

Bene BeraqRank 26Cluster 2

Rosh HaAyinRank 111Cluster 5 Arad

Rank 130Cluster 6

Modi’in IllitRank 150Cluster 7

Be’er Ya’aqovRank 172Cluster 8

Giv’at Ze’evRank 188Cluster 9

BasmaRank 196Cluster 10

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2. Summary Tables

Table 1 presents the Index of Compactness values, ranks, and clusters of the 197 municipalities and local councils, as well as the values of the 7 original variables, their standard scores, and ranks, in alphabetical order of the Hebrew names of the localities. Two additional values are presented for each locality: the population size, and the number of outlying nodes.

Table 2 presents a list of the municipalities and local councils in ascending order of the Index of Compactness values, their ranks (from 1, the most compact, to 197, the least compact) and allocation to clusters (from 1 to 10). The three largest cities – Jerusalem, Tel Aviv-Yafo, and Haifa – are allocated to cluster 10 (the lowest level of compactness).

Table 3 presents the mean values of the 7 original variables in each cluster, and the mean values of the variables for all municipalities and local councils. In addition, the average number of outlying nodes, mean population size, and total population size are presented for each cluster and for all of the municipalities and local councils. This table reveals the changes in the means of the variables across the clusters, as well as the gaps between the low and high clusters. Notably, the change that occurs from one cluster to another is not necessarily monotonous, nor is it always gradual. For example, the mean size of the urban fabric area increases monotonically from cluster 1 to cluster 7, decreases from cluster 7 to cluster 8, and then increases sharply in clusters 9 and 10.

Table 4 presents the minimum and the maximum Index values, as well as the range, the mean, and the standard deviation of the Index values in each cluster. These data reflect the variability of values of the Index of Compactness for the local authorities in each cluster.

Table 5 presents the distribution of municipalities and local councils by cluster of compactness and by the size of their urban fabric area. The following are the main findings in the table: Among the localities in which the area of the urban fabric is less than 2 square km, 70%

are allocated to the low clusters (1-3), which include the most compact localities. All of the localities in which the area of the urban fabric is 10 square km and more are

allocated to the middle and high clusters (5-10), which include the localities with moderate and low levels of compactness.

The size of the urban fabric area does not unambiguously determine the level of compactness of the locality, so that a considerable number of localities with a small urban fabric area are allocated to the high clusters (7-10), which include the least compact localities.

Table 6 presents the distribution of municipalities and local councils by cluster of compactness and by population size. This table shows no relation between the population size and the level of compactness of the locality. Thus, a substantial number of large localities are allocated to the low clusters, which include the compact localities, and a substantial number of small localities are allocated to the high clusters, which include non-compact localities.

Table 7 presents the distribution of municipalities and local councils by cluster of compactness and by district. This table shows that the high clusters (8 to 10), which include the non-compact localities, mainly comprise the localities from the Northern District, the Haifa District, and the Judea and Samaria Area.

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