Volume 6, Number 3, 2019, R1–R57 journal homepage: region.ersa.org
DOI: 10.18335/region.v6i3.267

REAT: A Regional Economic Analysis Toolbox for R

Thomas Wieland¹

¹ Karlsruhe Institute of Technology, Karlsruhe, Germany Received: 7 June 2019/Accepted: 4 November 2019

1 Introduction

Methods of regional economic analysis (or regional analysis) are used frequently in theory-based, empirical studies from regional and urban economics as well as (quantitative) economic geography. These methods aim at analyzing some of the most important issues in the mentioned research fields, including (but not limited to) the existence and evolution of agglomerations, regional economic growth and regional disparities (Capello, Nijkamp, 2009; Dinc, 2015; Farhauer, Kröll, 2014; Schätzl, 2000). In any of the mentioned fields, a growing amount of quantitative data has to be processed when using traditional or novel methods and models of regional analysis. This paper introduces the package (add-on) REAT (Regional Economic Analysis Toolbox) (Wieland, 2019) for the programming environment R (R Core Team, 2018a). The package provides a collection of mathematical regional analysis applications, designed in a relatively user-friendly way.

The main topics in the regional analysis context can be summarized as follows, showing also the structure of the present paper with respect to the presented approaches and their application in REAT:

1. Identifying regional inequality (or regional disparities) using indicators of concentration and/or dispersion (Section 2)
2. Regional disparities over time leading to the concept of beta and sigma convergence (Section 3)
3. Measuring agglomerations, which means the specialization of regions and the spatial concentration of industries as well as more complex cluster indices (Section 4)
4. Point-based measures of clustering and accessibility (Section 5)
5. Regional growth, especially shift-share analysis (Section 6)

Note that, in its original form, the open source software R is a command-line environment including a lot of mathematical and statistical features. For the installation of R and its packages as well as the basics of navigation and implemented statistical functions, see the R documentations (R Core Team, 2018b). A good supplement for working with R is RStudio (RStudio Team, 2016). The REAT package deals with several R data types: The most functions require and calculate numeric vectors, but, in some cases, also objects of type matrix, data frame and list, depending on the complexity of calculation. For a quick introduction to the data types in R and their properties, see e.g. Kabacoff (2017).

2 Concentration, dispersion and regional disparities

2.1 Indicators of concentration and dispersion

Regional disparities are a frequent topic in economic geography and regional economics. The spatial inequality with respect to e.g. regional output, income or employment is an essential element of polarization theory (Myrdal, 1957) and ”New Economic Geography” (Krugman, 1991; Fujita et al., 2001). Assessing regional disparities is possible using concentration and dispersion indicators, which belong to the univariate and descriptive analysis in statistics. Apart from regional economics, these measures are used in several contexts, such as competition economics (market concentration of firms) or welfare economics (income inequality). For a review of the most common indicators with respect to regional inequality, see Portnov, Felsenstein (2010), for studies comparing different indicators in the regional economic context using empirical data, see e.g. Gluschenko (2018); Habánik et al. (2013); Huang, Leung (2009); Palan (2017); Petrakos, Psycharis (2016).

Concentration is operationalized as the discrepancy between an empirical distribution of a variable x (e.g. annual turnover, income, gross domestic product [GDP]) with n observations or objects (e.g. competing firms, households, regions) and a (theoretical) equal distribution or a reference distribution (e.g. population distribution). Dispersion indicators aim at the deviation from the arithmetic mean of x, x. In this context, Portnov, Felsenstein (2005, 2010) distinguish between measures of deprivation and variation.

Typical measures of regional disparities are the Gini coefficient, the Herfindahl-Hirschman index and the coefficient of variation (Lessmann, 2005). The most popular measure of concentration is the Gini coefficient (Gini, 1912) in combination with the Lorenz curve (Lorenz, 1905). There are several calculation approaches for the Gini coefficient, all producing the same result. The Lorenz curve is a graphical indicator, showing the deviation of the empirical shares of the regarded variable x from a (theoretical) equal distribution. Another well-known indicator is the Herfindahl-Hirschman index, which was developed independently by Hirschman (1945) and Herfindahl (1950), both in the context of competition economics. Several other concentration indicators are also applied in the fields of regional economics with respect to regional disparities, such as the Hoover coefficient (Hoover, 1936) and the Theil coefficient (Theil, 1967).

Except for the standard deviation, whose unit is equal to the unit of x, all common indicators are dimensionless. Most of them (except for standard deviation and coefficient of variation) have a fixed value range, normally between zero (indicating complete equality/dispersion) and one (indicating complete inequality/concentration).

Most of the common indicators are mathematically formulated in an unweighted and in a weighted form, while, in the context of regional disparities, the latter is mostly done using the regions’ proportion of the total (e.g. national) population (Doran, Jordan 2013; Lessmann 2014; Mussini 2017; Petrakos, Psycharis 2016; for a critical discussion of weighting these coefficients, see Gluschenko 2018). In the literature, there are different formulations where the weighted coefficients also include a weighted arithmetic mean. Note that, in the case of the population-weighted Gini coefficient, a weighted arithmetic mean is mandatory to keep the indicators’ value range.

Especially when dealing with GDP per capita as an indicator of regional economic output, several recent studies use dispersion measures rather than concentration measures, especially the (weighted) coefficient of variation (e.g. Lessmann 2005, 2014, 2016; Lessmann, Seidel 2017; Petrakos, Psycharis 2016). This dispersion indicator is a dimensionless normalization of the standard deviation. Weighting the coefficient of variation with population shares was introduced by Williamson (1965), which has led to calling this coefficient the Williamson index. As regional incomes or outputs are not normally distributed in most cases, resulting in biased arithmetic means used in the calculation of dispersion measures, the regarded variable may be log-transformed, which means replacing x_i with log(x_i) in the calculations.

Table 1 shows the common indicators, including their (population-)weighted and their normalized form (if there exist any) and the corresponding value ranges. The formulae are shown in a way that includes several ways of application. The regarded variable is always named x_i, while the (population) weighting is called w_i. Some indicators, such as the Hoover or the Coulter coefficient, require a variable representing a reference distribution the shares of x_i are compared to. This reference is not a weighting. However, in many studies, the regional population is also used for the reference distribution. In these cases, reference and weighting are the same data. The reference distribution may also be equal to 1∕n.

Several indicators are also used for the analysis of regional specialization or the spatial concentration of industries, such as the Hoover coefficient or the Herfindahl-Hirschman index or its inverse (1∕HHI; also known as the “equivalent number” in the competition context). Other coefficients of concentration and specialization are discussed in Section 4. The last coefficient in Table 1, the mean square successive difference (von Neumann et al., 1941) is a measure for time variability not originating from but also transferable to regional economics.

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Table 1: Indicators of concentration and dispersion for analyzing regional disparities

Indicator	Unweighted	Weighted	Normalized
Gini	G = ∑ i=1n∑ j=1n	Gw = ∑ i=1n∑ j=1nw iwj	G* = G
	0 ≤ G ≤ 1 -	0 ≤ G ≤ 1 -	0 ≤ G*≤ 1
HHI	HHI = ∑ i=1n()2		HHI* =
	≤ HHI ≤ 1		0 ≤ HHI*≤ 1
Hoover	HC =	HCw =
	[∑ i=1n| -|]	[∑ i=1nw i| -|]
	0 ≤ HC ≤ 1	0 ≤ HC ≤ 1
Theil	TC = ∑ i=1n ln()	TCw = ∑ i=1nw i ln()
	0 ≤ TC ≤ 1	0 ≤ TCw ≤ 1
Coulter		CC =
		0 ≤ CC ≤ 1
Atkinson	AI = 1 - [∑ i=1nx i1-ϵ]
	0 ≤ AI ≤ 1
Dalton	δ =
	0 ≤ δ ≤∞
SD	s =	sw =	see CV
	0 ≤ s ≤∞	0 ≤ s ≤∞
CV	v =	see Williamson	v* =
	0 ≤ v ≤∞		0 ≤ v*≤ 1
Williamson		WI =
		0 ≤ v ≤∞
MSSD	MSSD =

Notes: x_i is the i-th observation of the regarded variable x (e.g. GDP [per capita] in region i), x_j is the value of the same variable with respect to object j, r_i is the i-th observation of a reference variable (e.g. population), n is the number of objects (e.g. regions), x is the arithmetic mean of x: x = ∑ _i=1ⁿx _i, x^w is the weighted arithmetic mean of x: x^w = ∑ _i=1ⁿw _ix_i, w_i and w_j are the population weightings: P_i∕∑ _i=1ⁿP _i and P_j∕∑ _j=1ⁿP _j, where P_i and P_j are the population sizes of regions i and j, respectively, ϵ is an inequality aversion parameter (0 < ϵ < ∞) for the Atkinson index, t is a given time period and T is the number all regarded time periods.

Compiled from: Charles-Coll (2011); Cracau, Durán Lima (2016); Damgaard, Weiner (2000); Gluschenko (2018); Heinemann (2008); Kohn, Öztürk (2013); Portnov, Felsenstein (2005, 2010); Taylor, Cihon (2004); Schätzl (2000); Störmann (2009)

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2.2 Application in REAT

2.2.1 REAT functions for concentration and dispersion indicators

Table 2 shows the functions for concentration and dispersion measures implemented in the REAT package. All functions require at least one argument, a numeric vector with a length equal to n, containing the regarded variable x (e.g. income) with i observations (e.g. regions), where i = 1,...,n. This data may be a single vector or a column of a data frame or matrix.

An optional weighting of the vector x can be done using the function argument weighting which is also a numeric vector of length n. By default, the functions remove missing (NA) values. The hoover() function always needs a reference distribution (see the Hoover coefficient formula in Table 1), which is stated via the ref argument, also requiring a numeric vector of length n. If no reference variable is stated (ref = NULL), the reference is set to 1∕n.

All functions (except for disp()) return the single value of the computed coefficient. In the relevant cases (gini(), gini2(), herf() and cv()), a normalization of the coefficient is possible using the function argument coefnorm = TRUE, returning the normalized coefficient instead of the raw coefficient. The function disp() is a wrapper for all mentioned functions, calculating all coefficients (except for the MSSD) at once for one vector x or a set of variables/columns from a data frame or matrix.

Note that there are two functions for the Gini coefficient, gini() and gini2(), both producing the same result in the unweighted case. The former function is designed for income inequality, where the weighting option is designed for the calculation of the Gini coefficient for groups (e.g. income classes), where the weighting represents the group mean. The function gini2() is designed for the population-weighted analysis of regional inequality.

2.2.2 Application example: Small-scale regional disparities in health care provision

Regional inequality with respect to health care providers is a topic of high societal significance. In Germany, the health care planning system (Kassenärztliche Bedarfsplanung) attempts to flatten the disparities of local health care provision (Kassenärztliche Bundesvereinigung, 2013). Here, we analyze small-scale regional disparities in health care provision in two neighboring German counties (Göttingen and Northeim) using the data on medical practices and local population from Wieland, Dittrich (2016). The data is stored in the datasets GoettingenHealth1 and GoettingenHealth2, both included as example datasets in the REAT package. The study area is segmented into 420 districts, representing either city districts of larger cities or villages and hamlets.

The dataset GoettingenHealth2 contains these 420 regions with an individual ID (column district) and geographic coordinates (columns lat and lon, respectively) and the number of general practitioners, psychotherapists and pharmacies located there (columns phys_gen, psych and pharm, respectively) as well as the local population (column pop). First, we load the dataset:

Now, we investigate how the health care providers are dispersed over the whole area. In the first step, we calculate the Gini coefficient for the concentration of general practitioners using the REAT function gini():

The empirical Gini coefficient is equal to 0.839, indicating a relatively strong concentration. If we want to calculate the normalized (unbiased) indicator instead, we use the same function with the optional argument coefnorm = TRUE:

In the same way, we calculate e.g. the Herfindahl-Hirschman index, non-normalized and normalized:

Remember that the minimum of HHI is 1∕n (here: 1∕420 ≈ 0.00238) and the minimum of HHI^* is equal to zero.

If we want to inspect the concentration graphically, we could use the Lorenz curve, which can be plotted using either the functions gini() or lorenz(). Here, we use gini(), tell the function to plot the curve (lc = TRUE), and include several graphical parameters (such as lc.col for the color of the Lorenz curve or lcx and lcy for the x/y axes labels). As we want to compare the population distribution to the location distribution, we start by plotting the Lorenz curve for the local population:

Now, we overlay the Lorenz curves of general practitioners and psychotherapists, which means adding two more curves (function argument add.lc = TRUE):

Our commands result in the output of Figure 1, showing three Lorenz curves (population, general practitioners and psychotherapists) and the line of equality (diagonal). All three empirical distributions differ from an equal distribution. In about 72% of the regions, representing about 23% of the whole population (orange curve; G ≈ 0.584), no general practitioner is located (red curve; G ≈ 0.839). But the psychotherapists are more concentrated, as they are located only in about 13% of all districts (blue curve; G ≈ 0.933). As we can see, the physicians are more concentrated than the inhabitants but the psychotherapists are more concentrated than the physicians.

Now, we calculate all mentionened concentration and dispersion coefficients at once for all three types of providers using the function disp(), including a population weighting:

Our output is:

We conclude that any concentration/dispersion measure is the highest for psychotherapists and the lowest for the general practitioners, while the values for pharmacies lie between them. The regional disparities with respect to pharmacies are higher than those with respect to general practitioners, while the most unequal distribution is that of psychotherapists. In other words: The pharmacies are more spatially concentrated than the general practitioners and the psychotherapists are the most concentrated health locations here.

In most cases, population weighting reduces the coefficient values. That is, because districts with a large (small) population have a high (low) impact on the resulting coefficient and the districts without health service providers are also small districts. Furthermore, as the regarded variables contain zero values (which means no health service locations), the Theil coefficient (including the term ln(x∕x_i)) and the Dalton coefficient (including the n-th root) cannot be computed, resulting in an output of NA.

The visible output of any function presented above can be saved in a new R object:

We can simply access our result:

The function disp() returns a matrix with 13 rows (when only unweighted coefficients are computed) or 19 rows (in the case of additional weighted coefficients) and one column for each regarded variable:

We call our results:

3 Regional convergence

3.1 The concept of beta and sigma convergence

Regional convergence is derived from (regional) growth theory (for an extensive survey, see Barro, Sala-i Martin 2004) and means the decline of regional disparities over time. The neoclassical growth model states that a region’s economic output (e.g. GDP per capita) depends on its stock of factors of production, capital and labor (aggregate production function), on condition of constant returns to scale and diminishing marginal product of the factor inputs. As a consequence, regions with a high (low) initial level of factor input grow slower (faster) than “poor” (“rich”) regions, what is called beta convergence. It is assumed that all regions converge to the same regional output level (steady-state). Sigma convergence means the decline of regional inequality with respect to regional output over time itself (Allington, McCombie, 2007; Capello, Nijkamp, 2009).

Both types of convergence can be tested empirically, as presented in Table 3. When testing for beta convergence, the natural logarithms of output growth over T time periods in i regions is regressed against the natural logarithms of the initial output values at time t. The original convergence formula was presented by Barro, Sala-i Martin (2004) using a nonlinear least squares (NLS) estimation approach. But in many cases, a linear transformation is used which allows for ordinary least squares (OLS) estimation (Allington, McCombie, 2007; Dapena et al., 2016; Schmidt, 1997; Young et al., 2008). The outcome variable of the convergence equation can be the regional growth between two years (e.g. Young et al. 2008) or the average growth rate per year (e.g. Goecke, Hüther 2016; Puente 2017; Weddige-Haaf, Kool 2017). Significance tests are carried out with t-tests for the regression coefficients and, in the OLS case, the F-test for the significance of R².

The estimated parameter of interest is the slope of the model, here denoted β (that is why the modeled process is called beta convergence): If β < 0 and statistically significant, there is absolute beta convergence. If additional variables (conditional variables) are included into the convergence equation, we have a test for conditional beta convergence. A further interpretation of the β coefficient is possible using the speed of convergence, λ, and H, the so-called half-life, which means the time (measured in the regarded time periods) to reduce the regional disparities by one half (Allington, McCombie, 2007; Schmidt, 1997).

Sigma convergence (which is named after the Greek letter for the standard deviation, σ) can be tested in two ways depending on the number of time periods: The regional inequality between all regions at time t is measured using the standard deviation, σ_t, or the coefficient of variation, cv_t, for the GDP per capita in its original or natural-logged form. If only two years are regarded, the quotient of both parameters is computed. If e.g. σ_t1 > σ_t2, the regional inequality has declined from t1 to t2. A significance test can be applied with a simple ANOVA (analysis of variance), where the test statistic is the quotient of the underlying variances (σ²) (Furceri, 2005; Schmidt, 1997; Young et al., 2008). Within a time series, the dispersion parameter is regressed (and plotted) against time. If the slope coefficient of time is negative, there is sigma convergence (Goecke, Hüther, 2016; Huang, Leung, 2009; Schmidt, 1997).

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Table 3: Beta and sigma convergence

Type of convergence	Two time periods	More than two time periods
Beta convergence	absolute
and estimation type	NLS	NLS
	ln() =	∑ t=1T ln() =
	α - [] ln(Y i,t1) + ϵ	α - [] ln(Y i,t1) + ϵ
	OLS	OLS
	ln() =	∑ t=1T ln() =
	α + β ln(Y i,t1) + ϵ	α + β ln(Y i,t1) + ϵ
	conditional
	NLS	NLS
	ln() =	∑ t=1T ln() =
	α - [] ln(Y i,t1) + θXi + ϵ	α - [] ln(Y i,t1) + θXi + ϵ
	OLS	OLS
	ln() =	∑ t=1T ln() =
	α + β ln(Y i,t1) + θXi + ϵ	α + β ln(Y i,t1) + θXi + ϵ
	β < 0	β < 0
	Convergence speed: λ =
	Half-life: H =
Sigma convergence	σt = or
	cvt =
	> 1 or	σ = a + bt + ϵ or
	> 1	cv = a + bt + ϵ
	Test statistic:	b < 0

Notes: Y _i,t is the regional output (e.g. GDP per capita) of region i at time t, Y _t is the arithmetic mean of Y _i,t for all regions at time t, T is the number of regarded time periods (e.g. years), X_i is a set of other variables (conditions), σ_t is the standard deviation of the regional output of all regions, cv_t is the corresponding coefficient of variation, α, β, θ, a and b are estimated coefficients, ϵ is an error term and n is the number of regions.

Compiled from: Allington, McCombie (2007); Barro, Sala-i Martin (2004); Furceri (2005); Schmidt (1997)

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3.2 Application in REAT

3.2.1 REAT functions for beta and sigma convergence

Table 4 shows the functions for beta and sigma convergence as implemented in REAT. The analysis of beta convergence is provided by the functions betaconv.ols() and betaconv.nls() for OLS and NLS estimation, respectively. Speed of convergence and half-life can be computed with the function betaconv.speed(). The ratio test of sigma convergence for two time periods can be done using the function sigmaconv(), while a trend regression over time is implemented into the function sigmaconv.t(). Both convergence types can be analyzed at once with the function rca(), which is a wrapper for all functions mentioned above.

The functions require (at least) two numeric vectors, containing the regarded variable Y (e.g. GDP per capita) for at least two different time periods, e.g. from the same data frame. Also the start and end time periods (t₁ and t_T) have to be stated. Optionally, a graphical output can be generated (scatterplot for beta convergence, line plot for sigma convergence with respect to longitudinal data). Furthermore, when analyzing sigma convergence, the user can choose whether Y should be log-transformed or not and/or which sigma measure is computed (variance, standard deviation or coefficient of variation; weighted or non-weighted).

Note that, unlike the functions for regional inequality indicators (Section 2), the REAT functions for regional convergence distinguish between a visible and an invisible output. The latter can be saved as a new R object. While the visible output shows the main results, the invisible output goes beyond that: betaconv.ols(), betaconv.nls() and rca() return a list, which is the most flexible data type in R, because it consists of a non-predetermined number of different data objects. Apart from the model results, e.g. the (transformed) regression data is returned in this invisible output.

3.2.2 Application example: Beta and sigma convergence in Germany on the county level

In this example, we look at regional convergence in Germany. The REAT package includes the example dataset G.counties.gdp with the GDP (gross domestic product), the population and the GDP per capita for the 402 counties (“Kreise”) in Germany 1992 to 2014 (complete data only for 2000-2014). First, we load the dataset:

In our case, we prevent scientific notation of numbers in R and set a limit of 4 digits:

We need the columns named gdppcxxxx, containing the GDP per capita for each year, e.g. G.counties.gdp$gdppc2010 contains the GDP per capita for 2010. In the first step, we test absolute beta convergence comparing the years 2010 and 2014 with OLS estimation using the function betaconv.ols():.

The output is:

We see that both regression coefficients, α and β, are statistically significant (t ≈ 5.50 and -3.99, respectively, both p < 0.001) and the linear regression model is significant as a whole (F ≈ 15.92, p < 0.001). The negative sign of β shows that, on average, the higher the initial GDP per capita, the lower its growth, which indicates absolute beta convergence. However, the convergence process is very slow: The speed of convergence, represented by λ, shows a harmonization by 0.185% per year. This implies that the output gap will be reduced by 50% in approximately 375 years.

Now we check sigma convergence for the same time using the function sigmaconv(). We choose the coefficient of variation as measure, while using the GDP per capita values in their original form:

The output is:

The coefficient of variation is a little smaller in 2014, which means the spatial inequality declined between 2010 and 2014. The quotient of the variances is slightly above one (F = σ₂₀₁₀²∕σ₂₀₁₄² ≈ 1.04), but not statistically significant (p ≈ 0.71).

When analyzing regional convergence with REAT, it is preferable (and more convenient) to use the wrapper function rca(). Instead of repeating the results above, we test for (absolute) beta and sigma convergence between 2000 and 2014. The analysis of sigma convergence uses trend regression (function argument sigma.type = "trend") for the coefficient of variation (sigma.measure = "cv"). We also want plots for both convergence types (beta.plot = TRUE and sigma.plot = TRUE, respectively) with specific axis labels (e.g. beta.plotX = "Ln (initial GDP p.c.)"). Our code is:

This results in the following output:

This function also produces the plots in Figures 2a and 2b, both showing a declining curve, which is a first indication of both beta and sigma convergence. The beta convergence model is statistically significant (F ≈ 52.06, p < 0.001), as well as the coefficients α (t ≈ 9.63, p < 0.001) and β (t ≈-7.21, p < 0.001). Again, we find evidence for absolute beta convergence because of a negative slope (β ≈-0.007). The trend regression model for sigma convergence is significant (F ≈ 281.4, p < 0.001). The slope is significant and negative (b ≈-0.00026, t ≈ 17.91, p < 0.001), which indicates sigma convergence. However, both types of convergence can be regarded as very slow processes: The half-life value shows that, resulting from the beta convergence model, the regional disparities in GDP per capita will be halved in approximately 1,356 years. When looking at the trend regression, we see that the coefficient of variation declines only by 0.00026 per year. Another aspect is that we only regarded absolute beta convergence, ignoring other spatial effects or the impact of regional policy. The latter is also not considered in neoclassical regional growth theory.

Remembering German reunification, we want to test if there are average growth differences between West Germany and East Germany (former German Democratic Republic), which leads to conditional beta convergence. The dataset G.regions.emp contains the column regional, where the counties are attributed either to West or East Germany, expressed as character string ("West" or "East"). We need to include our condition into the convergence equation. Thus, we use the REAT function to.dummy() to create dummy variables (1/0) out of (nominal scaled) variables, and add the indicator for West Germany (1, otherwise 0) to our data:

Now, we test for conditional beta and sigma convergence, including the condition “West”, again using the rca() function, but without plots and using the standard deviation (default setting) instead of the cv for sigma convergence. This time, we save the results in an object:

The output is:

In the rca() output, we can compare the results of absolute and conditional beta convergence. In the conditional model, the explained variance increases from R² ≈ 0.12 to R² ≈ 0.18, which indicates an increased explanatory power of the model due to the added condition variable. Both models are statistically significant, also the β values are negative and significant (p < 0.001 in both cases). The condition “West” is significant (t ≈-5.50, p < 0.001) and negative, which means that, on average, the GDP per capita in West German counties grew slower than in East Germany. These results seem to support the convergence hypothesis from growth theory, but one should not forget that e.g. political aspects (such as the German and/or EU regional policy) are not considered in this simple analysis.

As we have saved the invisible function output, we can access specific parts of our analysis, such as the regression data for the absolute convergence model:

If we want to look at the single sigma values, we can address them via:

4 Specialization of regions and spatial concentration of industries

4.1 Indicators of regional specialization and industry concentration

Specialization of regions or countries and the spatial concentration of industries or firms are phenomena linked to several research fields in regional economics and economic geography: Specialization is a key point in traditional theories of international trade with respect to comparative advantages (Ricardo, 1821) as well as in the generation of the “New Trade Theory” (introduced by Krugman 1979). Spatial clustering of firms or industries due to agglomeration economies is a perennial issue in all spatial economic fields. It especially reemerged in the context of the “New Economic Geography” (e.g. Krugman 1991; Fujita et al. 2001) as well as through the work of Porter (1990) regarding clusters. The common indicators are broadly discussed in Farhauer, Kröll (2014) or Nakamura, Morrison Paul (2009). For studies comparing some different indicators, see e.g. Goschin et al. (2009); Moga, Constantin (2011); Palan (2017).

When looking at the family of indicators of regional specialization and industry concentration, we have to distinguish between indicators for aggregate data, such as regional employment data, and those requiring individual firm data. The first group, compiled in Table 5, can be differentiated into indicators of specialization and indicators of spatial concentration. As both types of agglomeration are closely linked to each other, so are the corresponding indicators. The empirical basis of all those measures is the employment e in industry i in region j, e_ij. This employment stock is compared to some reference, mostly including the total employment in region j, e_j, and/or the total employment in industry i, e_i, as well as the all-over employment e. The individual firm level indicators in Table 6 can be segmented into indicators for agglomeration of one industry due to localization economies and indicators for the coagglomeration of different industries due to urbanization economies.

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Table 5: Coefficients of regional specialization and industry concentration

Indicator	Specialization of region j	Spatial concentration of industry i
Hoover/Balassa	LQij = ≡ MRCAij =
	LQj = ∑ i=1ILQij	LQi = ∑ j=1JLQij
Extensions:
O’Donoghue-Gleave	SLQij =
Tian	SLLQij =
Hoen-Oosterhaven	ARCAij = -
Hoover	Hj = [∑ i=1I| -|]	Hi = [∑ j=1J| -|]
	0 ≤ Hj ≤ 1	0 ≤ Hi ≤ 1
Gini	Gj = ∑ i=1Iλi(Ri -R)	Gi = ∑ j=1Jλj(Cj -C)
	0 ≤ Gj ≤ 1	0 ≤ Gi ≤ 1
	where: Ri = ,	where: Cj = ,
	R = ∑ i=1IRi and	C = ∑ j=1JCj and
	λi = 1,...,I (λi < λi+1)	λj = 1,...,J (λj < λj+1)
Krugman	Kjl = ∑ i=1I|sijs - sils|	Kiu = ∑j=1J|sijc - sujc|
(J = 2, I = 2)	0 ≤ Kjl ≤ 2	0 ≤ Kiu ≤ 2
	where: sijs = and sils =	where: sijc = and sujc =
Extensions:
Midelfart et al.,	Kj = ∑ i=1I|sijs -sils|	Ki = ∑j=1J|sijc -sujc|
Vogiatzoglou	0 ≤ Kj ≤ 2	0 ≤ Ki ≤ 2
(J > 2, I > 2)	where: sijs = and	where: sijc = and
	sils = ∑ iJsils,	sujc = ∑ uIsujc,
	l≠j	u≠i
Duranton-Puga	RDIj =
	where: sijs = and si =
Litzenberger-Sternberg	CIij =
	where ISij = , IDij =
	and PSij =

Notes: e_ij and e_il equal the employment of industry i in regions j and l, respectively, e_i is the total employment in industry i, e_uj ist the employment of industry u in region j, e_j is the total employment in region j, e is the total employment in the whole economy, I is the number of industries, J is the number of regions, a_j is the area of region j, a is the total area in the whole economy, p_j is the population in region j, p is the total population, b_ij is the number of firms of industry i in region j and b_i is the number of firms in industry i.

Compiled from: Farhauer, Kröll (2014); Hoen, Oosterhaven (2006); Hoffmann et al. (2017); Nakamura, Morrison Paul (2009); O’Donoghue, Gleave (2004); Tian (2013); Schätzl (2000); Störmann (2009)

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Table 6: Coefficients of agglomeration and coagglomeration using individual firm data

Indicator	Agglomeration	Coagglomeration
Ellison-Glaeser	γi =	γc =
	where: Gi = ∑ j=1J(sijc - sj)2,	where: G = ∑j=1J(xj - sj)2,
	sijc = , sj = and	xj = ∑ i=1U, sj = , si =
	HHIi = ∑ k=1K()2	and HHIU = ∑ i=1Usi2HHIi
	z-standardization:
	zi =
	where: var(Gi) = 2HHIi2∑ j=1Jsj2
	-2 ∑ j=1Jsj3 + (∑j=1Jsj2)2-
	∑ k=1Kzik4∑ j=1Jsj2 - 4 ∑j=1Jsj3
	+3(∑ j=1Jsj2)2
Howard et al.		CLab =
		XCLab = CLab - CLabRND
		where: Ckl = 1 if firms k and l are located
		in the same region and Ckl = 0 otherwise

Notes: e_ij is the employment of industry i in region j, e_i is the total employment in industry i, e_j is the total employment in region j, e is the total employment in the whole economy, e_ik is the employment of firm k from industry i, k and l are indices for single firms, I is the number of industries, J is the number of regions, U is a subset of all I industries (U ≤ I), K is the number of firms and K_a and K_b are the numbers of firms in industry a and b.

Compiled from: Farhauer, Kröll (2014); Howard et al. (2016); Nakamura, Morrison Paul (2009)

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The most popular indicator is the Location Quotient (LQ), which is attributed to Hoover (1936) and mathematically equivalent to the Revealed Comparative Advantage (RCA) index, developed by Balassa (1965) in the context of international trade. The LQ is utilized in many studies (e.g. Bai et al. 2008; Kim 1995) as well as in the OECD Territorial Reviews (OECD, 2019). Following O’Donoghue, Gleave (2004) and Tian (2013), the original formulation can be extended: As the location quotient is not normalized, there is no cut-off value for defining a cluster, which leads to a standardization of the computed values via z-transformation. Hoen, Oosterhaven (2006) developed an additive alternative to the RCA index. The original LQ provides the main mathematical basis for several indicators developed later, such as the spatial Gini coefficients described below.

Some indicators which are known from the context of regional inequality (see Section 2) are also used for the analysis of agglomeration: A modification of the Gini coefficient is used for the spatial concentration of industries as well as regional specialization (e.g. Ceapraz 2008; Wieland, Fuchs 2018). As we can see in the calculation of R_i and C_j, respectively, the spatial Gini coefficient is based on the LQ. Another popular option for analyzing agglomeration is the Hoover coefficient, comparing the structure of an industry/a region to a reference structure of all industries/regions (e.g. Dixon, Freebairn 2009; Jiang et al. 2007). Both indicator types range between zero (no specialization/concentration) and one (total specialization/concentration). Also the Herfindahl-Hirschman index and its derivates are used to measure concentration, specialization and diversification (e.g. Duranton, Puga 2000; Goschin et al. 2009; Lehocký, Rusnák 2016).

Another type of specialization/concentration indicator was introduced by Krugman (1991), originally designed for comparing the specialization of two regions. An extension of this indicator was established by Midelfart-Knarvik et al. (2000) for the comparison of regional specialization/industry concentration with respect to the sum or mean of all regions/industries (furthermore used e.g. by Haas, Südekum 2005; Vogiatzoglou 2006). Unlike the Gini- or Hoover-type measures, the Krugman coefficients range between zero (no specialization/concentration) and two (total specialization/concentration).

The cluster index developed by Litzenberger, Sternberg (2006) goes beyond employment data and includes additional information about the industry-specific firm size, population density and region size. It is composed of three parts: the relative industrial stock with respect to industry i and region j, IS_ij, the relative industrial density, ID_ij, and the relative firm size, PS_ij. All three components are modified location quotients. This is done to control for small and monostructural regions, which are identified as clusters otherwise (which is a problem in the original LQ). The cluster index CI_ij has a potential range from zero to infinity. This extended indicator is used e.g. by Hoffmann et al. (2017) for the German food processing industry.

The cluster indicators by Ellison, Glaeser (1997) compare the empirical distribution of firms to an arbitrary location pattern where agglomeration economies are absent (often referred to as a dartboard approach). Ellison, Glaeser (1997) differentiate between the clustering of firms from one industry (agglomeration) due to localization economies and the clustering of multiple industries (coagglomeration) due to urbanization economies. Their indices also take into account the industry-specific structure of the firms by including the Herfindahl-Hirschman index, HHI_i, for the employment concentration in industry i. This is the reason why individual firm-level data is required for the computation. The Herfindahl-Hirschman indicator is included to control the raw measures of spatial concentration, G_i and G, for firm employment concentration, which occurs especially when there are just a few firms with many employees. The Ellison-Glaeser (EG) index for agglomeration, γ_i, is designed for identifying the clustering of industry i, while the coagglomeration index, γ_c aims at the clustering of a set of U industries, where U ≤ I. Values of γ equal to zero imply the absence of agglomeration economies, while values above zero indicate positive effects due to spatial clustering. When γ is negative, firm locations are less spatially concentrated than expected on condition of the dartboard approach, which indicates negative agglomeration economies. The EG index is used in several current regional economic studies (e.g. Dauth et al. 2015, 2018; Yamamura, Goto 2018).

In contrast, Howard et al. (2016) argue that agglomeration economies should not be analyzed regarding employment but the firms itself. Their colocation index, CL_ab, sums the colocation of K_i and K_q firms from two industries, i and q, controlling for all possible combinations. This colocation measure is compared to a counterfactual location structure constructed via bootstrapping; specifically the arithmetic mean of a number of (e.g. 50) random assignments of the regarded firms to the locations. The value of the resulting excess colocation index, XCL_ab, ranges between -1 and 1.

4.2 Application in REAT

4.2.1 REAT functions for regional specialization and industry concentration

Table 7 shows the REAT functions for agglomeration measures based on aggregate (employment) data. All functions require at least information about the employment in one or more regions j in one or more industries i, e_ij. The Herfindahl-Hirschman index (function herf()) for measuring regional diversity is not displayed as it is used exactly in the same way as described in Section 2, replacing x_i with e_ij.

Location quotients for one region and one or more industries are computed by the function locq(), including the option for an additive indicator instead of the multiplicative. When calculating the LQ for a set of J regions and I industries, one can use function locq2(), which is a kind of batch processing extension of locq(). As the dimension of the Litzenberger-Sternberg cluster index is the same as in the LQ (a single value for each combination of region j and industry i), the related functions litzenberger() and litzenberger2() work in the same way. When using locq2() or litzenberger2(), the user may choose the type of function output: either a matrix with I columns and J rows or a data frame with I * J rows.

The Hoover-, Gini- and Krugman-type indicators require the same kind of input data. The hoover() function was already explained in Section 2, as it can be also used for measuring spatial concentration of industries or the specialization of regions with all-over employment vectors, e_i and e_j, respectively, as reference distributions. The spatial Gini coefficients are available through functions gini.spec() for regional specialization and gini.conc() for spatial concentration. The Krugman coefficients are divided into functions for the comparison of two regions/industries (krugman.spec() and krugman.conc(), respectively) and for applying all regions/industries as reference (krugman.spec2() and krugman.conc2(), respectively). The functions spec() and conc() are wrapper functions providing a convenient way to compute Hoover, Gini and Krugman coefficients of a given set of J regions and I industries at once, e.g. originating from official statistics on regional employment.

Table 8 shows the functions operating on the level of individual firm data. The Ellison-Glaeser (EG) indices are available through the functions ellison.a() (agglomeration index for industry i) and ellison.a2() (agglomeration indices for I industries) as well as ellison.c() (coagglomeration index for U industries) and ellison.c2() (coagglomeration indices for I * I - I industry combinations). All functions require the firm size (e.g. no. of employees) for the k-th firm from industry i (numeric vector) and the region j the firm is located in. The functions incorporating more than one industry (all except for ellison.a()) require a vector containing the industry i. The data could e.g. be stored in a data frame with at least three columns (firm size, region, industry). Like some of the convergence functions (see Section 3), the EG agglomeration index functions in REAT also distinguish between a visible and an invisible output: ellison.a() and ellison.a2() show the value(s) auf γ_i but return an invisible matrix including the raw measure of concentration (G_i), the z-standardized results (z_i) and the related Herfindahl-Hirschman index for industry-specific firm concentration (HHI_i) as well as the number of firms in industry i (K_i).

The Howard-Newman-Tarp coagglomeration measure is distributed over the functions howard.cl() (calculation of the colocation index for one pair of industries a and b), howard.xcl() (calculation of the excess colocation index for industries a and b) and howard.xcl2() (calculation of the excess colocation index for I * I - I combinations of I industries). As this cluster index works with firms instead of employment, we only need a vector containing the IDs of the firms k, the corresponding industry i and the region j where the firm is located. When calculating this measure for one pair of industries, the user must state the IDs of industries a and b. Note that calculation time for this index increases heavily with the number of firms and/or industries.

4.2.2 Application example 1: Regional specialization of Göttingen

We use the German classification of economic activities (WZ2008) on the level of 21 sections (A-U) for the classification of industries in the following examples (see Table 9).

Starting with a simple example, we analyze the regional specialization of Göttingen, a city with a population of about 134,000 in Niedersachsen, Germany. The example dataset Goettingen, which is included in REAT, contains the dependent employees in Göttingen and Germany for 2008 to 2017 in industries A to R (rows 2 to 16; row 1 contains the all-over employment). First, we load the data:

Using the REAT function locq(), we calculate a location quotient for Göttingen with respect to the manufacturing industry (”Verarbeitendes Gewerbe”), which is represented by letter C:

The output is simply the LQ value (LQ_ij, where i is manufacturing and j is Göttingen). We see that the LQ is very low, indicating that manufacturing is underrepresented in Göttingen as compared to Germany. Now, we calculate LQ values for all industries (A-R), including a simple plot (function argument plot.results = TRUE):

The output is a matrix with one row for each industry:

The result is plotted in Figure 3. The function plots a vertical line at LQ_ij = 1 automatically. This is the (only) reference value for the LQ. It indicates a stock of the related industry equal to the whole economy. The highest LQ values can be found for the industries with letters P (education) and Q (health). This is because Göttingen is mainly characterized by a large university (about 30,000 students) with a university hospital with about 7,000 employees.

Now, we want to measure the specialization of Göttingen with a single indicator. First, we simply use the Herfindahl-Hirschman coefficient for both Göttingen and Germany using the function herf():

The HHI for Göttingen is slightly larger than for Germany, which indicates a higher specialization (or lower economic diversity) of the region. To combine this information in one indicator, we calculate the Hoover coefficient of specialization using the function hoover(), where the reference distribution is the German industry structure:

We finish our analysis of Göttingen’s regional specialization by calculating both the Gini and the Krugman coefficient of regional specialization with the same data, using the REAT functions gini.spec() and krugman.spec(), respectively. Note that, here, we use the Krugman coefficient to compare the industry structure of Göttingen to the structure of whole Germany (instead of another region within the country, for which this coefficient was originally formulated):

There seems to be some specialization in Göttingen, but, unfortunately, we do not have any real reference value to interpret the results.

4.2.3 Application example 2: Identifying clusters in Germany using aggregate data

In this example, we will compute indicators of regional specialization and industry concentration for a set of J regions and I industries at once. We load the included test dataset G.regions.industries containing employment and firms on the level of I = 17 industries (WZ2008 codes B-S) and J = 16 regions (“Bundesländer”) in Germany:

The number of employees in the column emp_all includes dependent employees and self-employed persons. The classification code of industries (see Table 9) can be found in column ind_code, while the region code (abbreviation of the region’s official name) is in column region_code. First, we want to detect the spatial concentration of the 17 industries in Germany by calculating Hoover, Gini and Krugman coefficients for all industries at once, applying the REAT function conc() which is a wrapper function for the mentioned indicators. We save our output in the matrix object conc_i:

The output is:

The function returns a matrix with 17 rows (one for each industry) and three columns: H i is the Hoover coefficient, G i is the Gini coefficient and K i is the Krugman coefficient for industry i. We cannot interpret or compare all of these results, but we may pick out some findings: The strongest spatial concentration is found with respect to mining and quarrying (WZ08-B), no matter which indicator is regarded, which may be interpreted with “natural advantages” due to the spatial distribution of mineral resources in Germany. Services (such as retailing) as well as education and health are least concentrated, as these industries are bound to regional demand and/or their locations are regulated by policy and planning authorities.

At a first glance, the three indicators seem to produce similar results. Now, we want to test the similarity between Hoover, Gini and Krugman coefficients of concentration. As we saved our result matrix, we now calculate Pearson correlation coefficients (r) for each pair of indicators using the basic R function cor(), which is implemented in the stats package (included automatically in any R release). The function is applied to the three columns of conc_i, producing a 3 * 3 correlation matrix:

As we can see, each combination of the three indicators shows a strong positive correlation (H_i vs. G_i: r ≈ 0.97, H_i vs. K_i: r ≈ 0.95, G_i vs. K_i: r ≈ 0.97). At least in this context, we may conclude that these indicators are interchangeable. However, we have to recognize that the analysis presented here is on a large-scale regional level (German “Bundesländer”) and all of the mentioned indicators are affected by the modifiable areal unit problem, which means that the results depend on the aggregation unit in the analysis (see e.g. Dapena et al. 2016 for a discussion of this effect).

Now, we do exactly the same with respect to regional specialization of the 16 regions, using the same data. Analogously, we use the wrapper function spec() for calculating Hoover, Gini and Krugman coefficients of regional specialization, also saving the resulting matrix:

The output is:

The strongest specialization can be found in the city states Berlin (BE) and Hamburg (HH), while Niedersachsen (NI) and Nordrhein-Westfalen (NW) show the lowest values in all three indicators. As already mentioned in the concentration example, we have to remember the large-scale aggregation unit. If we used smaller scale units (e.g. counties like in Section 3.2.2), our results would surely be more differentiated. Again, we check the correlation between the indicators:

Again, we find a strong positive correlation between the Hoover coefficient and both Gini and Krugman coefficient (H_j vs. G_j: r ≈ 0.92, H_j vs. K_j: r ≈ 0.93), while the third Pearson correlation coefficient is a little lower, but still showing the same direction (G_j vs. K_j: r ≈ 0.79).

Now we check for clusters in a combination of a specific industry and a specific region. First, we calculate location quotients for the dataset G.regions.industries using the REAT function locq2(). Here, the optional function argument LQ.norm could be used for computing z-standardized location quotients according to O’Donoghue, Gleave (2004) (LQ.norm = "OG") or z-standardized values of the natural-logged LQs according to Tian (2013) (LQ.norm = "T"). However, we produce the original LQs, since we need exactly the same columns as in the examples above:

The output is a matrix with J rows and I columns:

These I * J = 17 * 16 = 272 coefficients are too much information. Thus, we calculate them again using the optional argument LQ.output = "df", which produces a data frame with I * J rows and three columns (j_region: ID of region j, i_industry: ID of industry i and LQ: location quotient LQ_ij). We save the results in the object lqs:

As we forego an inspection of these singe values, the results are not displayed here. Instead, we only deal with the five highest LQs in our results (the “top five”). We sort the resulting data frame decreasing and take a look at the first five rows:

The highest LQ is found for the arts, entertainment, and recreation sector (WZ08-R) in the German capital Berlin. Note that this result is congruent with several studies about the “creative class”, showing a large stock of “creative” employment in Berlin (e.g. Martin 2015). We also find a strong concentration of mining and quarrying in two Eastern regions, Brandenburg and Sachsen-Anhalt. Note that the LQ is a relative measure with respect to the total regional employment as well as the total industry-specific employment and the employment in the whole economy, not considering other aspects of industry or spatial structure.

These deficiencies should be overcome with the Litzenberger-Sternberg cluster index, also taking into account area, population and firm size. This additional data is also included in our current dataset (columns area_sqkm, pop and firms). The functions litzenberger() and litzenberger2() work equivalently to locq() and locq2(). To compute cluster indices for all I * J combinations, we use the function litzenberger2():

Like in locq2(), the default output is a matrix with I rows and J columns:

Note that there is a value equal to NaN, which means “not a number”, due to a division by zero; this is because there is no mining and quarrying (WZ08-B) in Bremen (HB). However, we take a look at the “top five” again:

Again, we find the largest cluster value for the arts and entertainment sector in Berlin. Also the other four highest indicators are discovered in the largest city states Berlin and Hamburg, especially with respect to the information and communication industry (WZ08-J) and other knowledge-intensive services. Obviously, the results of the Litzenberger-Sternberg index differ in a noticeable way from those of the LQ, which can be attributed to the consideration of other spatial aspects, especially controlling for the size of the regions.

4.2.4 Application example 3: Identifying clusters using micro-data

In our last example about agglomerations, we use the Ellison-Glaeser indices and the Howard-Newman-Tarp colocation index, which both require individual firm data. As this kind of micro-data is sensitive and, of course, not available in official statistics, we have to use fictional data from the textbook by Farhauer, Kröll (2014).

At first, we compute the Ellison-Glaeser agglomeration index for one industry i, γ_i. We use the REAT function ellison.a(), which is designed for this purpose and requires three vectors: the size (employment) of firm k, e_ik, the IDs of the regions j each firm is located in, and the total regional employment, e_j. The numerical example in Farhauer, Kröll (2014), Table 14.11, contains ten firms in three regions (Wien, Linz, and Graz). We simply compile the data from the original table into separate vectors:

Now, we apply ellison.a() to this data:

The EG agglomeration index of γ_i ≈ 0.06, which is, by the way, the same result as in the textbook, indicates a stronger clustering than expected from a dartboard approach. Since this data is fictional, we refrain from interpreting this result.

The REAT package contains the dataset FK2014_EGC, which is compiled from the numerical example in Farhauer, Kröll (2014), Tables 14.14 to 14.17. There are k = 42 firms from I = 4 industries (clothing trade, forestry, textiles dyeing and textiles trade) in J = 3 regions (1, 2 and 3). We load this example data:

We compute γ_i for all industries in the dataset. This can be done with the function ellison.a2(), which requires vectors containing the size of firm k, the corresponding industry i, and region j. We save the results in the object ega:

Here, we see the output of the function:

We see a strong clustering of the forestry industry, which is attributed to localization economies, but spatial avoidance in the three other industries. The visible output of ellison.a2() contains the γ_i values only, but the invisible matrix output also includes the other information referring to the EG agglomeration index:

When looking at the forestry industry, we also see a high standardized value (z_i ≈ 1.37) and a relatively low firm concentration (HHI_i ≈ 0.09).

In the next step, we compute the EG coagglomeration index, γ^c, for the same data using the function ellison.c(). This function requires the same information as ellison.a2() plus the total employment in the regarded regions (column emp_region):

Congruent with the calculation in Farhauer, Kröll (2014), the function returns γ^c ≈ 12.07. This value is very large, which indicates urbanization economies in this fictional example.

If we want to analyze the coagglomeration of industry pairs instead, we may use the function ellison.c2(), which requires the same data:

The output is a matrix with I * I - I rows (one for each industry pair, omitting the combination of the same industry i):

If we want to focus on firm numbers instead of employment size, we may compute the Howard-Newman-Tarp excess colocation index, which is included in REAT through the functions howard.cl() for one colocation index for one pair of industries, howard.xcl() for the corresponding excess colocation index and howard.xcl2() for all combinations of I * I industries. Subsequent to the numerical example above, we calculate XCL_ab for all industry pairs in the dataset FK2014_EGC, where the firm ID of k is stored in the column firm:

The output has the same structure as the output from ellison.c2():

We see that the index by Howard et al. (2016) is structured differently than the indicators presented above: Although they are based on exactly the same data, the value for forestry and clothing trade (XCL ≈ 0.019) is not equal to the value for clothing trade and forestry (XCL ≈ 0.024). Why? The XCL_ab is the difference between the colocation index, CL_ab, and the mean of a set of bootstrap samples, CL_ab^RND (see Table 6). These random samples are drawn again each time a XCL value is computed, consequently, also the XCL value changes.

5 Proximity and accessibility

5.1 Distance-based measures of accessibility and proximity using individual point-level data

In this chapter, we mix two different concepts of indicators, accessibility and spatial proximity (see Table 10), both frequently used especially in the context of GIS (geographic information systems). Both concepts are discussed together because they have two aspects in common: 1) they are based on the geographical distance between point locations, in particular, the distance between an origin point i or several origin points (i = 1,...,n) and one or more destination points j (j = 1,...,m), and 2) for the calculation, they require geocoded (with geographical coordinates) individual point data.

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Table 10: Accessibility and proximity indicators using point-level data

Indicator	Non-normalized	Normalized
Accessibility/Market potential
Harris	Mj = ∑ i=1nOidij-1
	0 ≤ Mj ≤∞
Hansen	Ai = ∑ j=1mOjf(dij)	Ai* =
	i≠j	i≠j
	0 ≤ Ai ≤∞	0 ≤ Ai*≤ 1
	where: f(dij) = dij-λ or f(dij) = e-λ*dij
	or f(dij) =
Proximity
Count within buffer	Ni = ∑ i=1nI(dij ≤ t)
	i≠j
Weighted count within buffer	Niw = ∑ i=1nI(dij ≤ t)Oj
	i≠j
Ripley	Kt = ∑ i=1n	Lt =	Ht = Lt - t
	i≠j	i≠j	i≠j
	E(Kt) = πt2	E(Lt) = t	E(Ht) = 0
	where: λ =

Notes: d_ij is the distance from origin location i (i = 1,...,n) to destination location j (j = 1,...,m), O_j is a variable quantifying the size of destination j, t is a maximum search radius and I(d_ij ≤ t) is the indicator function taking the value of I = 1 if d_ij ≤ t, and I = 0 otherwise.

Compiled from: Kiskowski et al. (2009); Krider, Putler (2013); Peña Carrera (2002); Pooler (1987); Reggiani et al. (2011); Smith (2016)

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One popular indicator of accessibility is the Hansen accessibility, developed by Hansen (1959) in the context of land use theory. The basic idea is that “accessibility” equals the sum of opportunities outgoing from a specific origin i. These opportunities are spread over a set of m locations (j = 1,...,m). The summation is weighted with the distance between i and the j-th location. This distance, no matter how measured (e.g. street distance, Euclidean distance, driving time) is assumed to be perceived in a nonlinear way, which is operationalized by a nonlinear distance decay function (a.k.a. distance impedance function or response function), e.g. power, exponential or logistic. A similar concept was introduced by Harris (1954) attempting to model the market potential of locations. If we replace the inverse distance weighting in the Harris indicator with another type of distance weighting, we see that both concepts are mathematically equivalent. The only difference is that the Harris indicator is conceptualized from the supplier’s perspective j (e.g. market potential of a retail store) and the Hansen accessibility takes the demand location i as a starting point (Pooler, 1987; Reggiani et al., 2011). As these indicators are dimensionless and range from zero to infinity, a normalization with a range from zero to one can be computed by weighting the results with the opportunities without distance correction.

This accessibility/potential concept can be used in the regional economic context e.g. to quantify the over-regional job potential (e.g. Wieland, Fuchs 2018) or the clustering of point locations of a specific type, such as retail stores (e.g. Larsson, Öner 2014). The most common application of these indicators may be the context of transport economics and transport geography (e.g. Albacete et al. 2017).

In the GIS context, spatial proximity can be measured using concentric zones within a radius of t (buffers) around point i, where the number of the j points within this radius is counted (Longley et al., 2005). A systematic analysis of spatial proximity or cluster patterns is possible using Ripley’s K function (Ripley, 1976). It compares empirical point counts with expected values from a random spatial point process based on a Poisson distribution. Ripley’s K computes empirical values for each distance band with a maximum distance of t, which can be compared to the expected value. A more comprehensible (and linear) interpretation is provided when normalizing the K function in the form of the L or H function. Also, confidence intervals for the expected values can be calculated by bootstrapping (Kiskowski et al., 2009; Smith, 2016). All of these measures are based on a simple indicator function, I(d_ij ≤ t), which takes the value of I = 1 if point j is within a distance of t from point i or not (I = 0). Originating from natural sciences, especially Ripley’s K is frequently used when analyzing location patterns in spatial economic contexts, such as the clustering of retail stores (e.g. Krider, Putler 2013) or other types of firms assumed to be connected in a network (e.g. Espa et al. 2010).

5.2 Application in REAT

5.2.1 REAT functions for accessibility and proximity on the point level

Table 11 shows the REAT functions for the accessibility and proximity methods described above. A simple Euclidean distance matrix for georeferenced points (data frame with latitude and longitude) can be calculated using the function dist.mat(). The function dist.buf() computes a “count points within buffer”, where also a weighting, O_j, can be summarized (e.g., if the destination points are cities of a given population, one could count the number of cities within 50 kilometers and their corresponding population). The latter function uses dist.mat(), thus, it is not necessary to create a distance matrix before.