Functional Method for Analyzing Open-Space Ratios around Individual Buildings and Its Implementation with GIS

Abstract

An open-space ratio is often used as a first basic metric to examine the distribution of open space in urbanized areas. Originally, the open-space ratio was defined as the ratio of the area of open space (unbuilt area) to the area of its building site. In recent years, residents have become more concerned with the open-space ratios in the broader neighborhoods of their individual buildings than with their own building sites. To address this concern, this paper proposes a method for dealing with the open-space ratio in the variable $x$-meter buffer zone around each building, called the open-space ratio function, and implements it using standard GIS operators. The function and its implemented analytical tool can answer the following questions. First, this function shows how the ratio varies with respect to the bandwidth to discuss the modifiable area unit problem. Second, as the ratio changes, the function shows in which bandwidth zone the ratio is the highest, indicating the best open-space environment zone. Third, in the pairwise comparison for housing selection, the function shows in which bandwidth zone a specific house is better than another. Fourth, the function shows in which bandwidth zone the variance among all buildings in a region is the greatest. Fifth, in this zone, buildings are clustered in terms of open-space ratio. The resulting clusters are the most distinct. Sixth, to examine the open-space ratio around a clump of buildings (such as a housing complex), the function shows how to obtain clumps. Seventh, it is shown how the open-space function provides a wide range of applicability without changing the mathematical formulation. Finally, this paper shows how to implement the function in a simple computational method using operators and a processing modeler provided by the standard GIS without additional software.

1. Introduction

An open-space ratio is often used in the literature (e.g., urban geography [1], urban analysis [2], environmental studies [3], land use studies [4], public health studies [5], city planning [6], building studies [7], and urban morphology [8]) as a first basic metric to examine the distribution of open space in urbanized areas. Originally, the open-space ratio was defined in the 1960s (Felt [9]; Tough [10]) as the ratio of the area of open space (unbuilt area) to the area of its building site. In recent years, residents have become more concerned with the open-space ratios in the broader neighborhoods of their individual buildings than with their building sites in general. To address this concern, this paper proposes a method for dealing with the open-space ratio in the variable buffer zone around each building, called the open-space ratio function, and implements it using standard GIS operators.

Regarding the extension of the area in which the open space ratio is defined, several attempts can be found in the literature. For example, Hamaina et al. [11] and Schinmer et al. [12] used the area Voronoi diagram. Some researchers use multiple bandwidth zones around buildings, such as Steiniger et al. [13], Pont et al., and Boffet et al. [14]. Obviously, the open-space ratio varies with the choice of the bandwidth zones, known as the notorious modifiable area unit problem (MAUP) (Openshaw [15]). The results would have changed if different buffer widths had been used. To observe how the open-space ratio varies with bandwidth and to determine which bandwidth is appropriate, an open-space ratio with respect to a continuous bandwidth, that is, a function of an open-space ratio with respect to a continuous bandwidth, is very useful.

The open-space function can answer the following questions. Q1: How does the open-space ratio continuously vary across small to large bandwidth zones? The answer indicates whether the notorious MAUP exists. If the ratio changes, we must keep in mind that the results may change depending on the choice of a zone width. If the ratio changes, we can ask the following question. Q2: Which bandwidth buffer zone has the highest or lowest open-space ratio? The answer tells us which zones have the “best” or “worst” open space environments, as measured by the open-space ratio. When comparing two buildings, we can ask the following question. Q3: In which bandwidth buffer zone is the difference in open-space ratio the largest or smallest? The answer would be useful for a family who are looking for a rental house in order to choose the best house by comparing the current house with an alternative house. As for the open-space ratios of all buildings in a region, we can ask the following question. Q4: In what bandwidth zone is the variance of open-space ratios at its maximum in the region? The answer shows in which bandwidth zones the open space environment of the buildings in the region are very different. When we want to see the distribution of buildings with respect to different open-space ratios on a map, we want to classify buildings into several classes that are distinctively different from each other. Q5: In what bandwidth zone can we clearly see the distribution of different classes of buildings on a map? This map is useful for discussing the open space environment created by the mix of buildings with different open-space ratios in a district. If we want to examine the open-space ratio not only around individual buildings but also around a clump of closely spaced buildings, such as a housing complex, we can ask the following question. Q6: How do we find clumps of buildings? The answer is useful for evaluating the open-space environment around a clump of buildings in terms of the open-space ratio. Q7: How does the open-space ratio function provide a wide range applicability without changing the mathematical formulation?

This paper consists of four sections and Appendix A. Section 2 deals with the methods and their implementations. The first method is the open-space ratio function, which has four types. We formulate the local open-space ratio function as the open-space ratio in a buffer zone around a single building where the width of the buffer zone varies continuously. Next, we formulate the global open-space function, which is obtained by averaging the local open-space ratio functions of all buildings in a region. To observe the distribution of open space within a buffer zone, we formulate the incremental local open-space ratio function (the concept of “incremental” is also used in [16]). Explicitly, we decompose a buffer zone into buffer rings. The incremental local open-space function is formulated as the open-space ratio in each buffer ring surrounding a given building. The incremental global open-space function is obtained by averaging the incremental local open-space ratio functions of all buildings in a region. We show how to implement these functions using standard GIS operators. Once these functions are estimated from the data, the functions answer the questions Q1, Q2, and Q3 above.

The second method is how to answer the questions Q4 and Q5 using the local and global open-space ratio functions. We can illustrate the distribution with distinctly different open-space ratio classes of buildings on a map. The third method is how to answer Q6, i.e., how to find clumps of buildings to obtain the open-space ratio functions of a set of closely spaced buildings. Since the exact computation of this method is complex, we alternatively show a practical implementation using the local open-space ratio function. In Section 3, we first discuss a method of how the open-space function provides a wide range of applicability without changing the mathematical formulation. Second, we discuss the limitations of the method and how to overcome them. Finally, in Section 4, we summarize our main results and point out possible extensions.

2. Methods and Implementation

This section explicitly formulates the open-space ratio functions outlined in the introduction. For illustrative purposes, we use a small empirical example and assume that the open space is the area not covered by building footprints. The study area is located within the Shibuya ward in Tokyo. The area is almost a rectangle (364 m × 268 m) surrounded by streets, and its area is 9.5 hectares (the thick lines in Figure 1). The building footprints in the study area are indicated by the gray polygons. The white polygons outside the study area are used for edge correction [17,18]. Note that the gray and white buildings are mainly for residential use, and the total number of buildings is 406.

Let S be a study area (indicated by the bold polygon in Figure 1); let $B_1,\dots , B_n$ (the gray polygons) be building footprints in S; let $B_{n+1},\dots , B_{n*}$ (the white polygons) be building footprints outside the study area S used for edge correction; and let S* be the extended study area including $B_1,\dots , B_n$ and $B_{n+1},\dots , B_{n*}$ (the area including all the gray and white polygons). The crude open space O in S* is obtained by excluding buildings $B_1,\dots , B_{n*}$ from S*, i.e., O = S*$\setminus {\bigcup }_{j=1}^{n^{*}}{B}_j$ (the white area in S* in Figure 1).

2.1. Local Open-Space Ratio Function

We first consider the open-space ratio around a building, $B_i$. Let $B_f\left(x\right|B_i)$ be the buffer zone around building $B_i$ with a width of x. An example, $B_f\left(50\mathrm{m}\right|B_{21})$, is illustrated by the area enclosed by the circular hairline centered at $B_{21}$ in Figure 2a. Note that $B_f\left(x\right|B_i)$ includes $B_i$. To obtain the open space in $B_f\left(50\mathrm{m}\right|B_{21})$, we defined the buildings in $B_f\left(50\mathrm{m}\right|B_{21})$, indicated by the gray polygons in Figure 2b. Then, the open space is obtained by excluding the buildings ${\bigcup }_{j=1}^{n^{*}}{B}_j$ from the buffer zone $B_f\left(x\right|B_i)$, i.e., $O\left(x|B_i\right)=B_f\left(x|B_i\right)\setminus {\bigcup }_{j=1}^{n^{*}}{B}_j$ as in Figure 2c. We define the local open-space ratio function for the x-width buffer zone around Building $B_i$ as the ratio of the area of open space (e.g., the white area within the hairline circular area in Figure 2a) to the area of the x-width buffer zone around Building $B_i$ (e.g., the gray area in Figure 2c). Mathematically, the local open-space ratio function of building $B_i$ is written as:

$$L_{\mathrm{C}}\left(x\right|B_i)=[O\left(x|B_i\right)]/[B_f\left(x|B_i\right)], 0\le x {\le x}_{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(1)

where the symbol $\left[\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{g}\mathrm{o}\mathrm{n}\right]$ is the operator of measuring the area of the polygon, and $x_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum distance. This function is the function of x-width. Therefore, we can continuously observe open-space ratios across a narrow zone to a wide zone.

It should be noted that the open space in the local open-space ratio function defined by Equation (1) is cumulative concerning x-width. Therefore, we cannot directly determine in which part of the x-width buffer zone the open-space ratio is high or low. More explicitly, we consider buffer rings surrounding Building $B_i$. For example, in Figure 1, the circular hairlines indicate the buffer rings with 10 m intervals surrounding $B_{130}$ in the 50m width buffer zone. The j-th ring, $R(x_j,x_{j+1}|B_i)$, is obtained by excluding the $x_j$-width buffer zone (e.g., $B_f\left(x_4|B_i\right)$ in Figure 3a) from the $x_{j+1}$-width buffer zone (e.g., $B_f\left(x_3|B_i\right)$ in Figure 3b), i.e., $R(x_j,x_{j+1}|B_i)$ = $B_f\left(x_{j+1}|B_i\right)\setminus B_f\left(x_j|B_i\right)$, $j=\mathrm{0,1},\dots , m$, where $x_0=0$, and $x_m=x_{\mathrm{m}\mathrm{a}\mathrm{x}}$ (e.g., $R(x_3,x_4|B_{130})$ in Figure 3c is obtained by substracting $B_f\left(x_4|B_{130}\right)$ in Figure 3a from $B_f\left(x_3|B_i\right)$ in Figure 3b). Similarly, the open space, $O\left(x_j,x_{j+1}|B_i\right)$, in a ring $R(x_j,x_{j+1}|B_i)$ (e.g., $O\left(x_3,x_4|B_{130}\right)$ in Figure 3f) is obtained by excluding the open space in the $x_j$-width buffer zone (e.g., $O\left(x_3|B_{130}\right)$ in Figure 3e) from that in the $x_{j+1}$-width buffer zone (e.g., $O\left(x_4|B_{130}\right)$ in Figure 3d), i.e., $O(x_j,x_{j+1}|B_i)$ = $O\left(x_{j+1}|B_i\right)\setminus O\left(x_j|B_i\right)$. In these terms, the open-space ratio in a ring $R(x_j,x_{j+1}|B_i)$ is written as:

$$L_{\mathrm{I}}\left(x_j\right|B_i)=[O\left(x_j,x_{j+1}|B_i\right)]/\left[R(x_j,x_{j+1}|B_i)\right], j=0,1,\dots , m.$$

(2)

With this function, we can observe the open-space ratio in every ring in detail in the $x_{j+1}$-width buffer zone. Because the ring $R(x_j,x_{j+1}|B_i)$ = $B_f\left(x_{j+1}|B_i\right)\setminus B_f\left(x_j|B_i\right)$ is the increased area from the $x_j$-width buffer zone to the $x_{j+1}$-width buffer zone, we call the function defined by Equation (2) the local incremental open-space ratio function. To distinguish the local open-space ratio function given by Equation (1) from this function, we call the former function the local cumulative open-space ratio function. The terms cumulative and incremental are the same use as in the K function [16]. It is noted that the cumulative open-space ratio function deals with open space in a buffer zone of $B_i$ that includes Building $B_i$, whereas the incremental open-space ratio function deals with open space in a buffer ring surrounding Building $B_i$ which does not include Building $B_i$. These two functions show different aspects of the open-space ratio, and they are complementary.

2.2. Global Open-Space Ratio Function

In the preceding subsection, we formulated the local open-space ratio functions in terms of the open-space ratio around a specific building in a study area. In this subsection, we consider a function for examining open-space ratios of all buildings in the whole study area. We define it as the average of local open-space ratios in the whole study area. The average is obtained by averaging the local cumulative and incremental open-space ratio functions given by Equations (1) and (2). In mathematical terms, they are written as:

$$G_{\mathrm{C}}\left(x|B_1, \dots , B_n\right)=\frac{1}{n}{\sum }_{i=1}^n\left[O\left(x|B_i\right)\right]/\left[B_f\left(x|B_i\right)\right] ,0\le x\le x_{\mathrm{m}\mathrm{a}\mathrm{x}},$$

(3)

$$G_{\mathrm{I}}\left(x_j|B_1, \dots , B_n\right) =\frac{1}{n}{\sum }_{i=1}^n\left[O\left(x_j,x_{j+1}|B_i\right)\right]/\left[R\left(x_j,x_{j+1}|B_i\right)\right], j=\mathrm{0,1},\dots , m.$$

(4)

We call $G_{\mathrm{C}}\left(x|B_1, \dots , B_n\right)$ the global cumulative open-space ratio function and $G_{\mathrm{I}}\left(x_j|B_1, \dots , B_n\right)$ the global incremental open-space ratio function, respectively. Note that the terms local and global are also used in the K function [19,20]. Because these global functions are obtained from the sum of local functions (variables), we can obtain their standard deviations as a function of the x-width buffer zone or ring. These functions are useful to determine which width zone or in which ring the difference among local open-space ratios of buildings is small or large.

The implementation of the local and global open-space ratio functions is illustrated by the flows in Figure 2 and Figure 3. These figures illustrate the calculation of a single building. In practice, we deal with hundreds of buildings. Such a calculation is very difficult to do by hand. Fortunately, the standard GIS provides a processing modeler, such as the graphic modeler of QGIS and the model builder of ArcGIS. The steps are shown in Appendix A. Using this computational procedure with the empirical data shown in Figure 1, we test whether the open-space ratio functions are obtained in practice and consider the implications of the results with respect to the questions Q1, Q2, and Q3.

The thick black curve in Figure 4a indicates the cumulative local open-space ratio function of Building $B_{21}$ in Figure 1, and the thick gray curve indicates that of Building $B_{130}$. The vertical axis in Figure 4a shows the open-space ratio, and the horizontal axis shows the x-width (meter) of a buffer zone. The 0-width buffer zone means a building itself, so the 0-width buffer zone does not include any open space. Therefore, every local open-space ratio function starts at the origin (0, 0). The upper dotted horizontal straight line in Figure 4a shows an average open-space ratio of 0.72 in the study area.

As x-width becomes larger than 0, the local open-space ratio functions of $B_{21}$ and $B_{130}$ show different curves. However, it is interesting to note that both curves approach the average ratio of 0.72 when the width of their buffer zones is 50 m. As discussed in the following section, this phenomenon is the same for almost all buildings in this study area. It is a little bit surprising to know that in this study area, the open-space environment around each building measured by the open-space ratio is almost the same in the x > 50 m wide buffer zone. Such a critical bandwidth c exists in any region although the value of c varies from region to region. In the buffer zone where x > c, MAUProblem [15] does not exist. To examine the difference in the open-space ratio between buildings, we have to focus on the x < 50 m wide buffer zone. In this zone, the open-space ratio varies with respect to x. We should keep in mind that the result obtained from the analysis with a specific zone width varies according to the width (MAUProblem). The thick gray curve in Figure 4a shows that the open-space ratio of Building $B_{130}$ is the highest 0.88 in the 13 m bandwidth zone, which is much higher than the average open-space ratio. On the other hand, the thick black curve in Figure 4a shows that the open-space ratio of Building $B_{21}$ is always below the average open space. The buildings in this study area are almost all residential houses. Therefore, we may evaluate the open-space ratio viewing from a residential open-space control of the Building Standard Act. In this study area, the lower bound of the open-space ratio is 50%. Comparing the lower horizontal straight dotted line in Figure 4a with the open-space ratio functions of Buildings $B_{21}$ and $B_{130}$, both buildings satisfy this condition in the x > 3.5 m wide buffer zone. The open-space environment of both houses is fairly good in terms of the open-space ratio. In examining open space, we often want to compare differences in open-space ratio between paired buildings. For example, a family who are looking for a rental house chooses the best house by comparing their current house with an alternative house. For this purpose, Figure 4b is useful. The thick curved line indicates the difference in the open-space ratio between Buildings $B_{21}$ and $B_{130}$, and the horizontal dotted straight line indicates that the difference is 0.1. From this figure, a distinct difference in the open-space ratio is between the 3 m to 31 m bandwidth zones. The difference is negligibly small outside this zone.

Up to here, we discussed empirical findings obtained from the cumulative open-space ratio function defined by Equation (1). This function has a macroscopic perspective in the sense that it deals with the open-space ratio within a buffer zone around a building, but it does not show how open space is distributed over that zone. Alternatively, the incremental open-space ratio function defined by Equation (2) has a microscopic perspective in the sense that it deals with the open-space ratios within respective 1 m wide buffer rings constituting the 5 m wide buffer zone in Figure 1 and Figure 3. Figure 5 shows the incremental open-space ratio functions of Buildings $B_{21}$ and $B_{130}$. Comparing Figure 4 and Figure 5, we can visually see that the difference between the cumulative open-space ratio function and the incremental open-space ratio function is large, implying that these functions show different complementary aspects of open-space ratios.

Both curves in Figure 5a start at 1.0. This means that both buildings have no other buildings within the 1 m area surrounding Buildings $B_{21}$ and $B_{130}$. A distinct difference, which we cannot clearly see from the cumulative function in Figure 4, appears in the 2nd to 10th rings. The gray curve remains at 1.0 in these rings, indicating that there are no buildings in the 10 m wide zone around Building $B_{130}$. The open-space ratio of Building $B_{21}$ is lower than 50% in the 8th to 12th rings. This implies that the open-space environment in these rings is not good enough. In addition, Figure 5a shows that the gray and black curves are not so different in the 17th to 49th rings. In fact, Figure 5b shows that the difference is less than 0.1 in those rings. The distribution of open space is almost the same in the area farther than 49 m from these houses. Compared with the cumulative open-space function shown in Figure 4b, the graph in Figure 5b is useful for a family with a 5-year-old child who lives in House $B_{130}$ and is looking for a new house by paying close attention to open space around their current house.

2.3. Distinctly Different Classes of Open-Space Ratios

The open-space ratio varies from building to building in a region. When we study the open space environment created by the mix of buildings with different open-space ratios in a region, we classify the buildings into similar open-space ratios and observe the distribution of buildings belonging to the same class on a map. In this case, we want to have distinctly different classes of open-space ratios. In this subsection, we present a method for finding such classes.

Figure 6a shows the global cumulative open-space ratio function defined by Equation (3), i.e., the average of local cumulative open-space ratios of all buildings in the whole study area. If the standard deviation is less than 0.05, the difference in open-space ratios is negligibly small. The dotted horizontal line in Figure 6b shows that the open-space ratios in the x > 42 m wide buffer zones are almost the same as 0.72. Therefore, to examine the difference in the open-space ratio among buildings, the x < 42 m wide buffer zones must be examined. Figure 6b shows that the maximum standard deviation is achieved at the 15 m wide buffer zone. This means that diverse open-space ratios of buildings are observable in the 15 m wide buffer zone. Therefore, if buildings are classified by their open-space ratios in a 15 m wide buffer zone, the resulting classes are well distinguishable.

Given the open-space ratios in the 15 m wide buffer zone and the number of classes is four, we used the Ward hierarchical cluster method [21]. The resulting classified buildings are indicated by color in Figure 7. Buildings with the same color share similar local open-space ratios. The buildings colored red are those with the lowest open-space ratio and the buildings colored green are those with the highest open-space ratio. The map in Figure 7 is useful for discussing the open space environment created by the mix of buildings with different open-space ratios in a district.

2.4. Open-Space Ratio Function of a Clump of Buildings

Until now, we have considered the open-space ratio function of a single building. In this section, we show a method for obtaining the open-space ratio function of a clump of buildings, such as a housing complex represented by the gray polygons in Figure 1. The first step is to obtain the clumps of buildings. The exact method is as follows: first, to construct the area Voronoi diagram (Section 3.6 in [22]); second, to construct the area Delaunay diagram; third, to construct the MPT for polygons. Currently, the first computation cannot be carried out by the operators of standard GIS, although it can be performed by an open-source program [23]; however, there are very few easy-to-access software that are able to perform the second and third computations. Alternatively, we propose a simple method using a function of the open-space ratio function. We generate the buffer zone with $x$-width for $x=x_1, { x}_2, \cdots$ with equal intervals. The choice of the interval depends on the precision level. An example is illustrated in Figure 8. As the width increases (Figure 8a–h), the number of clumps decreases. In the end, all buildings form one clump (Figure 8f). This procedure results in hierarchical clumps. It is time-consuming work to generate $x=x_1, { x}_2, \cdots$ by hand but can be easily performed by a processing modeler provided by standard GIS. The outline of this procedure is shown in Appendix A. Using this method, we obtained the clumps of buildings in the case of 6 m wide buffer zones in Figure 9. The incremental open-space function can be obtained from Equation (2) by just replacing $B_i$ with the union of the buildings forming a clump. The result is illustrated in Figure 10. As expected, the open-space ratio is high (1.0) within 10 m around the clump.

3. Discussion

In this section, we discuss the board applicability of the open-space ratio functions. We also discuss how to overcome the limitations of the open-space ratio functions. In the previous sections, we have explained the methods of the open-space ratio functions, assuming that the open space is defined as the area not covered by building footprints, called the crude open space. The crude open space consists of many types of open spaces, and specific open-space ratios are defined. Examples include green open-space ratio [24,25], green coverage ratio [26,27], public open-space ratio [28,29], space ratio [30], park ratio [31,32], water surface ratio [33,34], river ratio [35,36], and road ratio [37,38]. These open-space ratio functions are easily computed using the crude open-space ratio functions formulated in the previous section. Let O_k,$k=1, \dots , n$ be a specific type of open space, such as roads O₁, parking lots O₂, parks O₃, playgrounds O₄, groves O₅, lakes O₆, rivers O₇, and so on. Suppose that the k-th type open space consists $m_k$ polygons O_k = ${\bigcup }_{j=1}^{m_k}{O}_{kj}$, e.g., the park open space consists of $m_k$ parks. Then, the k-th type open-space ratio functions $L_{\mathrm{C}k}\left(x|B_i\right),$ $k=1, \dots , n$ are obtained by simply replacing O = S*$\setminus {\bigcup }_{j=1}^{n^{*}}{B}_j$ with O_k = ${\bigcup }_{j=1}^{m_k}{O}_{kj}$ in Equations (1)–(4). The computational method is the same.

The open-space function is a univariate function for a certain type of open space. To comprehensively analyze the open-space environment, we should consider the open space of several types and examine their interactions. For this analysis, we can readily formulate a multi-variate functions of n_k types ${\mathit{L}}_{\mathrm{C}}\left(x\right|B_i$) = ($L_{\mathrm{C}1}\left(x|B_i\right),\cdots , L_{\mathrm{C}{n}_k}\left(x\right|B_i\left)\right)$. Using this function, open-space environments can be discussed more comprehensively through its covariance matrix.

In addition to many kinds of the open-space ratio functions, our method can also be directly applied to the building-coverage ratio, which is often used in building studies. Since [building-coverage ratio] = 1 − [open-space ratio], the building-coverage ratio function is formulated by simply replacing a given open space with buildings. This function allows us to analyze the distribution of the building-coverage area around individual buildings. Buildings have many uses such as offices, retail, and public facilities. The valuation of open space varies from use to use. Our method can directly analyze the distribution of parking lots (a specific open space) around buildings of a certain use by simply restricting buildings in terms of their use, such as the parking space around supermarkets and hospitals. Furthermore, the base polygons $B_j, j=1, \dots , n$ are not necessarily buildings but also other facilities. For example, the buildings $B_j, j=1, \dots , n$ in the original function can be replaced by roads (long narrow polygons) without changing the mathematical equations of the original function. With this function, we can analyze the distribution of open space on the side of roads. Also, by replacing a certain type of open space with a certain type of building, we can examine the area covered by retail stores around each house.

In recent years, residents have been concerned about walkability to open space, and many studies analyze walkability. For example, Yen et al. [39] analyzed open spaces for sports within walking distance. Khatavkar et al. [40] examined walking access to parks, playgrounds, and streets as play areas near their homes. Yang et al. [41] surveyed the distribution of parks and green space around a community of elderly people. Kesarovski et al. [42] examined 10 min isochrones for walking and cycling centered on homes. These studies do not use the Euclidean distance but the shortest-path distance. Since our method assumes the Euclidean distance, the method cannot be directly applied to these studies. It is expected that the open-space ratio function will be developed assuming the shortest-path distance using the method developed by Okabe et al. [43].

Our method considers open space in the two-dimensional space. To discuss the open space environment in built-up areas, three-dimensional open space surrounding buildings is an important factor, as shown in Cherian et al. [6], Gamero-Salinas et al. [7], and Usui [44]. A challenging attempt is to extend the two-dimensional open-space function to the three-dimensional open-space function.

4. Conclusions

In the studies of open space in urbanized areas, an open-space ratio is often used as a first basic metric to examine the distribution of open space. Originally, the open-space ratio was defined as the ratio of the area of open space (unbuilt area) to the area of its corresponding building. In recent years, residents have become more concerned with the open-space ratio in the broader neighborhoods of their individual buildings than with the open-space ratio in their own building sites. To address this concern, we have formulated the local and global open-space ratio functions and implemented it using standard GIS operators. The local and global open-space ratio functions analyze the following problems.

First, because this function shows the change in the open-space ratio across a zone from small to large width, we can check whether the notorious modifiable area unit problem [15] exists. If the ratio changes, we must keep in mind that the results may change depending on the choice of a zone width. Second, as the ratio changes, the function shows in which bandwidth zone the ratio is the highest or lowest. This tells us which zones have the best or worst open space environment as measured by the open-space ratio. Third, when comparing between two buildings, the function shows in which bandwidth zone the difference is the largest or smallest. We can discuss the difference in open space between buildings in detail. This is useful for a family who are looking for a rental house as they can choose the best house by comparing their current house with an alternative house. Fourth, the function shows in which bandwidth zone the variance of the open-space ratios among all buildings in a region is the largest. This result shows us in which bandwidth zone the open space environment differs most among buildings in a region. Fifth, in the resulting bandwidth zone, we classify buildings into several classes according to the similarity of their open-space ratios and show their spatial distribution on a map. This map is useful for discussing the open space environment created by the mix of buildings with different open-space ratios in a district. Sixth, our method can deal with the open-space ratio not only around each building but also the around a clump of closely placed buildings, such as a housing complex. We can evaluate the open space environment around a housing complex in terms of the open-space ratio. Seventh, the open-space ratio function has broad applicability. It can be applied to specific open spaces, such as green spaces, public open spaces, parks, water surfaces, roads, and so on using the same computational method.

The open-space ratio function has limitations. Since the function assumes a Euclidean distance, it cannot analyze the walking access to open space. Since the function assumes two-dimensional open space, it cannot analyze the volumetric open space around a building in high-rise areas. We expect these limitation to be overcome in the future.