# Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Models

#### 2.1. Generalized Entropy and Fractal Dimension

^{b}f(x), where f(x) represents a function of variable x,

**T**denotes an operator of dilation-contraction transform (scaling transform), λ refers to scale factors, and b is the scaling exponent. In mathematics, if a transform

**T**is applied to a function f(x), and the result is the function f(x) multiplied by a constant C (e.g., C = λ

^{b}), then we will say that the function f(x) is the eigenfunction under the transform

**T**, and the constant C is the corresponding eigenvalue. The concept of eigenfunctions is a generalization of eigenvector in linear algebra. This implies that a fractal model is just an eigenfunction of scaling transform, and the fractal dimension is associated with the eigenvalue λ

^{b}. The solution to the functional equation is always a power function. Thus, a fractal is often formulated by a power law.

_{1}r

^{−D}, in which r is the scale of measurement, e.g., the linear size of boxes, N(r) is the number of fractal copies based on the scale r, e.g., the number of non-empty boxes, N

_{1}refers to the proportionality coefficient, and D refers to the fractal dimension. Based on the box-counting method, the fractal parameter satisfies the following condition: d

_{T}< D < d

_{E}, where d

_{T}refers to the topological dimension of a fractal object, and d

_{E}is the Euclidean dimension of the embedding space in which the fractal object exists.

_{i}is the growth probability of the ith fractal unit, r

_{i}is the linear size of the ith fractal unit, q denotes the order of moment, and the exponent D

_{q}represents the generalized correlation dimension [18]. For a monofractal, i.e., a simple self-similar fractal, we have, D

_{q}≡ D

_{0}; for a self-affine fractal, different directions have different fractal dimension values, and for a given direction, we have D

_{q}≡ D

_{0}. However, a multifractal system is more complex. Different parts of a multifractal system have different characters, and can be described with different fractal dimension values. To simplify the process of spatial measurement, the varied linear scales r

_{i}can be substituted with a unified scale r. For example, based on the box-counting method, the unified scale, r, can be represented by the linear sizes of boxes. Thus, Equation (5) can be re-written as [17,18]:

_{q}(r) represents the qth order Renyi entropy based on scale r, that is:

_{q}is termed a generalized correlation dimension describing the global features of multifractal sets, then f(α) can be termed the generalized information dimension reflecting the local features of the multifractals. It can be proved that the Renyi entropy, generalized Shannon entropy, and mixed entropy can be connected by the Legendre transform.

#### 2.2. Scale Dependence and Entropy Conservation of Fractal Urban Systems

_{i}values. For a homogeneous system (say, a regular monofractal object), if we enlarge the size of the study area, the entropy value will increase, but the location has no significant influence on the result. Meanwhile, for a heterogeneous system (say, a random multifractal object), both the size and location of the study area will impact the entropy values: different area sizes indicate different element numbers (N), and different locations imply different probability distribution patterns of elements (P

_{i}).

**Step 1**: entropy H = 0; fractal dimension D = 0. For a point, the fractal dimension value can be obtained by L’Hospital’s rule.**Step 2**: entropy H = ln(5) =1.6094 nat; fractal dimension D = ln(5)/ln(3) = 1.465.**Step 3**: entropy H = ln(25) = 3.2189 nat; fractal dimension D = ln(25)/ln(9) = 1.465.**Step 4**: entropy H = ln(125) = 4.8283 nat; fractal dimension D = ln(125)/ln(27) = 1.465.- ….

**Step 1**: entropy H = 0; fractal dimension D = 2. For a surface, the fractal dimension can be obtained by L’Hospital’s rule.**Step 2**: entropy H = ln(5) = 1.6094 nat; fractal dimension D = −ln(5)/ln(1/3) = 1.465.**Step 3**: entropy H = ln(25) = 3.2189 nat; fractal dimension D = −ln(25)/ln(1/9) = 1.465.**Step 4**: entropy H = ln(125) = 4.8283 nat; fractal dimension D = −ln(125)/ln(1/27) = 1.465.- ….

**Step 1**: entropy H = 0; fractal dimension D = 0.**Step 2**: entropy H = −ln(1/17)/17 − 4 × 4 × ln(4/17)/17 = 1.5285 nat; box dimension D = ln(1/17)/ln(1/5) = 1.7604.**Step 3**: entropy H = −ln(1/289)/289 − 8 × 4 × ln(4/289)/289 − 16 × 16 × ln(16/289)/289 = 3.0569 nat; box dimension D = ln(1/289)/ln(1/25) = 1.7604.

#### 2.3. Entropy-Based Fractal Dimension Analysis

_{q}into global correlation dimension D

_{q}and mass exponent τ(q). It is easy to define global multifractal dimensions based on Renyi entropy, which are applied to global spatial analyses. The global parameters comprise the generalized correlation dimension and mass exponent. See Equations (1)–(5). (2) Convert the global parameters into local multifractal parameters by Legendre transform. The local parameters, including the local fractal dimension f(α) and the corresponding singularity exponent α(q), can be used to make partial spatial analysis. See Equations (6)–(10). (3) Substitute the spatial analysis by moment order analysis. In practice, it is difficult to distinguish the different spatial units of a random multifractal object from one another. A clever solution is to use moment analysis to replace local analysis. Mapping the parameter information of different spatial units into different orders of moment, q, we will have multifractal parameter spectrums. A multifractal spectrum based on moment orders can be treated as the result of local scanning and sorting for a complex system [11,34].

## 3. Empirical Analysis

#### 3.1. Study Area and Methods

**Step 1**: Defining an urban boundary based on the recent image. The most recent material we used was the remote sensing image of 2015. Based on this image, the boundary of Beijing city can be identified by using the “City Clustering Algorithm” (CCA) developed by Rozenfeld et al. [40,41]. The urban boundary can be called an urban envelope [15,32]. Then, a measure area can be determined in terms of the urban envelope [8].**Step 2**: Extracting the spatial datasets using the function box-counting method. First of all, we can extract the dataset from the image of the recent year (2015). A set of boxes is actually a grid of rectangular squares, each of which has an area of urban land use. The area may be represented by the pixel number. Therefore, in the dataset, each number represents a value of land-use area of the urban pattern falling into a box (square). Changing the linear size of the boxes, we will have different datasets. The box system forms a hierarchy of grids, which yield a hierarchy of spatial datasets. Applying the system of boxes to the images in different years, we have different datasets for calculating spatial entropy and fractal dimensions.**Step 3**: Calculate the spatial Renyi entropy and generalized Shannon entropy. Using Equations (5)–(8), we can compute the generalized entropies of urban land use based on a given linear size of functional boxes. For each linear size of boxes, we can obtain a Renyi entropy value or a generalized Shannon entropy value for Beijing’s urban form. For each year, we have a number of sets of entropy values based on different linear sizes of boxes. If the entropy values based on different box sizes have no significant differences, we can utilize the generalized entropy values to conduct a spatial analysis of urban form and growth.**Step 4**: Computing the multifractal parameter spectrums. If the entropy values depend heavily on the linear sizes of boxes, we should transform the Renyi entropy into the generalized correlation dimension using Equation (4). For different linear sizes of boxes r, we have different Renyi entropy values, which are defined as M_{q}(r). As shown by Equation (5), there is a linear relation between ln(r) and M_{q}(r). Similarly, we can convert the generalized Shannon entropy values into local multifractal dimension using Equations (7) and (8). By using Legendre transform, as shown in Equations (11) and (12), a complete set of multifractal parameters can be obtained, and multifractal spectrums can be generated. The computational and analytical process can be illustrated as follows (Figure 4).

_{q}, which is just the slope of the semi-logarithmic equation. It should be noted that the regression equation has no intercept [34]. If q = 1, Equations (4) and (5) will be invalid. In this case, according to the L’Hospitale rule, the Renyi entropy will be replaced by the Shannon entropy, that is:

#### 3.2. Results and Findings

_{0}(r) value depends significantly on the linear sizes of boxes r (Figure 5, Table 3). In other words, the spatial Renyi entropy values of Beijing urban land use rely on the scales of measurement. Based on different linear sizes of boxes, the entropy values are different. In particular, the average value of the spatial entropy is invalid, because the mean depends on the size of datasets. That is to say, changing the range of the linear sizes of boxes yields different average values of Renyi entropy. Using Legendre transform, we can evaluate the corresponding generalized Shannon entropy and the mixed entropy.

_{0}(r) values into a capacity dimension D

_{0}value; For q = 1, we can transform a series of Shannon information entropy M

_{1}(r) values into an information dimension D

_{1}value; for q = 2, we can transform a series of Renyi correlation entropy M

_{2}(r) values into a correlation dimension D

_{2}value. For an arbitrary order of moment q, we can transform Renyi’s generalized entropy M

_{q}(r) values into a set of generalized correlation dimension D

_{q}values. Apparently, for a given order of moment, say, q = 1, the fractal dimension D

_{1}value is independent of the linear sizes of boxes r (Figure 6). Using Equation (3), we can convert the generalized correlation dimension D

_{q}values into the mass exponent τ

_{q}values. As indicated above, the generalized correlation dimension D

_{q}and mass exponent τ

_{q}belong to the global parameters of multifractal models. By means of Legendre transform, Equations (11) and (12), we can transform the global parameters into local parameters, including the singularity exponent α(q) and the corresponding fractal dimension f(α(q)) (Table 4). Based on the global parameters, we have the global multifractal spectrum, i.e., D

_{q}-q spectrums (Figure 6); based on the local parameters, we have the local multifractal curves (Figure 7), and f(α)−α spectrums (Figure 8). The local spectrum is often termed an f(α) curve in the literature [18]. In practice, we can compute the local parameter values by using the normalized measure method first [26,27]. Then, using Legendre transform, we can convert the local parameter values into the global parameter values [29,34,43].

**First, Beijing’s space-filling speed was too fast, and space-filling extent was too high.**From 1984 to 1994 to 2006 and then to 2015, the capacity dimension D_{0}values increased from 1.6932 to 1.8011 and 1.8877 to 1.9346. By means of the formula v = D_{0}/2, we can calculate the space-filling rate of urban form [31,44], v; the results were 0.8466, 0.9005, 0.9439, and 0.9673. In recent years, the level of space filling is close to the upper limit of one.**Second, spatial heterogeneity became weaker and weaker**. From 1984 to 1994 and 2006 to 2015, the information dimension D_{1}values went up from 1.6048 to 1.7467 and 1.8468 to 1.9081. Using the formula u = 1 − D_{1}/2, we can calculate the spatial redundancy rate of urban form, u; the results were 0.1976, 0.1267, 0.0766, and 0.0459. The spatial redundancy rate is in fact an index of spatial heterogeneity. A reduction of redundancy indicates a weakening process of spatial heterogeneity.**Third, the urban growth of Beijing is characterized by stages.**In the mass, the space-filling speed in the central area was obviously faster than that of the edge area (Figure 6 and Figure 7). Where the global feature is concerned, the characteristics are as below: From 1984 to 1994, the land-use speed in the central urban area was significantly higher than that in the fringe area; From 1994 to 2006, the gap of land-use speed between the central and peripheral areas decreased; from 2006 to 2015, the speed of land use in the central and peripheral areas was close to equilibrium (Figure 6b). Where the local level is concerned, the features are as below: From 1984 to 1994, the land-use speed in high-density areas was significantly higher than that in low-density areas. From 1994 to 2006, the situation reversed, and the land-use speed in low-density areas was significantly higher than that in high-density areas. From 2006 to 2015, the land-use speed in high-density areas was once again higher than that in low-density areas (Figure 7b).**Fourth, the growth of Beijing city is of outward expansion**. On the whole, the closer to the center area, the faster the space-filling speed will be. In terms of local fractal spectrums, city development can be classified into two types: one is central aggregation, and the other is peripheral expansion [43]. The difference can be reflected by the local multifractal spectrums. The unbalance of urban spatial expansion leads to the asymmetry of f(α) curves. If the urban development is centralized, the peak of the spectral curve tilts to the right; on the contrary, if the urban development is characterized by periphery diffusion, the peak of the spectrum inclines to the left [43]. The peak values of Beijing’s f(α) curves are obviously left-sided, which imply that Beijing’s development is mainly a process of expanding to the periphery (Figure 8).**Fifth, there was redundant correlation in Beijing’s urban fringe**. Generally speaking, the generalized correlation dimension value lies between zero and two. However, when the order of moment q approaches negative infinity, the D_{q}values exceeded two, and became bigger and bigger (Figure 6). This suggests that there are too many messy patches of land use to fill the urban fringe.**Sixth, the quality of spatial structure of Beijing city declined**. A local multifractal spectrum is supposed to be a smooth single-peak curve. In 1984, the local fractal dimension spectral lines were regular. However, from 1995 to 2015, the f(α) curves deviated more and more from the normative spectral line (Figure 8).

## 4. Discussion

_{0}, information dimension D

_{1}, and correlation dimension D

_{2}, are discussed in this research, but the results have not been generalized to multifractal parameter spectrums. In another companion paper, based on area radius scaling, the normalized Renyi entropy is generalized to multifractal spectrums [25]. Two sets of multifractal indicators are proposed to describe urban growth and form. The mathematical modeling based on characteristic scales and the spatial analysis based on scaling are integrated into a logical framework. Compared with the previous studies, this work bears three new points. First, the scale dependence of spatial Renyi entropy and generalized Shannon entropy is illustrated by the box-counting method. Changing the linear sizes of boxes yields different entropy spectral curves for the Renyi entropy and generalized Shannon entropy. It is complicated to conduct a spatial analysis of cities using these curves of entropy spectrums. Second, the solution to the scale-dependence problem of spatial entropies is clarified. Transforming the Renyi entropy into global multifractal parameters and converting the generalized Shannon entropy into local multifractal parameters, the different entropy values based on different measurement scales will be replaced by two fractal dimension values, which are actually characteristic values of generalized spatial entropies and independent of scales of measurement. Third, similarities and differences between spatial entropy and fractal dimension spectrums are illustrated. Spatial entropy is simple and easy to understand, but it cannot be used to describe the spatial heterogeneity of city systems. In contrast, using multifractal parameter spectrums, we can characterize the spatial heterogeneity of urban forms and urban systems. Unfortunately, multifractal spectrums are not suitable for non-fractal systems. The main shortcomings of this work rest with two aspects. First, the empirical analysis is chiefly based on the box-counting method. The other methods, such as the sandbox method, growing cluster method, and so on, are not taken into account for the time being. All of these methods can be applied to the studies on fractal cities. Second, the research method is confined to the fractal cities that are defined in the two-dimensional embedding space. If we take the third dimension of urban space, the measurements and subsequent calculations are significantly limited. The solution to this problem is to develop a three-dimensional box-counting method of fractal dimension estimation. What is more, the uncertainty of fractal dimension is not discussed. The fractal dimension values of urban form and urban systems depend on the size and central location of a study area.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A regular growing monofractal that bears an analogy with urban growth. (

**a**) Monofractal growth; (

**b**) Monofractal generation. Note: A monofractal possesses only one scaling process, and is also termed “unifractal” in literature. Figure 1a represents the variable scale of measurement based on the variable size of the study area, and Figure 1b represents the variable scale of measurement based on a fixed size of the study area.

**Figure 2.**A regular growing multifractal that bears an analogy with urban growth. Note: To illustrate the multifractal, Vicsek [13] proposed this fractal, with two different scales in the generator.

**Figure 3.**Four typical images of Beijing’s urban land-use patterns. (

**a**) 1984; (

**b**) 1994; (

**c**) 2006; (

**d**) 2015.

**Figure 4.**A flow chart of spatial analysis for cities from spatial entropy to multifractal spectrums. Note: Spatial entropy can be used to make spatial analysis of cities based on characteristic scales, while multifractal spectrums can be employed to make spatial analysis based on scaling in cities.

**Figure 5.**The Renyi entropy spectrums based on moment order parameter and different spatial scales of measurement. Note: From the bottom to the top, the linear size of functional boxes are r = 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, and 1/256, respectively. Different linear sizes of the boxes represent different spatial scales of measurements, and different Renyi entropy spectral lines based on different box sizes reflect the scale-dependence of spatial entropy. (

**a**) 1984; (

**b**) 1994; (

**c**) 2006; and (

**d**) 2015.

**Figure 6.**The global multifractal spectrums based on moment order parameter and change curves of global dimensions. Note: Based on different linear sizes of functional boxes, i.e., r=1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, and 1/256, and the corresponding Renyi entropy M

_{q}(r) values, the generalized correlation dimension D

_{q}can be evaluated. Multifractal parameter values depend on moment order q, but are independent of spatial scale r. (

**a**) Global multifractal dimension spectrums; (

**b**) Global fractal dimension changes.

**Figure 7.**The local multifractal spectrums based on moment order parameter and change curves of local dimensions. Note: Based on different linear sizes of functional boxes, i.e., r = 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, 1/128, and 1/256, and the corresponding generalized Shannon entropy H

_{q}(r) values, the generalized information dimension f(α) can be evaluated. (

**a**) Local multifractal dimension curves; (

**b**) Local fractal dimension changes.

**Figure 8.**The local multifractal spectrums based on singularity exponent, i.e., the f(α) curves. Note: A local multifractal spectrum is a unimodal curve, which can be used to reflect the aggregation or diffusion of a city’s evolution. (

**a**) 1984; (

**b**) 1994; (

**c**) 2006; (

**d**) 2015.

**Figure 9.**Two cases of spatial entropy analyses transformed into fractal dimension analyses. Note: The problem of scale dependence of entropy measurement can be solved by transforming the generalized entropies into fractal dimensions by means of scaling relation, and the property of spatial heterogeneity can be characterized by multifractal parameter spectrums.

Step for Fractal Generation | Figure 1a: Variable Size of Study Area and Measurement Scale | Figure 1b: Fixed Size of Study Area and Variable Measurement Scale | ||
---|---|---|---|---|

Entropy (nat) | Fractal Dimension | Entropy (nat) | Fractal Dimension | |

1 (outlier) | 0 | 0 | 0 | 2 |

2 | 1.6094 | 1.4650 | 1.6094 | 1.4650 |

3 | 3.2189 | 1.4650 | 3.2189 | 1.4650 |

4 | 4.8283 | 1.4650 | 4.8283 | 1.4650 |

… | … | … | … | … |

m | ln(5^{m}^{−1}) | ln(5^{m}^{−1})/ln(3^{m}^{−1}) | ln(5^{m}^{−1}) | ln(5^{m}^{−1})/ln(3^{m}^{−1}) |

**Note**: Different steps reflect different levels in a fractal hierarchy.

Step for Fractal Generation | Global Feature | Local Features | ||||
---|---|---|---|---|---|---|

Central Part | Peripheral Parts | |||||

Entropy (nat) | Fractal Dimension | Entropy (nat) | Fractal Dimension | Entropy (nat) | Fractal Dimension | |

1 (outlier) | 0 | 0 | 0 | - | 0 | - |

2 | 1.5285 | 1.7604 | 0.1667 | 1.7604 | 1.3618 | 1.5791 |

3 | 3.0569 | 1.7604 | 1.5285 | 1.7604 | 1.5285 | 1.5791 |

Moment Order q | Generalized Entropy Based on Different Scales r | |||||||
---|---|---|---|---|---|---|---|---|

r = 1/2 | r = 1/4 | r = 1/8 | r = 1/16 | r = 1/32 | r = 1/64 | r = 1/128 | r = 1/256 | |

−20 | 1.8225 | 3.6449 | 5.4674 | 7.2898 | 9.1123 | 10.9347 | 12.7572 | 14.5797 |

−15 | 1.8045 | 3.6089 | 5.4134 | 7.2179 | 9.0223 | 10.8268 | 12.6313 | 14.4357 |

−10 | 1.7702 | 3.5405 | 5.3107 | 7.0809 | 8.8511 | 10.6214 | 12.3916 | 14.1618 |

−5 | 1.6806 | 3.3612 | 5.0417 | 6.7223 | 8.4029 | 10.0835 | 11.7640 | 13.4446 |

−4 | 1.6430 | 3.2859 | 4.9289 | 6.5718 | 8.2148 | 9.8578 | 11.5007 | 13.1437 |

−3 | 1.5901 | 3.1801 | 4.7702 | 6.3602 | 7.9503 | 9.5403 | 11.1304 | 12.7204 |

−2 | 1.5123 | 3.0246 | 4.5369 | 6.0493 | 7.5616 | 9.0739 | 10.5862 | 12.0985 |

−1 | 1.4058 | 2.8116 | 4.2174 | 5.6232 | 7.0290 | 8.4348 | 9.8406 | 11.2464 |

0 | 1.3410 | 2.6820 | 4.0230 | 5.3640 | 6.7050 | 8.0460 | 9.3870 | 10.7280 |

1 | 1.3226 | 2.6452 | 3.9679 | 5.2905 | 6.6131 | 7.9357 | 9.2583 | 10.5809 |

2 | 1.3148 | 2.6296 | 3.9443 | 5.2591 | 6.5739 | 7.8887 | 9.2035 | 10.5183 |

3 | 1.3105 | 2.6209 | 3.9314 | 5.2419 | 6.5523 | 7.8628 | 9.1732 | 10.4837 |

4 | 1.3078 | 2.6155 | 3.9233 | 5.2311 | 6.5388 | 7.8466 | 9.1544 | 10.4621 |

5 | 1.3059 | 2.6119 | 3.9178 | 5.2237 | 6.5297 | 7.8356 | 9.1416 | 10.4475 |

10 | 1.3017 | 2.6035 | 3.9052 | 5.2069 | 6.5087 | 7.8104 | 9.1122 | 10.4139 |

15 | 1.3001 | 2.6003 | 3.9004 | 5.2006 | 6.5007 | 7.8008 | 9.1010 | 10.4011 |

20 | 1.2993 | 2.5986 | 3.8979 | 5.1972 | 6.4965 | 7.7957 | 9.0950 | 10.3943 |

**Note**: For q = 1, the numbers represent Shannon’s information entropy values.

Moment Order q | Fractal Parameter and Goodness of Fit | ||||||
---|---|---|---|---|---|---|---|

D_{q} | R^{2} | τ_{q} | α(q) | R^{2} | f(α) | R^{2} | |

−20 | 2.6293 | 0.8308 | −55.2143 | 2.7124 | 0.8113 | 0.9664 | 0.7092 |

−15 | 2.6033 | 0.8369 | −41.6527 | 2.7122 | 0.8115 | 0.9698 | 0.7112 |

−10 | 2.5539 | 0.8484 | −28.0929 | 2.7116 | 0.8120 | 0.9771 | 0.7159 |

−5 | 2.4246 | 0.8790 | −14.5473 | 2.7015 | 0.8116 | 1.0397 | 0.7361 |

−4 | 2.3703 | 0.8927 | −11.8515 | 2.6884 | 0.8082 | 1.0979 | 0.7423 |

−3 | 2.2940 | 0.9137 | −9.1759 | 2.6592 | 0.8012 | 1.1983 | 0.7535 |

−2 | 2.1818 | 0.9470 | −6.5454 | 2.5898 | 0.7984 | 1.3658 | 0.8074 |

−1 | 2.0281 | 0.9874 | −4.0563 | 2.3288 | 0.8782 | 1.7275 | 0.9683 |

0 | 1.9346 | 0.9992 | −1.9346 | 1.9791 | 0.9951 | 1.9346 | 0.9992 |

1 | 1.9081 | 1.0000 | 0.0000 | 1.9081 | 1.0000 | 1.9081 | 1.0000 |

2 | 1.8968 | 0.9998 | 1.8968 | 1.8888 | 0.9994 | 1.8807 | 0.9987 |

3 | 1.8906 | 0.9995 | 3.7812 | 1.8810 | 0.9986 | 1.8618 | 0.9959 |

4 | 1.8867 | 0.9992 | 5.6601 | 1.8773 | 0.9982 | 1.8489 | 0.9927 |

5 | 1.8841 | 0.9989 | 7.5363 | 1.8752 | 0.9979 | 1.8399 | 0.9900 |

10 | 1.8780 | 0.9982 | 16.9021 | 1.8720 | 0.9974 | 1.8181 | 0.9824 |

15 | 1.8757 | 0.9979 | 26.2599 | 1.8712 | 0.9973 | 1.8086 | 0.9784 |

20 | 1.8745 | 0.9978 | 35.6151 | 1.8709 | 0.9972 | 1.8031 | 0.9757 |

**Note**: The global parameter values are estimated using Equations (6)–(8), while the local parameters are estimated by means of the μ-weight method.

**Table 5.**The correspondence relationships between entropies and fractal dimensions for the spatial analysis of cities.

Parameter Level | Entropy | Fractal Dimension | |

Global parameter | General | Renyi entropy spectrum | Global multifractal dimension spectrum |

Special | Hartley entropy | Capacity dimension | |

Shannon entropy | Information dimension | ||

The second-order Renyi entropy | Correlation dimension | ||

Connection | Legendre Transform | ||

Local parameter | General | Generalized Shannon entropy spectrum | Local multifractal dimension spectrum |

Mixed entropy | Singularity exponent spectrum | ||

Special | Hartley entropy | Capacity dimension | |

Shannon entropy | Information dimension |

**Note**: The Hartley entropy and Shannon entropy represent the intersection of the Renyi entropy and the generalized Shannon entropy; thus, the capacity dimension and information dimension form the intersection of the global multifractal dimensions and local multifractal dimensions.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Chen, Y.; Huang, L.
Spatial Measures of Urban Systems: from Entropy to Fractal Dimension. *Entropy* **2018**, *20*, 991.
https://doi.org/10.3390/e20120991

**AMA Style**

Chen Y, Huang L.
Spatial Measures of Urban Systems: from Entropy to Fractal Dimension. *Entropy*. 2018; 20(12):991.
https://doi.org/10.3390/e20120991

**Chicago/Turabian Style**

Chen, Yanguang, and Linshan Huang.
2018. "Spatial Measures of Urban Systems: from Entropy to Fractal Dimension" *Entropy* 20, no. 12: 991.
https://doi.org/10.3390/e20120991