# Urban Dynamics, Fractals and Generalized Entropy

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## Abstract

**:**

## 1. Introduction

^{2}cells], establish a relation between these patterns and the set of social, economic and political changes that have taken place in NMAL - Northern Margin of the Metropolitan Area of Lisbon (and Portugal), during the last 40 years.

## 2. Fractal Dimension of Built-up Areas and GLSE

^{2}using the standard box-counting method. Here we explore the relation between the local fractal dimension (classified through the GLSE function) and the development of the built-up area of the Northern Margin of the Metropolitan Area of Lisbon (NMAL), for the period between 1960 and 2004. For clarity of reasoning we first summarize in the following sub-sections some methodological procedures already depicted in [8].

#### 2.1. Data Acquisition

^{2}) (see [8] for more details).

#### 2.2. Choice of Cell Size for NMAL

#### 2.3. Fractal Dimension of Built-Up Areas

_{k}squares (boxes) of side ε

_{k}= 2

^{k}(pixels, 1 pixel = 10 m) that cover the built-up area inside of the cell was computed. After m = 5 iterations k ϵ [0,…,m], the fractal dimension D will emerge as the slope of a linear regression of the duplets {N

_{k},ε

_{k}}, given that $\mathrm{log}{N}_{k}=-D\mathrm{log}{\epsilon}_{k}+c$, where c is constant (see [1] for a discussion of the quality of the fractal analysis). Given that the area occupied by the N

_{k}boxes reads ${A}_{k}={N}_{k}{\epsilon}_{k}^{2}$ it is possible to obtain the recurrence relation ${A}_{k}={A}_{k+1}{2}^{D-2}$. Taking the limit m into account, we obtain ${A}_{0}={A}_{m}{({2}^{m})}^{D-2}$. This can be used to write down the formula for the upper bound of the built-up area as a function of $D:U(D)={n}^{2}{({2}^{m})}^{D-2}$—where n is the side of each square cell, i.e., n = 100 pixels (or 1 km). The lower bound (L(D)) will be proportional to the upper bound U(D), such that L(D) = αU(D). Since L(0) = 1, we have that $\alpha ={({2}^{m}/n)}^{2}$ and $L(D)={2}^{mD}$.

#### 2.4. Computation of a Generalized Local Spatial Entropy Function (GLSE)

^{th}stage of the box-counting algorithm reads ${N}_{k}=A/{\epsilon}_{k}^{D}$ (assuming that c grows linearly with logA). Let us call $\Omega (A,D)$ the number of possible configurations within each cell characterized by a total built-up area A and fractal dimension D. In [8] it is shown that:

#### 2.5. Urban Growth through Fractal Dimension and a Generalized Entropy Function

**Figure 1.**Types of built-up areas. (Red line: S(D); Black line: S’(D); D: Fractal dimension. Each region type is illustrated in the insets (after [8]).

**Figure 2.**Region types growth rates (Left) and average annual growth rates (Right), for both time periods.

## 3. Types of Growth in NMAL

- (a)
- 286.2% growth in built-up area in existing patches (growth-type 2).
- (b)
- The decrease in the number of patches (−78.3%) by growth-type 3 (growth and aggregation), but with a 167.5% increase in built-up area.

- (a)
- 56.3% growth in built-up area in existing patches (growth-type 2).
- (b)
- The decrease in the number of patches (−78.8%) in growth-type 3 but with an increase in built-up area by 44.6%.
- (c)
- The number of new patches that have emerged (accounting for only 11% of total patches in 2004), that show a much weaker fragmentation process.

1960–1990 | N.° of patches | % | % of change | Built-up (m^{2}—millions) | % | % of change | ||||
---|---|---|---|---|---|---|---|---|---|---|

1960 | 1990 | 1960 | 1990 | 1960 | 1990 | 1960 | 1990 | |||

No change | 459 | 459 | 9.9 | 3.3 | 0.0 | 0.5 | 0.5 | 1.1 | 0.3 | 0.0 |

New patches (1) | − | 11,102 | − | 80.6 | − | − | 26.7 | − | 16.3 | − |

Growth (2) | 1666 | 1666 | 36.1 | 12.1 | 0.0 | 4.1 | 15.7 | 8.2 | 9.6 | 286.2 |

(4.6) | ||||||||||

Growth & Aggregation (3) | 2497 | 542 | 54.0 | 3.9 | −78.3 | 45.2 | 120.8 | 90.7 | 73.8 | 167.5 |

(−5.0) | (3.3) | |||||||||

Total | 4622 | 13,769 | 100 | 100 | 197.9 | 49.8 | 163.8 | 100 | 100 | 229.1 |

(3.7) | (4.1) | |||||||||

1990–2004 | N. ° of patches | % | % of change | Built-up (m^{2}—millions) | % | % of change | ||||

1990 | 2004 | 1990 | 2004 | 1990 | 2004 | 1990 | 2004 | |||

No change | 9432 | 9432 | 68.5 | 73.5 | 0.0 | 16.0 | 16.0 | 9.8 | 6.8 | 0.0 |

New patches (1) | − | 1406 | − | 11.0 | − | − | 6.0 | − | 2.5 | − |

Growth (2) | 1367 | 1367 | 9.9 | 10.7 | 0.0 | 14.0 | 21.9 | 8.6 | 9.2 | 56.3 |

(3.2) | ||||||||||

Growth & Aggregation (3) | 2970 | 629 | 21.6 | 4.9 | −78.8 | 133.7 | 193.4 | 81.7 | 81.5 | 44.6 |

(−10.5) | (2.7) | |||||||||

Total | 13,769 | 12,834 | 100 | 100 | −6.8 | 163.8 | 237.4 | 100 | 100 | 44.9 |

(−0.5) | (2.7) |

*****The values in ( ) show the average annual growth rate.

Types | % N.º patches | % of Built-up area | ||||
---|---|---|---|---|---|---|

1960 | 1990 | 2004 | 1960 | 1990 | 2004 | |

1 | 32.7 | 10.7 | 7.9 | 4.2 | 1.2 | 0.6 |

2 | 32.3 | 29.4 | 18.0 | 11.1 | 7.0 | 2.9 |

3 | 25.9 | 38.4 | 41.9 | 25.7 | 20.5 | 15.7 |

4 | 7.3 | 19.2 | 28.9 | 24.8 | 37.7 | 33.0 |

5 | 1.8 | 2.4 | 6.4 | 34.2 | 33.6 | 47.8 |

Total | 100 | 100 | 100 | 100 | 100 | 100 |

#### The Planning System...Neither Just in Case, Nor Just in Time

- (a)
- The first regional plan for the entire Metropolitan Area of Lisbon was the Plano Director de Desenvolvimento Urbanístico da Região de Lisboa - PDRL (Regional Master Plan). Its scope was defined by a new diploma in 1959, but it was never approved.
- (b)
- The non-approval of the PDRL weakened the idea of a regional strategy for urban growth [21] in that the administration limited itself to the evaluation and approval of allotment projects. Without the necessary planning instruments, suburbanization grew “without any supervision and guidance, boosted by improving accessibility and increasing rates of motorization, thus sharpening their effects within” [21].
- (c)
- Parallel to the non-approval of the PDRL, a new law was published in 1965 which enabled the participation of the private sector in the urban allotment processes. By enabling allotment processes in not only urban areas but also in rural areas it triggered the sprawl of built-up areas throughout Lisbon peripheries. Without a regional strategic plan that could orient their co-development, these areas grew based mainly on the strategies of housing market agents that sought profit maximization in the process of land conversion.
- (d)
- In 1976, Law No. 794/76 (Lei dos Solos), still in effect today, aimed at avoiding urban speculation and to solve the housing shortage problem. Although ambitious in some measures, as for example the delineation of areas aimed to control future land use changes near urban centers, it was never fully enforced.
- (e)
- In 1982, the Municipal Master Plan (Plano Director Municipal—PDM) was enacted, which would encompass and regulate, through zoning regulations, the entire territory of a municipality (previously only urban areas would have been the object of any kind of an urbanization plan).
- (f)
- However, before 1990 few municipalities in Portugal had their Master Plans approved. But the scenario changed when in that same year, Law No. 69/90 made them mandatory. Simultaneously, a municipality could not apply to European funds without an approved Master Plan. Following the general national trend, several Master Plans were approved for NMAL between 1993 and 1999. By 2000, the Metropolitan Area of Lisbon was finally covered by municipal plans although with little or no interrelationship among them, not even between neighboring municipalities.
- (g)
- In 1992, a new Regional Plan was developed but again not approved, as a result of political decisions. It was only in 2002 that a Regional Plan for the Metropolitan Area of Lisbon (PROTAML) was approved. In its strategic vision, the plan included several scales within MAL and between MAL and the national context [22].

## 4. Spatial Transitions among Region Types

**Figure 4.**Growth rates (Left) and average annual growth rates (Right), for built-up and non-built areas, by region type, 1960–1990 and 1990–2004.

- (a)
- Between 1960 and 1990, built-up areas in Types 2, 3 and 4 grew much more than non-built areas, as opposed to Types 1 and 5. These kinds of changes reflect mostly external transitions between region types rather than internal ones (see Table 3).
- (b)
- Between 1990 and 2004 and as NMAL entered a period of consolidation and compactification, both categories show similar growth rates, apart from Type 5 where non-built areas grew more than built-up ones, as new cells not completely developed entered this region type.

Area m^{2} | 1990 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

11 | 21 | 31 | 41 | 51 | 10 | 20 | 30 | 40 | 50 | Total | ||

1960 | 11 | 0.2 | 0.9 | 0.7 | 0.3 | 0.02 | 2.1 | |||||

21 | 0.01 | 0.7 | 2.9 | 1.4 | 0.4 | 5.5 | ||||||

31 | 0.1 | 3.3 | 7.8 | 1.8 | 13.0 | |||||||

41 | 0.1 | 4.4 | 8.3 | 12.8 | ||||||||

51 | 16.4 | 16.4 | ||||||||||

10 | 1.9 | 8.0 | 12.5 | 12.7 | 2.0 | 249.5 | 245.5 | 122.7 | 33.6 | 1.9 | 690.4 | |

20 | 0.004 | 1.8 | 10.9 | 12.7 | 6.3 | 1.0 | 57.9 | 113.0 | 33.2 | 5.2 | 242.1 | |

30 | 0.1 | 3.6 | 19.7 | 8.1 | 3.8 | 49.3 | 60.5 | 6.0 | 151.2 | |||

40 | 0.1 | 2.8 | 8.2 | 1.8 | 10.4 | 9.4 | 32.8 | |||||

50 | 2.4 | 3.6 | 6.0 | |||||||||

Total | 2058 | 2.1 | 11.6 | 34.2 | 61.8 | 54.1 | 250.5 | 307.2 | 286.8 | 137.8 | 26.2 |

Area m^{2} | 2004 | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

11 | 21 | 31 | 41 | 51 | 10 | 20 | 30 | 40 | 50 | Total | ||

1990 | 11 | 1.4 | 0.6 | 0.1 | 0.01 | 2.1 | ||||||

21 | 4.6 | 6.1 | 0.8 | 0.1 | 11.6 | |||||||

31 | 16.9 | 16.7 | 0.6 | 34.2 | ||||||||

41 | 0.1 | 30.2 | 31.6 | 61.8 | ||||||||

51 | 54.1 | 54.1 | ||||||||||

10 | 0.3 | 0.8 | 1.4 | 0.7 | 179.9 | 47.8 | 17.5 | 2.0 | 250.5 | |||

20 | 1.0 | 6.0 | 3.8 | 0.7 | 152.4 | 128.2 | 14.2 | 0.8 | 307.2 | |||

30 | 7.2 | 16.8 | 1.7 | 171.2 | 87.5 | 2.2 | 286.8 | |||||

40 | 0.01 | 10.1 | 18.1 | 0.9 | 70.3 | 38.3 | 137.8 | |||||

50 | 4.6 | 21.6 | 26.2 | |||||||||

Total | 1.7 | 7.0 | 37.9 | 79.1 | 111.5 | 179.9 | 200.2 | 317.9 | 174.1 | 63.0 | 1172.2 |

**Figure 5.**Two examples of cells that grew in built-up area but changed to a less compact type (4→3).

1960–1990 (%) | 1 | 2 | 3 | 4 | 5 | Total | 1990–2004 (%) | 1 | 2 | 3 | 4 | 5 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 36.3 | 36.7 | 19.6 | 6.7 | 0.6 | 100 | 1 | 71.9 | 19.4 | 7.6 | 1.1 | 100 | |

2 | 0.4 | 24.4 | 51.2 | 19.1 | 4.8 | 100 | 2 | 49.6 | 44.0 | 5.9 | 0.5 | 100 | |

3 | 2.4 | 34.2 | 53.7 | 9.7 | 100 | 3 | 60.9 | 37.7 | 1.4 | 100 | |||

4 | 4.4 | 38.7 | 56.9 | 100 | 4 | 55.4 | 44.1 | 100 | |||||

5 | 100 | 100 | 5 | 100 | 100 |

60–90 (%) | 11 | 21 | 31 | 41 | 51 | Total | 90–04 (%) | 11 | 21 | 31 | 41 | 51 | Total |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

10 | 5.0 | 21.5 | 33.8 | 34.2 | 5.5 | 100 | 10 | 10.7 | 23.8 | 44.6 | 20.9 | 100 | |

20 | 0.01 | 5.8 | 34.3 | 40.0 | 19.8 | 100 | 20 | 8.8 | 51.8 | 33.2 | 6.3 | 100 | |

30 | 0.2 | 11.5 | 62.6 | 25.7 | 100 | 30 | 28.1 | 65.2 | 6.8 | 100 | |||

40 | 1.0 | 25.4 | 73.6 | 100 | 40 | 35.7 | 64.3 | 100 | |||||

50 | 100 | 100 | 50 | 100 | 100 |

**Figure 10.**Informal settlements in NMAL, outside Lisbon city, by 1971 (after [24]).

## 5. Discussion and Conclusions

## Acknowledgments

## Conflict of Interest

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**MDPI and ACS Style**

Encarnação, S.; Gaudiano, M.; Santos, F.C.; Tenedório, J.A.; Pacheco, J.M. Urban Dynamics, Fractals and Generalized Entropy. *Entropy* **2013**, *15*, 2679-2697.
https://doi.org/10.3390/e15072679

**AMA Style**

Encarnação S, Gaudiano M, Santos FC, Tenedório JA, Pacheco JM. Urban Dynamics, Fractals and Generalized Entropy. *Entropy*. 2013; 15(7):2679-2697.
https://doi.org/10.3390/e15072679

**Chicago/Turabian Style**

Encarnação, Sara, Marcos Gaudiano, Francisco C. Santos, José A. Tenedório, and Jorge M. Pacheco. 2013. "Urban Dynamics, Fractals and Generalized Entropy" *Entropy* 15, no. 7: 2679-2697.
https://doi.org/10.3390/e15072679