# An Urban Cellular Automata Model for Simulating Dynamic States on a Local Scale

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Background

#### 2.1. Urban Models

#### 2.2. The Scale

#### 2.3. Clustering

#### 2.4. Dynamic Cellular States and Entropy

#### 2.5. Modifying CA

## 3. The Proposed Model

#### 3.1. The Conceptual Framework

#### 3.2. The Model Configurations

#### 3.2.1. Relaxation

#### 3.2.2. The Neighborhood and Cell States

_{i,u}is the total floor area for all uses (U1–U6) on the site j, and ej is the floor area ratio (ratio of the total floor area of the building to the size of the site) on the site j. Aj is the total area of the site j.

#### 3.2.3. Transformation Rules

## 4. Cases and Data

#### 4.1. The Case of Nekala

#### 4.2. The Case of Vaasa

#### 4.3. Data

## 5. Simulation Runs

#### 5.1. Performance of the Model

#### 5.1.1. Static States

#### 5.1.2. Dynamic States

_{(1–6)}) towards interaction between office/industrial uses (U4–U5) (see Figure A1 in Appendix A), the behavior of the model changed. First, the volumes started to gradually increase and decrease over time for all activities, resulting in a certain type of coherent yet unpredictable pulse emerging from phases of higher and lower utilization rate on the sites. A certain order seemed to emerge within the system, with measurable cycle length. The changes in the rule set (matrix values) have a marked influence over the dynamics of these periods: With certain rule sets (see the optimum configurations in Table 9) the system gravitated towards a periodic, non-uniform state. The period length was in flux, mostly oscillating between 10 and 12 time steps, revealing dynamics far more diverse than before. Some of these cyclical states started with a stochastic phase, soon settling onto predictable periods (see, e.g., Simulations 207, 212; Supplementary material, Figures S3–S8).

#### 5.2. Validation

_{all}is the number of different cycles in that run. The resulting entropy values are presented in Table 10. This equation describes the overall entropy of the simulation after the runs are completed, providing an estimated level of complexity in regards of time steps between changes in utilization of building right. (For example, for a periodic run 160, the cycle of 10 occurred 72 times out of a total of 178 different cycles. Hence, for run 160, ${s}_{j}$ = 72:178 = 0.040449 and consequently, ${\mathrm{log}}_{2}{s}_{j}$ = −1.3058. Thus ${s}_{j}{\mathrm{log}}_{2}{s}_{j}$ = 0.5282. This calculation was carried out for each cycle (10, 11, 12, 16, 22, etc.) for the total sum, yielding the entropy value of run 160).

_{n}

_{+1}− t

_{n}) between changes, it results in a fairly good representation of the overall entropy of the dynamics. The static states were not included since no measurable period occurred.

#### 5.3. Discussion

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Figure A1.**Weights for proximity preferences among activity types resulting in different dynamic states: U1, housing; U2, retail; U3, services; U4, offices; U5, light industry; U6, warehouses. (

**a**) Matrix values for static states; (

**b**) Matrix values for the complex states.

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**Figure 1.**Conceptual model. Interactions between variables; temporal (broken lines), stable (solid lines). Feedback from pattern to actors is implied in decay of overcrowded clusters—typically of CA, the model does not observe explicitly the global level patterns.

**Figure 2.**Degree of interaction between activities and their environment for classification of activities: U1, housing; U2, retail; U3, services; U4, offices; U5, light industry; and U6, warehouses. Local Access refers to the local interaction between the site and its environment—how easy it is to access the site, for example, from the street. Interference refers to the level of “disturbance” it tolerates—for example, regarding noise or air quality; and Flow to moving of goods and people to/from the site, implying global accessibility by car, truck etc. The classifications were made on the basis of these assumed relationships. (For example, the requirements for housing regarding disturbance (environmental “interference”) due to noise, smells or heavy traffic differ from those for retail or warehouses. Similarly, some activities need easy access from the street with less privacy, while others benefit from being part of the higher-scale networks, providing constant flows of customers, goods, or material).

**Figure 4.**Modes of cell transformation according to their utilization rates. P-1: “empty”, FAR = 0–0.1; P-2: “nearly-empty”, FAR = 0.1–0.3; P-3: “nearly-full”, FAR = 0.3–0.7; P-4: “full”, FAR = 0.7–1. For example, an almost empty cell is likely to be filled more, but also to be reconstructed—at presumably fairly low demolition costs of smaller buildings, whereas nearly full sites might be considered the most resistant to physical changes, but the new additions or uses may occupy these sites easily (see also Table 6).

**Table 1.**Wolfram’s [10] classification of evolution of dynamic cellular states.

1 | Homogeneous state |

2 | Simple stable or cyclical/periodic structures |

3 | Chaotic pattern |

4 | Complex localized structures |

**Table 2.**Classification of the evolution of dynamic cellular states by Braga et al. [11].

1 | Patterns disappear after a finite transient |

2 | All patterns stay limited under iteration of the global transition function |

3 | At least one pattern grows indefinitely |

**Table 3.**Analogies between cellular states and dynamic systems. The periodic and cyclical are used in this paper interchangeably.

CA Dynamics | Dynamic Systems Analogue |
---|---|

A spatially homogeneous state | Limit points |

A sequence of simple stable/periodic structures | Limit cycles |

Chaotic behavior | Chaotic (strange) attractors |

Complicated localized structures | Unspecified |

**Table 4.**Wuenche’s classification of evolution of dynamic cellular states. Entropy level increases from ordered to complex and chaotic states—complex having intermediate state of entropy.

1 | Ordered | Low degree of entropy in system |

2 | Complex | Intermediate degree of entropy in system |

3 | Chaotic | High degree of entropy in system |

**Table 5.**Relations and directions of interaction between variables (see also Figure 1). In this study, the plan is considered static and unresponsive (the “PLAN” column is empty), unlike in some cases in the reality.

Entity | Site (Cell) | Pattern | Use | Volume | Border | Plan |
---|---|---|---|---|---|---|

Site | bottom up | top down | top down | |||

Pattern | feedback | feedback | ||||

Use | bottom up | interaction | interaction | |||

Volume | bottom up | interaction | interaction | |||

Border | top down | top down | ||||

Plan | top down | top down | top down | top down |

State of the Site | Most Probable Procedure | The Motive |
---|---|---|

vacant | build a new building | to use the building right |

nearly-empty <10% | demolish (fill up) | to use the building right more effectively: low demolition costs |

nearly-full | fill up (change) | to use the building right more effectively: demolition costs above the threshold (It is assumed that there is a threshold value defining the shifts from one mode of transformation to another. E.g., a limit when it becomes more profitable to reconstruct the site, taking into account the demolition costs/m^{3} and new/old FAR. The demolition costs could be calculated, see e.g., DiPasquale and Wheaton [54], p. 85. In this theoretical approach, the classification of the sites is based on estimates.) |

full | remain/change/fill/reconstruct, e.g., 0.6/0.3/0.01/0.99 (These values can be changed to fit the circumstances depending on the case at issue.) | certain inertia on the full site; however, once the site is full, it will eventually be developed and reconstructed (no more space for additions). tendency to change if one use starts to dominate the neighborhood = high FA |

**Table 7.**Preference matrix which serves as a planner’s user interface: the values increase the likelihood of the two activities being located near to each other. Changing the values makes it possible to learn from their impact on the model dynamics.

U1 | U2 | U3 | U4 | U5 | U6 | |
---|---|---|---|---|---|---|

U1 | ${\mu}_{1}$ | ${\mu}_{2}$ | ${\mu}_{3}$ | ${\mu}_{4}$ | ${\mu}_{5}$ | ${\mu}_{6}$ |

U2 | ${\mu}_{7}$ | ${\mu}_{8}$ | ${\mu}_{9}$ | ${\mu}_{10}$ | ||

U3 | ${\mu}_{11}$ | ${\mu}_{12}$ | ${\mu}_{13}$ | |||

U4 | ${\mu}_{14}$ | ${\mu}_{15}$ | ||||

U5 | ${\mu}_{20}$ | |||||

U6 |

**Table 8.**Dynamic states of the model. In an oscillating system less complex than periodic state usually two or three values take turns.

Type 1 | Type 2 | |
---|---|---|

static | stagnation | oscillation |

dynamic | cyclical/periodic | complexity |

**Table 9.**Optimum rule sets resulting in different dynamic states. The values (1 to 20) represent the relative attraction between those activities. For example, in rule set 1, attraction is fairly equal. For rule set 2, office/industry is stressed. In rule set 3, in addition to that, the housing is restricted. (Note that the states with rule set 1 and 2 were remarkably resistant to changing matrix values, for the rule set 3 yielding complex dynamics the configuration was unique—only one configuration of matrix values yielded complex dynamics).

Emphasis | Matrix Configuration | Resulting Dynamics | ||||||
---|---|---|---|---|---|---|---|---|

rule set 1. optimum example | all uses: values range from low to moderate (1–8) | 6 | 4 | 4 | 4 | 1 | 1 | stagnating/oscillating dynamics; oscillation increased as the U1 × Un_{(1–6)} (attraction between housing and other activities) values decreased |

4 | 8 | 4 | 2 | 1 | 1 | |||

8 | 2 | 6 | 4 | 1 | 2 | |||

8 | 4 | 6 | 4 | 4 | 2 | |||

6 | 1 | 2 | 4 | 2 | 1 | |||

1 | 1 | 2 | 1 | 1 | 1 | |||

rule set 2. optimum example | U5 × U5 (small industry) and U4 × U4 (services) are high (µ > 10), other values are moderate (µ 2–8) | 2 | 4 | 2 | 4 | 1 | 1 | continuous, periodic (cyclical) dynamics (for all activities) |

4 | 4 | 2 | 2 | 1 | 1 | |||

2 | 2 | 6 | 4 | 1 | 2 | |||

2 | 4 | 6 | 10 | 16 | 2 | |||

1 | 1 | 2 | 8 | 12 | 1 | |||

1 | 1 | 2 | 1 | 1 | 1 | |||

rule set 3. unique configuration | U1 × U1 (housing) is low (µ = 1), and U4 × U4 (services) and U5 × U5 (small industry) are high (µ > 10), other values: moderate (µ 2–8) | 1 | 1 | 2 | 4 | 1 | 1 | continuous dynamics: - For housing (U1) complex, - For other uses (U2–U6) periodic (cyclical) |

1 | 4 | 4 | 2 | 1 | 1 | |||

2 | 2 | 6 | 4 | 1 | 2 | |||

2 | 4 | 6 | 10 | 16 | 2 | |||

1 | 1 | 2 | 8 | 12 | 1 | |||

1 | 1 | 2 | 1 | 1 | 1 |

**Table 10.**Degrees of entropy, random samples from complex and periodic/cyclical series, compared to a stochastic set.

periodic/cyclical | R = 159 | R = 160 | R = 162 | R = 163 | R = 212 | R = 207 |

2.85134 | 2.11139 | 2.20135 | 2.22823 | 2.32039 | 2.02637 | |

complex | R = 202 | R = 177 | R = 158 | R = 170 | R = 208 | R = 190 |

4.3864 | 4.17142 | 4.82893 | 4.1556 | 4.39262 | 3.81511 | |

(stochastic/hypothetical) | R = 202 | R = 177 | R = 158 | R = 170 | R = 208 | R = 190 |

5.8579 | 5.7279 | 5.90689 | 5.88264 | 5.72792 | 6.285 |

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Partanen, J.
An Urban Cellular Automata Model for Simulating Dynamic States on a Local Scale. *Entropy* **2017**, *19*, 12.
https://doi.org/10.3390/e19010012

**AMA Style**

Partanen J.
An Urban Cellular Automata Model for Simulating Dynamic States on a Local Scale. *Entropy*. 2017; 19(1):12.
https://doi.org/10.3390/e19010012

**Chicago/Turabian Style**

Partanen, Jenni.
2017. "An Urban Cellular Automata Model for Simulating Dynamic States on a Local Scale" *Entropy* 19, no. 1: 12.
https://doi.org/10.3390/e19010012