Estructura de la base de datos

The segregation metrics proposed in this thesis are based on the aggregated tra- jectories of groups of individual agents, which are here referred to as collective activity spaces. Figure 4.11 shows trajectories from home to work of a sample of individuals belonging to hypothetical groups Red and Blue. In real-world stud- ies, individuals can be grouped according to socioeconomic characteristics such as income, ethnicity, or level of education.

The activity and movement patterns of many individuals, such as the ones illustrated in figure 4.11, highlight sections of the urban space that are shared among many groups, as well as areas where a single group is predominant. Those areas are marked on the map as single-colour dashed circles, representing encounters and possible interactions between individuals of the same group, and dual-colour dashed circles representing possible interactions between individuals of both groups.

Two types of segregation measures are proposed in this thesis, based on the groups’ collective activity spaces: measures of diversity at street level, and measures of copresence in space and time. Those measures are detailed in this section.

Diversity

This thesis proposes adapted versions of Theil’s (1971) information theory index H as indicators of diversity at street level. In this study, Theil’s index H is calculated on the AxS model’s raster environment, quantifying the diversity of agents that moved through each cell based on the cell’s aggregated flow counts.

Figure 4.11: Hypothetical activity spaces of two groups of individuals.

diversity. Hence, local and global entropy indices were also adapted from Theil and Finizza (1971) to measure diversity on the AxS model’s aggregated flows. The global entropy index E summarises the diversity of the entire study area’s flows, as per equation 4.13. In that equation, τm is the proportion of flows of

population group m in the study area, and M denotes the number of population groups. E = M X m=1 (τm)ln  ₁ τm  (4.13)

Entropy can also be calculated locally, measuring the diversity of flows on each cell of the model’s environment. The local entropy index Ec of cell c is

calculated as per equation 4.14. In that equation, τcm is the proportion of flows

of population group m in cell c, and M denotes the number of population groups.

Ec= M X m=1 (τcm)ln  ₁ τcm  (4.14)

Global and local entropy values range from 0, indicating all flows in the study area (for global entropy) or the cell (for local entropy) belong to a single group, to log(M ), which indicates aggregated flows of all groups are evenly distributed in the urban area (or in the cell, in the case of local entropy).

The information theory index H is calculated by comparing the entropy of the entire study area to the entropy of the local areas (in the case of this study, individual cells). The global index H measures the average deviation between each cell’s entropy and the grid’s global entropy, as per equation 4.15. In that equation, H denotes the global index H of the study area, the total grid’s flow count is represented by F , while the local flow count of cell c is denoted by Fc,

and C indicates the number of cells in the grid. Global and local entropy values are represented by E and Ec, respectively.

H = C X c=1 " Fc(E − Ec) EF # (4.15)

Global index H values vary between 0, when each cell has the same entropy as the entire grid (maximum integration), and 1, when the city is totally segregated and each cell contains flows of only one group.

Local index H measures how much each cell is more or less diverse than the study area, as per equation 4.16. The notation used in equation 4.16 is the same as the previous one (4.15), apart from hc that denotes the local index H of

cell c.

hc=

Fj(E − Ec)

EF (4.16)

Negative values of the local index H indicate the cell has higher entropy than the grid, hence it is more diverse. Positive values indicate the opposite: the grid (city) is more diverse than that particular cell.

The segregation indices introduced here are indicators of the evenness/clustering dimension of segregation. They measure how evenly distributed are the aggre-

gated flows of different population groups in the study area. They can also be interpreted as measuring segregation from the perspective of a shop (or any other activity location) on a street. In this sense, entropy and index H can measure the diversity of that shop’s potential customers, who are the people who walk in front of that shop on a daily basis.

Copresence

As previously discussed, in time geography, the temporal dimension is as impor- tant as the spatial dimension in determining possibilities of interaction. This means that actual or potential interactions between individuals require those individuals to be present at the same place at the same time. In this study, en- counters and possible interactions between individual agents in space and time are referred to as copresence, and a series of copresence metrics were proposed to quantify those interactions.

During the simulation, the model keeps track of all encounters between agents in the model’s environment. In a dynamic and spatially explicit agent- based model such as the AxS model, identifying when and where agents encounter each other is relatively straightforward. Encounters, in the model, happen when two or more agents share the same cell during the same iteration. The total number of encounters that took place during a simulation is named absolute copresence. In the following equations, Cabs _{denotes the absolute copresence,}

while Cabs

mn denotes the absolute copresence between agents of groups m and n.

Proportional and relative copresence measures are derived from the ab- solute copresence, as follows. The proportional copresence C_mnprop (equation 4.17) measures the proportion of encounters between agents of groups m and n rela- tively to the absolute copresence. In a situation where all population groups are perfectly integrated, the values of relative copresence are expected to match the proportion of the populations of each pair of groups in the study area.

C_mnprop = C

abs mn

Cabs (4.17)

The proportional copresence is easier to interpret than the absolute co- presence, because the latter is highly dependent on a series of factors such as the size of the population sample and the number of active agents in the simulation.

The relative copresence Crel

mn (equation 4.18) measures how much the

proportional copresence between groups m and n deviates from the proportion τmnof the same groups in the study area’s population. Negative values of relative

copresence mean the probability of encounter is lower than expected, given the proportion of the groups in the study area.

C_mnrel = C

prop mn − τmn

τmn

(4.18)

The lower limit of relative copresence is -1, meaning no encounter hap- pened between agents of the two groups in question. Positive values indicate the

number of encounters is higher than expected given the groups’ presence in the study area, indicating such groups are more integrated.

Local copresence indices are also proposed in this thesis. In the model’s context, those indices measure the number of encounters among agents that hap- pened on each cell. The total number of encounters that took place during the simulation at cell c is named local absolute copresence, and is denoted by Cabs

c

in the following equations. Similarly, Cabs

cmn denotes the local absolute copresence

between agents of groups m and n in cell c. In order to better interpret the model’s results, measures of local expected and relative copresence were derived, as follows.

The local expected copresence Cexp

cmn indicates the expected absolute num-

ber of encounters between agents of groups m and n at cell c, given the proportion τmn of the groups’ populations in the study area and the cell’s local absolute co-

presence Cabs

c , as per equation 4.19.

C_cmnexp = C_cabs× τmn (4.19)

The local relative copresence Crel

cmn indicates the difference between the

absolute and expected copresence indices of groups m and n at cell c, as per equation 4.20.

C_cmnrel = C_cmnabs − Cexp

cmn (4.20)

Positive values of local relative copresence indicate higher than expected number of encounters has taken place at the cell, while negative values indicate the opposite trend. This makes the local relative copresence the easier to interpret of the local copresence indices proposed here.

The copresence indicators proposed in this thesis can be considered mea- sures of the exposure/isolation dimension of segregation, as they measure the probabilities of interaction between individuals. In fact, copresence measures ac- tual interactions in the model, which represent potential interactions in the real world. Obviously, it cannot be guaranteed an encounter between two individuals in the model’s environment will be translated into meaningful interactions in the real world. This is even more relevant when the relatively large cell sizes used in the model are considered, as agents can be hundreds of metres apart from each other and still occupy the same cell. However, it can be said the minimal time geographic conditions for such interactions were met: at some point during their trajectories in space and time, two individuals were in close proximity to each other.