# Methods in transport geography

## Today

• 15 minute window into statistical methods

• Transport geography

1. Spatial network from the graph theory perspective

2. Spatial Interaction and Gravity Model

3. Accessibility

## 15 minute window into statistical methods

• Regression/Linear Regression

## 15 minute window into statistical methods

Regression/Linear Regression - describes the strength and character of an association between two or more variables

## 15 minute window into statistical methods

Regression/Linear Regression

$Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i$

## Regression assumptions

1. Linearity - The relationship between X and the mean of Y is linear.

2. Homoscedasticity - The variance of residual is the same for any value of X.

3. Independence - Observations are independent of each other

## Transport geography: quantitative dimension

• Why transport in this module?
1. Fundamental for social and economic activities

2. All about networks & interactions

## Transport geography: key elements

• Distance

• Accessibility: the capacity of a location to be reached by, or to reach different locations

• Spatial interaction: movement of people, freight or information between an origin and destination

• Transportation and land use models

Source: Rodrigue (2020)

## Spatial network from the graph theory perspective

Source: Wu et al. (2019)

## Spatial network from the graph theory perspective

Terminology

• terminal = node = vertex

• Sub-graph - Loop (buckle)

More types of graphs

• Planar graph vs Non-planar graph

• Cycle, circuit

Source: Rodrigue (2020)

## Plannar graph analysis

• Proportion between nodes and edges: $\beta = e / v$, where $e$ is the number of links & $v$ the number of nodes

• Network density: $Gamma$ AKA (number of liks / maximum number of links)

• planar: $\gamma = \frac{e}{3(v-2)}$

• non-planar: $\gamma = \frac{2e}{v(v-1)}$

• Structure vs Topology

## Plannar graph analysis

• Degree centrality, Betweenness, Eigenvector centrality,â€¦

• Eccentricity: the distance from a given node to the farthest node from it in the network

• Shimbel index, or nodal accessibility, or Fareness (see Lecture 3

• $c_đť‘–= \sum_j d_{ij}$

• This is a purely topological accessibility metric, remember this for later.

## Graph analysis

Closeness centrality (from Lecture 3)

• Which node has the shortest distance to other nodes

• Instead of focusing on the number of links, the focus turns to the network distances

• Different definitions:

• Closeness, $c_{i} = 1/\sum_{j} d_{ij}$

• Fareness, $c_{i} = \sum_{j} d_{ij}$

• igraph calculates closeness

## Graph analysis

The Gini coefficient

• Measure of dispersion often used as Inequality measure

• 0: perfect equality
• 1 :perfect inequality
• Ordered X and Y, cumulative percentage

• Mostly used for income inequalities, but can be more widely used

• $Gini = A / (A + B)$

Source: Rodrigue (2020)

Source: Rodrigue (2020)

Source: Rodrigue (2020)

Example: measuring traffic concentration

Temporal variations of the Gini coefficient reflect changes in the comparative advantages of a location within the transport system

## Spatial interactions and the gravity model

• A spatial interaction is a realised movement of people, goods or information between an origin and a destination

• It is a transport demand/supply relationship expressed over geographical space.

## Spatial interactions and the gravity model

Conditions for spatial interaction to be materialised

Source: Rodrigue (2020)

## Spatial interactions and the gravity model

Gravity model

Analogy of Gravity model

$Force_{ij} = G \frac{Mass_i Mass_j}{Dist_ij}$

$Flow_{ij} = \frac{Attribute_i Attribute_j}{Separation_{ij}}$

## Spatial interactions and the gravity model

$T_{ij} = f(V_i, W_j, S_{ij})$

• Flows are a function of the attributes of the locations of origin, the attributes of the locations of destination and the friction of distance between the concerned origins and the destinations

• $T_{ij}$: Interaction between location $i$ (origin) and location $j$ (destination)

• $V_i$: Attributes of the location of origin $i$ (e.g. population, number of jobs available, industrial output, GDP); push factors; the potential of origins

## Spatial interactions and the gravity model

$T_{ij} = f(V_i, W_j, S_{ij})$

• $W_j$: Attributes of the location of destination $j$, pull factors; attractiveness of destinations

• $S_{ij}$: Attributes of separation between $i$ and $j$ (e.g. distance, transport costs, or travel time); cost of overcoming the separation between origins and destinations

## Spatial interactions and the gravity model

$T_{ij} = k\frac{V_i^\lambda W_j^\alpha}{d_{ij}^\beta}$

• $\beta$: transport friction parameter

• $\lambda$: Potential to generate movements

• $\alpha$: Potential to attract movements

# What can we do with this?

## 1. Calculate flows (naive)

$T_{ij} = k\frac{V_i^\lambda W_j^\alpha}{d_{ij}^\beta}$

• Known: $V$, $W$, and $d$

• Define: $\lambda=1$, $\alpha=1$, $\beta=2$, and $k=0.00001$

• These are some standard results from past empirical studies

• Big unknown: $T$

• Example from Tranos, Gheasi, and Nijkamp (2015)

## 1. Calculate flows (naive)

• Fairly good estimation of reality for such an oversimplified model, butâ€¦

• â€¦ not good enough.

## 2. Estimate $\lambda$, $\alpha$, $\beta$ and $k$

• Known: $T$, $V$, $W$, $d$ and $k$

• Estiamte: $\lambda=1$, $\alpha=1$, and $\beta=2$

• Why? To understand the effect of distance, pull and push factors

• How? Regression

## 2. Estimate $\lambda$, $\alpha$, $\beta$ and $k$

• $T_{ij} = k\frac{V_i^\lambda W_j^\alpha}{d_{ij}^\beta}$

• Look up properties of logarithms, e.g. here

• $T_{ij} = kV_i^\lambda W_j^\alpha d_{ij}^{-\beta}$

• $ln T_{ij} = ln (kV_i^\lambda W_j^\alpha d_{ij}^{-\beta})$

• $ln T_{ij} = lnk + lnV_i^\lambda + ln W_j^\alpha + lnd_{ij}^{-\beta}$

• $\color{red}{ln T_{ij}} = \color{blue}{lnk} + \lambda \color{green}{lnV_i} + \alpha \color{orange}{ln W_j} -\beta \color{purple}{lnd_{ij}}$

• $\color{red}{y} = \color{blue}{c} + \lambda \color{green}{x_1} + \alpha \color{orange}{x_2} + \beta \color{purple}{x_3}$

• Multivariate linear regression

## 2. Estimate $\lambda$, $\alpha$, $\beta$ and $k$

• $c = lnk = 13.84$

• $\lambda = lnV_i = 0.727$

• $\alpha = lnW_j = 0.464$

• $\beta = lnd_{ij} = -0.624$

• What did we learn?

• How can we use these coefficients

## 3. Estimate accessibility indicators

• The potential of opportunities for interaction

• Ease of spatial interaction

• Attractiveness of a node in a network taking into account the mass of other nodes and the costs to reach those nodes via the network

## 3. Estimate accessibility indicators

Different typologies

1. Network access

• distance to access the network

• travel opportunities

2. Travel cost measures

• network access + distance/time travelled on the network
3. Market potential accessibility

• destinations at greater distance provide diminishing opportunities
Source: Bruinsma and Rietveld (1998)

## 3. Estimate accessibility indicators

• $Acc_{i} = \sum_j \frac{W_j}{d_{ij}^2}$
Source:Rodrigue (2020)

## 3. Estimate accessibility indicators

• Geographical indicator

• Spatial structure (e.g. distance) and economic activities (e.g. population)

• The potential for interaction

• Opportunities

## Not the only way to define accessibility on networks!

Discussion: What is accessibility to you?

## Accessibility of locations from routing perspective

VrabkovĂˇ, ErtingerovĂˇ, and KukuliaÄŤ (2021)

VrabkovĂˇ, ErtingerovĂˇ, and KukuliaÄŤ (2021)

## References

Bruinsma, Frank, and Pieter Rietveld. 1998. â€śThe Accessibility of European Cities: Theoretical Framework and Comparison of Approaches.â€ť Environment and Planning A 30 (3): 499â€“521.
Holl, Adelheid. 2007. â€śTwenty Years of Accessibility Improvements. The Case of the Spanish Motorway Building Programme.â€ť Journal of Transport Geography 15 (4): 286â€“97. https://doi.org/https://doi.org/10.1016/j.jtrangeo.2006.09.003.
Rodrigue, Jean-Paul. 2020. The Geography of Transport Systems. Routledge. https://transportgeography.org/.
Tranos, Emmanouil, Masood Gheasi, and Peter Nijkamp. 2015. â€śInternational Migration: A Global Complex Network.â€ť Environment and Planning B: Planning and Design 42 (1): 4â€“22.
VrabkovĂˇ, Iveta, Izabela ErtingerovĂˇ, and Pavel KukuliaÄŤ. 2021. â€śDetermination of Gaps in the Spatial Accessibility of Nursing Services for Persons over the Age of 65 with Lowered Self-Sufficiency: Evidence from the Czech Republic.â€ť Edited by Dragan Pamucar. PLOS ONE 16 (1): e0244991. https://doi.org/10.1371/journal.pone.0244991.
Wu, Mincheng, Shibo He, Yongtao Zhang, Jiming Chen, Youxian Sun, Yang-Yu Liu, Junshan Zhang, and H. Vincent Poor. 2019. â€śA General Framework of Studying Eigenvector Multicentrality in Multilayer Networks.â€ť Proceedings of the National Academy of Sciences 116 (31): 15407â€“13. https://doi.org/10.1073/pnas.1801378116.