15 minute window into statistical methods

Transport geography

Spatial network from the graph theory perspective

Spatial Interaction and Gravity Model

Accessibility

- Regression/Linear Regression

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

Regression/Linear Regression

\(Y_i = \beta_0 + \beta_1 X_i + \varepsilon_i\)

**Linearity**- The relationship between X and the mean of Y is linear.**Homoscedasticity**- The variance of residual is the same for any value of X.**Independence**- Observations are independent of each other

- Why transport in this module?

Fundamental for social and economic activities

All about networks & interactions

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)

Source: Wu et al. (2019)

Terminology

terminal = node = vertex

link = edge

Sub-graph - Loop (buckle)

More types of graphs

Planar graph vs Non-planar graph

Cycle, circuit

Source: Rodrigue (2020)

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

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.

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

**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

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.

Conditions for spatial interaction to be materialised

Source: Rodrigue (2020)

Gravity model

Analogy of Gravity model

\(Force_{ij} = G \frac{Mass_i Mass_j}{Dist_ij}\)

\(Flow_{ij} = \frac{Attribute_i Attribute_j}{Separation_{ij}}\)

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

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

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

\(W_j\): 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

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

\(\beta\): transport friction parameter

\(\lambda\): Potential to generate movements

\(\alpha\): Potential to attract movements

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

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)

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

â€¦ not good enough.

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 factorsHow? Regression

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

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

\(c = lnk = 13.84\)

\(\lambda = lnV_i = 0.727\)

\(\alpha = lnW_j = 0.464\)

\(\beta = lnd_{ij} = -0.624\)

What did we learn?

How can we use these coefficients

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

Different typologies (Holl 2007)

Network access

distance to access the network

travel opportunities

Travel cost measures

- network access + distance/time travelled on the network

Market potential accessibility

- destinations at greater distance provide diminishing opportunities

- \(Acc_{i} = \sum_j \frac{W_j}{d_{ij}^2}\)

Geographical indicator

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

The potential for interaction

Opportunities

Discussion: What is accessibility to you?

VrabkovĂˇ, ErtingerovĂˇ, and KukuliaÄŤ (2021)

VrabkovĂˇ, ErtingerovĂˇ, and KukuliaÄŤ (2021)

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.