Tag: math

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Urban

Urban Street Network Orientation

My new article, Urban Spatial Order: Street Network Orientation, Configuration, and Entropy, has just been published in one of my favorite journals: Applied Network Science (download free PDF). This study explores the spatial signatures of urban evolution and central planning. It examines street network orientation, connectivity, granularity, and entropy in 100 cities around the world using OpenStreetMap data and OSMnx for modeling and visualization:

City street network grid orientations, order, disorder, entropy, rose plot, polar histogram made with Python, OSMnx, OpenStreetMap, matplotlib.

So, who’s got a grid and who doesn’t? Each of the cities above is represented by a polar histogram (aka rose diagram) depicting how its streets orient. Each bar’s direction represents the compass bearings of the streets (in that histogram bin) and its length represents the relative frequency of streets with those bearings. The cities above are in alphabetical order. Here they are again, re-sorted from most-ordered/gridded city (Chicago) to most-disordered (Charlotte):

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Data

US Street Network Models and Measures

My new article, “Street Network Models and Measures for Every U.S. City, County, Urbanized Area, Census Tract, and Zillow-Defined Neighborhood” has been published in Urban Science. This paper reports results from a broader project that collected raw street network data from OpenStreetMap using the Python-based OSMnx software for every U.S. city and town, county, urbanized area, census tract, and Zillow-defined neighborhood boundary. It constructed nonplanar directed multigraphs for each and analyzed their structural and morphological characteristics.

The resulting public data repository contains over 110,000 processed, cleaned street network graphs (which in turn comprise over 55 million nodes and over 137 million edges) at various scales—comprehensively covering the entire U.S.—archived as reusable open-source GraphML files, node/edge lists, and ESRI shapefiles that can be immediately loaded and analyzed in standard tools such as ArcGIS, QGIS, NetworkX, graph-tool, igraph, or Gephi.

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Data

Street Network Analysis in a Docker Container

Containerization is the way of the future present. I’ve heard feedback from some folks over the past few months who would like to play around with OSMnx for street network analysis, transport modeling, and urban design—but can’t because they can’t install Python and its data science stack on their computers. Furthermore, it would be nice to have a consistent reference environment to deploy on AWS or elsewhere in the cloud.

So, I’ve created a docker image containing OSMnx, Jupyter, and the rest of the Python geospatial data science stack, available on docker hub alongside additional usage instructions. If you’re starting from scratch, you can get started in four simple steps:

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Categories
Planning

New Chapter: Street Network Morphology

My chapter The Morphology and Circuity of Walkable and Drivable Street Networks is now in-press for publication in the forthcoming book The Mathematics of Urban Morphology (download free PDF). The book integrates recent theoretical and empirical work from urban planning, geography, sociology, architecture, economics, and mathematics around the theme of how we model and understand the urban form’s physical patterns and shaping processes. Fellow authors in this volume include Michael Batty, Diane Davis, Keith Clarke, Bin Jiang, Kay Axhausen, Carlo Ratti, and Stephen Marshall. The book itself can be purchased here.

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Categories
Planning

New Article: Planar Models of Street Networks

My article, “Planarity and Street Network Representation in Urban Form Analysis,” was recently published in Environment and Planning B: Urban Analytics and City Science. Models of street networks underlie research in urban travel behavior, accessibility, design patterns, and morphology. These models are commonly defined as planar, meaning they can be represented in two dimensions without any underpasses or overpasses. However, real-world urban street networks exist in three-dimensional space and frequently feature grade separation such as bridges and tunnels: planar simplifications can be useful but they also impact the results of real-world street network analysis. This study measures the nonplanarity of drivable and walkable street networks in the centers of 50 cities worldwide, then examines the variation of nonplanarity across a single city. While some street networks are approximately planar, I empirically quantify how planar models can inconsistently but drastically misrepresent intersection density, street lengths, routing, and connectivity.

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Urban

New Article: Complexity in Urban Form and Design

My article, Measuring the Complexity of Urban Form and Design, is now in-press for publication at Urban Design International (download free PDF). Cities are complex systems composed of many human agents interacting in physical urban space. This paper develops a typology of measures and indicators for assessing the physical complexity of the built environment at the scale of urban design. It extends quantitative measures from city planning, network science, ecosystems studies, fractal geometry, statistical physics, and information theory to the analysis of urban form and qualitative human experience.

The Mandelbrot set, a mathematical fractal. Venice's fractal urban form and fabric. The Eiffel Tower's fractal architecture in Paris.

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Categories
Data

Urban Street Network Centrality

Check out the journal article about OSMnx.

We can measure and visualize how “important” a node or an edge is in a network by calculating its centrality. Lots of flavors of centrality exist in network science, including closeness, betweenness, degree, eigenvector, and PageRank. Closeness centrality measures the average shortest path between each node in the network and every other node: more central nodes are closer to all other nodes. We can calculate this easily with OSMnx, as seen in this GitHub demo. For example, here is the node closeness centrality for Piedmont, California:

Urban street network graph node closeness and betweenness centrality

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Data

Animating the Lorenz Attractor with Python

Edward Lorenz, the father of chaos theory, once described chaos as “when the present determines the future, but the approximate present does not approximately determine the future.”

Lorenz first discovered chaos by accident while developing a simple mathematical model of atmospheric convection, using three ordinary differential equations. He found that nearly indistinguishable initial conditions could produce completely divergent outcomes, rendering weather prediction impossible beyond a time horizon of about a fortnight.

Lorenz system attractor animated GIF created with Python matplotlib scipy numpy PIL

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Categories
Data

Visualizing Chaos and Randomness

3-D Poincare plot of the logistic map's chaotic regime - this is time series data embedded in three dimensional state space

Download/cite the paper here!

In a previous post, I discussed chaos theory, fractals, and strange attractors – and their implications for knowledge and prediction of systems. I also briefly touched on how phase diagrams (or Poincaré plots) can help us visualize system attractors and differentiate chaotic behavior from true randomness.

In this post (adapted from this paper), I provide more detail on constructing and interpreting phase diagrams. These methods are particularly useful for discovering deterministic chaos in otherwise random-appearing time series data, as they visualize strange attractors. I’m using Python for all of these visualizations and the source code is available in this GitHub repo.

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Categories
Data

Chaos Theory and the Logistic Map

Logistic map bifurcation diagram showing the period-doubling path to chaosUsing Python to visualize chaos, fractals, and self-similarity to better understand the limits of knowledge and prediction. Download/cite the article here and try pynamical yourself.

Chaos theory is a branch of mathematics that deals with nonlinear dynamical systems. A system is just a set of interacting components that form a larger whole. Nonlinear means that due to feedback or multiplicative effects between the components, the whole becomes something greater than just adding up the individual parts. Lastly, dynamical means the system changes over time based on its current state. In the following piece (adapted from this article), I break down some of this jargon, visualize interesting characteristics of chaos, and discuss its implications for knowledge and prediction.

Chaotic systems are a simple sub-type of nonlinear dynamical systems. They may contain very few interacting parts and these may follow very simple rules, but these systems all have a very sensitive dependence on their initial conditions. Despite their deterministic simplicity, over time these systems can produce totally unpredictable and wildly divergent (aka, chaotic) behavior. Edward Lorenz, the father of chaos theory, described chaos as “when the present determines the future, but the approximate present does not approximately determine the future.”

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