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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OSMnx: Python for Street Networks

OSMnx: New York City urban street network visualized and analyzed with Python and OpenStreetMap dataOSMnx is a Python package for downloading administrative boundary shapes and street networks from OpenStreetMap. It allows you to easily construct, project, visualize, and analyze complex street networks in Python with NetworkX. You can get a city’s or neighborhood’s walking, driving, or biking network with a single line of Python code. Then you can simply visualize cul-de-sacs or one-way streets, plot shortest-path routes, or calculate stats like intersection density, average node connectivity, or betweenness centrality. You can download/cite the paper here.

In a single line of code, OSMnx lets you download, construct, and visualize the street network for, say, Modena Italy:

import osmnx as ox
ox.plot_graph(ox.graph_from_place('Modena, Italy'))

OSMnx: Modena Italy networkx street network in Python from OpenStreetMap Continue reading OSMnx: Python for Street Networks

R-tree Spatial Indexing with Python

r-tree spatial index with python geopandas: Thumbnail of Walnut Creek, California city boundary and street intersections inside and outside city limitsA spatial index such as R-tree can drastically speed up GIS operations like intersections and joins. Spatial indices are key features of spatial databases like PostGIS, but they’re also available for DIY coding in Python. I’ll introduce how R-trees work and how to use them in Python and its geopandas library. All of my code is in this notebook in this urban data science GitHub repo.

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Craigslist and U.S. Rental Housing Markets

This is a summary of our JPER journal article (available here) about Craigslist rental listings’ insights into U.S. housing markets.

Small map of 1.5 million Craigslist rental listings in the contiguous U.S., divided into quintiles by each listing's rent per square footRentals make up a significant portion of the U.S. housing market, but much of this market activity is poorly understood due to its informal characteristics and historically minimal data trail. The UC Berkeley Urban Analytics Lab collected, validated, and analyzed 11 million Craigslist rental listings to discover fine-grained patterns across metropolitan housing markets in the United States. I’ll summarize our findings below and explain the methodology at the bottom.

But first, 4 key takeaways:

  1. There are incredibly few rental units below fair market rent in the hottest housing markets. Some metro areas like New York and Boston have only single-digit percentages of Craigslist rental listings below fair market rent. That’s really low.
  2. This problem doesn’t exclusively affect the poor: the share of its income that the typical household would spend on the typical rent in cities like New York and San Francisco exceeds the threshold for “rent burden.”
  3. Rents are more “compressed” in soft markets. For example, in Detroit, most of the listed units are concentrated within a very narrow band of rent/ft² values, but in San Francisco rents are much more dispersed. Housing vouchers may end up working very differently in high-cost vs low-cost areas.
  4. Craigslist listings correspond reasonably well with Dept of Housing and Urban Development (HUD) estimates, but provide up-to-date data including unit characteristics, from neighborhood to national scales. For example, we can see how rents are changing, neighborhood by neighborhood, in San Francisco in a given month.

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Scientific Python for Raspberry Pi

Raspberry Pi 3 Model BA guide to setting up the Python scientific stack, well-suited for geospatial analysis, on a Raspberry Pi 3. The whole process takes just a few minutes.

The Raspberry Pi 3 was announced two weeks ago and presents a substantial step up in computational power over its predecessors. It can serve as a functional Wi-Fi connected Linux desktop computer, albeit underpowered. However it’s perfectly capable of running the Python scientific computing stack including Jupyter, pandas, matplotlib, scipy, scikit-learn, and OSMnx.

Despite (or because of?) its low power, it’s ideal for low-overhead and repetitive tasks that researchers and engineers often face, including geocoding, web scraping, scheduled API calls, or recurring statistical or spatial analyses (with small-ish data sets). It’s also a great way to set up a simple server or experiment with Linux. This guide is aimed at newcomers to the world of Raspberry Pi and Linux, but who have an interest in setting up a Python environment on these $35 credit card sized computers. We’ll run through everything you need to do to get started (if your Pi is already up and running, skip steps 1 and 2). Continue reading Scientific Python for Raspberry Pi

Urban Informatics and Visualization at UC Berkeley

The fall semester begins next week at UC Berkeley. For the third year in a row, Paul Waddell and I will be teaching CP255: Urban Informatics and Visualization.

This masters-level course trains students to analyze urban data, develop indicators, conduct spatial analyses, create data visualizations, and build Paris open datainteractive web maps. To do this, we use the Python programming language, open source analysis and visualization tools, and public data.

This course is designed to provide future city planners with a toolkit of technical skills for quantitative problem solving. We don’t require any prior programming experience – we teach this from the ground up – but we do expect prior knowledge of basic statistics and GIS.

Our teaching materials, including IPython Notebooks, tutorials, and guides are available in this GitHub repo, updated as the semester progresses.

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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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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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