Category Archives: physics

A data-driven study of the patterns of life for 180,000 people

Here at the Computational Story Lab, some of us commute by foot, some by car, and a few deliver themselves by bike, even in the middle of our cold, snowful Vermont winter.  Occasionally, we transport ourselves over very long distances in magic flying tubes with wings to attend conferences, to see family, or for travel.  So what do our movement patterns look like over time?  Are there distinct kinds of movement patterns as we look across populations, or are they variations on a single theme?

Inspired by an analysis of mobile phone data by Marta Gonzalez at MIT, James Bagrow at Northwestern, and colleagues, we used 37 million geotagged tweets to characterize the movement patterns of 180,000 people during their 2011 travels. We used the standard deviation in their position, a.k.a. radius of gyration, as a reflection of their movement. As an example, below we plot a dot for each geotagged tweet we found posted in the San Francisco Bay area, colored by the author’s radius of gyration.

The Bay Area is shown with a dot for each tweet, colored by the radius of gyration of its author.

The Bay Area is shown with a dot for each tweet, colored by the radius of gyration of its author. The color scale is logarithmic, so we can compare people with very different habits.

You can see from the picture that there are many people with a radius near 100km tweeting from downtown San Francisco. This pattern could reflect a concentration of tourists visiting the area, or individuals who live downtown and travel for work or pleasure. Images for New York City, Chicago, and Los Angeles are also quite beautiful.

In the image below, we rotated every individual’s movement pattern so that the origin represents their average location, and the horizontal line heading to the left represents their principle axis (most likely the path from home to work). We also stretched or shrunk the vertical and horizontal axes for each individual, so that everyone could fit on the same picture. Basically, we have a heatmap of collective movement, with each individual in their own intrinsic reference frame.  The immediate good news for these kinds of data-driven studies is that we see a very similar form to those found for mobile phone data sets.  Apart from being a different social signal, Geotagged Tweets also have much better spatial resolution than mobile phone calls which are referenced by the nearest cellphone tower.

Movement pattern exhibited by 180,000 individuals in 2011, as inferred from 37 million geolocated tweets. Colormap shows the probability density in log10. Note that despite the resemblance, this image is neither a nested rainbow horseshoe crab, nor the Mandelbrot set.

Movement pattern exhibited by 180,000 individuals in 2011, as inferred from 37 million geolocated tweets. Colormap shows the probability density in log10. Note that despite the resemblance, this image is neither a nested rainbow horseshoe crab, nor the Mandelbrot set.

Several features of the map reveal interesting patterns. First, the teardrop shape of the contours demonstrates that people travel predominantly along their principle axis, with deviations becoming shorter and less frequent as they move farther away. Second, the appearance of two spatially distinct yellow regions suggests that people spend the vast majority of their time near two locations. We refer to these locations as the work and home locales, where the home locale is centered on the dark red region right of the origin, and the work locale is centered just left of the origin.

Finally, we see a clear horizontal asymmetry indicating the increasingly isotropic variation in movement surrounding the home locale, as compared to the work locale. We suspect this to be a reflection of the tendency to be more familiar with the surroundings of one’s home, and to explore these surroundings in a more social context. The up-down symmetry demonstrates the remarkable consistency of the movement patterns revealed by the data.

We see a clear separation between the most likely and second most likely position.

We see a clear separation between the most likely and second most likely position.

Looking just at the messages posted along the work-home corridor, the distribution is skewed left, with movement from home in a heading opposite work seen to be highly unlikely.

The isotropy ratio shows the change in the probability density's shape as a function of radius.

The isotropy ratio shows the change in the probability density’s shape as a function of radius.

Above we see that individuals who move around a lot have a much larger variation in their positions along their principle axis, exhibiting a less circular pattern of life than people who stay close to home. Remarkably, the isotropy ratio decays logarithmically with radius.

Finally, we grabbed messages from the most prolific tweople, those 300 champions who had posted more than 10,000 geotagged messages in 2011. We received 10% of these messages through our gardenhose feed from Twitter. Below, we plot the times during the week that they post from their most frequently visited location. These folks most likely have the geotag switch on for all messages, and exhibit a very regular routine.

A robust diurnal cycle is observed in the hourly time of day at which statuses are updated, with those from the mode location (black curve) occurring more often than other locations (red curve) in the morning and evening.

A robust diurnal cycle is observed in the hourly time of day at which statuses are updated, with those from the mode location (black curve) occurring more often than other locations (red curve) in the morning and evening.

Peaks in activity are seen in the morning (8-10am) and evening (10pm-midnight), separated by lulls in the afternoon (2-4pm) and overnight (2-4am) hours.  As we and our friend Captain Obvious would expect, people tend to tweet more from their home locale than any other locale (red curve) in the morning and evening.

Bottom line: Despite our seemingly different patterns of life, we are remarkably similar in the way we move around. Our walks are a far cry from random.

Next up: We’ll examine the emotional content of tweets as a function of distance.  Is home where the heart is?

For more details on these results, see our paper Happiness and the Patterns of Life: A Study of Geolocated Tweets.

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Filed under networks, physics, prediction, social phenomena

Chaos in an Atmosphere Hanging on a Wall

This month marks the 50th anniversary of the 1963 publication of Ed Lorenz’s groundbreaking paper, Deterministic Nonperiodic Flow, by the Journal of Atmospheric Science. This seminal work, now cited more than 11,000 times, inspired a generation of mathematicians and physicists to bravely relax their linear assumptions about reality, and embrace the nonlinearity governing our complex world. Quoting from the abstract of his paper:

`A simple system representing cellular convection is solved numerically. All of the solutions are found to be unstable, and almost all of them are nonperiodic.’

While many scientists had observed and characterized nonlinear behavior before, Lorenz was the first to simulate this remarkable phenomenon in a simple set of differential equations using a computer. He went on to demonstrate the limit of predictability of the atmosphere to be roughly 2 weeks, the time it takes for two virtually indistinguishable weather patterns to become completely different. No matter how accurate our satellite measurements get, no matter how fast our computers become, we will never be able to predict the likelihood of rain beyond 14 days. This phenomenon became known as the butterfly effect, popularized in James Gleick’s book Chaos.

lorenz-sketch

Lorenz’s sketch of the attractor for his system.

Inspired by the work of Lorenz and colleagues, in our lab at the University of Vermont we’re using Computational Fluid Dynamics (CFD) simulations to understand the flow behaviors observed in a physical experiment. It’s a testbed for developing mathematical techniques to improve the predictions made by weather and climate models. Here you’ll find a brief video describing the experiment analogous to the model developed by Lorenz:

And below you’ll find a CFD simulation of the dynamics observed in the experiment:

What is most remarkable about Lorenz’s 1963 model is its relevance to the state-of-the-art in weather prediction today, despite the enormous advances that have been made in theoretical, observational, and computational studies of the Earth’s atmosphere. Every PhD student working in the field of weather prediction cuts their teeth testing data assimilation schemes on simple models proposed by Lorenz, his influence is incalculable.

In 2005, while I was a PhD student in Applied Mathematics at the University of Maryland, the legendary Lorenz visited my advisor Eugenia Kalnay in her office in the Department of Atmospheric & Oceanic Science. At some point during his stay, he penned the following on a piece of paper:

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.’

Even near the end of his career, Lorenz was still searching for the essence of nonlinearity, seeking to describe this incredibly complicated phenomenon in the simplest of terms.

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*Note: this post also appeared as part of the Mathematics of Planet Earth 2013 daily blog.

Taming Atmospheric Chaos with Big Data, a talk I gave at the 2011 UVM TEDx Conference Big Data, Big Stories:

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Chaos in an Experimental Toy Climate

In the 1960’s, MIT meteorologist Edward Lorenz was investigating the effects of nonlinearity on short-term weather prediction in a model of convection. In his ground-breaking paper “Deterministic Nonperiodic Flow,” Lorenz showed that numerical solutions of the model exhibit sensitive dependence on their initial position, leading virtually indistinguishable states to diverge quickly. This phenomenon, which became known as chaos, is a major contributor to inaccuracies in weather and climate forecasts.

The thermal convection loop is an experimental analog of Lorenz’s system in the form of a hula-hoop shaped tube, filled with fluid, and oriented vertically like a wheel. The bottom half of the tube is warmed uniformly by a bath of hot water and the top half is cooled. Under certain conditions, a steady state is never reached, and the fluid switches direction in an unpredictable pattern.

In the past few years, we have used Computational Fluid Dynamics (CFD) simulations of the loop as a testbed for data assimilation, ensemble forecasting, and model error experiments in weather and climate prediction. Our team is developing algorithms to improve forecasts and uncertainty quantification using this simple but realistic toy climate. Successful techniques are then implemented on more realistic weather and climate models.

Details:

K. D. Harris, E.-H. Ridouane, D. L. Hitt, C. M. Danforth. 2012. Predicting Flow Reversals in Chaotic Natural Convection using Data Assimilation. Tellus A, 64, 17598. [pdf]

N. Allgaier, K. D. Harris, C. M. Danforth. 2012. Empirical Correction of a Toy Climate Model. Physical Review E. 85, 026201. [pdf]

R. Lieb-Lappen, C. M. Danforth. 2012. Aggressive Shadowing of a Low-Dimensional Model of Atmospheric Dynamics. Physica D. Volume 241, Issue 6, Pages 637–648. [pdf]

E.-H. Ridouane, C. M. Danforth, D. L. Hitt. 2009. A 2-D Numerical Study Of Chaotic Flow In A Natural Convection Loop. International Journal of Heat and Mass Transfer. [pdf]

and a lecture on the topic given by Danforth to the Applied Dynamics graduate course at UNC Chapel Hill:

Funding from the project comes from NASA and NSF through the Mathematics and Climate Research Network.

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Ooblexity

Vibrating cornstarch and water in slow motion, narration by xtranormal.

A google search for ‘ooblexity’ returns “Did you mean: complexity?”  Maybe I did.

More info on the experiment, and a 10 log-decade spread in material costs:


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