ENTRIES TAGGED "math"
Lessons from the design community for developing data-driven applications
When you hear someone say, “that is a nice infographic” or “check out this sweet dashboard,” many people infer that they are “well-designed.” Creating accessible (or for the cynical, “pretty”) content is only part of what makes good design powerful. The design process is geared toward solving specific problems. This process has been formalized in many ways (e.g., IDEO’s Human Centered Design, Marc Hassenzahl’s User Experience Design, or Braden Kowitz’s Story-Centered Design), but the basic idea is that you have to explore the breadth of the possible before you can isolate truly innovative ideas. We, at Datascope Analytics, argue that the same is true of designing effective data science tools, dashboards, engines, etc — in order to design effective dashboards, you must know what is possible.
One of the chapters of Think Bayes is based on a class project two of my students worked on last semester. It presents “The Red Line Problem,” which is the problem of predicting the time until the next train arrives, based on the number of passengers on the platform.
Here’s the introduction:
In Boston, the Red Line is a subway that runs between Cambridge and Boston. When I was working in Cambridge I took the Red Line from Kendall Square to South Station and caught the commuter rail to Needham. During rush hour Red Line trains run every 7–8 minutes, on average.
When I arrived at the station, I could estimate the time until the next train based on the number of passengers on the platform. If there were only a few people, I inferred that I just missed a train and expected to wait about 7 minutes. If there were more passengers, I expected the train to arrive sooner. But if there were a large number of passengers, I suspected that trains were not running on schedule, so I would go back to the street level and get a taxi.
While I was waiting for trains, I thought about how Bayesian estimation could help predict my wait time and decide when I should give up and take a taxi. This chapter presents the analysis I came up with.
Sadly, this problem has been overtaken by history: the Red Line now provides real-time estimates for the arrival of the next train. But I think the analysis is interesting, and still applies for subway systems that don’t provide estimates.
One of the frequently-asked questions over at the statistics subreddit (reddit.com/r/statistics) is how to test whether a dataset is drawn from a particular distribution, most often the normal distribution.
There are standard tests for this sort of thing, many with double-barreled names like Anderson-Darling, Kolmogorov-Smirnov, Shapiro-Wilk, Ryan-Joiner, etc.
But these tests are almost never what you really want. When people ask these questions, what they really want to know (most of the time) is whether a particular distribution is a good model for a dataset. And that’s not a statistical test; it is a modeling decision.
All statistical analysis is based on models, and all models are based on simplifications. Models are only useful if they are simpler than the real world, which means you have to decide which aspects of the real world to include in the model, and which things you can leave out.
For example, the normal distribution is a good model for many physical quantities. The distribution of human height is approximately normal (see this previous blog post). But human heights are not normally distributed. For one thing, human heights are bounded within a narrow range, and the normal distribution goes to infinity in both directions. But even ignoring the non-physical tails (which have very low probability anyway), the distribution of human heights deviates in systematic ways from a normal distribution.
An interview with Allen Downey, the author of Think Bayes
When Mike first discussed Allen Downey’s Think Bayes book project with me, I remember nodding a lot. As the data editor, I spend a lot of time thinking about the different people within our Strata audience and how we can provide what I refer to “bridge resources”. We need to know and understand the environments that our users are the most comfortable in and provide them with the appropriate bridges in order to learn a new technique, language, tool, or …even math. I’ve also been very clear that almost everyone will need to improve their math skills should they decide to pursue a career in data science. So when Mike mentioned that Allen’s approach was to teach math not using math…but using Python, I immediately indicated my support for the project. Once the book was written, I contacted Allen about an interview and he graciously took some time away from the start of the semester to answer a few questions about his approach, teaching, and writing.
How did the “Think” series come about? What led you to start the series?
Allen Downey: A lot of it comes from my experience teaching at Olin College. All of our students take a basic programming class in the first semester, and I discovered that I could use their programming skills as a pedagogic wedge. What I mean is if you know how to program, you can use that skill to learn everything else.
I started with Think Stats because statistics is an area that has really suffered from the mathematical approach. At a lot of colleges, students take a mathematical statistics class that really doesn’t prepare them to work with real data. By taking a computational approach I was able to explain things more clearly (at least I think so). And more importantly, the computational approach lets students dive in and work with real data right away.
At this point there are four books in the series and I’m working on the fifth. Think Python covers Python programming–it’s the prerequisite for all the other books. But once you’ve got basic Python skills, you can read the others in any order.
Areas concerned with shapes, invariants, and dynamics, in high-dimensions, are proving useful in data analysis
I’ve been noticing unlikely areas of mathematics pop-up in data analysis. While signal processing is a natural fit, topology, differential and algebraic geometry aren’t exactly areas you associate with data science. But upon further reflection perhaps it shouldn’t be so surprising that areas that deal in shapes, invariants, and dynamics, in high-dimensions, would have something to contribute to the analysis of large data sets. Without further ado, here are a few examples that stood out for me. (If you know of other examples of recent applications of math in data analysis, please share them in the comments.)
Compressed sensing is a signal processing technique which makes efficient data collection possible. As an example using compressed sensing images can be reconstructed from small amounts of data. Idealized Sampling is used to collect information to measure the most important components. By vastly decreasing the number of measurements to be collected, less data needs to stored, and one reduces the amount of time and energy1 needed to collect signals. Already there have been applications in medical imaging and mobile phones.
The problem is you don’t know ahead of time which signals/components are important. A series of numerical experiments led Emanuel Candes to believe that random samples may be the answer. The theoretical foundation as to why a random set of signals would work, where laid down in a series of papers by Candes and Fields Medalist Terence Tao2.
Design compels. Math is proof. Both sides will defend their domains at Strata's next Great Debate.
At Strata Santa Clara later this month, we’re reprising what has become a tradition: Great Debates. These Oxford-style debates pit two teams against one another to argue a hot topic in the fields of big data, ubiquitous computing, and emerging interfaces.
Part of the fun is the scoring: attendees vote on whether they agree with the proposal before the debaters; and after both sides have said their piece, the audience votes again. Whoever moves the needle wins.
This year’s proposition — that design matters more than math — is sure to inspire some vigorous discussion. The argument for math is pretty strong. Math is proof. Given enough data — and today, we have plenty — we can know. “The right information in the right place just changes your life,” said Stewart Brand. Properly harnessed, the power of data analysis and modeling can fix cities, predict epidemics, and revitalize education. Abused, it can invade our lives, undermine economies, and steal elections. Surely the algorithms of big data matter!
But your life won’t change by itself. Bruce Mau defines design as “the human capacity to plan and produce desired outcomes.” Math informs; design compels. Without design, math can’t do its thing. Poorly designed experiments collect the wrong data. And if the data can’t be understood and acted upon, it may as well not have been crunched in the first place.
This is the question we’ll be putting to our debaters: Which matters more? A well-designed collection of flawed information — or an opaque, hard-to-parse, but unerringly accurate model? From mobile handsets to social policy, we need both good math and good design. Which is more critical? Read more…
Practical advice for those considering a career in data science
When I was a youngster in college I found myself dissatisfied after I took a stats class from the math department. So I decided to take another stats class. Classmates thought I was crazy. Let’s be real, what precocious over-achieving teenager majoring in English lit seeks to retake a math class? And not because of a grade but because they were dissatisfied with what they didn’t get out of it? After a bit of research, I decided to take the stats class offered by the psych department.
It made a significant difference.
Thinking about math from the perspectives of research design methodology and how data can be used to manipulate people made quite an impact on my teenage worldview. This experience also reinforced my belief that education is what you decide it will be. There is always more than one way to learn and education doesn’t necessarily have to happen in a physical classroom. Growing up in the San Francisco Bay Area where friends and loved ones decided to forgo traditional higher ed completely to start their own companies or immediately work in jobs in technology also contributed to this belief.
While full time students who are looking at a career in data science may have the time to do seemingly nutty things like take overlapping math classes, this is not something that most people with full time jobs are able to do. When people with full time jobs ask me about what they need to do to move into data science, I probe them about the kind of job in data science they want and about their analytical and empathy skills. Then, I immediately follow up with “So, how are your math skills?.” Interestingly enough, I get a lot people saying how they don’t have time to physically go into a classroom or that it has been, like, forever since they’ve used statistics and/or linear algebra for data analysis. Even more interesting is how often people don’t realize just how many resources are available to learn math outside of the physical-attendance-in-a-classroom-model.
Huh. Read more…
One of the largest gatherings of mathematicians, the joint meetings of the AMS/MAA/SIAM, took place last week in San Francisco. Knowing that there were going to be over 6,000 pure and applied mathematicians at Moscone West, I took some time off from work and attended several sessions. Below are a few (somewhat technical) highlights. It’s the only conference I’ve attended where the person managing the press room, was also working on some equations in-between helping the media.