Matlab and Python

I use Matlab/Octave, quite a bit, for pre- and post- processing. About 3-4 years ago, I started dabbling with python, and gradually began using it to handle increasing parts of my workflow.

Last week, I chanced upon this webpage at Mathworks, which tries to argue why Matlab is superior to python. Here are the key advantages:

• The matrix-based MATLAB language lets you express math directly: This is definitely true. Even now, when my work predominantly involves linear algebra, I sometimes use Matlab/Octave. The notation is natural and concise. But this is Matrix Laboratory after all.

• Engineers and scientists deserve tools that fit the way they work. They shouldn’t have to adapt the way they work to fit their tools: It is true that Matlab documentation feels like it was written with engineers and scientists in mind, while python documentation has a computer-sciency feel. But, the other reasons bundled under this heading don't seem like they apply to me.

• Proven MATLAB toolboxes provide the functions and capabilities you need. Period.: However, many toolboxes are have to be bought separately. The standard scientific python stack is fairly mature at this point. Even the library overall landscape for python is richer for python in my mind. It reminds of this xkcd cartoon.

• MATLAB apps let you complete tasks more easily than with custom programming
• MATLAB helps automate the entire path – from research to production
• You can trust the results you get in MATLAB: I lumped these three together, because for some reason, they are non-issues for me. I enjoy prototyping in python. If I am hardware limited, and I have to squeeze performance, then I usually code in C++ or Fortran.

• MATLAB runs your programs faster – meaning you can try more ideas and solve bigger problems: This is probably true, but ignores the existence of JIT compilation (Numba) or integrating with C++/Fortran (Cython) etc.

Some of the advantages of python over Matlab are listed here, here, and here. If I had to make an intermediate term prediction (10 years), I think python will become more popular than Matlab, among scientists and engineers.

Tyranny of Metrics

EconTalk recently had Jerry Muller on the podcast discussing his book "Tyranny of Metrics". He laments over our fixation with our obsession with measurement.

Metrics are fine as a diagnostic tool, he argues, but when they are used as surrogates for success and attached to rewards, things go haywire. There are only a handful of metrics that retain their usefulness once they are widely adopted. The power of incentives prods people to game the statistic.

Muller gives several examples from various fields:

Measuring the success rates of surgeons performing certain operations and making them available seems like a good idea. Undoubtedly, transparency would helps patients make better choices. However, after the scorecards became public, surgeons began avoiding complicated cases, which would lower their batting average.

Then there is the phenomenon of goal diversion. A great deal of K-12 education has been distorted by the emphasis that teachers are forced to place on preparing students for standardized tests of English and math, where the results of the tests influence teacher retention or school closings. Teachers are instructed to focus class time on the elements of the subject that are tested (such as reading short prose passages), while ignoring those elements that are not (such as novels). Subjects that are not tested—including civics, art, and history—receive little attention. 

Or, to take an example from the world of business. In 2011 the Wells Fargo bank set high quotas for its employees to sign up customers who were interested in one of its products (say, a deposit account) for additional services, such as overdraft coverage or credit cards. For the bank’s employees, failure to reach the quota meant working additional hours without pay and the threat of termination. The result: to reach their quotas, thousands of bankers resorted to low-level fraud, with disastrous effects for the bank. It was forced to pay a fortune in fines, and its stock price dropped.

Here is yet another book review in Science. 

Diffusion in Higher Dimensions

In the previous posts (1 and 2), we wrote down the probability or concentration distribution of a bunch of Brownian diffusors initially at \(x = 0\) (delta function), \[p_{1D}(x, t) = \dfrac{1}{\sqrt{4 \pi Dt}} \exp\left(-\dfrac{x^2}{4Dt}\right)\]

The PDF is normalized on the domain \(x \in [-\infty, \infty]\) so that, \[\int_{-\infty}^{\infty} p_{1D}(x,t)\, dx = 1.\] In 2D, \(\langle r^2(t) \rangle = \langle x^2(t) \rangle + \langle y^2(t) \rangle\). If diffusion is isotropic, then \(\langle r^2(t) \rangle = 2Dt + 2Dt = 4Dt\). In this case,
\begin{align}
p_{2D}(r, t) & = p_{1D}(x, t) \, p_{1D}(y, t)\\
& = \dfrac{1}{\sqrt{4 \pi Dt}} \dfrac{1}{\sqrt{4 \pi Dt}} \exp\left(-\dfrac{1}{2} \dfrac{x^2+y^2}{2Dt}\right)\\
& =\dfrac{1}{4 \pi Dt} \exp\left(-\dfrac{r^2}{4Dt}\right)
\end{align}

The PDF is normalized such that, \[\int_{0}^{\infty} (2\pi r) \, p_{2D}(r,t)\, dr = 1.\]
Finally, for isotropic 3D diffusion, \[p_{3D}(r, t) = \left(\dfrac{1}{4 \pi Dt}\right)^{3/2} \exp\left(-\dfrac{r^2}{4Dt}\right).\] The PDF is normalized such that, \[\int_{0}^{\infty} (4\pi r^2) \, p_{3D}(r,t)\, dr = 1.\] In summary, for \(d\) = 1, 2, or 3 dimensions
\[p_{dD}(r, t) = \left(\dfrac{1}{4 \pi Dt}\right)^{d/2} \exp\left(-\dfrac{r^2}{4Dt}\right).\]

Notebooks and Exploration

The Atlantic has a nice article on genesis and evolution of Mathematica and Jupyter notebooks, and how the latter was inspired by the former. It is provocatively (unfortunately) titled, "The Scientific Paper is Obsolete".

The article itself is more thoughtful and nuanced.

It is a reflection on the use of notebooks as exploratory vehicles, and as computational essays. This is indeed how I use Jupyter notebooks these days. I use them as a pre-processing tool (exploratory mode) when I have to design a new lecture or lab, or plan a set of new calculations. I also use them as a post-processing tool, especially in my research. Once all the raw computation is done, I can play with the results interactively, and eventually interleave a narrative and charts. This notebook often becomes the starting point of the "Results and Discussion" section of any resulting paper.

Here are some passages from the article that I found interesting or appealing:

The notebook interface was the brainchild of Theodore Gray, who was inspired while working with an old Apple code editor. Where most programming environments either had you run code one line at a time, or all at once as a big blob, the Apple editor let you highlight any part of your code and run just that part. Gray brought the same basic concept to Mathematica, with help refining the design from none other than Steve Jobs.

“I’ve noticed an interesting trend,” Wolfram wrote in a blog post. “Pick any field X, from archeology to zoology. There either is now a ‘computational X’ or there soon will be. And it’s widely viewed as the future of the field.” 

A 1997 essay by Eric S. Raymond titled “The Cathedral and the Bazaar,” in some sense the founding document of the modern open-source movement, challenged the notion that complex software had to be built like a cathedral, “carefully crafted by individual wizards or small bands of mages working in splendid isolation.” Raymond’s experience as one of the stewards of the Linux kernel (a piece of open-source software that powers all of the world’s 500 most powerful supercomputers, and the vast majority of mobile devices) taught him that the “great babbling bazaar of differing agendas and approaches” that defined open-source projects was actually a strength. “The fact that this bazaar style seemed to work, and work well, came as a distinct shock,” he wrote.

The Mathematica notebook is the more coherently designed, more polished product—in large part because every decision that went into building it emanated from the mind of a single, opinionated genius. “I see these Jupyter guys,” Wolfram said to me, “they are about on a par with what we had in the early 1990s.” They’ve taken shortcuts, he said. “We actually want to try and do it right.”

Writing Technical Papers

Here is some decent advise on how to improve the quality of technical writing:

Ten simple rules for structuring papers
PLOS Computational Biology, 2017.

Whitesides’ Group: Writing a Paper
Advanced Materials, 2004.

Writing a Research Paper in the Natural Sciences
Graduate Writing Lab, Yale University, 2015.

10 Tips on How to Write Less Badly
Michael Munger, CHE, 2010.

Diffusion and Random Walks

In the previous post, we saw how the probability distribution \(p(x,N)\) after \(N\) random steps on a unit lattice is given by, \[p(x, N) = \dfrac{1}{\sqrt{2 \pi N}} \exp\left(-\dfrac{x^2}{2N}\right)\] If the average step size is \(b\) instead of \(b=1\), then we can generalize, and write the formula as:
\[p(x, N) = \dfrac{1}{\sqrt{2 \pi Nb^2}} \exp\left(-\dfrac{x^2}{2Nb^2}\right)\] Now consider a Gaussian random walk in 1D. Suppose the stepsize at each step is drawn from a normal distribution \(\mathcal{N}(0, 1)\). While it has the same average stepsize as a walk on the lattice, an individual step may be shorter or longer than b=1.

In polymer physics, where a Gaussian coil is often used as a model for polymer conformations, \(b\) is called the Kuhn length, and \(N\) is proportional to the molecular weight.

Due to the connection between Brownian motion and random walks, the mean squared distance travelled by a particle in 1D with self-diffusivity \(D\) is \(\langle x^2(t) \rangle = 2Dt\). Similarly, the mean end-to-end squared distance of a Gaussian random walk is given by, \[\langle x^2(N) \rangle = \int_{-\infty}^{\infty} x^2 p(x, N) dx = Nb^2 \equiv 2Dt = \langle x^2(t) \rangle.\] This allows us to re-parameterize the equation for the position of a Brownian diffusors. \[p(x, t) = \dfrac{1}{\sqrt{4 \pi Dt}} \exp\left(-\dfrac{x^2}{4Dt}\right)\] Look at the correspondence between \(t\) and \(N\), and \(b\) and \(\sqrt{2D}\).