resources
visualizations
presentations
lunch break
Distractions and amusements, with a sandwich and coffee.
visualization + design
The art of Pi (`\pi`), Phi (`\phi`) and `e`
This section contains various art work based on `\pi`, `\phi` and `e` that I created over the years.
Some of the numerical art reveals interesting and unexpected observations. For example, the sequence 999999 in π at digit 762 called the Feynman Point. Or that if you calculate π to 13,099,586 digits you will find love.
`\pi` day art and `\pi` approximation day art is kept separate.
For some time I have been thinking about creating minimalist typographical art based on the digits of `\pi`. The `i`-ness `\pi` project was one of my first forays into this kind of art.
the `i`-ness of `\pi`
In the `i`-ness of `\pi` poster shown above, the average is mapped onto a color and the standard deviation onto size.
the `i`-ness of `\phi` and `e`
Compare the `i`-ness of `\pi` to that of the other famous transcendental number, `e`, and the mysterious but attractive Golden Ratio, `phi`.
the 4-ness of `\pi`
These posters show the difference between each digit and 4,
Spring and fall color themes of the posters are also available.
curiously humorous variant
I assure you—`\pi` has a lot of 4s. Why, in the first 19,528 digits there are 2,000 of them! That's a lot. Here they are.
If you stare at them long enough, they even appear to move. Amazing.
news + thoughts
Curse(s) of dimensionality
There is such a thing as too much of a good thing.We discuss the many ways in which analysis can be confounded when data has a large number of dimensions (variables). Collectively, these are called the "curses of dimensionality".
Some of these are unintuitive, such as the fact that the volume of the hypersphere increases and then shrinks beyond about 7 dimensions, while the volume of the hypercube always increases. This means that high-dimensional space is "mostly corners" and the distance between points increases greatly with dimension. This has consequences on correlation and classification.
Statistics vs Machine Learning
We conclude our series on Machine Learning with a comparison of two approaches: classical statistical inference and machine learning. The boundary between them is subject to debate, but important generalizations can be made.Inference creates a mathematical model of the datageneration process to formalize understanding or test a hypothesis about how the system behaves. Prediction aims at forecasting unobserved outcomes or future behavior. Typically we want to do both and know how biological processes work and what will happen next. Inference and ML are complementary in pointing us to biologically meaningful conclusions.
Statistics asks us to choose a model that incorporates our knowledge of the system, and ML requires us to choose a predictive algorithm by relying on its empirical capabilities. Justification for an inference model typically rests on whether we feel it adequately captures the essence of the system. The choice of pattern-learning algorithms often depends on measures of past performance in similar scenarios.
Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Statistics vs machine learning. Nature Methods 15:233–234.
Background reading
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: supervised methods. Nature Methods 15:5–6.
Happy 2018 `\pi` Day—Boonies, burbs and boutiques of `\pi`
Celebrate `\pi` Day (March 14th) and go to brand new places. Together with Jake Lever, this year we shrink the world and play with road maps.
Streets are seamlessly streets from across the world. Finally, a halva shop on the same block!
Intriguing and personal patterns of urban development for each city appear in the Boonies, Burbs and Boutiques series.
No color—just lines. Lines from Marrakesh, Prague, Istanbul, Nice and other destinations for the mind and the heart.
The art is featured in the Pi City on the Scientific American SA Visual blog.
Check out art from previous years: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, 2016 `\pi` Day and 2017 `\pi` Day.
Machine learning: supervised methods (SVM & kNN)
Supervised learning algorithms extract general principles from observed examples guided by a specific prediction objective.We examine two very common supervised machine learning methods: linear support vector machines (SVM) and k-nearest neighbors (kNN).
SVM is often less computationally demanding than kNN and is easier to interpret, but it can identify only a limited set of patterns. On the other hand, kNN can find very complex patterns, but its output is more challenging to interpret.
We illustrate SVM using a data set in which points fall into two categories, which are separated in SVM by a straight line "margin". SVM can be tuned using a parameter that influences the width and location of the margin, permitting points to fall within the margin or on the wrong side of the margin. We then show how kNN relaxes explicit boundary definitions, such as the straight line in SVM, and how kNN too can be tuned to create more robust classification.
Bzdok, D., Krzywinski, M. & Altman, N. (2018) Points of Significance: Machine learning: a primer. Nature Methods 15:5–6.
Background reading
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.
Human Versus Machine
Balancing subjective design with objective optimization.In a Nature graphics blog article, I present my process behind designing the stark black-and-white Nature 10 cover.
Nature 10, 18 December 2017
Machine learning: a primer
Machine learning extracts patterns from data without explicit instructions.In this primer, we focus on essential ML principles— a modeling strategy to let the data speak for themselves, to the extent possible.
The benefits of ML arise from its use of a large number of tuning parameters or weights, which control the algorithm’s complexity and are estimated from the data using numerical optimization. Often ML algorithms are motivated by heuristics such as models of interacting neurons or natural evolution—even if the underlying mechanism of the biological system being studied is substantially different. The utility of ML algorithms is typically assessed empirically by how well extracted patterns generalize to new observations.
We present a data scenario in which we fit to a model with 5 predictors using polynomials and show what to expect from ML when noise and sample size vary. We also demonstrate the consequences of excluding an important predictor or including a spurious one.
Bzdok, D., Krzywinski, M. & Altman, N. (2017) Points of Significance: Machine learning: a primer. Nature Methods 14:1119–1120.