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visualization + design
`\pi` Approximation Day Art Posters
The never-repeating digits of `\pi` can be approximated by 22/7 = 3.142857 to within 0.04%. These pages artistically and mathematically explore rational approximations to `\pi`. This 22/7 ratio is celebrated each year on July 22nd. If you like hand waving or back-of-envelope mathematics, this day is for you: `\pi` approximation day!
getting it mostly right
Curiously, the 22/7 rational approximation of `\pi` is more accurate (to within 0.04%) than using the first three digits 3.14, which are accurate to 0.05%.
It seems that `\pi` Approximation Day is 20% more accurate (verify on Wolfram Alpha)! And therefore definitely worth celebrating. $$ \frac{(\pi-3.14)-(22/7-\pi)}{\pi-3.14} = 0.206 $$
art of `\pi` rational approximation
The poster shows the accuracy of 10,000 rational approximations of `\pi` for each `m/n` and `m=1...10000`. Read about the details of the method.
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art of `\pi`-approximate packing
These posters show warped circles, which embody the 22/7 approximation of `\pi`, using a retro 1970's color scheme. Read about the details of the method.
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upcoming
news + thoughts
Model Selection and Overfitting
With four parameters I can fit an elephant and with five I can make him wiggle his trunk. —John von Neumann.
By increasing the complexity of a model, it is easy to make it fit to data perfectly. Does this mean that the model is perfectly suitable? No.
When a model has a relatively large number of parameters, it is likely to be influenced by the noise in the data, which varies across observations, as much as any underlying trend, which remains the same. Such a model is overfitted—it matches training data well but does not generalize to new observations.
We discuss the use of training, validation and testing data sets and how they can be used, with methods such as cross-validation, to avoid overfitting.
Altman, N. & Krzywinski, M. (2016) Points of Significance: Model Selection and Overfitting. Nature Methods 13:703-704.
Background reading
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Classifier evaluation. Nature Methods 13:603-604.
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. Nature Methods 13:541-542.
Classifier Evaluation
It is important to understand both what a classification metric expresses and what it hides.
We examine various metrics use to assess the performance of a classifier. We show that a single metric is insufficient to capture performance—for any metric, a variety of scenarios yield the same value.
We also discuss ROC and AUC curves and how their interpretation changes based on class balance.
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Classifier evaluation. Nature Methods 13:603-604.
Background reading
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. Nature Methods 13:541-542.
Happy 2016 `\pi` Approximation, roughly speaking
Today is the day and it's hardly an approximation. In fact, `22/7` is 20% more accurate of a representation of `\pi` than `3.14`!
Time to celebrate, graphically. This year I do so with perfect packing of circles that embody the approximation.
By warping the circle by 8% along one axis, we can create a shape whose ratio of circumference to diameter, taken as twice the average radius, is 22/7.
If you prefer something more accurate, check out art from previous `\pi` days: 2013 `\pi` Day and 2014 `\pi` Day, 2015 `\pi` Day, and 2016 `\pi` Day.
Logistic Regression
Regression can be used on categorical responses to estimate probabilities and to classify.
The next column in our series on regression deals with how to classify categorical data.
We show how linear regression can be used for classification and demonstrate that it can be unreliable in the presence of outliers. Using a logistic regression, which fits a linear model to the log odds ratio, improves robustness.
Logistic regression is solved numerically and in most cases, the maximum-likelihood estimates are unique and optimal. However, when the classes are perfectly separable, the numerical approach fails because there is an infinite number of solutions.
Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. Nature Methods 13:541-542.
Background reading
Altman, N. & Krzywinski, M. (2016) Points of Significance: Regression diagnostics? Nature Methods 13:385-386.
Altman, N. & Krzywinski, M. (2015) Points of Significance: Multiple Linear Regression Nature Methods 12:1103-1104.
Altman, N. & Krzywinski, M. (2015) Points of significance: Simple Linear Regression Nature Methods 12:999-1000.
Visualizing Clonal Evolution in Cancer
Genomic instability is one of the defining characteristics of cancer and within a tumor, which is an ever-evolving population of cells, there are many genomes. Mutations accumulate and propagate to create subpopulations and these groups of cells, called clones, may respond differently to treatment.
It is now possible to sequence individual cells within a tumor to create a profile of genomes. This profile changes with time, both in the kinds of mutation that are found and in their proportion in the overall population.
Clone evolution diagrams visualize these data. These diagrams can be qualitative, showing only trends, or quantitative, showing temporal and population changes to scale. In this Molecular Cell forum article I provide guidelines for drawing these diagrams, focusing with how to use color and navigational elements, such as grids, to clarify the relationships between clones.
I'd like to thank Maia Smith and Cydney Nielsen for assistance in preparing some of the figures in the paper.
Krzywinski, M. (2016) Visualizing Clonal Evolution in Cancer. Mol Cell 62:652-656.
Binning High-Resolution Data
Limitations in print resolution and visual acuity impose limits on data density and detail.
Your printer can print at 1,200 or 2,400 dots per inch. At reading distance, your reader can resolve about 200–300 lines per inch. This large gap—how finely we can print and how well we can see—can create problems when we don't take visual acuity into account.
The column provides some guidelines—particularly relevant when showing whole-genome data, where the scale of elements of interest such as genes is below the visual acuity limit—for binning data so that they are represented by elements that can be comfortably discerned.
Krzywinski, M. (2016) Points of view: Binning high-resolution data. Nature Methods 13:463.