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EMBO Practical Course: Bioinformatics and Genome Analysis, 5–17 June 2017.


fun + amusement

The Ptolemaic Clock — A Proposal

the standard clock

Consider the lowly wall clock. It's practical and generally tells the correct time. It's the same clock everywhere and after a while it gets boring pretty quickly—maybe now?

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
In a standard clock, the bezel is fixed and the hands rotate.

In the regular clock the face bezels stay in place and the hands move. Why am I telling you this? Well, maybe you see where I'm going.

the Ptolemaic Clock

Who says it's the hands that have to rotate? Instead of rotating hands and a stationary bezel, consider the clock with stationary hands rotating bezels.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
In the Ptolemaic clock, the hands stay in place while independent minute and hour hand bezels rotate to simulate the movement of the hands.

In the Ptolemaic clock there are two independent bezels and two independent hands. The bezels rotate counterclockwise to simulate the standard clockwise motion of the hands. The hands are not moving but in the frame of reference of the bezels, it's the hands that are rotating. The position of the bezel is always related to the current time and the position of its corresponding hand.

The bezel can move clockwise.

Thanks to Rodrigo Goya for suggesting the name for this kind of clock—Ptolemaic Clock, named so after the geocentric Ptolemaic model of the solar system.

telling time on the Ptolemaic clock

To tell the time on the Ptolemaic clock is a process identical to using the standard clock. You look at the bezel numbers at the ends of the hour and minute hands.

On the fixed bezel layout, most people will take a short cut and tell the time by the position of the hands. This works as long as you have a standard clock. On a Ptolemaic clock the position of the hands tells you nothing.

Here is a Ptolemaic clock telling us it is 6:30. It uses the same position of hands as in the figures above.

You know this because the blue hour hand points to midway between 6 and 7 on the inner hour bezel and the grey minute hand points to 30 on the outer minute bezel.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
It is 6:30 on this Ptolemaic clock.

After 15 minutes, it's 6:45 and our Ptolemaic clock bezels have moved a little bit.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
It is 6:45 on this Ptolemaic clock.

Can you tell what time it is on the Ptolemaic clock below?

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
If you answered 8:50, you are correct. It is 8:50.

customizing the Ptolemaic clock

Customizing your Ptolemaic clock is easy. Simply adjust the hands to desired positions and set the time by moving the bezels. The clock below shows the same time as the clock in the above figure — both show 8:50.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
This clock tells us it's 8:50. Compare this to the clock in the figure above, which also tells the same time.

ptolemaic clock — hard layout

In the clock design shown here, the hands are the same size and only differ by color. To make things less confusing, emphasize the hour hand.

To make things more confusing, remove all color and number cues, keeping only a single symbol on each of the bezels to indicate 12 o'clock and 0 minutes. This is shown in the clock below.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
In the hard layout of a Ptolemaic clock, there are fewer cues. I think it's 8:50.

news room parodies

Spice it up with multiple Ptolemaic clocks side-by-side telling the same time with different hand positions.

Suppose it is 2:30 in Vancouver—this is my location. The clocks below all show 2:30, but with hands set to 5:30, 11:30 and 7:30.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Looks like a wall of clocks in a newsroom. Except these Ptolemaic clocks tell us that it's 2:30, three times over in Vancouver.

These hand positions are those that would appear on a standard clock showing the times in New York (5:30), Paris (11:30) and Tokyo (7:30).

Let's now use the Ptolemaic clock to show times at these three locations but with the hand set to the curiously satisfying layout of 10ish minutes to 2.

Non-standard clock with rotating bezel. / Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A challenging panel of Ptolemaic clocks.

TIP

Set both hand positions to 12 o'clock and then remove the hands; to tell time, read the numbers on the hour and minute bezels at the apex of the clock.

EXTENSION

Sophisticated implementations of the Ptolemaic clock could periodically randomize hand positions to keep things interesting; by the time you've figured out the time in the morning, you're wide awake.

Every minute the clock randomly resets its hand positions. The movement is smooth and the bezels follow.

hardware implementation

If you would like to implement the Ptolemaic clock, I would be happy to hear from you. One should be able to take a regular wall clock, reverse the direction of the hand mechanism and rig a freely moving bezel to each of the minute and hour mechanism. The hands should not move and can be fixed to the front glass plate, for example.

conclusions

It should now be clear that the Ptolemaic clock is superior to the standard clock. The reasons are

  • it's much harder to tell time on the Ptolemaic clock, which makes your brain do more work
  • it tips its hat off to a simpler time when we didn't know anything and hints at the possibility of regression anytime
    • it will confuse everyone
    • you have a great excuse for being late
    • return to geocentric values!
  • you can customize your own Ptolemaic clock by moving the hands to arbitrary locations
    • two Ptolemaic clocks can have their hands and bezels at different positions but still be telling the same time
    • two Ptolemaic clocks can have their hands at the same position but be telling different times
VIEW ALL

news + thoughts

Machine learning: a primer

Tue 05-12-2017
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.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Machine learning: a primer. (read)

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.",

...more about the Points of Significance column

Snowflake simulation

Tue 14-11-2017
Symmetric, beautiful and unique.

Just in time for the season, I've simulated a snow-pile of snowflakes based on the Gravner-Griffeath model.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
A few of the beautiful snowflakes generated by the Gravner-Griffeath model. (explore)

Gravner, J. & Griffeath, D. (2007) Modeling Snow Crystal Growth II: A mesoscopic lattice map with plausible dynamics.

Genes that make us sick

Thu 02-11-2017
Where disease hides in the genome.

My illustration of the location of genes in the human genome that are implicated in disease appears in The Objects that Power the Global Economy, a book by Quartz.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
The location of genes implicated in disease in the human genome, shown here as a spiral. (more...)

Ensemble methods: Bagging and random forests

Mon 16-10-2017
Many heads are better than one.

We introduce two common ensemble methods: bagging and random forests. Both of these methods repeat a statistical analysis on a bootstrap sample to improve the accuracy of the predictor. Our column shows these methods as applied to Classification and Regression Trees.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Ensemble methods: Bagging and random forests. (read)

For example, we can sample the space of values more finely when using bagging with regression trees because each sample has potentially different boundaries at which the tree splits.

Random forests generate a large number of trees by not only generating bootstrap samples but also randomly choosing which predictor variables are considered at each split in the tree.

Krzywinski, M. & Altman, N. (2017) Points of Significance: Ensemble methods: bagging and random forests. Nature Methods 14:933–934.

Background reading

Krzywinski, M. & Altman, N. (2017) Points of Significance: Classification and regression trees. Nature Methods 14:757–758.

...more about the Points of Significance column

Classification and regression trees

Mon 16-10-2017
Decision trees are a powerful but simple prediction method.

Decision trees classify data by splitting it along the predictor axes into partitions with homogeneous values of the dependent variable. Unlike logistic or linear regression, CART does not develop a prediction equation. Instead, data are predicted by a series of binary decisions based on the boundaries of the splits. Decision trees are very effective and the resulting rules are readily interpreted.

Trees can be built using different metrics that measure how well the splits divide up the data classes: Gini index, entropy or misclassification error.

Martin Krzywinski @MKrzywinski mkweb.bcgsc.ca
Nature Methods Points of Significance column: Classification and decision trees. (read)

When the predictor variable is quantitative and not categorical, regression trees are used. Here, the data are still split but now the predictor variable is estimated by the average within the split boundaries. Tree growth can be controlled using the complexity parameter, a measure of the relative improvement of each new split.

Individual trees can be very sensitive to minor changes in the data and even better prediction can be achieved by exploiting this variability. Using ensemble methods, we can grow multiple trees from the same data.

Krzywinski, M. & Altman, N. (2017) Points of Significance: Classification and regression trees. Nature Methods 14:757–758.

Background reading

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Logistic regression. Nature Methods 13:541-542.

Altman, N. & Krzywinski, M. (2015) Points of Significance: Multiple Linear Regression Nature Methods 12:1103-1104.

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: Model Selection and Overfitting. Nature Methods 13:703-704.

Lever, J., Krzywinski, M. & Altman, N. (2016) Points of Significance: Regularization. Nature Methods 13:803-804.

...more about the Points of Significance column



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