Projections…a loss of information.

Workout notes Weights then a 4 mile treadmill walk. In all honesty, I walked on the treadmill because I was lazy; the roads aren’t that bad.

rotator cuff
pull ups: 5 sets of 10
superset: abs and incline bench press: 10 x 135, 4 x 155, 5 x 150, 6 x 145
superset rows and dumbbell bench press: rows: 3 sets of 10 x 65, bench press: 2 sets of 10 x 65, 1 set of 10 x 70
superset: military press, pull down, curl: military press: 2 sets of 15 x 50 dumbbell, 1 set of 10 x 70 (each) machine,
pull down: 3 sets of 10 with 160 (one with a rotated grip), curl: 3 sets of 10 x 70 (machine).

walking: treadmill, 2 miles at 15 mpm where I upped the incline from 1 to 8 (every two minutes) and then back down, then with the incline decreasing from 5 to eventually 1, increased the speed to 12 mpm. Second 2 miles took 25:10. Again, I was too lazy to put on cold weather gear.

But it paid off as it turns out that Barbara was leaving HER workout (third one in the last few days) and so I caught her in time for us to eat at the Indian buffet.

I admit that this is a cool video:

So what is going on? This is an example of mathematical projection: the image you are seeing is a projection of objects in 3 space being projected onto a 2 dimensional screen (sort of). Our brain attempt to decode what it sees.

Schematically, it is something like this:


Imagine a light at the origin and a vertical screen 3 units away. In between you can place objects; here you see the “semi-circle” (looks too bent to be a circle) and the line segment are different objects at different distances from the screen, yet BOTH objects cast the same “shadow” or “image”. You could NOT distinguish these two objects by the image they cast on the screen.

In short, “projection” loses information, by design.

There are all sorts of examples and implications of this concept. One of the most interesting ones comes from statistics and the famous Anscombe data sets. These are 4 very different data sets which, when one applies simple linear regression, yields not only the same regression line, but also the same correlation coefficient r and the same confidence intervals for the regression coefficients (up to a tiny difference).


Mathematically, think of it this way: a linear regression in this case can be thought of as a map from the set of, in this case, 11 different vectors in R^2 (denoted by (x_i, y_i) ) to 5 real numbers: \sum x_i, \sum y_i, \sum (x_i)(y_i), \sum x_i^2, \sum y_i^2 . (linear algebra exercise: write these 5 quantities in terms of the inner products of the various vectors and vector components). Or, one could see this as a projection from the sets of two 11 dimensional vectors (one consisting of the x components, one of the y components) onto the same set of numbers.

Why? Remember that the linear regression coefficients, the confidence intervals and the correlation coefficients are completely determined by these 5 quantities. So, a linear regression is a type of projection, and yes, you lose information when you do it.

December 28, 2012 - Posted by | mathematics, statistics, walking, weight training | , , ,

1 Comment »

  1. […] wrote about the loss of information that comes with projecting something onto a “smaller” set of numbers (or onto fewer dimensions) […]

    Pingback by Another example of projection …. « blueollie | March 13, 2013 | Reply

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