Wikipedia:Reference desk/Archives/Mathematics/2017 January 26

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January 26

Best way to count with fingers ?

(For the following Q, I consider thumbs to be a type of finger.)

• There's the common way of not considering the place and just counting each finger as 1 (up) and 0 (down), which allows for easy counting from 0-10.

• There's assigning each a positional binary digit, which allows for counting from 0-10271023, but can require some painful finger positions. You could even attempt to have 3 or more possible positions for each finger, but that we be even more painful.

• Base 6 seems like a promising compromise, where the fingers on one hand, in any order, total to give you the ones digit, and the total of the other hand's fingers is the 6's digit. No painful manipulations, and you can count 0-35:

,,,,, ,,,,, =  0  |,,,, ,,,,, =  1  ||,,, ,,,,, =  2  |||,, ,,,,, =  3  ||||, ,,,,, =  4  ||||| ,,,,, =  5  
,,,,, |,,,, =  6  |,,,, |,,,, =  7  ||,,, |,,,, =  8  |||,, |,,,, =  9  ||||, |,,,, = 10  ||||| |,,,, = 11  
,,,,, ||,,, = 12  |,,,, ||,,, = 13  ||,,, ||,,, = 14  |||,, ||,,, = 15  ||||, ||,,, = 16  ||||| ||,,, = 17  
,,,,, |||,, = 18  |,,,, |||,, = 19  ||,,, |||,, = 20  |||,, |||,, = 21  ||||, |||,, = 22  ||||| |||,, = 23  
,,,,, ||||, = 24  |,,,, ||||, = 25  ||,,, ||||, = 26  |||,, ||||, = 27  ||||, ||||, = 28  ||||| ||||, = 29  
,,,,, ||||| = 30  |,,,, ||||| = 31  ||,,, ||||| = 32  |||,, ||||| = 33  ||||, ||||| = 34  ||||| ||||| = 35  

So, are there other methods that beat this ? I imagine you can get around the painful finger positions by not allowing certain combos, still using each finger as a binary digit, but this seems too complex to use. Moving the 3rd and 4th fingers as a unit is less painful, and still allows counting from 0-255, using binary positional notation. Is this system used much ? StuRat (talk) 04:12, 26 January 2017 (UTC)

0 to 99, with one hand doing units and other tens, T is thumb

,,,,, = 0  ,|,,, = 1  ,||,, = 2  ,|||, = 3  ,|||| = 4    (or ,,||| for 3)
T,,,, = 5  T|,,, = 6  T||,, = 7  T|||, = 8  T|||| = 9    (or T,||| for 8)

-- SGBailey (talk) 07:06, 26 January 2017 (UTC)

Chisanbop is a method of counting 0-99 on both hands. There are other methods as well. Optimist on the run (talk) 09:26, 26 January 2017 (UTC)

Whether or not finger positions are painful seems to depend a great deal on your hands: I find finger binary and finger ternary are not that painful for me. I suppose moving the 3rd and 4th fingers as a unit has the advantage that you are then expressing binary one byte at a time, though. Double sharp (talk) 10:15, 26 January 2017 (UTC)

Yes. Is there a name for this method ? StuRat (talk) 00:51, 27 January 2017 (UTC)

Not that I know of, but I would be very surprised if it had not already been invented by someone else given its utility. Double sharp (talk) 04:43, 28 January 2017 (UTC)

Resampling with autocorrelation

I have a time series (equal steps) with some amount of temporal autocorrelation. For a simulation, I now wish to create a new time series by resampling from this source series, while taking the autocorrelation into account. Frankly, I have no idea how to approach this. My naive hope would be that there is some way of using the correlation coefficients computed for the source series, to weight the probability of a new random value x_t being chosen based on x_t-1 (e.g., by forcing a resample if the new value does not fall into an interval from the previous value that is defined by the coefficient). - Is there a standard method for doing this? -- Elmidae (talk · contribs) 10:16, 26 January 2017 (UTC)

Add: I'm thinking correlation range 1 here (sensible for these data), otherwise I guess one would need some kind of on-the-fly re-analysis of the evolving time series on each step. -- Elmidae (talk · contribs) 10:25, 26 January 2017 (UTC)

I don't see anything about it in the article Resampling (statistics), though you might want to check the articles it links to, such as Bootstrapping Bootstrapping (statistics). I'll see if I can think of anything, but it seems like a hard problem. Loraof (talk) 21:02, 26 January 2017 (UTC)

• Bootstrapping (statistics)#Time series: Moving block bootstrap has something, but I'm not sure it will be helpful to you. Loraof (talk) 21:14, 26 January 2017 (UTC)
• The only thing I can think of is original research and probably not rigorously valid for whatever use you have in mind. From a set of n data points that have been expressed as deviations from their mean, maybe you could draw one value, then multiply it by your computed correlation coefficient rho and draw the value from the full set of n values the one that is closest to the product of the multiplication, giving you your second value in your new series; then multiply that second value by rho and find the value from the full set of n values that is closest to your computed product; etc. Sounds kind of ad hoc though. Loraof (talk) 02:06, 27 January 2017 (UTC)

Thank you, very helpful! Your ad hoc approach basically gels with what I thought one could do based on pre-computed coefficients. Ad hoc is fine by me - this needn't be very rigorous, I'm just trying to improve on complete random sampling (although rigorous would of course be best). - The moving block bootstrap angle seems to suggest that something like a random walk might work - force autocorrelation in resamples by resampling not values but time points, and those within a restricted distance (possibly 1) from each other. Humm-homm. -- Elmidae (talk · contribs) 07:54, 27 January 2017 (UTC)

Given an autocorrelated series $y_{i}$ with correlation coefficient r, you can create a decorrelated series $w_{i}$ by taking $w_{i}={\frac {y_{i}-r\,y_{i-1}}{\sqrt {1-r^{2}}}}$. You could resample the $w_{i}$ values and then recreate an autocorrelated sequence $v_{i}={\sqrt {1-r^{2}}}w_{i}+r\,v_{i-1}$. The $v_{i}$ would then be a simulation of the original $y_{i}$. Dragons flight (talk) 15:17, 27 January 2017 (UTC)

Dragons flight, thank you, I like it a lot! It's quite elegant. The only possible issue is that I'm not sure whether the rest of the simulation will be happy with a series of values that preserves the same correlation structure as the source but is composed of different values. I may instead have to shoot for something that preserves the same pool of values but arranges them in a similar (but not identical) structure. In which case I think I may have to go back to blockwise resampling / random walks. Otherwise, this is a nice solution. -- Elmidae (talk · contribs) 14:20, 28 January 2017 (UTC)