# Galton’s board all askew

**R – Xi'an's Og**, and kindly contributed to R-bloggers]. (You can report issue about the content on this page here)

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**S**ince Galton’s quincunx has fascinated me since the (early) days when I saw a model of it as a teenager in an industry museum near Birmingham, I jumped on the challenge to build an uneven nail version where the probabilities to end up in one of the boxes were not the Binomial ones. For instance, producing a uniform distribution with the maximum number of nails with probability ½ to turn right. And I obviously chose to try simulated annealing to figure out the probabilities, facing as usual the unpleasant task of setting the objective function, calibrating the moves and the temperature schedule. Plus, less usually, a choice of the space where the optimisation takes place, i.e., deciding on a common denominator for the (rational) probabilities. Should it be 2⁸?! Or more (since the solution with two levels also involves 1/3)? Using the functions

evol<-function(P){ Q=matrix(0,7,8) Q[1,1]=P[1,1];Q[1,2]=1-P[1,1] for (i in 2:7){ Q[i,1]=Q[i-1,1]*P[i,1] for (j in 2:i) Q[i,j]=Q[i-1,j-1]*(1-P[i,j-1])+Q[i-1,j]*P[i,j] Q[i,i+1]=Q[i-1,i]*(1-P[i,i]) Q[i,]=Q[i,]/sum(Q[i,])} return(Q)}

and

temper<-function(T=1e3){ bestar=tarP=targ(P<-matrix(1/2,7,7)) temp=.01 while (sum(abs(8*evol(R<-P)[7,]-1))>.01){ for (i in 2:7) R[i,sample(rep(1:i,2),1)]=sample(0:deno,1)/deno if (log(runif(1))/tempI first tried running my simulated annealing code with a target function like

targ<-function(P)(1+.1*sum(!(2*P==1)))*sum(abs(8*evol(P)[7,]-1))where P is the 7×7 lower triangular matrix of nail probabilities, all with a 2⁸ denominator, reaching

60

126 35

107 81 20

104 71 22 0

126 44 266914

61123 113 92 91 38

109607 19 44 74 50for 128P. With four entries close to 64, i.e. ½’s. Reducing the denominator to 16 produced once

8

12 1

13 11 3

16 7 6 2

14 13 16 15 0

15 15 2 7 7 4

80898168

as 16P, with five ½’s (8). But none of the solutions had exactly a uniform probability of 1/8 to reach all endpoints. Success (with exact 1/8’s and a denominator of 4) was met with the new target

(1+,1*sum(!(2*P==1)))*(.01+sum(!(8*evol(P)[7,]==1)))imposing precisely 1/8 on the final line. With a solution with 11 ½’s

0.5

1.0 0.0

1.0 0.0 0.0

1.00.51.00.5

0.50.51.0 0.0 0.0

1.0 0.00.50.00.50.0

0.5 0.5 0.51.0 1.0 1.00.5and another one with 12 ½’s:

0.5

1.0 0.0

1.0 .375 0.0

1.0 1.0 .6250.5

0.50.5 0.5 0.50.0

1.0 0.00.5 0.50.00.5

0.51.00.50.0 1.00.50.0Incidentally, Michael Proschan and my good friend Jeff Rosenthal have an 2009 American Statistician paper on another modification of the quincunx they call the uncunx! Playing a wee bit further with the annealing, and using a denominator of 840 let to a 2P with 14 ½’s out of 28

.5

60 0

60 1 0

30 30 300

30 30 30 30 30

60 60 60 0 60 0

6030030306030Toleave a commentfor the author, please follow the link and comment on their blog:R – Xi'an's Og.

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