1.3-galtonBoard-Multiplicative.nlogo

;; The MIT License (MIT)
;;
;; Copyright (c) 2011-24 David O'Sullivan and George Perry
;;
;; Permission is hereby granted, free of charge, to any person
;; obtaining a copy of this software and associated documentation
;; files (the "Software"), to deal in the Software without restriction,
;; including without limitation the rights to use, copy, modify, merge,
;; publish, distribute, sublicense, and/or sell copies of the Software,
;; and to  permit persons to whom the Software is furnished to do so,
;; subject to the following conditions:
;;
;; The above copyright notice and this permission notice shall be included
;; in all copies or substantial portions of the Software.
;;
;; THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
;; OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
;; FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
;; THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
;; LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
;; FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
;; DEALINGS IN THE SOFTWARE.

;; extensions [r]

breed [pegs peg]
breed [balls ball]

turtles-own [
  real-x ; keep track of the real x coordinate not the NetLogo 'world' xcor
]

globals [
  column-heights
  off-the-edge
  bottom-edge
  x-start-position
  x-coords
]

to setup
  clear-all

  set column-heights n-values world-width [0]
  set off-the-edge 0
  set bottom-edge 40
  set x-start-position max-pxcor / 10

  draw-pegs
  ;; r:setPlotDevice
  reset-ticks
end

to draw-pegs
  ;; common source location in x direction
  let xs max-pxcor / 10
  ;; y row starts from 1 goes to the bottom edge
  let y-row n-values (max-pycor - bottom-edge + 1) [ ?1 -> ?1 ]
  set x-coords (list x-start-position)
  foreach y-row [ y ->
    foreach x-coords [ x ->
      create-pegs 1 [
        set shape "dot"
        set size 1.2
        set color red
        setxy x (max-pycor - y)
      ]
    ]
    set column-heights x-coords
    ; each row starts a factor of c to the left
    set x-coords map [ x -> x * c ] x-coords
    ; and extends one peg to the right
    set x-coords lput (last x-coords / c / c) x-coords
    ; throw away any x-coords off the edge of the screen
    set x-coords filter [ x -> x < max-pxcor ] x-coords
  ]
  set x-coords map [ x -> precision x 4 ] column-heights
  set column-heights map [0] column-heights
end

to go
  drop-one
  if max [ycor] of balls >= bottom-edge [
    stop
  ]
end

to drop-one
  create-balls 1 [
    set-starting-position
    drop
    move-to-final-position
  ]
  tick
end

to set-starting-position
  set shape "circle"
  set color gray
  setxy x-start-position (max-pycor)
  set real-x xcor
  pen-down
end

to drop
  ;; multiplicative -> to build a log-normalish distribution
  while [pycor > bottom-edge] [
    set real-x real-x * one-of (list c (1 / c))
    ;; this updates the column-heights list and colours them in as appropriate
    ifelse real-x <= max-pxcor
    [ setxy real-x (pycor - 1) ]
    [ ;; ball falling off the right hand side
      set  off-the-edge off-the-edge + 1
      ;; turn it into a line so we can roughly show where it went
      ;; face towards the next peg
      let x-dist-to-next-peg xcor / c - xcor
      set heading atan x-dist-to-next-peg -1
      fd floor x-dist-to-next-peg
      die
    ]
  ]
end

to move-to-final-position
  ;; update the tally for this column and colour the patches appropriately
  let column position (precision xcor 4) x-coords ; floor xcor
  let height item column column-heights
  ;; change the ball to a line scaled
  ;; to reflect where it is located
  ;; and spaced vertically by the inverse of that
  ;; this makes an approximate histogram
  let factor xcor / c - xcor * c
  setxy xcor min-pxcor + (height / factor)
  set shape "line"
  set heading 90
  set size factor
  set color white
  set column-heights replace-item column column-heights (height + 1)
end

;; R plotting code
;;to plot-r-hist
;;  r:put "x" [xcor] of turtles
;;  r:put "max.x" max-pxcor
;;  r:eval("hist(x, xlab = 'X', ylab = 'Frequency', main = '', las = 1, breaks = seq(0,300,20))")
;;
;;end
;;
;;to plot-r-density
;;  r:put "x" [xcor] of turtles
;;  r:put "max.x" max-pxcor
;;  r:eval( "plot(density(x), las = 1, xlab = 'X', main = '' )")
;;end
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Note that the lattice discretizes the log-normal distribution that emerges from the multiplicative process.  This produces funny results for the low values, where the grid resolution cannot track counts properly in this simple implementation. Plot the data in R to get a better sense of the process.
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As c -> 0, the right-skew on the 'triangles' in the Galton board increases
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off-the-edge
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BUTTON
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clear
clear-drawing\nask balls [die]\nset column-heights map [0] column-heights
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OBSERVER
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1

@#$#@#$#@
## WHAT IS IT?

This model is an implementation of a multiplicative Galton board as discussed in Chapter 1 of

+   O'Sullivan D and Perry GLW 2013 _Spatial Simulation: Exploring Pattern and Process_. Wiley, Chichester, England.

You should consult that book for more information and details of the model.

## THINGS TO TRY

It is useful to slow this model down so that you can see how individual 'balls' arrive at their final locations.  Also, use the **drop-one** button to see how one ball drops through the peg board.

## THINGS TO NOTICE

This models is very similar to the previous model **1.2-galtonBoard-Additive.nlogo**, the sole difference being that at each downward step in the dropping of the balls, they are _multiplicatively_ shifted to the left or right, so that the resulting distribution is approximately _log-normal_.

Set up of the peg board makes use of iteration over nested lists of `y-rows` from the top of the models space, and `x-coords`. The latter list begins with only one entry at the x coordinate starting location. Each iteration of the `y-rows` the items in the list are shifted a factor of `c` to the left, and a new item is added to the end of the list a factor `(1 / c ^ 2)` to the right. Since the list has already moved to the left, this right-most element in the list will be a factor of `(1 / c)` to the right of the last peg in the previous row.

## HOW TO CITE

If you mention this model in a publication, please include these citations for the model itself and for NetLogo

+   O'Sullivan D and Perry GLW 2013 _Spatial Simulation: Exploring Pattern and Process_. Wiley, Chichester, England.
+   Wilensky U 1999 NetLogo. http://ccl.northwestern.edu/netlogo/. Center for Connected Learning and Computer-Based Modeling, Northwestern University, Evanston, IL.

## COPYRIGHT AND LICENSE

The MIT License (MIT)

Copyright &copy; 2011-24 David O'Sullivan and George Perry

Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to  permit persons to whom the Software is furnished to do so, subject to the following conditions:

The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
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