globals [ top-row ; used by drop-marbles field ; the patches that are not border patches border ; the border patches bins ; the patches that make up the bottom bins pegs ; the patches that are pegs Fill ; Use to start/stop filling Field-color ; color of the background (field) Marble-color ; color of the marbles Peg-color ; normal grid color Bin-color ; color for bins Right-color ; pegs biased to the right Left-color ; pegs biased to the left Border-color ; color of the border patches Distr-color ; color to mark distribution on bins Hole-size ; how big is the hole from which marbles fall? Bin-height ; how tall are the bins? Pegs-top ; where at the top do the pegs begin? Pegs-bottom ; where at the bottom do the pegs end? total-error ; ] breed [ marbles ] patches-own [ void? ; true means patch is empty, and not an obstacle error ; used by distribution patches ] to setup ca ask patches [ set void? true ] if use-rseed [ random-seed r-seed ] set-color-scheme set Bin-height 0.4 * max-pycor set Pegs-top 0.95 * max-pycor set Pegs-bottom max-pycor * -0.5 reset-screen-regions set top-row patches with [ pycor = max-pycor ] set Hole-size 2 set Fill true make-hole make-pegs make-bins print "Initialization complete" end to go ; if using mouse, all else pauses ifelse mouse-down? [ do-mouse-action ] [ ; evaluate the run-time effect switches if Fill = true [ drop-marbles ] if Distribution != "None" [ show-dist ] ; marbles fall without-interruption [ ask marbles [ fall-down ] ] ] end to set-color-scheme set Field-color black set Marble-color 89 set Bin-color 62 set Peg-color 3 set Border-color sky set Right-color 16 set Left-color lime set Distr-color 15 end to define-field set field patches with [ abs pxcor < max-pxcor and pycor > min-pycor ] end to define-border set border patches with [ abs pxcor = max-pxcor or pycor = min-pycor ] end to define-bins set bins patches with [ pycor <= min-pycor + Bin-height and pxcor mod 3 = 0 and pycor != min-pycor and abs pxcor != max-pxcor ] end to define-pegs set pegs patches with [ pycor >= Pegs-bottom and pycor <= Pegs-top and (pycor + 1) mod 2 = 0 and floor (pxcor + pycor / 2) mod 2 = 0 and pycor != min-pycor and abs pxcor != max-pxcor ] end to-report field-not-defined? report not is-patch-set? field end to-report border-not-defined? report not is-patch-set? border end to-report bins-not-defined? report not is-patch-set? bins end to-report pegs-not-defined? report not is-patch-set? pegs end to define-screen-regions if field-not-defined? [ define-field ] if bins-not-defined? [ define-bins ] if border-not-defined? [ define-border ] if pegs-not-defined? [ define-pegs ] end to reset-screen-regions define-screen-regions clear-field set-border end to clear-field ask marbles [ pop ] ask field [ clear-grid ] end to set-border ask border [ set void? false set pcolor Border-color ] end to set-fill ifelse Fill [ set Fill false] [ set Fill true ] end to drop-marbles let open-patches top-row with [ void? = true ] if random 100 < drop-rate and count marbles < max-marbles [ if any? open-patches [ ask one-of open-patches [ make-marble ] ] ] end to make-marble ; patch procedure sprout 1 [ set breed marbles set shape "circle" set heading 180 set void? false set color Marble-color ] end to fall-down let below patch-at 0 -1 let below-left patch-at -1 -1 let below-right patch-at 1 -1 let beside patches at-points [ [ -1 0 ] [ 1 0 ] ] ; if open space directly below, fall straight down if [void?] of below = true [ set void? true set ycor ycor - 1 set void? false stop ] ; check if there is a biased grid point below. if [pcolor] of below = Right-color and [void?] of below-right = true and random 100 < bias-strength [ set void? true ; set xcor pxcor-of below-right ; set ycor pycor-of below-right setxy [pxcor] of below-right [pycor] of below-right set void? false stop ] if [pcolor] of below = Left-color and [void?] of below-left = true and random 100 < bias-strength [ set void? true ; set xcor pxcor-of below-left ; set ycor pycor-of below-left setxy [pxcor] of below-left [pycor] of below-left set void? false stop ] ; if void?-of below = false ; [ set beside beside with [ void? = true and [void?] of patch-at 0 -1 = true ] if count beside > 0 [ set beside one-of beside set void? true setxy [pxcor] of beside ( pycor - 1 ) set void? false stop ; ] ] end ; this procedure can do two things: set some of the pins to have a bias, or draw a distribution on the pegs to do-mouse-action ifelse Distribution != "Manual" or mouse-ycor >= Pegs-bottom [ let biased pegs with [ abs ( mouse-xcor - pxcor ) <= Bias-radius and abs ( mouse-ycor - pycor ) <= Bias-radius ] ask biased [ ifelse Bias-direction = "Left" [ set pcolor Left-color ] [ ifelse Bias-direction = "Right" [ set pcolor Right-color ] [ set pcolor Peg-color ] ] ] ; ask biased ] [ let marked bins with [ abs ( pxcor - mouse-xcor ) < 1 ]; and abs ( pycor - mouse-ycor ) < 1 ] ask marked [ ifelse pycor = round mouse-ycor [ set pcolor Distr-color ] [ set pcolor Bin-color] ] ] end to clear-marbles without-interruption [ ask marbles [ pop ] ] end to clear-bias without-interruption [ ask pegs [ set-color-grid Peg-color ] ] end to clear-distr without-interruption [ ask bins with [ pcolor = Distr-color ] [ set-color-grid Bin-color ] ] end to make-hole ask field with [ pycor = max-pycor and abs ( pxcor ) > Hole-size ] [ set-color-grid Border-color ] end to make-pegs ask pegs [ set-color-grid Peg-color ] end to make-bins ask bins [ set-color-grid Bin-color ] end ; superimpose the distribution on the bin columns to show-dist set total-error 0 ask bins [ if Distribution = "Dynamic" [ ifelse ( [void?] of patch-at -1 0 = false or [void?] of patch-at 1 0 = false ) and ( [void?] of patch-at -1 1 = true and [void?] of patch-at 1 1 = true or [pcolor] of patch-at 0 1 = Field-color ) [ set pcolor Distr-color ] [ set pcolor Bin-color ] stop ] if Distribution = "Gaussian" [ ifelse pycor = min-pycor + floor ( d-amp * Bin-height * exp ( - ( pxcor - d-mean ) * ( pxcor - d-mean ) / ( d-stdev * d-stdev ) ) ) [ set pcolor Distr-color set error 0 let left-col min-pycor let right-col min-pycor let not-found true ] [ set pcolor Bin-color set error 0 ] stop ] ] end to clear-grid set pcolor Field-color set void? true end to set-color-grid [ gcolor ] set pcolor gcolor set void? false end to pop hide-turtle set void? true die end @#$#@#$#@ GRAPHICS-WINDOW 234 10 512 509 33 58 4.0 1 10 1 1 1 0 1 1 1 -33 33 -58 58 0 0 1 ticks CC-WINDOW 5 533 659 628 Command Center 0 BUTTON 10 12 76 45 Init Setup NIL 1 T OBSERVER NIL I NIL NIL BUTTON 83 12 146 45 Run go T 1 T OBSERVER NIL R NIL NIL MONITOR 544 474 650 519 # of marbles count marbles 3 1 11 SLIDER 10 120 180 153 drop-rate drop-rate 0 20 20 1 1 NIL HORIZONTAL BUTTON 10 84 98 117 Clear marbles clear-marbles NIL 1 T OBSERVER NIL C NIL NIL SLIDER 11 249 183 282 Bias-radius Bias-radius 1 20 5 1 1 NIL HORIZONTAL BUTTON 10 201 98 246 Clear Pegs clear-bias NIL 1 T OBSERVER NIL P NIL NIL BUTTON 151 12 220 45 Drop set-fill NIL 1 T OBSERVER NIL M NIL NIL TEXTBOX 9 181 159 199 Bias settings\n 11 0.0 0 TEXTBOX 10 66 160 84 Marble settings 11 0.0 0 CHOOSER 101 201 223 246 Bias-direction Bias-direction "Left" "Right" "None" 0 BUTTON 13 370 110 415 Clear Distr clear-distr NIL 1 T OBSERVER NIL D NIL NIL TEXTBOX 13 352 163 370 Distribution 11 0.0 0 CHOOSER 116 370 212 415 Distribution Distribution "None" "Dynamic" "Gaussian" "Manual" 1 SLIDER 11 285 183 318 bias-strength bias-strength 0 100 100 1 1 % HORIZONTAL SLIDER 14 420 106 453 d-mean d-mean -21 21 0 1 1 NIL HORIZONTAL SLIDER 14 457 106 490 d-stdev d-stdev 0 50 10 1 1 NIL HORIZONTAL SLIDER 544 357 650 390 r-seed r-seed 0 10 0 1 1 NIL HORIZONTAL SLIDER 109 420 201 453 d-amp d-amp 0.0 1.0 1 0.1 1 NIL HORIZONTAL SWITCH 544 322 650 355 use-rseed use-rseed 1 1 -1000 SLIDER 101 84 220 117 max-marbles max-marbles 10 1000 250 10 1 NIL HORIZONTAL @#$#@#$#@ WHAT IS IT? ----------- This is a simulation of falling marbles to illustrate certain points about statistics and modeling. HOW IT WORKS ------------ Marbles fall from the top and bounce over pegs to the left or to the right. At the bottom they are collected into bins. The distribution of marbles across bins can be observed and modified by applying a "bias" to some of the pegs. When there are no biases, the resulting distribution approximates a Gaussian (or "Bell-shaped" or "Normal" distribution). Groups of pegs can be biased to the left or to the right by varying degrees. HOW TO USE IT ------------- === Main Control Buttons === - Click "Init" to reset the entire simulation and draw the pegs and the bins - Click "Go" to start/end the simulation. - Click "Drop" to start/stop dropping marbles through the hole at the top. === Marble Settings === - Click "Clear Marbles" to erase all Marbles. - Use the "max-marbles" slider to determine hoe many marbles to drop (good default is 250) - Adjust the "drop-rate" slider to determine how quickly marbles drop from the hole. === Bias Settings === Use these controls to change some of the pegs to be biased to the left or right. Click anywhere over the pegs to bias some of them, based on the various settings. - Use the "Bias-direction" drop-down to select Left, Right or None. - Use the "Bias-radius" slider to determine how large of a patch will be biased around the point where you click the mouse. - Use the "bias-strength" slider to set the bias. For example, when a marble hits a red peg (Right bias) with the slider set to 100%, the marble will always go to the right. At a 0% bias, the marble has a 50-50 chance of falling to the right. [Technically, the probability of the marble going to the right is 0.5*bias + 0.5.] === Distribution Settings === - Click "Clear Distr" to clear the red dots showing the distribution on the bins. - Use the "Distribution" drop-down to select one of the following modes: _* None: do not show distribution, or, if a distribution was being shown, do not change it. Once "Init" or "Clear Distr" are pressed, any remaining distribution will be erased. _* Dynamic: For each bin, place a dot corresponding to the highest marble stacked on its left (not ideal but gets the point across). This changes dynamically as the marbles drop to match the current distribution. _* Gaussian: Draw a Gaussian distribution based on the sliders below _* Manual: Use the mouse to draw a distribution by clicking on individual bins. Note that you can also click-and-drag to draw the distribution. - Use the "d-mean" "d-amp" and "d-stdev" sliders to set the mean, amplitude and standard deviation of the Gaussian distribution. === Additional controls === - The toggle "use-rseed" determines whether at initialization the random number generator uses a seed or not. If you want to replicate a run, turn this ON and select a desired seed. - Use the "r-seed" slider to set a random number seed between 0 and 10 - The "# of marbles" box is a counter showing how many marbles have been dropped. WHAT THIS IS ABOUT ------------------ The main point of this demo is to illustrate the difference between statistical approaches and agent-based or "bottom-up" approaches to modeling. When there are no biases, it can be shown mathematically that the distribution of falling marbles should approach a Gaussian distribution. In fact, if you run this with about 250 marbles and no biases, you should be able to fit the resulting distribution with a Gaussian with mean 0, amplitude 1 and standard deviation 10. However, suppose now that someone shows you a distribution which is not Gaussian. Perhaps it is skewed to one side, or even bimodal. A statistical modeling approach would try to find a different mathematical function (perhaps a gamma or beta distribution) and adjust parameters until it minimizes the error between theoretical and empirical distributions. However, this approach doesn't really say anything about WHY the distribution should be the way it is, nor is there any relationship between the mathematical expression for the distribution and the underlying behavior. In contrast, using a bottom-up modeling approach, we can make speculations about WHY a certain distribution arises. For instance, if you see a distribution where most marbles end up on the right, you can assume that some pegs must be biased to the right. By selecting different pegs you can make adjustments until you get close to the observed distribution. Conversely, you can make hypotheses about the impact of different biases. For instance, what happens if all the pegs on the right are biased to the right and all the pegs on the left are biased to the left? Is there a difference between biasing the first five rows of pegs versus biasing the bottom five rows of pegs? Now imagine the pegs representing people expressing opinions about a product. Each "person" decides whether he or she thinks the product is good (right) or bad (left). Each person then expresses that opinion to another person. Now imagine one person is biased in one direction. If that person can also influence many of its friends through word-of-mouth, a big cluster of biased individuals can cause the opinion of the entire population to shift. Modeling these kinds of effects is nearly impossible with statistical approaches. You might even imagine a more complex situation, in which the bias of individuals is changed gradually over time by watching what other people decided. Bottom-up modeling gives you the ability to test different conditions, and to do so based on direct, intuitive terms, such as "what happens if this group is biased?" or "where is the most effective group to bias?" By comparison, with a statistical approach you might be able to ask questions like "what if the standard deviation of the distribution increases?" or "what if I use a Gamma distribution instead of a Gaussian?" In a nutshell, bottom-up modeling gives you the power to explore and understand causality, while statistical approaches at best can tell you about correlations between variables. BUGS ETC ------------------- The "manual" distribution should place dots at the average between the highest marbles on the left and right sides. It would be nice to let the bias change as a function of several other factros, such as what other patches are doing, or what information the marbles carry. COPYRIGHT --------- This work was created by Paolo Gaudiano and is copyright of Icosystem Corporation. This may not be used for commercial purposes without prior consent of Icosystem. 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