Dotting The Dot Map, Revisited. A. Jon Kimerling Dept. of Geosciences Oregon State University

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1 Dottig The Dot Map, Revisited A. Jo Kimerlig Dept. of Geoscieces Orego State Uiversity

2 Dot maps show the geographic distributio of features i a area by placig dots represetig a certai quatity of features where the features are most likely to occur.

3 Whe makig a dot map, we first select a dot diameter ad the determie the dot uit value. dot diameter dot uit value

4 The dot uit value is always greater tha oe. dot uit value > 1

5 Selectig the dot diameter is a subjective decisio.

6 Selectig the dot uit value is doe by trial ad error or usig the Mackay omograph. J. Ross Mackay, 1949 om o graph ou a graph, usually cotaiig three parallel scales graduated for differet variables, desiged to solve a equatio; also called a aligmet graph or calculatio graph.

7 Usig the Mackay omograph to fid the dot uit value

8 Mark the selected dot diameter

9 Draw a lie from the origi to your mark

10 Fid where the lie crosses the zoe of coalescig dots

11 Draw a vertical lie dow to the x-axis

12 Note the umber of dots per square cetimeter. 52

13 The, fid the map area of the regio that will have the highest desity of dots 1.2 sq cm WHEAT

14 Multiply the map area by the umber of dots per sq. cm. from the omograph to fid the umber of dots to place i that area WHEAT 52 dots / sq cm * 1.2 sq cm = 62 dots

15 Divide the quatity i the desest area by the umber of dots to fid the dot uit value WHEAT 62,000 acres 1997 STATE TOTAL Washigto: 2,422,506 Idaho: 1.410,978 Orego: 882,862 Each dot represets 1000 acres 62,000 acres / 62 dots = 1000 acres / dot

16 Issues with the Mackay omograph pt 3.5 pt 3 pt 2.5 pt mathematically imprecise zoe of coalescece 2 pt 1.5 pt poit size rage limits??? 1 pt

17 Let s look carefully at the omograph A 2 pt dot with diameter cm has a area of cm 2, so 128 dots without overlap would have a aggregate area of oly 0.50 cm 2 o overlap! 2 pt

18 Let s look carefully at the omograph The MacKay omograph is a graph of : dot area dots per sq cm = aggregate area of dots, with a zoe of coalescig dots overlaid. 2 pt

19 We eed a theoretically soud mathematical basis for dot coalescece.

20 Modelig dot coalescece usig the Uificatio Equatio from probability theory is the aswer!

21 ( ) ( ) ( ) ( ) ( ) ( ) < < < + < < < = = + + = k... j i k j i k j i j i j i i i i i E... E E E P... E E E P E E P E P E P U The Dot Coalescece Model A 1 cm 2 square has a probability P of 1.0. The area of each of dot (Ei) i proportio to the area of the square is its probability p = P(E i ). The Uificatio Equatio Sice the dots i the square are the same size, E i = E j = = E ad p = P(Ei) = P(Ej) = = P(E)

22 ( ) ( ) < < < + < < < = = + + = k... j i k j i j i i i p... p p p p P U = = i p p 0 Summatios are umbers of dot combiatios The umber of dot combiatios i each summatio are foud by: ( )! k k!! ( ) = U i p P 0 is the aggregate dot area i cm 2 The Uificatio Equatio

23 So P U i= 0 ( p) = p 2!! p +! ( 2)! 3! ( 3)! p... + ( 1) k!! ( k)! p k There is a term i the series for each dot added, but I trucated the series at k = 10 without affectig the results for up to: P U i= 0 ( p) = with up to 1,000 poits.

24 ( ) ( ) ( ) ( ) ( ) p!!!... p!!! p!!! p p P i + + = = U Or doig the factorials ( ) ( ) p,, ) )( )( )( )( )( )( )( )( )( (... p ) )( ( p ) ( p p P i + + = = U So

25 Usig the equatio with ESRI dots ESRI dots are i 0.5 Postscript poit icremets from 0.5 to 11 poits, although sizes smaller tha 1 or larger tha 5 poits would ot ormally be used for dot mappig. Give that 1 Postscript poit = mm, the followig table gives values of p (dot areas) for this dot size rage. Poit Size p (cm 2 ) Poit Size p (cm 2 )

26 Ruig the equatio with ESRI dots i 10 dot icremets gave the results plotted o this graph.

27 Usig the graph Example: Havig half the 1 cm 2 area covered by 2 pt dots takes 177 radomly placed dots

28 This compares with 128 dots from the MacKay omograph. 2 pt

29 Amout of dot overlap is aother measure of coalescece. Percet Overlap = p P U i= 0 ( p) 100

30 Ruig the equatio with ESRI dots i 10 dot icremets gave the results plotted o this graph. equal overlap poits

31 Equal overlap poits defie lies of costat dot overlap a more precise form of zoe of coalescig dots. equal overlap poits

32 Testig the model

33 Step 1. Geerate lots of radom umbers from 0.0 mm to 10.0 mm. #iclude <stdio.h> #iclude <stdlib.h> mai() { FILE*ofil; char oame[80]; it seed; double r; /* radom value i rage [0,1) */ it cout; seed = 10000; /* choose a seed value */ srad(seed); /*iitialize radom umber geerator*/ pritf("eter output file ame: "); scaf ("%s",oame); if((ofil = fope(oame,"wt")) == NULL) { pritf("error: caot ope output file\"); retur 1; } for(cout=1; cout<=3000; ++cout) { r = 10.0*(rad() / ((RAND_MAX)+(1.0))); fpritf(ofil,"%lf\",r); } retur 0; } /*mai*/

34 Step 2. Split the umbers ito two Excel colums ad make a scatterplot.

35 Step 3. Import the scatterplot ito Freehad ad chage the dots to the desired diameter.

36 Step 4. Wrap the dots aroud the four edges.

37 Step 5. Import the graphic ito Photoshop ad crop the edge dots.

38 Step 6. Use the histogram tool to fid the proportio of the square covered by dots.

39 I made these dot proportio measuremets for the umber of dots predicted at differet poit sizes for 10-50% dot overlap.

40 How close did the measured aggregate areas match those predicted from the Uificatio Equatio?

41 I tested the equatio for 10% overlap of 3 pt dots 55 radom dots.

42 I also made a Dot Selectio Guide based o aggregate dot area.

43 What is the problem with radom dot placemet? Cartographers place dots maually i a pseudoradom fashio!

44 Let s look at pseudo-radomess i terms of maximum allowable overlap of idividual dots 200 dots are placed i each square

45 Idividual dot overlap is computed from the les area equatio. Area Les = d 2R R cos d 4R d

46 Graphig the les area equatio shows the oliear relatioship betwee the spacig of dot ceters ad the proportio of a dot overlapped by the les. example, 2 dots spaced at 1.4 times their radius will overlap by 0.2 (20% of the dot area)

47 This iformatio is the basis for a pseudo-radom dot geerator. 1. Select a dot radius ad maximum dot overlap. 2. Geerate the first radom dot positio. 3. For succeedig dots, compute the distace betwee the dot ceter ad all other dot ceters. 4. If all distaces are greater tha d, add the dot to the array of dots otherwise, discard the dot positio ad geerate aother radom positio. 5. Repeat util the umber of dots you eed is geerated, or util a maximum umber of tries is reached.

48 The procedure I wrote a C program to ru the procedure 30 times for 10%, 20%, ad 30% maximum idividual dot overlaps, i steps of 25 or 50 dots with a maximum of 32,000 tries for each ru. Oce I had a pseudo-radom dot positio array, I could check the distace of each dot agaist all other dots, ad compute the les area if a dot was less tha a dot diameter away. Summig the les areas gave the total dot overlap area, ad hece the aggregate proportio or percetage covered by dots, assumig that there were o triple dot overlaps.

49 1 pt dots maximum aggregate area reached i less tha 32,000 tries

50 What is happeig with mutually exclusive dot placemet? rigid radom packig for 1 pt dots i a 1 x 1 cm square

51 Do the graphs for 1.0 pt to 2.5 pt dots look similar?

52 Aggregate area equatios Mutually exclusive dots: Area = p Area = p 2!! Totally radom dots: 2! 3 p + p... +! ( 2)! 3! ( 3) ( 1) 10 10!! ( 10)! p 10 My guess is that the equatio for itermediate pseudo-radom dots is a liear combiatio of the two boudig equatios above.

53 A geeral aggregate area equatio Area = kp + ( 1 k) ( p 2!! p 2 +! ( 2)! 3! ( 3)! p ( 1) 10 10!! ( 10)! p 10 ) or Area = p + ( 1 k) ( 2!! p 2 +! ( 2)! 3! ( 3)! p ( 1) 10 10!! ( 10)! p 10 ) where k rages from 0 (totally radom) to 1 (mutually exclusive).

54 Guesses as to what k is proportioal to? 1. k = les proportio 2. k = d/2 totally radom mutually exclusive

55 The secod possibility for k fit the data better about twice as good a fit.

56 Average values for a umber of tries of 30 rus of creatig pseudo-radom are plotted o these graphs. 1 pt dot example

57 From these values we ca make a pseudoradom dot selectio guide.

58 Pseudo-radom placemet of 1 pt dots with less tha 5,000 tries

59 Pseudo-radom Dot Selectio Guide for 1 pt dots with variable maximum dot overlap

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65 mappigceter.esri.com Other Resources

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