polar plots manual

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Polar Plots and Circular Statistics

Last updated 23 June 2014Jeff Jenness


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Manual: Polar Plots ArcGIS ExtensionLast Modified: June 23, 2014

About the Author

Jeff Jenness is an independent GIS consultant specializing in developing analytical applications for a wide variety oftopics, although he most enjoys ecological and wildlife-related projects. He spent 16 years as a wildlife biologist with theUSFS Rocky Mountain Research Station in Flagstaff, Arizona, mostly working on Mexican spotted owl research. Sincestarting his consulting business in 2000, he has worked with universities, businesses and governmental agencies aroundthe world, including a long-term contract with the United Nations Food and Agriculture Organization (FAO) for which herelocated to Rome, Italy for 3 months. His free ArcView tools have been downloaded from his website and the ESRI

ArcScripts site over 190,000 times.


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NAME: PolarPlotsandCircularStatistics

InstallFile: PolarPlots.exe

Lastmodified: June23,2014

TOPICS: Polarplot,Roseplot,CircularStatistics,AngularDeviation,AngularVariance,MeanResultantLength,Circular

StandardDeviation,CircularVariance

AUTHOR: JeffJenness

WildlifeBiologist,GISAnalyst

JennessEnterprises

3020N.ScheveneBlvd.

Flagstaff,AZ86004 USA

Email:[email protected]

WebSite: http://www.jennessent.com)

Phone: 19286074638

Description: Thisextensionincludestwotools. BothtoolsareavailableattheArcView(aka

ArcGISBasic)licenselevel.

1. TheShapestoSegmentstoolconvertspolygonorpolylinefeaturesintopolyline

featuresofindividualvertextovertexsegments,withoptionsonvariousgeometric

attributestocalculate(startingandendingcoordinates,length,azimuth).

2. ThePolarPlotstoolcreatesagraphicintheMXDlayoutillustratingthedistributionof

directionvaluesinafeatureclass(suchasthosegeneratedintheShapestoSegments

tool)ortable. ThePolarPlotstoolalsohasanoptiontocalculatebasiccircular

descriptivestatisticsondata.

Output:

One

tool

produces

polyline

feature

classes

and

the

other

tool

produces

a

graphic

plot

intheMXDLayout.

Requires: ArcGIS9.1orbetter,atanylicenselevel.

RevisionHistoryonp.47.

RecommendedCitationFormat: Forthosewhowishtocitethisextension,theauthor

recommendssomethingsimilarto:

Jenness,J. 2014. PolarPlotsandCircularStatistics:ExtensionforArcGIS. JennessEnterprises.

Availableat: http://www.jennessent.com/arcgis/Polar_Plots.htm

Pleaseletmeknowifyoucitethisextensioninapublication([email protected]).Iwill

updatethecitationlisttoincludeanypublicationsthatIamtoldabout.


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4

Table

of

Contents

TABLEOFCONTENTS............................................................................................................................ 4

INSTALLINGTHEPOLARPLOTSANDCIRCULARSTATISTICSEXTENSION........................................................... 5

For ArcGIS 9.x .................................................................................................................................................................. 5

For ArcGIS 10.0 ............................................................................................................................................................... 5Viewing the Tools ............................................................................................................................................................. 8

UNINSTALLINGTHEPOLARPLOTSANDCIRCULARSTATISTICSEXTENSION.................................................... 10

For ArcGIS 9.x. ............................................................................................................................................................... 10

For ArcGIS 10.0 ............................................................................................................................................................. 10

TROUBLESHOOTING............................................................................................................................ 13

If Any of the Tools Crash ................................................................................................................................................ 13

Object variable or With block variable not set Error: .................................................................................................... 13

RICHTX32.OCX Error (also comct332.ocx, comdlg32.ocx, mscomct2.ocx, mscomctl.ocx, msstdfmt.dll errors): ......... 13

ISSUESWITHDIRECTIONALDATA.......................................................................................................... 17

Azimuths and Geodesic Curves ..................................................................................................................................... 17

Graphically Representing Directional Data .................................................................................................................... 18

Analyzing Directional Data: Circular Statistics ............................................................................................................... 21

MeanDirectionandMeanResultantLength........................................................................................................22

VarianceandStandardDeviation:........................................................................................................................24

Other Methods to Analyze Directional Data ................................................................................................................... 25

ClassificationofDirectionalValues.......................................................................................................................26

DeviationsfromaBearing....................................................................................................................................26

SineandCosineTransformations.........................................................................................................................26

SPECIALCONSIDERATIONSFORASPECT.................................................................................................. 27

How Aspect is Calculated ............................................................................................................................................... 27

Isthisthebestmethodtocalculateaspect?.........................................................................................................29

Issues with Aspect .......................................................................................................................................................... 29

Solar Insolation: An alternative to Aspect ................ ................ ................ ................ ................ ................. ................ ..... 30

ESRIsSolarRadiationTool...................................................................................................................................31

Hillshade...............................................................................................................................................................32

USINGTHETOOLS.............................................................................................................................. 33

Convert Shapes to Segments ........................................................................................................................................ 33

Generate Circular Plots .................................................................................................................................................. 36

PlotStyle...............................................................................................................................................................37

DataSourceOptions:............................................................................................................................................38

PlotandTitleOptions:..........................................................................................................................................39

ReferenceLineandLabelOptions:.......................................................................................................................42

Calculate Descriptive Circular Statistics ......................................................................................................................... 45

REVISIONS........................................................................................................................................ 47

REFERENCES...................................................................................................................................... 48


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5

InstallingthePolarPlotsandCircularStatisticsExtension

ForArcGIS9.x

FirstcloseArcGISifitisopen. ToolsdonotinstallproperlyifArcGISisrunningduringthe

installation.

InstallthePolarPlotsandCircularStatisticsextensionbydoubleclickingonthefile

PolarPlots.exeandfollowingtheinstructions. Theinstallationroutinewillregisterthe

PolarPlots.dllwithalltherequiredArcMapcomponents.

ThedefaultinstallfolderfortheextensionisnamedPolar_Plotsandislocatedinsidethe

folderProgramFiles\Jennessent. Thisfolderwillalsoincludesomeadditionalfilesandthis

document.

ForArcGIS10.0

Note: ThisfunctionwillonlyworkifyouhaveArcGIS10installed.

1. FirstcloseArcGISifitisopen. ThetoolsdonotinstallproperlyifArcGISisrunningduring

theinstallation.

2. InstallthePolarPlotsandCircularStatisticsextensionbydoubleclickingonthefile

PolarPlots.exeandfollowingtheinstructions. Thisinstallationroutinewillinstallthe

PolarPlots.dllandseveralancillaryfilesonyourharddrive,butwillnotregisterthetoolwith

ArcGIS.

3. UseWindowsExplorertoopenyourinstallationfolder. Ifyouusedthedefaultvalues,then

thisfolderwillbelocatedatProgramFiles\Jennessent\Polar_Plots\. Thisfolderwillalso

includesomeadditionalfilesandthismanual.

4. ForWindowsXP: DoubleclickthefileMake_Batch_Files.exetocreateregistrationand

unregistrationbatchfilesthatareproperlyformattedtoyoursystem.


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6

a. DoubleclickthenewbatchfileRegister_Polar_Plots_Tool.battoregisterthe

toolwithArcGIS10.0.

b. Iftheregistrationissuccessful,thenyoushouldseeaRegistrationSucceeded

notice.


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7

5. ForWindowsVistaorWindows7: RightclickthefileMake_Batch_Files.exe,andthen

chooseRunasAdministratortocreateregistrationandunregistrationbatchfilesthatare

properlyformattedtoyoursystem.


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8

6. RightclickthenewbatchfileRegister_Polar_Plots_Tool.bat,andthenchooseRunas

AdministratortoregisterallthetoolswithArcGIS10.0.

7. Iftheregistrationissuccessful,thenyoushouldseeaRegistrationSucceedednotice.

Note: Fortheconcernedorcurious,thebatchfileRegister_Polar_Plots_Tool.batcontainsa

singlelineoftextthatlookssimilartothefollowing:

"C:\Program Files (x86)\Common Files\ArcGIS\bin\ESRIRegAsm.exe" /p:Desktop"C:\Program Files (x86)\Jennessent\Polar_Plots\PolarPlots.dll" /f:"C:\ProgramFiles (x86)\Jennessent\Polar_Plots\PolarPlots.reg"

ItdirectstheESRIinstallerESRIRegAsmtoregistertheextensionDLLPolarPlots.dllwithin

ArcGIS,usingGUIDandClassIDvaluesfromtheregistryfilePolarPlots.reg(alsolocatedin

yourinstallationdirectory). BothRegister_Polar_Plots_Tool.batandPolarPlots.regmaybe

openedandviewedusingstandardtexteditorssuchasNotepadorWordPad.

ViewingtheTools

ThistoolisinstalledasanextensioninArcMap,butitisatypeofextensionthatisautomatically

loaded. YouwillnotseethisextensionintheExtensionsdialogavailableintheArcGISTools

menu. ItisnotdependentonanyotherextensionsoranyArcGISlicenselevel.

Thisextensiondoesnotincludeaseparatetoolbarsoyouwillneedtomanuallyputthetoolsin

anexistingtoolbar.


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9

Todothis,openyourCustomizetoolbyeither:

1) DoubleclickingonablankpartoftheArcMaptoolbar,or

2) ForArcGIS9,clicktheToolsmenu,thenCustomize,or

3) ForArcGIS10,clicktheCustomizemenu,thenCustomizeMode

IntheCustomizedialog,clicktheCommandstabandscrolldowntoselectJennessEnterprises:

Finally,simplydragtheConvertShapestoSegmentsand/orthePolarPlotcommandsoutofthe

CustomizedialogupintoanyoftheexistingArcGIStoolbars.


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10

UninstallingthePolarPlotsandCircularStatisticsExtension

ForArcGIS9.x.

1) CloseArcGISifitisopen.

2) ClicktheStartbutton.

3) OpenyourControlPanel.

4) DoubleclickAddorRemovePrograms.

5) ScrolldowntofindandselectPolarPlotsandCircularStatistics.

6) ClicktheRemovebuttonandfollowthedirections.

For

ArcGIS

10.01) CloseArcGISifitisopen.

2) UseWindowsExplorertoopenyourinstallationfolder. Ifyouusedthedefaultvalues,

thenthisfolderwillbelocatedatProgramFiles\Jennessent\Polar_Plots\. Thisfolder

willalsoincludesomeadditionalfilesandthismanual.


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11

3) FindthefileUnregister_Polar_Plots_Tool.bat. IFYOUDONOTSEETHISFILE,usethe

Make_Batch_Files.exetooltocreatethebatchfile. RefertoStep4inInstallingthe

PolarPlotsandCircularStatisticsExtensionaboveforinstructionsonhowtousethis

tool.

4) ForWindowsXP: DoubleclickthefileUnregister_Polar_Plots_Tool.battounregister

thetoolwithArcGIS10.0.

ForWindowsVistaandWindows7: RightclickthefileUnregister_Polar_Plots_Tool.bat

andselectRunasAdministratortounregisterthetoolwithArcGIS10.0.

Iftheunregistrationissuccessful,thenyoushouldseeanUnregistrationSucceeded

notice.

5) ClicktheStartbutton.

6) OpenyourControlPanel.

7) DoubleclickAddorRemovePrograms.

8) ScrolldowntofindandselectPolarPlotsandCircularStatistics10.

9) ClicktheUninstallbuttonandfollowthedirections.


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12

Note: Fortheconcernedorcurious,thebatchfileUnregister_Polar_Plots_Tool.batcontains

somethingsimilartothefollowingsinglelineoftext:

"C:\Program Files\Common Files\ArcGIS\bin\ESRIRegAsm.exe" /p:Desktop /u "C:\ProgramFiles\Jennessent\Polar_Plots\PolarPlots.dll"

ItdirectstheESRIinstallerESRIRegAsmtounregistertheDLLPolarPLots.dllwithinArcGIS.

Unregister_Polar_Plots_Tool.batmaybeopenedandviewedusingstandardtexteditorssuchas

NotepadorWordPad.


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13

Troubleshooting

IfAnyoftheToolsCrash

Ifatoolcrashes,youshouldseeadialogthattellsuswhatscriptcrashedandwhereitcrashed. I

wouldappreciateitifyoucouldcopythetextinthatdialog,orsimplytakescreenshotsofthe

[email protected]. Note:

Pleasemakesurethattheline

numbersarevisibleinthescreenshots! Thelinenumbersarelocatedonthefarrightsideofthe

text. Usethescrollbaratthebottomofthedialogtomakethelinenumbersvisible.

ObjectvariableorWithblockvariablenotsetError:

IfyouopenArcMapandimmediatelyseetheerrordialogappearwithoneormoreerror

messagesstatingthatObjectvariableorWithblockvariablenotset,then90%ofthetimeitis

becauseArcGISwasrunningwhenyouinstalledtheextension. TheObjectvariablebeing

referredtoistheExtensionobject,andArcGISonlysetsthatvariablewhenitisinitially

opened.

ThesolutionisusuallytosimplycloseArcGISandrestartit. Ifthatdoesnotwork,then:

1) CloseArcGIS

2) Reinstalltheextension

3) TurnArcGISbackon.

RICHTX32.OCXError(alsocomct332.ocx,comdlg32.ocx,mscomct2.ocx,

mscomctl.ocx,msstdfmt.dllerrors):

Ifyouseealineintheerrordialogstating:

Component 'RICHTX32.OCX' or one of its dependencies not correctly registered: a file ismissing or invalid

Orifyouseeasimilarerrorstatingthatoneormoreofthefilescomct332.ocx,comdlg32.ocx,

mscomct2.ocx,mscomctl.ocxormsstdfmt.dllaremissingorinvalid,thensimplyfollowthe

instructionsforRICHTX32.OCXbelow,butsubstitutetheappropriatefileforRICHTX32.OCX.

ThiserrorisalmostalwaysduetothefactthatnewinstallationsofWindows7andWindows

Vistadonotincludeafilethattheextensionexpectstofind. Forexample,thefilerichtx32.ocx

isactuallytheRichTextBoxcontrolthatappearsonsomeoftheextensiondialogs. Theother

OCXfilesrefertoothercommoncontrolsthatmightappearonthevariousextensiondialogs.

Thesolutionistomanuallyinstallthemissingfile(richtx32.ocx)yourself. Hereishowtodoit:

1) OpenWindowsExplorerandlocatethefilerichtx32.ocxinyourextensioninstallation

file.

2) Ifyouarerunninga32bitversionofWindows,thencopyrichtx32.ocxtothedirectory

C:\Windows\System32\

Ifyouarerunninga64bitversionofWindows,thencopyrichtx32.ocxtothedirectory

C:\Windows\SysWOW64\

3) OpenanElevatedCommandPromptwindow. ThisisthestandardWindows

CommandPromptwindow,butwithadministrativeprivilegesenabled. Youneedthese


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privilegesenabledinordertoregistertheOCXwithWindows. Note: TheElevated

CommandPromptopensupinthe..\windows\system32directory,notthe

..\Users\[UserName]directory. Thewindowtitlewillalsobeginwiththeword

Administrator:

a. Method1: ClicktheStartbutton,thenAllPrograms,thenAccessoriesand

thenrightclickonCommandPromptandselectRunasAdministrator.

b. Method2: ClicktheStartbutton,andthenclickontheSearchProgramsand

Filesbox. TypecmdandthenclickCONTROL+SHIFT+ENTERtoopenthe

CommandwindowwithAdministratorprivileges.

FormorehelponopeninganElevatedCommandPrompt,pleasereferto:

http://www.sevenforums.com/tutorials/783elevatedcommandprompt.html


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15

http://www.winhelponline.com/articles/158/1/HowtoopenanelevatedCommand

PromptinWindowsVista.html

OrsimplydoasearchforElevatedCommandPrompt.

4) Registerthefilerichtx32.ocxusingtheWindowsRegSvrfunction:

a. Ifusinga32bit

version

of

Windows,typetheline

regsvr32.exe c:\windows\system32\richtx32.ocx

b. Ifusinga64bitversionofWindows,typetheline

regsvr32.exe %windir%\syswow64\richtx32.ocx

c. Click[ENTER]andyoushouldseeamessagethattheregistrationsucceeded.


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17

Issues

with

Directional

Data

Manycharacteristicsofwildlifehabitatandbehaviorincludeadirectionalcomponent. Animal

movementmaytendtoangletowardorawaysomelandscapefeature,andthestrengthofthe

attractionorrepulsioneffectmaychangewithproximitytothatobject. Aspect(i.e. the

azimuththelandscapefacesinthedirectionofsteepestslope)isanintuitivelyvaluablehabitat

variablebecause northfacingslopesaretypicallycoolerandmoremesicwhilesouthfacingslopesaregenerallywarmerandmorexeric(atleastinthenorthernhemisphere!). Habitatscan

bedramaticallydifferentdependingonwhichsideofahillyoustandon.

Directioncanbedifficulttoanalyzestatisticallybecauseofitscircularnature. Thedifference

between1degreeand360degreesisthesameasthedifferencebetween1degreeand2

degrees,makingitdifficulttoplugintoatypicalstatisticalmodel. Thisarticlewilldiscusssome

issuesweoftenfacewithdirectionaldataplussomebasicideastodescribeandanalyzeit.

Manyoftheconceptscanalsobeappliedtootherperiodicdatasuchastime,butthisarticlewill

focusondirection.

Anadditionalcomplicationisthatsometimesweconsiderdirectionasasinglevalue(the

forwarddirection)andsometimeswecountboththeforwardandreversedirections. In

wildlifeanalysis,directionisusuallytreatedasasinglevalue. Weareinterestedinwherean

animalisgoingandwhyitchoosestogointhedirectionitdoes. Orweareinterestedinthe

generalaspectofahabitatblock,andwhatthatmeanstoourspeciesofinterest. Insomecases,

however,weneedtoconsiderboththedirectionanditsoppositevalue. Analysisofgeologic

fracturelinesonthelandscapeshouldbeconsideredtogoinbothdirections. Analysisoftrails

orpathsshouldbetreatedsimilarly,exceptintherarecaseofonewaytrails.

AzimuthsandGeodesicCurves

Inmostcasesthedirectionorazimuthvalueiseasytodefine. Theaspectatapointcanbe

derivedwithalittlemathfromaDEM(seeSpecialConsiderationsforAspectonp.27)orwith

acompass. Theazimuthofamovementsegmentissimplytheazimuthfromthestartingtothe

endingpointsofthatsegment.

Complicationsariseifweareinterestedinazimuthsoverlongdistances. Thestraightline

segmentoveralongdistanceisbestapproximatedbyageodesiccurverepresentingthegreat

circledistance(i.e.Asthecrowflies)betweentwopoints. Theproblemhereisthat,witha

fewrareexceptions,azimuthvalueschangeconstantlyoverthecourseofageodesiccurve.

ConsiderthegeodesiccurveconnectingNewYorkCity,USAtoMoscow,Russia:


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18

ThelinefromNewYorkstartsatabearingof34,continuesinastraightlinetheentireway,but

arrivesinMoscowonabearingof130.

Formovementsegmentslessthanafewhundredmiles,andsufficientlyfarawayfromthe

poles,thedifferencebetweenstartingandendingazimuthstendstobenegligible. Forlong

distancesandforanalysisnearthepoles,however,weshouldbeawareofthisphenomenon

andbecarefulabouthowwedefineourdirectionalvalues.

GraphicallyRepresentingDirectionalData

Becauseofthecircularnatureofdirectionaldata,wetypicallyusecircularplotstodisplaythe

distribution. Mostofthesecircularplotsarevariationsonthestandardhistogram,except

wrappedinacirclesothatthemaximumpossibleXaxisvaluewrapsbacktotouchthe

minimumpossiblevalue.

OneofthemorefamousandhistoricallyinterestingoftheseistheRosePlotorRoseDiagram,

namedassuchbecausethehistogrambinsresembleflowerpetals. Thistypeofdiagramwas

originallypopularizedbynurseandstatisticianFlorenceNightingale,whousedittoshow

seasonalpatternsinsoldiermortality(seehttp://en.wikipedia.org/wiki/Florence_Nightingalefor

thesourceofthefamousimagebelow). Nightingaleactuallycalledthesediagramscoxcombs

becausetheyresemblethecombofarooster.


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19

Nightingalesplotsshowedvariationbyseason,butthismethodalsoworksverywellfor

displayingdirectionaldata. Forexample,givenasetoftrailsintheSanFranciscoBayarea,we

canusearoseplottoillustratethegeneralNorthwest/Southeastorientationofthosetrails:

CommonvariationsonthetraditionalroseplotincludesymbolizingthehistogrambinsasPeaks

andValleys,andputtingthebinsontheoutsideofthecircleratherthantheinside.


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20

Whenonlyafewdirectionvaluesneedtobedisplayed,youcansimplyshowthebearingsina

circularplot.

Anotherinterestingmethodisapiecharttypeimagewheretheslicesareshadedaccordingto

theproportionofthedatathatfacesthatparticulardirection:


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21

Althoughthislastmethod islesseffectiveatshowingtheshapeofthedistributionasother

methods,itisstillaninterestingstyleofillustrationbecauseitissovisuallyintuitive. Inthe

imageabove,itiseasytoseethattheobservedroostlocationspredominatelyfacetowardthe

southeast.

Thisextensionprovidesmethodstocreatealltheplottypesillustratedabove(seep.36fora

descriptionofthetool).AdditionalsoftwaretoolscanbefoundwithaGooglesearch. Fisher

(1995:1530)alsodiscussesalternativemethodsforgraphingcirculardata.

AnalyzingDirectionalData:CircularStatistics

Beforebeginningthissection,Iwanttoemphasizetwoimportantpointsthatpeopleveryoften

makemistakeswith:

1) Donotcalculatethemeandirectionusingthearithmeticmean! Thisisespecially

frustratingbecausethearithmeticmeanissometimescorrectandsometimeswildly

incorrect. Forexample,themeandirectionof90and180is135,whichcoincidentally

isequaltothearithmeticmean 90

1352

. Butwhatisthemeandirectionof359

and1? Theyarebothpointingalmostduenorth(1),andclearlythetruemean

directionisexactlyduenorth. However,thearithmeticmeangivesus180

359180

2

,whichisduesouthandexactlytheoppositeofthecorrectanswer. The

correctwaytocalculatethemeandirectionisdescribedbelow.

2) IfyouapplyaSineorCosinetransformation,makesuretoconvertthevaluesto

radiansfirst! Mostanalyticalsoftwareandprogramminglanguageshavesineand

cosinefunctions,butthesefunctionsusuallyassumethevaluesareinradians,not

degrees. Thereareexactly2 (~6.28)radiansinacircle. Thismeansthatthesoftware

willassumethatadifferenceof3.14(i.e.)unitsisequivalenttogoinghalfwayaround

thecircle. Ifyourdataareindegrees,thenthesoftwarewillinterpretachangein2to


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22

beroughlyequivalenttogoingathirdofthewayaroundthecircle. Fortunatelyitis

easytoconverttoradiansusingthefollowingformula:

3 14159265358979

180 180

.DegreesDegreesRadians

DegreestoRadians:

Itispossiblethatyoursoftwarehasafunctionthatallowsyoutocalculatesinesand

cosinesfromdegrees(manycalculatorsdo),buteveninthiscaseyoumustremember

tosettheswitchcorrectly.

Fortunatelytherearewellestablishedmethodsavailableforanalyzingcircularorperiodicdata

suchasmovementdirectionoraspect. Thecircularnatureofthedataleadtoveryspecificand

interestingapproachestocalculatingmeasuresofcentraltendencyanddispersion(seep.45for

howtousethisextensiontocalculatecircularstatistics). Someofthebasicdescriptivestatistics

are:

Mean Direction and Mean Resultant Length

2 2 2

1 1

1

1

1

2

0 0

0

2 0 0

0 02

0 02

cos sin

tan ,

tan

tan ,

,

,

n n

i ii iC S R C S

S C S C

S C C

S C S C

S C

S C

R R

Where: DirectioninRadians

MeanDirection:

ResultantLength:

MeanResultantLen

R

R ngth:

Theequationsformeandirectionlookalittleconfusing,butthelogicisactuallyveryintuitive. It

issimplyaprocessofvectoraddition,whereeachdirectionvalueisasinglevector. Vector

additionessentiallyconnectsallthedirectionvectorsintoapath,andthemeandirectionisjust

thedirectionfromtheorigintothelastpointonthepath.

Forexample,considerascenariowith4directionvaluesat45,75,120and220:


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23

Weconnectthe4bearingsinapath(vectoradditionjustaddsuptheXandYcomponentsof

eachvector,whichisthesameastreatingeachbearingasasegmentinapath). Itdoesnot

matterwhatorderweconnectthevectorsin;theywillalwaysendupatthesamepoint. The

MeanDirectionisthebearingfromthestartofthepathtotheendofthepath.

Onasidenote,thisisalsothewaytocalculatethemeandirectionofanactualobserved

movementpath. IfyouhaveaseriesoflocationsfromaGPScollaronanelk,forexample,and

youwonderwhataveragedirectiontheanimalmovedovertheday,thenthataveragedirection

issimplythedirectionfromthefirstGPSlocationofthedaytothelast.

TheMeanResultantLengthR isthebasisforseveralvaluesofdispersion(analogousto

varianceorstandarddeviation),andiscalculatedasthestraightlinedistancefromthestarting

pointtotheendingpointofthepathdividedbythetotallengthofthepath. Ifthesegmentsare

unitvectors(e.g.aspectvaluesfromanaspectraster,whereeachaspectvaluehasthesame

weightasanyother),thenthiscanbesimplifiedtodividingbythenumberofsegmentsor

observations.

Noticethatthemeanresultantlengthhasapotentialrangeof0to1. Ifallthevectorspointedinexactlythesamedirection,theresultantlengthwouldthenbeequaltothetotalpathlength

andthemeanresultantlengthwouldbeequalto1. Thisisthescenariowiththeminimum

possiblevarianceordispersioninthevectors. Themorethepathwandersaround,theshorter

boththeresultantlengthandthemeanresultantlengthwillbe. Ifthepathendedbackatthe

origin,thenbothvalueswouldbeequalto0.


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Manual: Polar Plots ArcGIS Extension

Last Modified: June 23, 2014

24

Variance and Standard Deviation:

MardiaandJupp(2000),Fisher(1995)andBatschelet(1981)describecircularstatistics

analogoustovarianceandstandarddeviations,allbasedonthemeanresultantlength R.

2

2

1

2 1

2

2 1

ln

R

s R

v R

s s R

Circular

Variance: V (Mardia

&

Jupp,

Fisher)

Angular

Variance: (Batschelet)

Circular

Standard

Deviation:

In

Radians (Mardia

&

Jupp,

Fisher)

Angular

Deviation:

In

Radians (Batschelet)

The

Circular

Standard

Deviation

and

Angular

Deviation

are

both

in

units

of

Radians,

but

these

caneasilybeconvertedtoDegrees.

180

Degrees Radians

Radians

to

Degrees:

Batcheletpointsoutthatcircularstandarddeviationandtheangulardeviationtendtobeclose

to

each

other

for

high

values

of

R

(i.e.

near

1).

However,

as

R

tends

toward

0,

circularstandarddeviationtendstoward whileangulardeviationtendstowardamaximumvalueof

2. Batchelet,citingSeyfarthandBarth(1972),presentsageometricderivationofangular

deviationwhichalsoprovidesavisualsenseofwhattheconceptmeans.

GivenaunitcirclewithRadius=1,andwith:

PointAdefinedasonthecirclehorizontalfromtheOriginO

PointCdefinedas RdistancealongsegmentOA(rememberthat Rwillalwaysbe

between0and1)

PointBdefinedasonthecircleverticalfromPointC

Thens(angulardeviation)isjustthelengthofthelineconnectingpointsAandB(alsoknownas

theChordof AOB ). Theangulardeviationcanthereforebecalculatedbyapplyingthe

Pythagoreantheoremto BCO and :ABC


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25

2 2 2

2 2

22 2

2

22 2

2

2 2

1

1

1

1

1

1 1 1

1 2 1

2 2

2 1

2 1

:

:

:

BCO

AB

R

R

R

C

R R R

R R

Rs

s

s

R

R

R

R

x

x

xx

R

FromthePythagoreanTheoremon

FromthePythagoreanTheoremon

Substitutingfor

Basedonthisillustration,itiseasytoseethat 0s when 1R ,andthat 2s when 0R

Note: Ameanresultantlength R near1alwaysimpliesatightlyfocusedsetofdirections,butameanresultantlengthnear0doesnotnecessarilyimplyahighamountofvariationor

dispersion. Allitimpliesisthatthedirectionsbalanceeachotherout. Thiscanoccurwitha

uniformdistributionofdirections,inwhichcasetheretrulywouldbehighdispersion. Itcanalso

occur,however,ifyouhaveanumberofbearingsinonedirectionandanequalnumberinthe

oppositedirection. Forexample,perhapsyouhaveabirdwithdistinctandseparateroostand

foragelocations. Everydaythebirdtravelstotheforagelocationandthenreturnstotheroost

location. Inthisscenario,thebirdonlygoesintwodistinctandoppositedirections(tothe

foragelocation,andthenbacktotheroostlocation). Thebirdsmovementdirectionsarehighly

focusedandpredictable,butthemeanresultantlengthwouldbe0andconsequentlythe

variancewouldbehigh. Aswithmostsituations,plottingthedistributionofthedataisagood

waytounderstandwhatisreallyhappening.

Justaswithstandardstatistics,thereareanumberofcirculardistributionsandsophisticated

analyticaltechniquesavailable. Thesegobeyondthescopeofthisdiscussion,butpleaserefer

toJammalamadakaandSenGupta(2001),MardiaandJupp(2000),Zar(1999;seeespeciallych.

26and27),Fisher(1995)andBatschelet(1981)forsomegoodtextsoncircularstatistics,

distributions(i.e.theFisher,VonMisesandWrappedNormaldistributions),circularhypothesis

testingandotheranalyticaltechniques. ThereisalsoagoodcircularstatisticalpackageforR,

originallybasedonJammalamadakaandSenGuptastext. AsofJune,2012,themanualforthis

packagecanbeviewedathttp://cran.wustl.edu/web/packages/circular/circular.pdf.

OtherMethodstoAnalyzeDirectionalData

Ifwewanttoincludedirectionasoneofseveralindependentpredictorvariablesinatraditional

statisticalanalysis,thenwemusttransformthevalueintosomethingapproachingalinear

measure. Thereareafewstraightforwardwaystodothis,althoughanytransformationwill

probablyviolatesomeoftheassumptionsofmosttraditionalstatisticaltests.


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Classification of Directional Values

Probablytheeasiesttransformationistosimplygroupyourdirectionvaluesintogeneraland

intuitiveranges(forexample,N=31545,E=45135,S=135225,andW=225

315),creatingacategoricaldatasetwhichmaybeappropriateforsomeanalyses.

Deviations from a Bearing

Asimpleandbasictransformationistoconvertyourdirectionvaluesintodeviationsfroma

directionofinterest. Forexample,ifyoufeltthattheobjectofyourstudywaslikelytobe

affectedbythenorth vs.southfacingaspectphenomenon,thenyoumightdefineyour

directionvaluesintermsofDeviationfromNorthwhereeachvaluewouldreflectthe

distance,indegrees,fromdueNorth. Yourfullsetoftransformedvalueswouldrangefrom0to

180(seefigurebelow). Thisoptionhastheadvantageofmaintainingaconstantinterval

betweenunits,suchthatthedifferenceindirectionbetween0and1degreeisthesameasthe

differencebetween90and91degrees.

Sine and Cosine Transformations

Aspectvaluesareoftenconvertedtosineandcosinevalues,essentiallydecomposingtheminto

northsouthandeastwestcomponents. Rememberto

convert

your

direction

values

to

radians

beforedoingthistransformation! Sinevaluesrangefrom 1(atduewest)to1(atdueeast),

whilecosinevaluesrangefrom 1(atduesouth)to1(atduenorth). Notethatthismethoddoes

notmaintainaconstantintervalbetweenunits. Thesineandcosinevalueschangebyavariable

amountdependingonthedirection,suchthatachangeinsinecorrespondingtoachangeof1

degree=0.00015whengoingfrom90to91degrees,butincreasesbymorethan2ordersof

magnitudeto0.017whengoingfrom180to181degrees. Thisissuemaybeimportantinyour

statisticalanalysisifyourmethodassumesthatyourdataareintervallevel.


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27

TrimbleandWeitzman(1956)andBeersetal.(1966)suggestaninterestingalternative

combiningtwooftheapproachesabove,rescalingaspectvaluesbasedonanoptimumbearing

(135forTrimble&Weitzman,andageneralequationforanyaspectofinterestforBeersetal.)

thentakingthesineoftherescaledvalues,thenadding1,whichtheyputtoextensiveuseinsiteproductivityresearchfortimberstands. Beersetal.sgeneralequationis:

90 1

1

TransformedAspectCode,inDegrees

Where:

Theaspectwhichistobeassignedthehighestnumericalvalueonthetransformscale

sin

cos

Max

Max

Max

A

A A

A A

A

Note: Beersetal.sequationpresumesthatyoucancalculatethesineorcosineofavaluein

degrees. Manycalculatorscandothis,butmanyprogramminglanguagesrequireyourvaluesto

beinradians. Ifyouneedtoconvertyourvalues,theDegreestoRadiansconversionformulaisonp.22ofthismanual.

Special

Considerations

for

Aspect

HowAspectisCalculated

AspectiscalculatedfromthedirectionalEastWestandNorthSouthgradientsataparticular

point. Therearevariousmethodsforestimatingthesedirectionalgradientsfroma3x3setof


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28

elevationpoints,butingeneralaspectisdefinedasthedirectionofmaximumslope. Giventhat

G=easttowestgradientandH=northtosouthgradient,thenaspectiscalculatedasfollows:

180

2atan ,H G

MathematicalDirection(indegrees)

Theatan2functionisamethodofcalculatingthearctangent 1tan usingtwovalues(XandY),andithasapossiblerangerepresentingafullcircle( to+ radians,or 180to+180).

Thestandardarctangentfunctioncanonlyproducevaluesrangingfrom2

to

2

,or 90to

+90. Thisatan2functionisthesameoneusedtocalculatemeandirectionabove,andis

calculatedasfollows:

1

1

1

0 0

0 0

0 0

0 02

0 02

0 0

tan ,

tan ,

, tan ,

,

,

,

yy x

x

yy x

x

y

y x y xx

y x

y x

y x

atan2 :

Undefined i.e.nomovement

Mathematicaldirectionisdifferentthancompassdirection. Inthemathematicalpolar

coordinatesystem,0isequivalenttodueeastandpolardirectionsincreaseinacounter

clockwisedirection. Compassdirectionstartsatnorthandincreasesinaclockwisedirection.

Thereforewemustconvertthemathematicaldirectiontoacompassdirectionaccordingtothe

followingrules:

ifMathematicalDirection>90thenCompassDirection=(450MathematicalDirection)

OtherwiseCompassDirection=(90MathematicalDirection)

ArcGISusesHornsmethod(Horn1981,BurroughandMcDonnell1998)tocalculatetheeast

west(G)andnorthsouth(H)gradientsfroma3x3arrayofelevationpoints. Thefigurebelow

showsthe8elevationvaluesthatareusedtocalculateaspectatthecentralcellXX(notethat

theformuladoesnotusetheelevationvalueatthecentralcell):

Z1 ---- Z2 ---- Z3| | || | || | |Z4 ---- XX ---- Z6| | || | || | |Z7 ---- Z8 ---- Z9

InHornsMethod,thenorthsouthgradientiscalculatedfromallthreecolumnsinthe3x3array,

withthecentralcolumnweightedtwiceashighasthetwooutercolumns. Theeastwest

gradientiscalculatedinasimilarmanner.


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29

3 6 9 1 4 7

1 2 3 7 8 9

2 2

8

2 2

8

Z Z Z Z Z Z G

x

Z Z Z Z Z Z H

y

EastWestGradient

NorthSouthGradient

Is this the best method to calculate aspect?

Interestingly,Jones(1998)comparedseveralslopeandaspectalgorithmsonamathematical

surfaceinwhichthetrueexactslopeandaspectatanypointcouldbecalculated. Hefoundthat

Hornsmethoddidbetterthanmostmethods,butwasmarginallyworsethanamethodcalled

the4cellmethod. HethentestedallmethodsonatruelandscapeDEMandfoundthatthe

rankingsamongmethodswereconsistent,soHornsmethodstillcamein2ndplace.

Anothertidbit: MichaelHodgson(1995)publishedapaperusingthesamemathematical

surfacethatJones(1998)did,inwhichhedemonstratesthatthearearepresentedbycomputedslopeandaspectisactuallylargerthantherasterDEMcellsize. 8cellalgorithms(suchasHorns

method)moreaccuratelydescribeslopeandaspectforanareaapproximatelytwicethecell

size,while 4cellalgorithms(suchasthe4Cellmethod)describeanareaapproximately1.6X

thecellsize.

Insum,amethodcalledthe4cellmethodissimpler,slightlyquickertocalculate,moreprecise

andmarginallymoreaccuratethanHornsmethod. However,thetimeandaccuracylostusing

Hornsmethodisprobablyveryminorinmostcases.

Forthosewhoareinterested,theauthoroffersafreeArcGISextensionthatwillcalculateslope

andaspectusingavarietyofmethodsincludingbothHornsmethodandthe4cellmethod(see

http://www.jennessent.com/arcgis/surface_area.htm). ThisextensiondoesnotrequireSpatial

Analyst.

IssueswithAspect

AsaconsequenceofdeterminingaspectfromEastWestandNorthSouthcomponents,aspect

canbedifficulttocalculatefromgeographicallyprojected(latitude/longitude)data. The

equationsaboveallworkfor3x3arraysinwhichtheelevationlocationsareallequallyspaced

alongboththeX andYaxes. However,thisisnotthecasewithlatitude/longitudedata.


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30

Rastercellsthataresquareinunprojectedspaceactuallyformtrapezoidswhenviewedonthe

actualsurfaceoftheplanet.

Inthiscase,wecannottreatthecellsizeasasingleconstantvalue. Themethodstocorrectfor

latitude/longitudedatagobeyondthescopeofthisarticlebutaredescribedinJenness(2011;

seeespeciallyp.40 49ofthemanual). Forthosewhoareinterested,theauthorsfreeArcGIS

extensionDEMSurfaceToolswillautomaticallycorrectforlatitude/longitudedata(see

http://www.jennessent.com/arcgis/surface_area.htm).

SolarInsolation:AnalternativetoAspect

Aspecthasawellestablishedhistoryinhabitatanalysis. Itisbotheasytomeasureandagood

predictor

of

certain

habitat

characteristics.

However,

in

some

cases

we

are

really

interested

in

howmuchdirectsunlighthitsanarea(insolation),whichisafunctionofaspect,slope,nearby

topography,landscapereflectivityandatmosphericeffects,andassuchmaybeamore

importantdriverofhabitatcharacteristicsthanaspectalone. Insuchcases,itmaybe

worthwhiletoestimateinsolationdirectlyratherthanuseaspectasasurrogate. Therearea

fewapproachesyoucantake:


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31

ESRIs Solar Radiation Tool

Asofversion9.2ofArcGISandSpatialAnalyst,ESRIhasincludedatoolintheArcToolboxto

calculatesolarradiationforspecificpointsonthelandscape,orovertheentirelandscape.

Thisfunctionincorporatesbothdirectanddiffuseradiationandshadowsfromnearby

topography. Itincludesseveralusefuloptions,including:

1) Outputseithertheamountofenergyhittingtheground(inWattsperHour),orthetotal

numberofhoursinwhichthegroundisexposedtothesun.

2) Canbecalculatedforspecificdates,seasonsoryears.

3) Hasoptionalparameterswhereyoucanspecifythegeneralatmosphericconditionsinyourareaofinterest.

4) Hasoptionalparameterswhereyoucanspecifyhowintensivelyitexaminesthelocal

topographybeforedeterminingtheamountofradiationhittinganarea.

Ingeneral,thisisawonderfulandexcitingnewtool. Ihavenoticedtwominordrawbackstothe

tool:

1) Itisslowonlargegrids,andonoccasionIhaveneededtoletitrunforhoursordays.


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32

2) Idonotbelievethatitincorporatesreflectivityoffthelandscape. Thiswouldbehardto

model,ofcourse,andwoulddependonexactlyhowreflectiveyourlandscapeis(snow

reflectsverydifferentlyfromlavarocks,forexample).

Despitethesetwominordrawbacks,Iexpectthatthistoolwillbecomeveryvaluableforhabitat

analysis.

Hillshade

ForthosewhodonothaveaccesstotheSolarAnalyst,youcanusethehillshadefunction

availableinmanyGISpackagesgetareasonableestimatefortherelativeamountofsunlight

thathitsthelandscapeatasinglepointintime. Valueswilltypicallyrangefrom0(meaningno

sunlighthitsthelandscape)to255(meaningtheangleofthelandscapeisfacingthesun

directly). HillshadesaredifficulttocalculateifyourDEMisingeographiccoordinates,butthe

authoroffersafreeArcGIStooltodothis(see

http://www.jennessent.com/arcgis/surface_area.htm)whichdoesnotrequireSpatialAnalyst.


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33

Using

the

Tools

ConvertShapestoSegments

Thisfunctiondecomposesacomplexfeatureintoasetofindividualsegments,makingiteasier

toanalyzethedirectionalcharacteristicsofthosesegments. Forexample,asinglepolyline

mightrepresenttheentiremovementhistoryofananimalduringitsmigration. Becauseitisasingleobjectinthedatabase,itisdifficulttodecomposeintoaformsuitablefordirectional

analysis.

Thisfunctiongeneratesapolylinefeatureclassrepresentingthevertextovertexsegmentsfrom

thefeaturesofapolygonorpolylinefeatureclass. Eachpolylinewillrepresentasinglestraight

linesegment. Youhavetheoptiontogeneratesegmentsforallfeatures,oronlytheselected

setoffeatures.

Thisfunctionoffersseveraloptionstocalculateadditionaldatadescribingeachsegment,

includingthestartingandendingcoordinates,theazimuthofthesegmentandthereverse

azimuthofthesegment.

Ifyourdataareinlatitude/longitudecoordinates,thenyoualsohavetheoptiontocalculateboththestartingandendingazimuthsofeachsegmentplusthereversestartingandending

azimuths. Ifthislastoptionisconfusing,recallthatgreatcirclegeodesiccurvesalmostalways

changetheirbearingconstantlyoverthelengthofthecurve(seeAzimuthsandGeodesic

Curvesonp.17).

ClicktheShapestoSegmentsbutton toopentheFeaturestoSegmentsdialog:


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34

Selectthelayertoconvert,anyattributefieldsyouwanttotransfertoyournewfeatureclass,

whethertodoallorselectedrecords,anyadditionalattributesdescribingthegeometryofeach

segment,andwheretosaveyournewfeatureclass. ClickOKandthetoolwillcreatethe

segmentfeatureclassandaddittoyourArcMapdocument.

Thetoolwillalsogenerateareportdescribingwhatitdid.


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35


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36

GenerateCircularPlots

Thistoolwillgenerate6typesofcircularplots,withnumerousoptionsregardingcolors,labels,

titlesandreferencelines.

ThetoolwillgeneratetheplotasagraphicintheMXDLayoutandthereforeyoumusthavethe

layoutactivewhenyougeneratetheplot. IfyoutrytogeneratetheplotwhiletheMapviewis

active,thetoolwilloffertoswitchyoutothelayoutviewautomatically.

Thegraphiccanbecopiedandpastedtoothersoftwarepackages(MicrosoftWordand

PowerPoint,forexample). However,forbestresultsIrecommendexportingthegraphicintoa

standardimagefileusingthestandardArcGISExportfunctions.

ThegraphicwillactuallybeaGroupedgraphicwhereeachcomponentisaseparategraphic. If

youcreateyourgraphicplotandwanttomodifyanysingleportionofit,simply:

1) SelecttheplotgraphicusingtheArcGISSelectElementstool

2) RightclickonthegraphicandchooseUngroup

3) SelecttheindividualelementyouwanttoeditusingtheSelectElementstool

4) RightclickonitandchooseProperties.


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37

Note: TheBearingsplottypeisonlyintendedforcaseswithlessthanafewhundredbearings.

Thisgraphictakesalongtimetodrawandcanbemiserabletoworkwithifyouhavemorethan

afewhundredbearings.

ClickthePolarPlotsbutton toopenthePolarPlotParametersdialog:

Therearealotofoptionsonthisdialog,sotheyarebrokendownintosubsetsandareaccessed

byclickingthesubsetbutton. Forexample,theShowDataSourcebuttonopensupthedata

sourceoptionsandletsyoupickthedatasettoplot,plusspecifythesourceforazimuthand

weightvalues.

Plot Style

ThePlotStylesectionisalwaysvisibleandshowsthesixplotstyleoptionsavailable. Toseealargerexampleofanyoftheplotstyles,clicktheShowExamplebutton. Thislargerexample

alsohasmostoftheplot,title,referencelineandlabeloptionspointedout:


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38

Data Source Options:

ClicktheShowDataSourcebuttontoopentheDataSourceoptions:

DataSource: Thelistincludesallfeatureclassesandtablesinyourcurrentfocusmap.Note: Thisversionofthetooldoesnotworkwithrasterdatasets. Ifyouwishtoplotthe

distributionofarasterdataset(ofaspectvalues,forexample),therearetwoapproaches

youmighttry:

1) Converttherastertoapointfeatureclass,thenselectthepointsthatliewithinyour

areaofinterest,andthenrunthistoolontheselectedpoints.

2) ConverttherastertoanIntegertyperaster(whichhasanassociatedattributetable

containingValueandCountfields),thenexporttheattributetabletoastandalone

table,andthenrunthetoolontheattributetable. UsetheValuefieldasthesource

ofazimuthvalues,andtheCountfieldasthesourceofweightvalues.

AzimuthSource: IfyourdatasourceisaTable,PointFeatureClassorPolygonFeatureClass,thenyoumustsetyoursourceofazimuthvaluestoanattributefieldintheattributetable.

IfyourdatasourceisaPolylineFeatureClass,thenyoumayderiveyourazimuthvalues

directionfromthepolylinefeatures. Inthiscase,azimuthisdefinedastheazimuthfromthe

startingpointtotheendingpointofthepolyline. Ifthepolylineisinlatitude/longitude

coordinates,thenthistoolusesthestartingazimuthvalue(seeAzimuthsandGeodesic

Curvesonp.17foranexplanationofstartingvs.endingazimuthvalues).


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39

WeightSource: Ifyourdirectionvalueshavedifferentweights(forexample,ifyourdatasourceisapolylinefeatureclassofsegmentsandeachsegmentmightbeadifferentlength),

thenyoumaychoosetoweightyourdirectionvaluesbysomeattributevalue. Ifyourdata

sourceisapolylinefeatureclass,thenyoualsohavetheoptiontoweighteachpolylineby

thelengthofthatpolyline.

AnalyzeAllvs.SelectedRecords: Ifanyofyourrecordsareselected,thenyouhavetheoptiontoonlyanalyzethatselectedset.

AddReverseAzimuths: Ifyourdatasetrepresentsaphenomenainwhichbothdirectionsshouldbeplotted(suchasgeologicfracturelinesonthelandscape),thenthisoptionwill

plotboththeobservedazimuthvalueforeachrecordplustheazimuthinthereverse

direction. Inaplotwithreversedazimuths,everybinvaluewillbereflectedbyabinofthe

exactsamelengthgoingintheoppositedirection.

ExcludeNegativeValues: Dependingonyourdata,negativevaluesmayreflectgenuinedirections(counterclockwisefromnorth)ortheymaybeaflagthatthevalueisaninvalid

azimuth. Forexample,aspectgeneratingalgorithmsoftenproducea 1valueiftheraster

cellisflat(i.e.ifithasnoaspect). Thisoptionletsyouexcludeanynegativevaluesfromthe

analysis.

Plot and Title Options:

Theseoptionsallowyoutosetthesymbolizationfortheplot,plusoptionallyaddandsymbolize

primaryandsecondarytitles.

ClicktheShowPlotandTitleOptionsbuttontoshowtheplotcolorandtitleoptions:


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40

Differentplotstyleswillhavedifferentoptionsavailable. Forexample,boththeExternalPeaks

andValleysandtheExternalCircularBinsrequireInsideCircleDiameterandOutsidePlot

Diametervalues,whileallotherplottypesonlyrequireasingleplotdiametervalue.

NumberofBins: Thisisthenumberofbinsaroundthecircle. Avalueof72meansthateachbinwillcover5. Avalueof180meansthateachbinwillcover2.

BinOutlineandInternalColors: Theinternalplotofbinshasbothaninternalcolorandanoutlinecolor. Eithercolorcanbesettotransparent.

BinOutlineWidth: Inpoints. Ifyoudonotwishtoseetheoutline,youmayeithersetthisvalueto0orsettheoutlinecolortransparent.

BearingLineColor(onlyforBearingsplot): Thisisthecolorofthebearinglinesintheplot.

BearingLineWidth(onlyforBearingsplot): Thewidthofthebearinglinesintheplot,inpoints. Thisvaluemustbe0.2.


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LowandHighDensityColors(onlyforShadebyDensityplot): ThebinwiththelowesttotalvaluewillbecoloredtheLowDensityColorvalueandthebinwiththehighestvalue

willbecoloredtheHighDensityColor. Intermediatebinswillbecoloredaccordingtoa

colorrampfromthelowtohighcolor.

PlotDiameter(onlyforPeaksandValleys,CircularBins,BearingsandShadeby

Densityplots): Thisisthediameterofthecircle,ininches,thatcontainsallthedata. It

doesnotincludeanylabelsoutsidethiscircle.

InsideCircleDiameter(onlyforExternalPeaksandValleysandExternalCircularBinsplots): Thisisthediameteroftheinnercircle,ininches,whichformsthebaseoftheplot

bins.

OutsidePlotDiameter(onlyforExternalPeaksandValleysandExternalCircularBinsplots): Thisisthediameterofthecircle,ininches,thatcontainsallthedata. Itdoesnot

includeanytitlesoutsidethiscircle.

PrimaryTitle: Checkthisoptionandspecifyatitleifyouwishtoaddatitletoyourgraphic.Youmayalsospecifythefontstyle,sizeandcolor.

SecondaryTitles: Allsecondarytitleswillusethesamefontstyle,sizeandcolor. Youhavetheoptiontoaddanyorallof4secondarytitles:

o DataSource,includingboththedatasetandthesourceofazimuthvalues.

o Countofvaluesanalyzed. Thismaybeeitherthetotalnumberofrecordsorthenumber

ofselectedrecordsdependingonyourchoice. Ifyouchosetoaddreverseazimuth


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42

values,thenthecountwillbedoubled. Ifyouchosetoexcludenegativevalues,then

thecountwillonlyincludetheazimuths0.

o TotalWeight/Length: Ifyouweightedyourazimuthvalues,thenthisoptionwillshow

theweightingsourceandthetotalcumulativeweight. Ifyoudidnotweightyour

azimuthvalues,thenthisoptionwilladdthelineValuesNotWeightedtoyourplot.

o MaximumBinValue(thisapplies

to

all

plot

types

except

for

Shade

by

Bearing

plots):

Thisisthemaximumbinvalue. Ifyouweightedyourazimuthvalues,thenthiswillbe

thelargestcumulativeweightinabin. Otherwisethiswillbethemaximumnumberof

azimuthvaluesinabin.

Reference Line and Label Options:

Theseoptionsallowyoutooptionallyadd,symbolizeandlabelreferencelines.

ClicktheShowReferenceLineandLabelOptionsbuttontoshowthevariousoptions:


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ReferencelinesaredividedintoPrimaryandSecondaryreferencelines. Thesetwotypescanby

symbolizeddifferently. Forexample,youmaywanttosetyourprimaryreferencelinesina

darkercolorandwithathickerlinethanyoursecondaryreferencelines. Primaryreferencelines

willalwaysbedrawnasasolidline,butsecondaryreferencelinesmayoptionallybedrawnas

solidlines,dots,dashesormixeddotsanddashes.

Bothprimaryandsecondaryreferencelinesaredrawnforbothazimuthsandvalues.Azimuth

referencelinesextendoutfromthecenterasrays,whileValuereferencelinesaredrawnas

progressivelylargercircles.

PrimaryAzimuthreferencelinesareonlydrawnattheNorth,East,SouthandWestdirections.

ThePrimaryAzimuthOuterCircleisdrawnatthemaximumbinvalue.

ThePrimaryAzimuthInnerCircleisonlyavailableintheExternalPeaksandValleysandtheExternalCircularBinsplottypes,andisdrawnatthebaseofthebins.


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44

Anynumber3SecondaryReferenceLineBearingsmaybeadded. Avalueof12meansthatsecondaryreferencelineswillbedrawnevery30. Avalueof15willproducereference

linesevery24,andavalueof18willproducelinesevery20.

Anynumber1SecondaryReferenceCirclesmaybeadded. ThisoptionappliestoallplottypesexceptforShadebyBearingplots. Thisisthenumberofcirclesbetweenthe0level

andthemaximumbinlevel. Avalueof1willputasinglecirclehalfwaybetween0andthemaximumbinlevel.

AzimuthLabelsarethenumericDegreevaluesaroundtheoutsideofthePeaksandValleys,CircularBins,BearingsandShadebyDensityplottypes,oraroundtheinsideoftheExternal

PeaksandValleysandExternalCircularBinsplottypes. Theseazimuthlabelswillbedrawn

atallprimaryandsecondaryreferenceazimuths.

InnerLabelsshowthebinvalueatallofthesecondaryreferencecirclelevels. TheseinnerlabelsmaybedrawnonanyoralloftheNorth,East,SouthandWestaxes. Thisoption

appliestoallplottypesexceptforShadebyBearingplots. Ifnosecondaryreference

circlesaredrawn,thentheInnerLabeloptionswillbedisabled. Note: Thereisnooptionto

add

the

bin

value

at

the

maximum

circle

level.

However,

there

is

an

option

to

add

this

maximumbinvaluetothetitle.


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45

CalculateDescriptiveCircularStatistics

Thistoolcalculatesthebasicsetofcircularstatsdescribingcentraltendencyanddispersion

describedonp.21,includingtheMeanDirection,ResultantLength,MeanResultantLength,

CircularVariance,CircularStandardDeviation(inRadiansandDegrees),AngularVarianceand

AngularDeviation(inRadiansandDegrees).

TheseCircularStatisticsareavailableinthesametoolthatgeneratesthepolarplots. Clickthe

PolarPlotsbutton toopenthePolarPlotParametersdialog:

ClicktheShowDataSourcebuttontoopentheDataSourceoptionssection:


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Selectthedatasettoanalyze,theazimuthsourceandoptionallyaweightingsource,andthen

clicktheCalculateStatisticsbuttonatthebottomofthedialog. Thetoolwillgiveyouthe

statisticsinaseparatewindow:


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47

Revisions

Version1.0.225(July6,2012):

InitialRelease

Version1.0.237(September12,2012):

FixedanerrorinwhichitwouldnotworkwithnonPolylinefeatureclasses,insteadgivingyouamessagestatingthatthespecifiedfeatureclasswasnotapolylinefeatureclass.

AlsoupdatedtheCalculateStatisticsfunctionsoitwouldtellyouhowmanyfeaturesorrowsitanalyzed.

Version1.0.250(September20,2013):

AddedVonMisesKappameasureofconcentrationtostatistics. Modifiedanalysiscount(inplotandinstatisticsreport)toshowcorrectnumberifanyrecordswere

excludedfromtheanalysis(negativevalues,nullfeatures,missingazimuthorweightvalues,etc.).

Afewcosmeticchanges.

FixedanerrorthatproducedthemessageInvalidprocedurecallorargumentatline1328offrmLayerID.frmwhentheAddReverseAzimuthsbuttonwasclickedwhennodatasourceswereavailable

inthemap.

Version1.0.253(June23,2014):

ModifiedFeaturestoSegmentstooltoclearupamysteriousbugthatcausedArcGIStoshutdowncompletelyinsomecases. Alsomodifiedtooltorunfasteringeneral,andespeciallywith

twovertexpolylines.


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References

Batschelet,Edward.1981.Circularstatisticsinbiology.London;NewYork:AcademicPress.

Beers,ThomasW.,PeterE.Dress,andLeeC.Wensel.1966.NotesandObservations:Aspect

TransformationinSiteProductivityResearch.JournalofForestry64(10):691692.

Burrough,PeterA,andRachaelAMcDonnell.1998.Principlesofgeographicalinformationsystems.

Oxford:OxfordUniversityPress.

Fisher,N.I.1995.Statisticalanalysisofcirculardata.Cambridge[England];NewYork:Cambridge

UniversityPress.

Hodgson,M.1995.WhatCellSizeDoestheComputedSlope/aspectAngleRepresent?Vol.61.

PhotogrammetricEngineeringandRemoteSensing.Bethesda,MD,ETATSUNIS:American

SocietyforPhotogrammetryandRemoteSensing.

Horn,B.K.P.1981.HillShadingandtheReflectanceMap.ProceedingsoftheIEEE69(1):1447.

doi:10.1109/PROC.1981.11918.

Jammalamadaka,S.Rao,andASenGupta.2001.Topicsincircularstatistics.RiverEdge,N.J.:World

Scientific.http://site.ebrary.com/id/10255767.

Jenness,Jeff.2011.DEMSurfaceTools:AnArcGISExtensionforAnalyzingRasterElevationDatasets.

JennessEnterprises.http://www.jennessent.com/arcgis/surface_area.htm.

Jones,KevinH.1998.AComparisonofAlgorithmsUsedtoComputeHillSlopeasaPropertyoftheDEM.

Computers&Geosciences24(4)(May15):315323.doi:10.1016/S00983004(98)000326.

Mardia,K.V,andP.EJupp.2000.Directionalstatistics.NewYork:Wiley.

Seyfarth,Ernst August,andFriedrichG.Barth.1972.CompoundSlitSenseOrgansontheSpiderLeg:

MechanoreceptorsInvolvedinKinestheticOrientation.JournalofComparativePhysiologyA:

Neuroethology,Sensory,Neural,andBehavioralPhysiology78(2)(June1):176191.

doi:10.1007/BF00693611.

Trimble,G.R.,andSidneyWeitzman.1956.SiteIndexStudiesofUplandOaksintheNorthernAppalachians.ForestScience2(3):162173.

Zar,JerroldH.1999.BiostatisticalAnalysis.4thed.PearsonEducation.

.

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