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Polar Plots and Circular Statistics
Last updated 23 June 2014Jeff Jenness
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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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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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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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a. DoubleclickthenewbatchfileRegister_Polar_Plots_Tool.battoregisterthe
toolwithArcGIS10.0.
b. Iftheregistrationissuccessful,thenyoushouldseeaRegistrationSucceeded
notice.
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5. ForWindowsVistaorWindows7: RightclickthefileMake_Batch_Files.exe,andthen
chooseRunasAdministratortocreateregistrationandunregistrationbatchfilesthatare
properlyformattedtoyoursystem.
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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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Todothis,openyourCustomizetoolbyeither:
1) DoubleclickingonablankpartoftheArcMaptoolbar,or
2) ForArcGIS9,clicktheToolsmenu,thenCustomize,or
3) ForArcGIS10,clicktheCustomizemenu,thenCustomizeMode
IntheCustomizedialog,clicktheCommandstabandscrolldowntoselectJennessEnterprises:
Finally,simplydragtheConvertShapestoSegmentsand/orthePolarPlotcommandsoutofthe
CustomizedialogupintoanyoftheexistingArcGIStoolbars.
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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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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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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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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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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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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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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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Nightingalesplotsshowedvariationbyseason,butthismethodalsoworksverywellfor
displayingdirectionaldata. Forexample,givenasetoftrailsintheSanFranciscoBayarea,we
canusearoseplottoillustratethegeneralNorthwest/Southeastorientationofthosetrails:
CommonvariationsonthetraditionalroseplotincludesymbolizingthehistogrambinsasPeaks
andValleys,andputtingthebinsontheoutsideofthecircleratherthantheinside.
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Whenonlyafewdirectionvaluesneedtobedisplayed,youcansimplyshowthebearingsina
circularplot.
Anotherinterestingmethodisapiecharttypeimagewheretheslicesareshadedaccordingto
theproportionofthedatathatfacesthatparticulardirection:
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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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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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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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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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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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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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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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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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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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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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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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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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Selectthelayertoconvert,anyattributefieldsyouwanttotransfertoyournewfeatureclass,
whethertodoallorselectedrecords,anyadditionalattributesdescribingthegeometryofeach
segment,andwheretosaveyournewfeatureclass. ClickOKandthetoolwillcreatethe
segmentfeatureclassandaddittoyourArcMapdocument.
Thetoolwillalsogenerateareportdescribingwhatitdid.
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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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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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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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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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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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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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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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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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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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