Outline. Point Pattern Analysis Part I. Revisit IRP/CSR
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1 Pot Patter Aalyss Part I Outle Revst IRP/CSR, frst- ad secod order effects What s pot patter aalyss (PPA)? Desty-based pot patter measures Dstace-based pot patter measures Revst IRP/CSR Equal probablty: ay pot has equal probablty of beg ay posto or, equvaletly, each small subarea of the map has a equal chace of recevg a pot Idepedece: the postog of ay pot s depedet of the postog of ay other pot k x P( k,, x) = k x x k λ e P( k) = k! λ k ad λ = x
2 Frst- & Secod Order Effects IRP/CSR s ot realstc geography Frst-Order effect: varatos the desty of a process across space due to varatos evromet propertes: No equal probablty Secod-Order effect: teracto betwee locatos: NO depedece What Is Pot Patter Aalyss (PPA) Pot patters, where the oly data are the locatos of a set of pot objects Represet the smplest possble spatal data Examples Hot-spot detecto (crme, dsease, geologcal actvtes) Dsease & evrometal relatos Freeway accdets Vegetato, archaeologcal studes What Is PPA (Cot.) I a pot process the basc propertes of the process are set by a sgle parameter, the probablty that ay small area wll receve a pot
3 What Is PPA (Cot.) Requremets for a set of evets to costtute a pot patter Mapable patter o the plae Objectvely determed study area A eumerato or cesus of all ettes of terest, ot a sample A oe-to-oe correspodece betwee objects the study area ad evets the patter Evet locatos must be proper (ot be the cetrods of polygos) Descrbg a Pot Patter Pot desty (frst-order or secod-order?) Pot separato (frst-order or secodorder?) Whe frst-order effects are marked, absolute locato s a mportat determat of observatos, ad a pot patter has clear varatos across space the umber of evets per ut area are observed Whe secod-order effects are strog, there s teracto betwee locatos, depedg o the dstace betwee them, ad thus relatve locato s mportat Descrbg a Pot Patter I practce t s close to mpossble to dstgush from varato the evromet or teracto Low desty larger separato Hgh desty smaller separato Frst-order or secod order?
4 Some Notatos s (x, y ) A set of locatos S wth evets S = { s, s2,..., s,..., s} Each evet s has two coordates: s = ( x, y ) The study rego A has a area a Descrbg a Pot Patter Mea Ceter: the mea X coordate ad the mea Y coordate x y = = s = ( µ = x, µ y ), Stadard Dstace: a measure of how dspersed the evets are aroud ther mea ceter d = = 2 2 [( x µ ) + ( y µ ) ] x y Descrbg a Pot Patter A summary crcle ca the be plotted for the pot patter, cetered at the mea ceter wth radus d (SD) If the stadard dstace s computed separately for each axs, a summary ellpse ca be obtaed (2 ds) Summary crcle Summary ellpse
5 Measurg Pot Patter Two basc famles Desty based measures: st order effect Dstace based measures: 2 d order effect Desty-Based Measures Frst-order effect Three methods Crude desty: Overall Quadrat cout: couts of evets each quadrat Desty estmato: cotuous Sestve to the defto of the study area: MAUP & Edge Effect Crude Desty Overall testy of evets the study area λ = o.( S A = ) a a Descrptve statstcs for summary purpose too smple
6 Quadrat Couts Cout the umbers of evets a set of equal-area quadrats ad record these couts as a frequecy dstrbuto Equally-spaced patters wll have most quadrats wth smlar couts Clustered patter wll have a few hgh cout quadrates ad may empty oes Two approaches: Exhaustve cesus Radom samplg Quadrat Couts Approach I: Exhaustve cesus of quadrats Completely fll the study rego wth o overlaps The choce of org, quadrat oretato, ad quadrat sze affects the observed frequecy dstrbuto MAUP Quadrat Couts (Cot.) Approach II: Radom samplg approach Possble to crease the sample sze smply by addg more quadrats (for sparse patters) May descrbe a pot patter wthout havg complete data o the etre patter
7 Quadrat Couts (Cot.) Other shapes of quadrats Desty Estmato Assume: the patter has a desty aywhere the study area ot just locatos where there s a evet Estmated by coutg the umber of evets a rego, or kerel, cetered at the locato where the estmate s to be made Varato of Quadrat Couts Two approaches Smple desty estmato Kerel desty estmato Smple Desty Estmato Smply cout the umber of evets of a patter a rego (usually, crcle) the calculate the desty by dvdg t wth the area of the rego Badwdth r Too large or too small? what s the proper sze? Desty does ot smoothly vary across space ) o.[ S C( p, r)] λ p = 2 πr C(p,r) s a crcle of radus e cetered at the locato of terest p
8 Kerel Desty Estmato Itroduce a dstace effect wth a kerel fucto Weght earby evets more heavly tha dstat oes estmatg the local desty e.g. a pot at half of badwdth r oly couts for ½ pot Kerel Desty Estmato (Cot.) Badwdth r mpact the smoothess of the resultg desty feld depeds Narrow kerels produce bumpy surfaces Wde kerels produce smooth surfaces Applcatos of Desty-Based Methods Vsualze a pot patter to detect hot spots Check whether or ot that process s frst-order statoary from the local testy varatos Lk pot objects to other geographc data (e.g. dsease ad polluto)
9 Summary Pot patter s the oly patter st order vs. 2 d order SV Descrptve summary of pot patter Mea ceter ad stadard dstace Desty measures Quadrat Couts Desty Estmato Dstace-Based Measures Secod-order effect Look at patters wth the dstaces amog pot evets Two approaches: Nearest Neghbor Dstace (NND) Dstace fuctos G fucto F fucto K fucto Nearest Neghbor Dstace NND: the dstace from a evet to the earest evet Usually Eucldea dstace Pythagorea Theorem: d( s, sj) = ( x x j ) + ( y y j ) d ( s ) = m( d( s, s )) m j Mea NND: Summarzes all the earest-eghbor dstaces by a sgle mea value: d s d = = m( ) m Way too smple lose a lot useful formato
10 Expected Behavor of Mea NND Clustered: All NND are short small mea NND Equally-spaced: Mmum dstaces are loger larger mea NND G Fucto G(d) exames the cumulatve frequecy dstrbuto of the NNDs btw the evet pots What fracto of all the NNDs the patter are less tha a specfed dstace (d) Smplest dstace fucto Illustrato
11 G fucto (radom) Expected Behavor of G(d).75 Clustered: Rapd.5 crease, starts at shorter dstace Equally-spaced: Also rapd crease but starts at a.25 much loger dstace G fucto (clustered) G fucto (eve-spacg) Dstace Dstace Dstace F Fucto F(d) s also the cumulatve frequecy dstrbuto of the certa NNDs BUT, NNDs are calculated btw a group of radom pots the study area ad evet pots Icreased sample sze smoother curve G(d) vs. F(d) Both use the dea of NNDs Oly NNDs are used F(d) s smoother tha G(d) Combe both more revealg. G ad F fuctos Dstace F fucto G fucto
12 G ad F fuctos (eve spacg).75.5 F fucto G fucto G ad F fuctos (radom) Dstace.5.75 G ad F fuctos (clustered) F fucto G fucto Dstace.25 F fucto G fucto Dstace Rpley s K Fucto K(d) makes use of all ter-evet dstace Makes t a complex fucto to calculate But, t cludes all possble dstace formato about the patter stead of NNDs Calculatg the K(d) For each dstace d:. Determe how may evets le wth that dstace (d) of each evet 2. Calculate a average umber of evets at the dstace d 3. Dvde by overall study area evet desty
13 Iterpretato of K(d) Revew (PPT Statstcs) Desty-based : Quadrat Couts Kerel desty estmato Dstace-based: Mea NND G & F fuctos K fucto
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