Map as Outcomes of Processes Outlie Defiitios: processes & patters A Startig poit: Complete Spatial Radomess More defiitio: Statioary Map as Outcomes of Processes The basic assumptio of Spatial Aalysis: Maps have the ability to show patters i the pheomea they represet Patters provide clues to a possible causal process Thus, maps ca be uderstood as outcomes of processes Processes Patters Map
What Is Process? Defiitio: A spatial process is a descriptio of how a spatial patter might be geerated Huma activities (residece, employmet, leisure) determie urba structures Geological forces form differet ladscapes Types of Processes: the chace of a specific outcome Determiistic: 00% Stochastic: < 00% Determiistic Processes Always the same outcome at a locatio z 2x + 3y z 2x + 3y Where x ad y are two spatial coordiates while z is the umerical value for a variable Stochastic Processes More ofte, geographic pheomea appear to be the result of a chace process Outcome is subject to variatio that caot be give precisely by a mathematical fuctio e.g. the idividual or collective results of huma decisios Some spatial patters are the results of determiistic physical laws, but they appear as if they are the results of chace process. z 2x + 3y + d Where d is a radomly chose value at each locatio, - or +.
Y Stochastic Processes (Cot.) Stochastic: two realizatios of z 2x + 3y ± 2 64,446,44,03,09,55,66 possible realizatios Exercise - 4 Dot map with radomly distributed poits Created 40 (20 each colum) Radom umbers from Excel It(0 * Rad()) Use them as x ad y coordiates for a plot Repeat this process 0 9 6 5 4 3 2 0 0 2 4 6 0 X SA & Processes Basic Questio of SA: How liely is this (observed ) patter a realizatio of that (hypothesized) process? Observed patter is oly oe potetial realizatio of a hypothesized process!!!
Complete Spatial Radomess (CSR) Suppose we assume that: Aythig ca happe at aywhere, with equal probability We have Complete Spatial Radomess (CSR) Meas that o Geographic effects The most commoly used Stadard process or ull hypothesis Also called Idepedet Radom Process (IRP) Illustratio: IRP/CSR Quadrat Aalysis Cout the umbers of evets i each quadrat Evet: a poit i the map, represetig a icidet. Quadrats: a set of equal-sized & ooverlappig areas A B Patter CSR Process Illustratio: IRP/CSR (Cot.) Assumptios: Equal probability Idepedece A B P (evet A i Yellow quadrat) / P (evet A ot i Yellow quadrat) / P (oly evet A i the Yellow quadrat) P (evet A i Yellow quadrat ad other evets ot i the Yellow quadrat) A B C D E F G H I J
Illustratio: IRP/CSR (Cot.) P (oe evet oly) P (evet A oly) + P (evet B oly) + + P (evet J oly) 0 P (evet A oly) 0 A B Illustratio: IRP/CSR (Cot.) A B P (evet A & B i Yellow quadrat) / / P (evet A & B i Yellow quadrat oly) P ((evet A & B i Yellow quadrat) ad (other evets ot i Yellow quadrat)) A B C D E F G H I J Illustratio: IRP/CSR (Cot.) P ( two evets i Yellow quadrat) P(A&B oly) + P(A&C oly) + + P(I&J oly) (o. possible combiatios of two evets) How may possible combiatios? 2 A B
Illustratio: IRP/CSR (Cot.) The formula for umber of possible combiatios of evets from a set of evets is give by C )!!(!... 2) ( ) (! I our case, 0, ad 2 A B Illustratio: IRP/CSR (Cot.) P ( evets) C 0 0 0 )!!(0 0! p p P ) ( ), ( p quadrat area / area of study regio A B Biomial Distributio Biomial distributio x x x x P ),, ( x is the umber of quadrats used is the umber of evets is the umber of evets i a quadrat A B
Poisso Distributio Biomial distributio is computatio itesive 50! 30,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000,000 Poisso distributio is a good approximatio of biomial distributio Reduced computatio burde λ λ e λ Itesity of evets P( ) x! e is a costat, 2.2 Poisso vs. Biomial Real Processes Differ From IRP/CSR The CSR is mathematically elegat ad forms a useful startig poit for spatial aalysis, but its use is ofte exceedigly aive ad urealistic. If real-world spatial patters were ideed geerated by ucostraied radomess, geography would have little meaig or iterest, ad most GIS operatios would be poitless
Real Processes Differ From IRP/CSR First-Order effect: variatios i the desity of a process across space due to variatios i eviromet properties: No equal probability e.g. plats are always clustered i the areas with favored soils e.g. the locatios of disease cases teds to cluster i more desely populated areas Secod-Order effect: iteractio betwee locatios: NO idepedece e.g. physicias ted to cluster aroud a major medical facility e.g. stores of McDoald ted to be far away from each other Real Processes Differ From IRP/CSR First-order ad secod-order effects shift a process from beig statioary to chagig over space st Order statioary: o variatio i the itesity of poit evets over space 2 d Order statioary: o iteractio i evets No-statioary: violate either st or 2 d a model ca ot uiversally apply Real Processes Differ From IRP/CSR BUT, i practice it is close to impossible to distiguish from variatio i the eviromet or iteractio by the aalysis of spatial data What cause the clusters of crime i a city? Eviromet factors: low icome, low employmet rate Or vulerability of populatio?
Summary Map ca be regarded as outcome of a process A patter is the result of a process thus able to offer clues for processes Determiistic vs. stochastic processes Spatial processes are more liely to be stochastic A startig poit: Complete Spatial Radomess (CSR) Equal probability: st order Idepedece: 2 d order Stadard SA: Compare observed patter to CSR Illustratio: IRP/CSR (Cot.) Observed vs. Predicted More Differetiatios Aisotropic: directioal effects i spatial variatio of data e.g. Dow-stream areas are polluted by up-stream sources Isotropic: NO directioal effects i spatial variatio of data e.g. the ifestatio rate of disease simply spread outward