Intensity Analysis of Spatial Point Patterns Geog 210C Introduction to Spatial Data Analysis Chris Funk Lecture 5
Topic Overview 1) Introduction/Unvariate Statistics 2) Bootstrapping/Monte Carlo Simulation/Kernel Estimation 3) Distance Matrices/Point Pattern Analysis 4) Bivariate/Multivariate/Spatial Regression 5) Spatial Covariance and Covariance Models 1) Spatial Stochastic Processes 6) Kriging and Spatial Estimation I 7) Kriging and Spatial Estimation II 8) Spatial Sampling Strategies 9) Principal Components and Ordination 10) Combining 1 st order and 2 nd effects 2
Spatial Point Patterns Definition Set of point locations with recorded events" within study region, e.g., locations of trees, disease or crime incidents point locations could correspond to all possible events or to subsets of them (mapped versus sampled point pattern) attribute values could have also been measured at event locations, e.g., tree diameter (marked point pattern) 3 Objective: Introduce statistical tools for quantifying spatial interaction of events,e.g., clustering versus randomness or regularity
1D Kernel Density Estimation Flowchart 1D Kernel Density Estimation Flowchart 1. choose a kernel function k(x-x i ), i.e., a PDF, and a bandwidth parameter b controlling kernel extent and consequently the smoothness" of the final estimated density profile f(x); this amounts to choosing a scaled kernel function k(x-x i ;b) 2. discretize 1D segment, i.e., choose a set of P x-coordinates {x p ; p = 1 P} at which the density function f(x) will be estimated 3. for each datum coordinate x i, evaluate the scaled kernel function k(x p -x i ;b) for all P x- values; this yields N scaled kernel profiles {k i (b); i = 1 N} each one stemming from a particular event coordinate x i 4. for each discretization coordinate x p, compute estimated density f(x p ) as the sum of the N scaled kernel values k(x p -x i ; b), after weighting each such value by 1/N: Output A (Px1) vector k(b) with estimated density values f(x) at the specified x-coordinates; the N scaled & weighted kernels {(1/N)k i (b); i = 1 N} can be regarded as N elementary profiles whose super-position builds up the final estimated density profile 4 Checkout http://parallel.vub.ac.be/research/causalmodels/tutorial/kde.html
Constructing A Separable 2D Kernel Two 1D Gaussian kernels for the x- and y-dimensions Replicated 1D Gaussian kernels and 2D separable composite 5 Anisotropic kernel = multidimensional kernel with different bandwidths along different directions
2D Kernel Intensity Estimation 1. center a circle C(u;b) of radius b at any arbitrary location u in D 2. estimate local intensity at u as: λ(u) =N(u; b)/ C(u;b) where N(u; b) = # of events within C(u; b) C(u; b) = kernel measure, b 2 in 2D. Note: steps 1 and 2 amount to choosing a 2D kernel function that plots like a cylinder with base radius b and height 1=(b 2 ) 3. repeat estimation for set of points (typically arranged at the grid nodes of a regular raster) in the study region to create an intensity map 6 Looping over # of grid nodes (P) instead over # of events (N), yields same results as kernel density estimation case
Recap (Lecture 3) Event intensity of spatial point patterns λ(u): mean # of events over a unit area centered at u estimated overall intensity λ = N/ D local intensity via quadrat counts or kernel density estimation Kernel intensity estimation conversion of point data (events) to raster format (intensity surface) statistical multidimensional (multivariate) density estimation methods are used to estimate local intensity f(u). Note: Density surface integrates to 1, so multiply every such estimate f(u) by N to convert it to an intensity value f(u) resulting intensity surface depends on: (i) kernel type, and (ii) bandwidth; the latter is more influential alternative approaches for non-parametric multivariate density estimation include: k-nearest neighbor and mixture of Gaussian densities methods intensity surface can be linked (via regression models) to explanatory variables, e.g., disease occurrence intensity as function of air quality variables 7
Outline-Lecture 5 8 Concepts & Notation Distance & Distance Matrices Distances Involved in Spatial Point Patterns Statistical Tests Sampling Distributions Bootstrap example Quadrat count example Quantifying Spatial Interaction: Nearest Neighbor Distance Metrics G Function Proportion of minimum event-to-nearest-event distances no greater than given distance cutoff d F Function Proportion of minimum point-to-nearest-event distances no greater than given distance cutoff d Measures based on the distribution of event intensities The K function Quadrat counts and KS-test? Quantifying Spatial Interaction: K Function Points To Remember
Some Notation Point events Set of N locations of events occurring in a study area: Variable of interest y(s) = number of events (a count) within arbitrary domain or support s with measure (length, area, volume) s ; support s is centered at an arbitrary location u and can also be denoted as s(u); in statistics, y(s) is treated as a realization of a random variable (RV) Y(s) Objective Quantify interaction, e.g., covariation, between outcomes of any two RVs Y(s) and Y(s ). To do so, all RVs must lie in the same environment"; in other words, the long-term average (expectation) of RV Y(s) should be similar to that of Y(s ) 9
Intensity of Events Local intensity λ(u) Mean number of events per unit area at an arbitrary location or point u, formally defined as: Overall intensity λ First order stationarity Any RV should have the same long-term average, for a fixed areal unit s. This implies a constant intensity: 1 0 The expected number of events with a region s is just a function of s : E{Y(s)} =λ s
Interaction Between Count RVs Second-order intensity Long-term average (expectation) of products of counts per unit areas at any two arbitrary points u and u, formally defined as: Some terminology second-order stationarity: expectation of all RVs is constant (first-order stationarity), and second-order intensity is a function of separation vector between any two locations u and u isotropy: only distance (not orientation) of separation vector matters 11 Outlook Quantifying interaction in spatial point patterns within the above assumptions or working hypotheses amounts to studying distances between events
Distance A measure of proximity (typically along a crow's flight path) between any two locations or spatial entities Euclidean distance Consider two points in a 2D (geographical or other) space with coordinates u i = (x i ;y i ) and u j = (x j ;y j ). The Euclidean distance d ij between points u i and u j is computed via Pythagoras's theorem as: 1 2
Distance Metric Formal characteristics of a distance metric A measure d ij of proximity between locations u i and u j is a valid distance metric if it satisfies the following requirements: distance between a point and itself is always zero: d ii = 0 distance between a point and another one is always positive: d ij > 0 distance between two points is the same no matter which point you consider first: d ij = d ji the triangular inequality holds: sum of length of two sides of a triangle cannot be smaller than length of third side: d ij d il + d lj 1 3 A metric d ij need not always be Euclidean, and hence should be checked to ensure that it is a valid distance metric
Non-Euclidean Distances Alternative distance" measures (i) over a road, or railway, (ii) along a river, (ii) over a network Even more exotic distance" measures (i) travel time over a network, (ii) perceived travel time between urban landmarks, (iii) volume of exports/imports 1 4 Euclidean distances between network nodes actual or perceived distances on the network the latter might not even be formal distance metrics, i.e. d ij d ji
Minkowski's Generalized Distance Definition Consider two points in a K-dimensional (geographical or other) space R K with coordinate vectors u i = [u i1 u ik u ik ] and u j = [u j1 u jk u jk ]. The Minkowski distance of order p (with p > 1), denoted as d ij (p), between points u i and u j is computed as: Particular cases Manhattan or city-block distance: Euclidean distance Distances computed from points in multidimensional spaces are routinely used in statistical pattern recognition; points represent objects or cases, each described by K attribute values 1 5
Euclidean Distance Matrix: Single Set of Points Definition Consider a set of N points {u 1 u i u N } in a K-dimensional (geographical or other) space. The distance matrix D is a square (NxN) matrix containing the distances {d(u i,u j ); i = 1 N; j = 1 N} between all NxN possible pairs of points in the set by convention, u 1 is the coordinate vector of the 1st point in the set (1st entry in data file) 1 6 i-th row (or column) contains distances between i-th point ui and all others (including itself) D is symmetric with zeros along its diagonal
Euclidean Distance Matrix: Two Sets of Points Consider 2 sets of points {u 1 u i u N } and {t 1 t i t M } in a K-dimensional (geographical or other) space. The distance matrix D is a (NxM) matrix containing the Euclidean distances {d(u i,t j ); i = 1 N; j = 1 M} between all N x M possible pairs formed by these two sets of points by convention, u 1 is the coordinate vector of the 1st datum in the data set #1, and similarly for t 1 1 7 i-th row contains distances between i-th point ui in set #1 and all points in set #2 j-th column contains distances between j-th point tj in set #2 and all points in set #1 D is not symmetric, i.e., d12 d21: pair {u1,t2} is not the same as pair {u2,t1}
Distances Between Events in A Point Pattern Event-to-event distance Distance d ij between event at location u i and another event at location u j Point-to-event distance Distance d pj between a randomly chosen point at location tp and an event at location u j : Event-to-nearest-event distance Distance d min (u i ) between an event at location u i and its nearest neighbor event: Point-to-nearest-event distance Distance d min (t p ) between a randomly chosen point at location t p and its nearest neighbor event: 1 8
Event-to-Nearest-Event Distances Some events might be nearest neighbors of each other: e.g., u 4,u 5, or have same nearest neighbor: e.g., u 2, u 3, u 4 are nearest neighbors of u 5 Mean nearest neighbor distance Average of all d min (u i ) values: 1 9 Drawback: single number does not suffice to describe point pattern
The G Function Definition Proportion of event-to-nearest-event distances d min (u i ) no greater than given distance cutoff d, estimated as: Cumulative distribution function (CDF) of all N event-tonearest-event distances Example 2 0
Event-to-Nearest-Event (E2NE) Distance Histograms 2 1 for evenly-spaced events, E2NE distances similar to spacing of events for clustered events, more small E2NE distances and fewer large distances
Point Pattern Analysis Metrics G hat function Cumulative distribution function (CDF) of all N event-to-nearest-event distances F hat function Cumulative distribution function (CDF) of all N point-to-nearest-event distances K hat function Relative number of events at distance D calculated around all events 2 2 event-to-nearest-event distances use the sample G function G(d) point-to-nearest-event distances use the sample F function F(d) event-to-event distances use the sample K function K(d)
Sample G Function Examples G hat =Cumulative distribution function (CDF) of all N event-tonearest-event distances 2 3 for evenly-spaced events, G(d) rises gradually up to the distance at which most events are spaced, and then increases rapidly for clustered events, G(d) rises rapidly at short distances, and then levels off at larger d-values
The F Function Definition Proportion of point-to-nearest-event distances d min (t j ) no greater than given distance cutoff d, estimated as: Cumulative distribution function (CDF) of all M point-to-nearest-event distances 2 4 for larger number M of random points, F(d) becomes even smoother Note: The F function provides information on event proximity to voids
Point-to-Nearest-Event (P2NE) Distance Histograms 2 5 for evenly-spaced events, there are more nearest events at small distances from randomly placed points for clustered events, P2NE distances are generally larger than the previous case, and there are a few large such distances
Sample F Function Examples 2 6 for evenly-spaced events, F(d) rises rapidly up to the distance at which most events are spaced, and then levels off (more nearest neighbors at small distances from randomly placed points) for clustered events, F(d) rises rapidly at short distances, and then levels off at larger d-values
The Sample K Function Concept building 1. construct set of concentric circles (of increasing radius d) around each event 2. count number of events in each distance band" 3. cumulative number of events up to radius d around all events = sample K function, K(d) Formal definition 2 7
Interpreting The Sample K Function Re-expressing In other words: Function K(d) is the sample cumulative distribution function (CDF) of all N 2 -N event-to-event distances, scaled by D 2 8 Note: Ignore bin at d = 0 (center plot) and point at d = 0 (right plot
Event-to-Event Distance Histograms 2 9 for evenly-spaced events, there are more medium-sized E2E distances than small or large such distances for clustered events, the distribution of E2E distances is multi-modal
Event-to-Event Distance CDFs 3 0 for clustered events, there are multiple bumps in the CDF of E2E distances due to the grouping of events in space
Sample K Function Examples 3 1 sample K function K(d) is monotonically increasing and is a scaled (by domain measure D version of the CDF of E2E distances
Recap Quantifying interaction in spatial point patterns event-to-nearest-event distances use the sample G function G(d) point-to-nearest-event distances use the sample F function F(d) event-to-event distances use the sample K function K(d) K function looks at information beyond nearest neighbors Caveats clustering is always a function of the overall intensity of a point pattern clustering might occur due to local intensity variations or due to interaction; it is very difficult to disentangle each contribution Watch out for boundaries and edge effects distance distortions due to map projections sampled versus mapped point patterns Interactions of 1 st versus second order stationarity 3 2