Lab #3 Background Material Quantifying Point and Gradient Patterns

Transcription

1 Lab #3 Background Material Quantifying Point and Gradient Patterns

2 Dispersion metrics Dispersion indices that measure the degree of non-randomness Plot-based metrics Distance-based metrics

3 First-order metrics Variance to mean ratio plot-based measure of dispersion at the scale of the plot Ratio of the variance-tomean count of events under the assumption of a Poisson process D > 1 = Clumped D < 1 = Uniform

4 First-order metrics Clark and Evans index (1954) distance-based measure of dispersion agnostic to scale Ratio of the mean nearest neighbor distance to the expected mean distance under the assumption of a Poisson process Nearest neighbor Nearest event D < 1 = Clumped D > 1 = Uniform

5 Second-order metrics Ripley s K distribution distance-based measure of dispersion across scales Random point pattern Expected number of points within circle of radius d from an arbitrary point: where K(d s ) is the area of a circle defined by radius d s Thus, under complete spatial randomness (csr):

6 Second-order metrics Ripley s K distribution distance-based measure of dispersion across scales Random point pattern If E(d s ) > E(d s [csr]) = Clumped If E(d s ) < E(d s [csr]) = Uniform

7 Second-order metrics Ripley s K distribution distance-based measure of dispersion across scales L(d s ) Besag s transformation d (distance) Under complete spatial randomness

8 Second-order metrics Ripley s K distribution distance-based measure of dispersion across scales L(d s ) - d Typical transformation Besag s transformation d (distance) Under complete spatial randomness

9 Second-order metrics Ripley s K distribution distance-based measure of dispersion across scales From D Urban

10 Second-order metrics Ripley s K distribution pattern intensity maps Modified Ripley s K for each point in the data set (Getis and Franklin 1987) Contours of point cluster intensity at each scale (neighbor distance) Shaded areas have L(d) values above the Poisson expectation

11 Dispersion metrics Important characteristics: Suitable for any feature that can be meaningfully treated as a point location (x,y) Samples Can be computed for sample or census of points Methods for plot- and distance-based sampling

12 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location

13 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location

14 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location

15 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location

16 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location

17 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location

18 Kernel intensity Probability distribution placed over each point to estimate the intensity of points at any location 100 m 200 m 400 m 800 m

19 Kernel intensity Important characteristics: Suitable for any feature that can be meaningfully treated as a point location (x,y) Kernels are extremely flexible in representing ecological processes due to flexibility in shape and bandwidth Kernels require census of points to be meaningful

20 Gradient Patterns Autocorrelation structure functions Spatial autocorrelation structure function based on the similarity/dissimilarity of a quantitative variable as a function of the (lag) distance between locations Topographic moisture index for Coweeta Experiment Station Transect From D Urban

21 Gradient Patterns Correlograms Moran s I statistic Transect Lag distance = 3 Lag distance = 1 Sample points/quadrats Moran s I statistic I( d) = n n n w ( y y)( y y) i= 1 j= 1 ij i j n W ( y y) i= 1 Moran s I(d) is the correlation (r) in the variable at lag distance d i 2

22 Gradient Patterns Correlograms Moran s I correlogram under artificial patterns

23 Gradient Patterns (semi)variograms Semivariance Transect Lag distance = 3 Lag distance = 1 Sample points/quadrats Semivariance γ ( d) = n n 2 wij ( yi y j ) i= 1 j= 1 2n d Sill Simivariance is the variance (γ) in the variable at lag distance d (divided by 2) Nugget Range

24 Gradient Patterns (semi)variograms Topographic moisture index Semivariogram (scaled) 1600 ha watershed 400 random samples From D Urban

25 Gradient Patterns Autocorrelation structure functions Important characteristics: Suitable for any quantitative variable that varies spatially (or temporally) Can be computed for sample or census data Intuitive interpretation due to familiarity with correlation and variance Correlogram for significance testing; Variogram for modeling Samples