LAB EXERCISE #3 Quantifying Point and Gradient Patterns

Transcription

1 LAB EXERCISE #3 Quantifying Point and Gradient Patterns Instructor: K. McGarigal Overview: In this exercise, you will learn to appreciate the challenges of quantifying point and gradient patterns and gain practical hands-on experience doing so with R. Specifically, you will: 1) establish a pattern-process objective and associated landscape definition, 2) quantify and interpret the landscape patterns using point pattern metrics and gradient metrics, and 3) discuss the challenges in choosing and interpreting the results of these analyses. Primary objectives To learn how to quantify point patterns and gradient patterns based on selected metrics using R. To gain an appreciation for the many challenges (and alternatives) in quantifying point patterns and gradient surfaces for a specific application. Part 1: Background information The case study landscape is the Massachusetts portion of the Lower Connecticut River watershed, a HUC6 (Hydrologic Unit Code) level watershed representing the lower half of the Connecticut River watershed. The landscape encompasses 704,882 ha and, as noted in lab1, supports a range of ecological systems and human land uses. Step 1. Establish the objective of the analysis Given a user-specified overall goal pertaining to a pattern-process of interest, the first step of any landscape pattern analysis is to establish a clear objective; specifically, a Specific, Measurable, Attainable, Relevant, and Time-based (SMART) statement of the pattern and process to quantify. Unfortunately, because of the data preparation required for landscape pattern analysis, it is not practical in this brief exercise for you to define your own unique goal and objective and associated landscape definition (step 2). Consequently, while there are numerous possibilities (as you hopefully discovered in lab 1), goal and objective of the analysis for this exercise were pre-established, as follows. Goal: Evaluate the distribution of potential vernal pools and the associated terrain among HUC8 watersheds within the Massachusetts portion of the lower Connecticut River watershed. Pattern: distribution of potential vernal pools and spatial structure of the terrain, as defined below, within HUC8 watersheds. Lab 3 Page 1

2 Process: terrain factors, as defined below, affecting the distribution of potential vernal pools within each HUC8 watershed in the project area. Analysis objective: quantify the spatial distribution of potential vernal pools and the spatial autocorrelation structure of terrain-derived indices of wetness and solar radiation within each HUC8 watershed within the Massachusetts portion of the lower Connecticut River watershed. Goal: Evaluate the distribution of trees and their establishment age and size within two forest stands in western Massachusetts [data from Helena Murray]. Pattern: distribution of trees and the spatial structure of their establishment age and size within the two forest stands. Process: abiotic, biotic and disturbance factors affecting the distribution of trees within these two forest stands. Analysis objective: quantify the spatial distribution of tree species and the spatial autocorrelation structure of their ages and sizes within the two forest stands. Step 2. Define the landscape The next step is to define the landscape in accordance with the objective. Conceptual model.-for this exercise, we will use the point pattern model for the potential vernal pools and the gradient model for the continuous terrain indices. Thematic content.-potential vernal pools for the point pattern and the topographic wetness index and solar radiation index derived from a digital elevation model. Thematic resolution.- the potential vernal pool layer has a single class of pools, whereas the thematic resolution of the terrain indices is defined by their measurement precision. Spatial grain.-the spatial grain of the point pattern is the precision in which the geographic coordinates of the pools were mapped, which we have no real way of knowing without conducting an accuracy assessment. The spatial grain of the terrain indices is based on the 30 m elevation raster from which they were derived. Spatial extent.-the landscape extent is the HUC8 watershed, including each of the five subbasins comprising the Massachusetts portion of the lower Connecticut River watershed, which encompasses 704,882 ha in total. Fragmenting features.-fragmenting features are not relevant to describing the spatial distribution of potential vernal pools or for describing the spatial autocorrelation structure of the terrain indices. Lab 3 Page 2

3 Landscape boundary and context.-the HUC8 watershed boundaries are relatively arbitrary with respect to the distribution of potential vernal pools and solar radiation; however, these watershed boundaries do loosely affect the topographic wetness index, as wetness is determined in part by the flow of surface water to each cell, which is constrained by the watershed boundaries. However, for all practical purposes, the landscape is "open" to the pattern-process under consideration. Nevertheless, the HUC8 landscapes are sufficiently large given the internal heterogeneity in vernal pool numbers and distribution and variability in terrain for inferences on pattern and process to be valid. Conceptual model.-for this exercise, we will use the point pattern model for the tree stems and the gradient model for the continuous age and size variables. Thematic content.-trees for the point pattern and tree age and size for the gradient patterns. Thematic resolution.- tree stems are classified into 8 different tree species; trees are resolved into age since establishment to the nearest year and diameter breast height (dbh) to a precision of 0.1 cm. Spatial grain.-the spatial grain of the tree stem pattern and associated measurements (i.e., species, age, dbh) is the precision in which the geographic coordinates of the trees were mapped, which was to the nearest 0.1 m. Spatial extent.-the landscape extent is a 50x50 m (0.25 ha) plot, in each of two different forest stands. Fragmenting features.-fragmenting features are not relevant to describing the spatial distribution of tree stems or for describing the spatial autocorrelation structure of their age and dbh. Landscape boundary and context.-the 0.25 ha plots are arbitrary with respect to the distribution of tree stems in these two forest stands. It is unclear how the abiotic template, biotic processes and disturbance processes at the scale of 0.25 ha plots might affect the distribution and structure of trees, but it seems likely that these factors are operating across a range of scales, including much broader scales than the 0.25 ha plot. Therefore, any inferences about pattern and process derived from this analysis should be done cautiously. Step 3. Inspect the GIS data Now that we have defined the landscape, take some time to familiarize yourself with the GIS data associated with the potential vernal pool application. Open up in ArcMap the following project file:...\exercises\lab3\lab3.mxd Lab 3 Page 3

4 Take some time to review each of the data layers. The ArcMap table of contents is organized into four groups of layers, from bottom to top, as follows: Lower Connecticut watershed - suite of layers encompassing the Lower Connecticut River HUC6 watershed. These layers provide a spatial context for the focal landscape: o hillshade.tif - backdrop for other layers if desired. o dslland_2010.tif - original 30m resolution land cover map from the DSL project containing 65 classes within the project area. o cover7_2010.tif - dslland reclassified into 7 major cover types based on ecological formation and human land use. Potential vernal pools - point shapefile depicting potential vernal pools; given for the entire lower Connecticut River watershed (PVP) and each of the five major named HUC8 watersheds. solar radiation index - 30 m resolution raster layers depicting the unitless index of incident solar radiation based on latitude, longitude, slope, aspect and topographic shading derived from a digital elevation; given for the entire lower Connecticut River watershed (sun.tif) and each of the five major named HUC8 watersheds. Topographic wetness index - 30 m resolution raster layers depicting the unitless index of topographic wetness derived from a digital elevation; given for the entire lower Connecticut River watershed (wet.tif) and each of the five named HUC8 watersheds. Vector overlays - a handful of shapefiles that may be useful as overlays, including: o DSLroads - roads layer with the symbology set to display road class. o NHDhighRes - streams layer derived from the National Hydrography data based on high resolution 1:24 k streams. o HUC8, 10, 12 and 6 level watersheds. o HUC8_? watershed for each of the five major named subbasins. o HUC6_Mass - boundary of the HUC6 lower Connecticut River watershed within Massachusetts. o HUC6 buffer - HUC6 lower Connecticut River watershed with a 10 km buffer to serve as the clip extent for the land cover layers. Part 2: Analysis of point patterns In this section, we will analyze the spatial point patterns in which each entity (either a potential vernal pool or tree stem) is considered a unique event with a geographic location within the landscape extent, and it is the spatial distribution of point locations (i.e., their x,y coordinates) that we are interested in. Lab 3 Page 4

5 The R scripting differs slightly between the two applications owing to differences in the source data. Therefore, the detailed steps below will be presented separately for each application. In both cases, however, you will need to source the landeco.r package and install (if need be) and load the other required libraries, as given in the provided lab3 R scripts. Step 2a. Data prep First, prepare the data for point pattern analysis in R. 1. First, create an object that points to the folder containing the GIS data: path<-'c:\\work\\landeco\\exercises\\lab3\\gisdata\\' 2. Next, run the following lines to load all of the HUC8 boundary shapefiles that we need into R in a special format (owin) required by the spatstat R package that we will use below to conduct some of the analyses: setwd(path) boundary_huc6_mass<-as(readogr('.','huc6_mass'),'owin') boundary_miller<-as(readogr('.','huc8_miller'),'owin') boundary_chicopee<-as(readogr('.','huc8_chicopee'),'owin') boundary_connecticut<-as(readogr('.','huc8_middle_connecticut'),'owin') boundary_deerfield<-as(readogr('.','huc8_deerfield'),'owin') boundary_westfield<-as(readogr('.','huc8_westfield'),'owin') 3. Next, run the following lines to load all of the PVP point shapefiles that we need into R in a special format (ppp) required by the spatstat R package that we will use below to conduct some of the analyses. Note, here we will also assign the corresponding HUC polygon as the boundary for the points. Without this, a rectangular bounding box around the points would be assigned as the boundary, and this would result in a major bias in the point pattern analysis below: HUC6_Mass<-as(readOGR('.','PVP'),'ppp')[boundary_HUC6_Mass] Miller<-as(readOGR('.','PVP_Miller'),'ppp')[boundary_Miller] Chicopee<-as(readOGR('.','PVP_Chicopee'),'ppp')[boundary_Chicopee] Connecticut<-as(readOGR('.','PVP_Middle_Connecticut'),'ppp') [boundary_connecticut] Deerfield<-as(readOGR('.','PVP_Deerfield'),'ppp')[boundary_Deerfield] Westfield<-as(readOGR('.','PVP_Westfield'),'ppp')[boundary_Westfield] 4. Lastly, plot each of the point patterns to verify that they fall correctly within their assigned polygon boundary. Note, here we will set use.marks=false to plot just the x,y location of each PVP: plot(huc6_mass,use.marks=false) Lab 3 Page 5

6 plot(miller,use.marks=false) plot(chicopee,use.marks=false) plot(connecticut,use.marks=false) plot(deerfield,use.marks=false) plot(westfield,use.marks=false) 1. First, create an object that points to the folder containing the data: path<-'c:\\work\\landeco\\exercises\\lab3\\gisdata\\' 2. Next, we need to create a polygon boundary object in R for the 50x50 m plot in a special format (owin) required by the spatstat R package that we will use below to conduct some of the analyses: boundary<-owin(xrange=c(0,50),yrange=c(0,50)) 3. Next, run the following lines to read in the tabular point data (.csv file) for each of the two forest stands, coerce the dataframes to a planar point pattern (ppp) objects (as required by the spatstat package), and then assign the boundary above as extent. Note, we will use the marks= argument to specify which fields in the dataframe represent "marks" containing additional attribute data about each point: temp<-read.csv(paste(path,'shelburne.csv',sep='')) Shelburne<-ppp(temp$x,temp$y,window=boundary,marks=temp[,4:8]) ShelburneHemlock<-Shelburne[Shelburne$marks$species=='hemlock'] temp<-read.csv(paste(path,'williams.csv',sep='')) Williams<-ppp(temp$x,temp$y,window=boundary,marks=temp[,4:8]) WilliamsHemlock<-Williams[Williams$marks$species=='hemlock'] 4. Lastly, plot each of the point patterns to verify that they look correct and fall within the assigned polygon boundary. Note, here we will set which.marks='species' to plot the stem locations by tree species, and will assign 'red' to the hemlock trees, leaving all other species as black: plot(shelburne,which.marks='species',cols=c(rep('black',4), 'red',rep('black',4))) plot(williams,which.marks='species',cols=c(rep('black',2), 'red',rep('black',6))) Step 2b. Clark and Evans dispersion index Next, compute the Clark and Evans dispersion index. Recall from lecture that the Clark and Evans dispersion index is a global index of point pattern that discerns clumped from random from uniform distributions, but is agnostic with regards to the scale of the Lab 3 Page 6

7 pattern. Index values equal 1 for a completely random distribution; >1 for a more regular or uniform distribution; and <1 for a more clumped distribution. The test implemented here does two things. First, it uses an edge correction method to adjust for the bias due to a limited and irregular sample area. Second, it conducts a Monte Carlo permutation test to determine significance. The Monte Carlo test involves randomly distributing the points with the boundary of the input data set and recalculating the Clark and Evans index, and repeating this many times (999 by default), then computing the proportion of the permutation distribution that is larger than (for an upper onesided test) or less than (for a lower one-sided test) or more different from 1 than (for a two-sided test) the observed value of the index for your dataset. Here, by default, we are conducting a two-sided test, allowing for distributions that are either more regular (index >1) or more clumped (index <1) than random. Compute the Clark and Evans Index for the potential vernal pools in each of the HUC8 watersheds. Note, the correction='cdf' uses the available edge correction method (see help for clarkevens function) for dealing with the bias associated with points close to the polygon boundary: clarkevans.test(miller,correction='cdf') clarkevans.test(chicopee,correction='cdf') clarkevans.test(connecticut,correction='cdf') clarkevans.test(deerfield,correction='cdf') clarkevans.test(westfield,correction='cdf') Compute the Clark and Evans Index for all trees and for hemlock trees as a subset for each of the two forest stands. Note, the correction='cdf' uses the available edge correction method (see help for clarkevens function) for dealing with the bias associated with points close to the polygon boundary: clarkevans.test(shelburne,correction='cdf') clarkevans.test(shelburnehemlock,correction='cdf') clarkevans.test(williams,correction='cdf') clarkevans.test(williamshemlock,correction='cdf')) Assignment: Q2b: What did you learn (or not learn) from the Clark and Evans index about the distribution of points (potential vernal pools or tree stems) among the landscapes (HUC8 watersheds or forest stands)? Step 2c. Ripley's K-distribution Next, compute the Ripley's K-distribution for each of the point patterns: Lab 3 Page 7

8 1. First, create a named list of the point pattern objects associated with each of the HUC8 watersheds (ppp files created above): infiles<- list(miller=miller,chicopee=chicopee, Connecticut=Connecticut,Deerfield=Deerfield, Westfield=Westfield) 2. Next, run the following line to compute the Ripley's K distribution for each of the point patterns in the list and to plot the results: ripleyk(infiles) 1. First, create a named list of the point pattern objects associated with the two forest stands (ppp files created above): infiles<-list(shelburne=shelburne, ShelburneHemlock=ShelburneHemlock, Williams=Williams, WillamsHemlock=WilliamsHemlock) 2. Next, run the following line to compute the Ripley's K distribution for each of the point patterns in the list and to plot the results: ripleyk(infiles) Assignment: Q2c: Interpret the Ripley's K plot. What exactly does it tell you? What did you learn about the distribution of points (potential vernal pools or tree stems) among the landscapes (HUC8 watersheds or forest stands) from the Ripley's K plot? Step 2d. Kernel intensity maps Next, derive kernel intensity maps for the points. Recall, there are many options for computing kernel intensity, including the shape of the kernel and the bandwidth. Here, we will stick with the default Gaussian kernel, but you can play with the bandwidth, which in this case represents the standard deviation (in meters in this case) of the Gaussian distribution. Derive a kernel intensity map for potential vernal pools across the entire Massachusetts portion of the lower Connecticut River watershed to get another perspective on the vernal pool distribution patterns. Remember, our raster resolution is 30 m, so bandwidths less than this will be meaningless. Lab 3 Page 8

9 1. First, compute the kernel intensity using the density() function in the spatstat library, which requires a ppp object as input. However, I prefer the format and the appearance of the plot when the density object is coerced to a raster package object, so I have wrapped the function call in the raster() function. Also, feel free to change the sigma= argument to play with different bandwidths: temp<-raster(density(huc6_mass,sigma=800)) 2. Next, plot the kernel intensity map: plot(temp) 3. Lastly, if you want to, you can write the raster object to disk and then add it to your ArcMap project to overlay it with the other layers: writeraster(temp,paste(path,'pvpkernel800.tif',sep=''), datatype='flt4s',overwrite=true) Derive a kernel intensity map for the tree stems in the two forest stands to get another perspective on the tree distribution patterns. Remember, our raster resolution is 1 m, so chose bandwidths accordingly. 1. First, compute the kernel intensity using the density() function in the spatstat library, which requires a ppp object as input. However, I prefer the format and the appearance of the plot when the density object is coerced to a raster package object, so I have wrapped the function call in the raster() function. Also, feel free to change the sigma= argument to play with different bandwidths: ShHemSigma2<-raster(density(ShelburneHemlock,sigma=2)) WiHemSigma2<-raster(density(WilliamsHemlock,sigma=2)) 2. Next, plot the kernel intensity map: plot(shhemsigma2) plot(wihemsigma2) 3. Lastly, if you want to, you can write the raster object to disk and then add it to your ArcMap project to overlay it with the other layers: writeraster(shhemsigma2,paste(path,'shhemsigma2.tif',sep=''), datatype='flt4s',overwrite=true) writeraster(wihemsigma2,paste(path,'wihemsigma2.tif',sep=''), datatype='flt4s',overwrite=true) Q2d: Interpret the kernel intensity map. What exactly does it tell you? Give one ecological/management justification for choosing a kernel bandwidth(s). There are many possibilities and they all depend on the specific objective and application, but come up with one plausible justification for the chosen bandwidth(s). Lab 3 Page 9

10 Part 3: Analysis of gradient patterns In this section, we will analyze the spatial autocorrelation structure of one or more quantitative marks associated with the points using two different structure functions: correlograms and variograms. Step 3a. Correlograms First, produce correlograms for the selected quantitative variables. Note, in the correlogram plots below, the solid points represent observations that differ significantly (at p<0.05) from the null or expected value of zero based on a Monte Carlo permutation test with 100 permutations by default (resamp=100; although this can be increased to achieve greater precision in the significance test); whereas the open circles are insignificant. Derive correlograms for the two terrain-based indices (index of solar radiation and index of topographic wetness) that may reflect terrain features associated with the occurrence and distribution of potential vernal pools. We already have GIS layers of both terrain indices (sun and wet) for each of the five major HUC8 watersheds. Here, we will use the landeco package correloraster() wrapper function for the correlog() function in ncf package to compute and plot correlograms for each of the HUC8 watersheds and for each of the terrain indices. Note, the input for this function is one or more rasters (geotiffs). The function will sample the input rasters at the specified sampling intensity and treat the data as a marked point pattern. The increment=60 argument specifies lag distance increments (i.e., bin size) to be 60 m; npnts=1000 specifies the number of random points to sample from the raster; and maxdist=1000 specifies the maximum lag distance (m) for computing and plotting the Moran's I correlation. Feel free to play with variations on these argument settings: correloraster(path='c:\\work\\landeco-umass\\exercises\\lab3\\gisdata\\', infiles=c('sun_miller','sun_chicopee', 'sun_middle_connecticut','sun_deerfield', 'sun_westfield'),increment=60,npnts=1000,maxdist=1000) correloraster(path='c:\\work\\landeco-umass\\exercises\\lab3\\gisdata\\', infiles=c('wet_miller','wet_chicopee', 'wet_middle_connecticut','wet_deerfield', 'wet_westfield'),increment=60,npnts=1000,maxdist=1000) Derive correlograms for the two forest stands based on the quantitative tree marks for year of establishment (year) and diameter (dbh). Lab 3 Page 10

11 1. Read in the raw tabular data (csv file) as a data.frame, and create a separate object for the subset of the data including only hemlock trees. Note, the csv file must contain the fields named 'x' and 'y' corresponding to the geographic coordinates of the points: ShelAll<-read.csv(paste(path,'Shelburne.csv',sep='')) ShelHemlock<-ShelAll[ShelAll$species=='hemlock',] WillAll<-read.csv(paste(path,'Williams.csv',sep='')) WillHemlock<-WillAll[WillAll$species=='hemlock',] 2. Create list objects containing the dataframe objects created above, including one for all trees and one for the hemlock only trees: Alltrees<-list(Shelburne=ShelAll,Williams=WillAll) Hemlock<-list(Shelburne=ShelHemlock,Williams=WillHemlock) 3. Lastly, use the landeco package correlopoints() wrapper function for the correlog() function in ncf package to compute and plot correlograms for each of the forest stands and for each of the quantitative marks. Note, the mark='year' or 'dbh' argument specifies the column name containing the quantitative mark; increment=3 specifies lag distance increments (i.e., bin size) to be 3 m; and maxdist=25 specifies the maximum lag distance (m) for computing and plotting the Moran's I correlation. Feel free to play with variations on these argument settings: correlopoints(infiles=alltrees, mark='year',increment=3,maxdist=25) correlopoints(infiles=hemlock, mark='year',increment=3,maxdist=25) correlopoints(infiles=alltrees, mark='dbh',increment=3,maxdist=25) correlopoints(infiles=hemlock, mark='dbh',increment=3,maxdist=25) Q3a: Interpret the correlogram. What exactly does it tell you? What did you learn about the spatial autocorrelation structure of the quantitative marks from the correlogram? Did the comparison among HUC8 watersheds or between forest stands differ between the two marks and, if so, why, and how does this relate, if at all, to the processes affecting the distribution of the points? Step 3b. (semi-)variograms Lastly, we will use the landeco package varioraster() and variopoints() wrapper functions for the variogram() function in gstat package to compute and plot variograms for each of the quantitative variables. Lab 3 Page 11

12 Derive variograms for each of the HUC8 watersheds and for each of the terrain indices. Note, in the varioraster() function below the infiles= argument specifies either a list of raster objects or, in this case, a vector of file names corresponding to geotiffs on disk; npnts=10000 specifies the number of random points to sample from the raster; maxdist=1000 specifies the maximum lag distance for computing and plotting the semivariance; and model='exp' specifies the theoretical variogram model to fit to the empirical variogram. Here, we are only using the fitted variogram model to produce an estimate of the "range" (i.e., the lag distance at which the "sill" is reached), although the fitted variogram has other potential uses, such as in kriging (i.e., interpolation based on the spatial autocorrelation structure to produce a complete surface of estimated values of the quantitative variable. Feel free to play with variations on these argument settings (see variogram help file for options, and vgm help file for alternative variogram models): varioraster(path='c:\\work\\landeco-umass\\exercises\\lab3\\gisdata\\', infiles=c('sun_miller','sun_chicopee', 'sun_middle_connecticut','sun_deerfield', 'sun_westfield'),npnts=10000,maxdist=1000,model='exp') varioraster(path='c:\\work\\landeco-umass\\exercises\\lab3\\gisdata\\', infiles=c('wet_miller','wet_chicopee', 'wet_middle_connecticut','wet_deerfield', 'wet_westfield'),npnts=10000,maxdist=1000,model='exp') Derive variograms for each of the forest stands and for each of the quantitative marks (year and dbh). Note, in the variopoints() function below the infiles= argument specifies either a list object containing a list of input data.frames, as in this case, or a vector of the csv files to read from disk; mark='year' or 'dbh' specifies the column name containing the quantitative mark; maxdist=1000 specifies the maximum lag distance for computing and plotting the semivariance; and model='exp' specifies the theoretical variogram model to fit to the empirical variogram. Here, we are only using the fitted variogram model to produce an estimate of the "range" (i.e., the lag distance at which the "sill" is reached), although the fitted variogram has other potential uses, such as in kriging (i.e., interpolation based on the spatial autocorrelation structure to produce a complete surface of estimated values of the quantitative variable. Feel free to play with variations on these argument settings (see variogram help file for options, and vgm help file for alternative variogram models): variopoints(infiles=alltrees,mark='year',maxdist=25) variopoints(infiles=hemlock,mark='year',maxdist=25) Q3b: Interpret the variogram. What exactly does it tell you. What did you learn about the spatial autocorrelation structure of the quantitative variables from the variogram? Did the comparison among the landscapes (watersheds or stands) differ between the Lab 3 Page 12

13 quantitative variables and, if so, why, and how does this relate, if at all, to the distribution of points? Did the variograms reveal anything different than the correlograms? Lab 3 Page 13