Home

S+SpatialStats Supplement

Contents

1. Having the estimates of intensity of the maple process on a Data Window helps you to continue with other visualization of the data For example 11 Select all three columns in the maple i nt Data window 12 Open the Plots3D palette Press the 32 Color Surface button to create a filled surface plot of intensity You may also rotate the resulting plot and get different views of its peaks and valleys in doing so Three methods are available to estimate the intensity of a spatial point pattern using the Intensity dialog in StSPATIALSTATS binning kerne and gauss2d These three methods estimate the intensity locally over the total region A and return a data frame containing smoothed intensity estimates which may vary over A as well as interpolated x and y values to facilitate plotting Several S PLUS 3D plot types can then be used to visualize this variation and to assess the hypothesis of a constant intensity throughout the sampling area 53 Chapter 5 Spatial Point Patterns 54 The binning method uses a two dimensional histogram to form rectangular bins The counts in these bins are smoothed using a loess smoothing algorithm Using the binning method for the maple data as explained in the sequence above yielded the following plot All of the intensity estimation and other visualization techniques used in this section show that the intensity of the maple trees in the Lansing Woods appears to vary more than wo
2. result print max abs a b lt le 14 summary spatial neighbor Summary Method summary spatial neighbor DESCRIPTION Returns a summary list for objects of class spatial neighbor USAGE summary spatial neighbor object REQUIRED ARGUMENTS 79 Appendix Data and Function Reference 80 object VALUE nregion symmetric minConnected maxConnected aveNumLinks aveWeight rowMissing colMissing islands DETAILS SEE ALSO EXAMPLES an object that inherits from class spatial neighbor an object of class summary spatial neighbor which is a list of lists one list for each unique value of matrix in object The sublists each contain the components that summarize the particular spatial neighbor matrix an integer indicating the number of regions object covers This is the same as attr object nre gion a logical value if TRUE the object is assumed to be symmetric This is the same as attr ob ject symmetric a named vector of the least connected regions The names are the row indices that have the smallest number of connections for the i th matrix in object The values all the same are the minimum num ber of neighbors a named vector of the most connected regions The names are the row indices that have the largest number of connections for the i th matrix in object The values all the same are the maximum number of neighbors a single value giving the mean number of neighbors each region has a
3. x 2 data coal ash covfun spher cov range 4 31 sill 0 14 nugget 0 89 predictions over default 30 x 30 grid pcoal lt predict kcoal 74 Appendix Data and Function Reference plot prediction surface wireframe fit x y data pcoal screen list z 300 x 60 y 0 drape T block kriging predictions with block of size 2 x 2 at 4 locations predict kcoal data frame x c 4 5 9 11 y c 7 13 9 18 blocksize c 2 2 nxy c 5 5 spatial neighbor Create a spatial neighbor Object spatial neighbor DESCRIPTION Function used to create an object of class spatial neighbor given its component parts USAGE spatial neighbor row id col id weights rep 1 length row id neighbor matrix nregion max c row id col id symmetric F matrix id lt lt see below gt gt REQUIRED ARGUMENTS row id an integer vector containing the row indices of the non zero elements of the neighbor weight matrix The i th element of row id and the i th element of col id specify two regions which are spatial neighbors Two regions are spatial neighbors if observations from the two regions have a non zero spa tial weight and vice versa row id can also be a two column matrix containing the row indices the first column and the column indices the second column This argument is ignored if neigh bor matrix is given col id integer vector of the same length as row id with the column indices of the non zero elements of
4. 1988 in some cases relThreshold the absolute magnitude an element must have to be considered as a pivot candidate except as a last re sort This should be set to a small fraction of the smallest absolute diagonal element diagPivoting if TRUE pivot selection should be confined to the diagonal if possible shareMemory if TRUE the in memory representation of the sparse matrix will be shared by other routines If memo ry is shared it needs to be released later One way to release the memory is to call c de stroy_sparse_matrix after the in memory representation of the matrix is no long needed Most users should use the default value FALSE VALUE a matrix or vector b solving the linear system Sb x DETAILS This routine uses the sparse matrix code of Kenneth Kundert and Alberto Sangiovanni Vincentelli 1988 The University of California Berkeley holds the copyright for these routines REFERENCES Kundert Kenneth S and Sangiovanni Vincentelli Alberto 1988 A Sparse Linear Equation Solver Department of EE and CS University of California Berkeley SEE ALSO spatial cg solve spatial multiply spatial neighbor spatial neighbor object EXAMPLES amp lt 134 row id lt 1 1 2 2 3 col id lt c 1 3 1 3 4 alpha lt 0 3 neighbor lt spatial neighbor row id row id col id col id symmetric T a lt solve diag attr neighbor nregion alpha spatial weights neighbor x b lt spatial solve neighbor x rho alpha
5. C New Data New Dat C Import File cen T T Show Dialog on Startup Cancel Apply if current Help You could also use the command line directive gt guiOpenView Name coal ash classname data frame 2 Proceed by selecting the first 2 columns of the data frame in the window 1 Cressie Noel A C 1993 Statistics for Spatial Data Revised Edition John Wiley and Sons New York E xploratory D ata Analysis 3 From the Plots2D palette choose a scatter plot by pressing the first button on the top left hand side corner 4 A plot of the sampling locations appears Notice the gridded pattern that these observations follow To superpose information about the percent coal ash at each sampling location we can overlay a contour map of coal ash or make the symbols at each sampling point vary in some way according to that variable coal For illustration purposes we will demonstrate both in this section The S SPATIALSTATS User s Manual contains detailed plots that show the distribution of coal ash percentages over the sampled area potential outliers and trend analysis We will refer to those results whenever necessary To vary symbols according to a third variable 1 Select the Graph Sheet with the points and click on the points themselves until a green knob appears at the bottom of the bulk of the data 2 Right click and select Data to Plot from the middle of the context menu that appears 3 Select
6. REFERENCES Friedman J Bentley J L and Finkel R A 1977 An algorithm for finding best matches in loga rithmic expected time ACM Transactions on Mathematical Software 3 209 226 SEE ALSO quad tree EXAMPLES x lt cbhind sidsSeasting sidsSnorthing sids nhbr lt find neighbor x max dist 30 Find the nearest neighbors for the Lansing hickories hickory lt lansing lansing 3 hickory 1 2 hickory nhbrl lt find neighbor hickory k 2 drop self T Now find the closest maple for each hickory maple lt lansing lansing 3 maple 1 2 hmn lt find neighbor hickory quad tree maple and plot the tree locations with lines joing the neighbors par pty s plot maple 1 maple 2 pch 16 points hickory 1 hickory 2 pch 1 col 2 segments hickory hmn 1 1 hickory hmn 1 2 maple hmn 2 1 maple hmn 2 2 Glasgow neighbor Neighbors for Glasgow Mortality Rate Data Glasgow neighbor SUMMARY An object of class spatial neighbor containing the neighbor specification among the 87 com munity medicine areas in Glasgow Scotland The standardized mortality rate SMR values for this data are contained in Glasgow SMR DATA DESCRIPTION Four hundred and fifty neighbor relationships are specified The neighbor relationships are not sym metric See spatial neighbor object for a description of the data within an object of class spatial neighbor SOURCE The data
7. The data frame contains the following columns row id the row index in the neighbor matrix that corresponds to a region or spatial unit This implies a num bering of regions from 1 to the total number of regions col id the column index in the neighbor matrix that corresponds to the neighbor of the region defined by the corresponding element of row id weights a numeric value giving the relative strength of the neighbor relationship The larger the value the stronger the relationship matrix if multiple types of neighbor matrices are possible this column contains the type of the neighbor this weight represents it gives a numeric identifier for each spatial neighbor contiguity matrix SPECIAL ATTRIBUTES nregion the number of total regions in the study The row and column identifiers given in row id and col id might not include ALL the spatial units in the area of interest This happens when units are isolated i e have no neighboring regions In this case nregion must be used to determine the total number of rows and columns in the contiguity matrix 71 Appendix Data and Function Reference symmetric logical flag It provides an indication of whether the contiguity matrix is symmetric TRUE or not FALSE If TRUE only the weights for the upper or lower triangle of the contiguity matrix need to be specified in the object Use the function spatial weights to expand the full symmetric weights ma trix DETAILS An object of cla
8. cos angle 2 ratio sin angle 2 A 1 2 1 ratio sin angle cos angle and A 2 2 sin angle 2 ratio cos angle 2 See Journel and Huijbregts 1978 pp 179 181 The variogram is then estimated using these corrected locations REFERENCES Cressie N and Hawkins D M 1980 Robust estimation of the variogram Mathematical Geology 12 115 125 Journel A G and Huijbregts Ch J 1978 Mining Geostatistics Academic Press New York Matheron G 1963 Principles of geostatistics Economic Geology 58 1246 1266 SEE ALSO loc variogram xyplot EXAMPLES anisotropy plot log tcatch 1 long lat data scallops lag 075 check islands Detect Isolated Spatial Regions check islands DESCRIPTION Given an object of class spatial neighbor detects spatial units that have no neighbors islands USAGE check islands x remap F REQUIRED ARGUMENTS x an object of class spatial neighbor OPTIONAL ARGUMENTS remap logical flag if there is an island should we recode the indexing of the spatial contiguity matrix to elim inate the rows and columns with all zeroes That is should we renumber components row id and col id of the spatial neighbor object VALUE 58 if remap FALSE the list of existing islands is returned Otherwise an object of class spa tial neighbor with remapped row idand col id Appendix Data and Function Reference SIDE EFFECTS the attribute nregion of the output may differ from that of x whenrem
9. module of S PLUS This new version features a Graphical User Interface to the functions already introduced in S SPATIALSTATS 1 0 plus exciting new functionality New features in version 1 5 include Block Kriging Variogram Fitting Summary and plot methods for spatial neighbor objects Simulation of Nonhomogenous Poisson Point Patterns New data set GI asgow SMR Numerous Bug fixes find neighbor quad tree sids dataset names The Graphical User Interface that is the menus and dialogs in S SPATIALSTATS 1 5 has been designed with your convenience in mind It also delivers the accuracy and high quality you expect from our product Use this supplemental guide to Get started using the dialogs in the Graphical User Interface of S SPATIALSTATS 1 5 to facilitate your spatial data analyses Learn to use other menus dialogs and graphical interface features in the general S PLUS environment to perform analysis of spatial data vi INTRODUCTION This guide describes how to use the S SPATIALSTATS 1 5 Graphical User Interface GUI It is a companion to the S SPATIALSTATS User s M anual That manual provides extensive detail regarding the various techniques available for spatial data analysis as well as information on how to perform such analyses using S PLUS commands In this guide you will also find descriptions of the features in S SPATIALSTATS that are new to version 1 5 specifically how to fit variograms perform block
10. summary sn Appendix Data and Function Reference triangulate Delaunay s Triangulation triangulate DESCRIPTION Calculate Delaunay s triangulation for points with given coordinates x and y USAGE triangulate x y plot it T shrink 0 1 REQUIRED ARGUMENTS x alist with components x and y a 2 column matrix or a vector containing the horizontal coordi nates of the vertices that form the polygon of interest OPTIONAL ARGUMENTS y if x is a vector of X coordinates then y must contain the corresponding vertical or Y coordinates plot it logical flag should the resulting triangulation be plotted Default is TRUE shrink fraction by which the triangles will be shrunken for better discrimination of the individual triangles in the plot no edges overlap if shrink gt 0 VALUE invisibly returns a list with 2 components ipt a matrix with 3 rows for each column the 3 row values can be used to index x and y and extract corre sponding triangle vertices This provides an ordering of the triangles as well ipl another integer matrix with 3 rows These are the point numbers of the end points of the border line segments and their corresponding triangle number SIDE EFFECTS if plot it TRUE a colorful representation of the triangulation is produced DETAILS A Delaunay triangulation of a point set is a triangulation whose vertices are the point set with the property that no point in the point set falls in the interior of the circ
11. 64 Appendix Data and Function Reference additional named arguments can be passed to covfun VALUE an object of class krige with components x the first spatial location vector i e the first argument in loc function call in formula y the second spatial location vector i e the second argument in loc function call in formula coefficients the vector of coefficients for the trend surface These are for the polynomial based on the scaled spatial location vectors see the DETAILS section residuals the vector of residuals from the trend surface call an image of the call that produced the object Other components are included that are used by predict krige for computing interpolations DETAILS The kriging system is solved using generalized least squares see Ripley 1981 The polynomial terms are scaled to 1 1 internally to avoid numeric problems the coefficients component returned is for these scaled terms This implementation of kriging does not handle multiple observations at a point Methods for objects of class krige include predict and print REFERENCES Cressie Noel A C 1993 Statistics for Spatial Data Revised Edition Wiley New York Ripley Brian D 1981 Spatial Statistics Wiley New York SEE ALSO exp cov loc predict krige EXAMPLES krige the Coal Ash data with a quadratic trend in the x direction using a spherical covariance function kcoal lt krige coal loc x y x x 2 data coal as
12. Block kriging is the general term used for the prediction of the average value of a random field over a segment surface or volume The term Point kriging refers to prediction of the field at a point In S SPATIALSTATS block kriging is computed by the predict method for objects of call krige predict krige and is implemented on the Predict tab of both Kriging dialogs Block kriging is restricted to prediction of the average value over a rectangular area The integral over the block rectangular is approximated by the average of the point predictions within the block You may control the number of points to be considered in the average as well as the block size 27 Chapter 3 Geostatistical Data To perform block kriging on the coal data 1 Choose Spatial gt Universal Kriging from the main menu 2 Use the rollback button at the bottom of the dialog to recover the settings used in performing Universal K riging on the coal data 3 Move to the Predict tab The dialog below appears Universal Kriging Xx Model Predict Plot Save Prediction Locations Save In coalBKpred Locations Type aid M Predictions a IV Standard Errors Grid Options Prediction Type Min Location 1 Prediction Type Block Max leze snih rts Block Length gt 1 0 0 Number of Bins 1 poo BlockWidhY P MinLocation2 C Num Pts 5 Max Location 2 poo Num Y Pts 5 Number of Bins 2 Cancel Apply KE current
13. Haining R 1990 Spatial Data Analysis in the Social and Environmental Sciences Cambridge Uni versity Press Cambridge SEE ALSO Glasgow neighbor 61 Appendix Data and Function Reference Kenv Compute Simulations of Khat Kenv DESCRIPTION Computes Khat Lhat for simulations of point processes Returns upper and lower bounds as well as the average of all simulated values USAGE Kenv object nsims 100 maxdist lt lt see below gt gt ndist 100 process binomial boundary bbox object add T Lenv object nsims 100 maxdist lt lt see below gt gt ndist 100 process binomial boundary bbox object add T REQUIRED ARGUMENTS object an object of class spp representing a spatial point pattern or a data frame or matrix with first two columns containing locations of a point pattern OPTIONAL ARGUMENTS nsims integer Number of desired simulations maxdist numeric value indicating the maximum distance at which Khat or Lhat should be estimated De faults to half the length of a diagonal of the sample s bounding box ndist desired number of default distances at which to compute Khat or Lhat Default is 100 process a character string with one of five possible processes for the spatial arrangement of the resulting pat tern This must be one of binomial poisson cluster Strauss or SSI See the help file for make pattern for information on parameters for each process add logical flag sho
14. Help 4 Enter coal BKpred as the name of the object to save the predictions in 5 Check both Predictions and Standard Errors to be saved 6 Choose Block as the Prediction Type 7 Specify a 1x 1 block by entering 1 as the Block Length X and the same as the Block Width Y or leaving the default values in 28 Ordinary and Universal Kriging 8 Specify 5 as the number of points in the X direction to be averaged for each block 9 Specify also 5 pointsin the Y direction 10 Click OK The predictions are calculated with the supplied prediction locations in the center of the block The block prediction will be the average of point predictions at 25 locations within each block The predicted values are very similar to those obtain with the default Point kriging when performing Universal Kriging That is to be expected The standard errors are much smaller for the block kriging since the predictions are averages se fit 0 200 0 300 0 400 0 500 1 Kriging Std Errors M fill GU ype Su My eee UL 29 Chapter 3 Geostatistical Data 30 LATTICE DATA Lattice data are observations from a random process observed over a countable collection of spatial regions and supplemented by a neighborhood structure The observation locations can be regular equally spaced grid or irregular and data at a particular location typically represent the entire region The data observed at each
15. constant term is always fit All terms on the right hand side must be entered with a sign The loc call can include arguments angle and ratio to correct for geometric anisotropy see the loc help file Note that an evaluated 1oc object cannot be used in formula covfun a function that returns the distanced based covariance between two points The first argument to the function must be the distance Additional parameters will be passed through the OPTIONAL ARGUMENTS data an optional data frame in which to find the objects mentioned in formula subset expression saying which subset of the rows of the data should be used in the fit This can be a logical vector which is replicated to have length equal to the number of observations or a numeric vector in dicating which observation numbers are to be included or a character vector of the row names to be in cluded na action a function to filter missing data This is applied to the data in formula after any subset argument has been used The default with na fail is to create an error if any missing values are found A possible alternative is na omit which deletes observations that contain one or more missing values nc the number of points to use internally by the algorithm in approximating the distance based covariance function Note this argument has nothing to do with the number of observed points used in computing the kriging All observed points are used in computing kriging predictions
16. of fixed sampling sites along the coast Lattice data are observations associated with spatial regions where the regions can be regularly or irregularly spaced The spatial regions can be any spatial collection and are not limited to a grid Generally neighborhood information for the spatial regions is available An example of regular lattice data is information obtained by remote Spatial Point Patterns Spatial Data sensing from satellites The earth s surface is divided into a series of small rectangles pixels and the data are received as a regular lattice in R2 An example of irregular lattice data is cancer rates corresponding to each county in a state M athematically a lattice is defined by a set of vertices and edges T he sites form the vertices which are then connected to neighboring sites by edges Since lattice data are defined for spatial regions a method of referencing sites must be determined sites are often referenced by the centroids of the regions A lattice is composed of an index set of sites with an associated set of neighbors Point pattern data arise when locations themselves are the variable of interest Spatial point patterns consist of a finite number of locations observed in a spatial region Identification of spatial randomness clustering or regularity is often the first analysis performed when looking at point patterns Examples of point pattern data include locations of a species of tree in a fores
17. per unit area Intensity plots display a smooth estimate of intensity for a spatial point pattern The intensity estimate may be saved and displayed in a Data window for further exploration using the point and click graphics To calculate intensity Choose Spatial gt Intensity from the main menu The dialog below appears Intensity i x Data Plots Data Set lansing M Contour Plot Location 1 p M Filled Contour Plot Location 2 y l Surface Plot Subset Rows with species maple Plot Options Omit Rows with Missing Values M Include Points Results Intensity Options Method Save In maple int binning Number ok Binet IV Intensity Information Number of Bins 2 Smoothing Param 0 25 OK Cancel F i K current Example Calculate and plot the intensity for the lansing data in S SPATIALSTATS 1 Launch Intensity dialog 2 Enterlansing as the Data Set of interest 3 Selectx andy asthe Location 1 and 2 variables respectively 52 Intensity 4 Type species maple in the Subset Rows with field This will subset those rows of the data frame that correspond to the maples only 5 Select binning asthe Method 6 Specify 0 25 asthe Smoothing Parameter 7 Check the Contour Plot and Filled Contour Plot boxes 8 Check Include Points 9 Typemaple int into the Save In field 10 Click OK A graph sheet appears with the intensity plots and a Data window with the intensity estimates
18. regions which border each other or as regions within a certain distance of each other The neighbor relationship is not necessarily symmetric For example the underlying process may flow in only one direction or a region that is very large might exhibit influence on but not be influenced by a smaller region Since the neighborhood structure is the basic structure for the covariance model for lattice data the careful definition of spatial neighbors is a crucial analysis step The Spatial Neighbors dialog provides a variety of ways to create a spatial neighbor object 35 Chapter 4 LatticeData To create a spatial neighbor object Choose Spatial gt Spatial Neighbors from the main menu The dialog below appears Spatial Neighbors ME x Source Nearest Neighbors Nearest Nhbors Metric Euclidean hd Row and Col ID Num of Neighbors fi Weight Matrix Max Dist jaq C Create Grid inaire Create Grid Data Data Set sids Variable 1 feasting Read File Variable 2 nothing Results Save In sids nhbr30 Cancel Apply LE current Help Example Model the spatial neighborhood structure for the sids dataset in S SPATIALSTATS 1 Launch Spatial Neighbors dialog 2 Select Nearest Nhbrs as the source of neighborhood information This implies that the neighbors will be defined by distances between point locations and so the location variables will need to be provided Enter si ds as t
19. scribes the relationship among spatial regions using a sparse representation of the Weight or Contiguity matrix or matrices It has columns row id col id weights and matrix determined by ma trix id Objects of class spatial neighbor are required by the spatial regression spatial correlation and other functions in S SPATIALSTATS Two methods for constructing a spatial neighbor object are available A matrix of weights where all weights are often 1 can be given as input and the spa tial neighbor object is constructed from its non negative elements In this case argument neigh bor matrix must be a square matrix such that if element c i j of the matrix is non zero then spa tial regions i and j are neighbors with weight given by the value of the element usually a 1 Another method for constructing an object of class spatial neighbor is by directly specifying the row and column numbers and the weight value of the non zero elements of the contiguity matrix which is usually a sparse matrix A sparse representation is usually preferred in practice In this case row id i gives the row of the i th non negative element of the neighbor matrix and the correspond ing element col id i gives its column index Thus each pair c row id i col id i repre sents a pair of neighboring spatial units The strength of their association can then be given by weights i Notice that row id and col id contain INDICES of the contiguity matrix a
20. site may be continuous or discrete Before modeling the spatial component of lattice data in S SPATIALSTATS we assume stationarity and multivariate normality of the small scale variation in the data that is of the error term This means that trend must be removed and transformations may be required to stabilize the variance and or to approximate normality The primary tools available for examining lattice data are Spatial Correlations and Spatial Regression These dialogs require a data set containing the observations at each location and a spatial neighbor object describing the spatial relationship between the observations The Spatial Neighbors dialog creates a spatial neighbor object This section describes the following dialogs Spatial Neighbors Spatial Correlations Spatial Regression 31 Chapter 4 LatticeData EXPLORATORY ANALYSIS The sample data frame sids contains spatial data collected on a lattice The collection points are counties in the state of North Carolina and the data are the rates of death from Sudden Infant Death Syndrome SID S for the years 1974 1978 C ressie 1993 The components of the SID S data frame are gt names sids 1 id easting northing 5 births nwbirths group 8 sid ft nwbirths ft gt D ata for the years 1979 1984 are also available in si ds 2 Seethesi ds help file for explanations of the individual variables To form a spatial lattice you mu
21. specifying that the Predictions and Standard Errors should be saved Enter coal kgpred in the Save In field 9 Leave the Locations Type set to its default value of Grid The predicted values will be on a grid 10 Move to the Plot tab Specify Surface Plots for both the predictions and their standard errors 11 PressOK You see a summary of the fitted object coal krige in a Report window and the plot on different pages of a Graph sheet and a Data window with the predictions A plot of both the predictions and their standard errors follows Kriging Predictions Kriging Std Errors I 4 Vo yy d Compare this with Figure 4 25 of the S SPATIALSTATS User s M anual O ther plots can be generated by having the prediction object coal kgpred open Select 3 columns x y and f i t and try different displays but pushing buttons on the Plots2D and Plots3D palettes You may want to rotate different graphs to look at the predictions from several different angles Block Kriging Ordinary and Universal Kriging Plotting the residuals from this fit produces a tighter fit about the 0 reference lines though there are still a few high values Residuals from Universal Kriging Fit a A A a aa N Ea aa Ea a a 2 a 4 N amp a a8 Ren gp BQ RA amp sedes saes Bape fe RE E E E EEE a ae Se re r An gt a ARR na wv A 4S a A A sd iy ag SA A A a a r 4 a A a 4 4 a 4 By a a uniResid
22. the neighbor weight matrix This is ignored if neighbor matrix is given or if row id is a matrix It is important to note that even if a pair of regions c row id i col id i are spatial neighbors the permuted pair c col id i row id i does not have to define spatial neighbors correspond ing contiguity matrix element can be zero For example consider two regions on a river and suppose that a region corresponding to row id i is downstream from the region in col id i and neigh bors By this definition downstream of the transpose pairing need not satisfy a neighbor relation ship See argument symmet ric below neighbor matrix a matrix of neighbor weights where all weights are often 1 from which the object of class spa tial neighbor is to be constructed This must be a square matrix such that if element i j is non zero then spatial regions i and j are considered neighbors and its value is used as a weight in measures of correlation or in further model fitting This is also known as the contiguity matrix OPTIONAL ARGUMENTS weights numeric vector of the same length as row id and col id weights i gives a weight for the corre sponding neighbor pair relationship given in c row id i col id i If weights is not speci fied and argument neighbor matrix is not used then the spatial weights are all set equal to 1 Each spatial weight defines the strength of the association between two neighbors This argument is ignored if n
23. the rectangular region For example for the unit square the boundary could be given as bbox x c 0 1 y c 0 1 the bounding box of two diagonally opposed points Defaults to a rectangle covering the range of points plot it logical flag should the resulting K estimates be plotted Default is TRUE VALUE a list containing components values atwo column matrix The first column called dist contains the distances at which Lhat was com puted and the second column called Lhat contains the values of L dist ndist number of distances returned This could be smaller than its input value if the extent of the distances is too large mindist minimum distance between any pair of points maxdev maximum deviation from L t t See DETAILS SIDE EFFECTS if plot it TRUE a plot of the value of L t against distance will be produced on the current graph ics device DETAILS Khat computes Ripley s 1976 estimate of K t for a spatial point pattern K t A E number of events lt distance t of an arbitrary event where 4 is the intensity of the spatial point pattern The theoretical K function for a Poisson completely spatially random process is K t t so L t VK t z is equal to t the distances The default plots L t versus t which should approximate a straight line for a homogeneous process with no spatial dependence See function Khat for estima tion of K t REFERENCES Ripley Brian D 1976 The second order analysis of stat
24. used in calculating a vari ogram value If np is less than minpairs that value is dropped from the variogram method a character string to select the method for estimating the variogram The possible values are clas sical for Matheron s 1963 estimate and robust for Cressie and Hawkins 1980 robust estima tor Only the first character of the string needs to be given smooth a logical flag if TRUE a loess smooth line is drawn for each variogram panel If panel is supplied then this value is ignored panel a panel function to be used in plotting the variograms If plot it FALSE this value is ignored plot it a logical flag if TRUE a plot of all the variogram is drawn additional arguments to be passed down to the panel function for plotting 57 Appendix Data and Function Reference VALUE a data frame with columns distance the average distance for pairs in the lag gamma the variogram estimate np the number of pairs in each lag angle a factor denoting the angle for the geometric anisotropy ratio a factor with levels denoting the ratio for the geometric anisotropy SIDE EFFECTS If plot it TRUE the default the variogram for each combination of angle and ratio is plotted The plot is drawn using xyplot DETAILS For each combination of angle and ratio the locations are corrected for geometric anisotropy The correction consists of multipling each location pair x i y i by the symmetric 2 x 2 matrix A where A 1 1
25. x coordinates of the data whose neighbor relations are defined in x ycoord a numeric vector containing the y coordinates of the data whose neighbor relations are defined in x Must be the same length as xcoord OPTIONAL ARGUMENTS line col a numeric value indicating the color to draw the lines connecting the points that are neighbors See the col parameter in the par help file line type a numeric value indicating the line type to use for the lines connecting the points that are neighbors See the 1t y parameter in the par help file line width a numeric value indicating the line width to use for the lines connecting the points that are neighbors See the 1wd parameter in the par help file matrix id a positive integer indicating which spatial neighbor matrix is to be plotted Only one spatial neighbor matrix can be plotted per call to the function but objects of class spatial neightbor can contain more than one matrix add a logical value if TRUE no initial plot is drawn only the lines joining the neighbors are added to the current plot arrows a logical value if TRUE arrows are drawn from each point to its neighbor if FALSE segments are drawn from each point to its neighbor Plotting with arrows can be useful when there are one way neighbor relations in x i e point B is a neighbor of point A but point A is not a neighbor of point B If x is a symmetric spatial neighbor object attr x symmetric is TRUE then all neighbor relations are bi di
26. LSTATS data sets From the analysis in the User s M anual we know that these data exhibit a strong East West trend The variogram values in the East West direction are likely influenced by the trend the stationarity 13 Chapter 3 Geostatistical Data 14 assumption is violated in the presence of trend We will restrict the variogram cloud computations to pointsin a North South direction by manipulating the azimuth and its tolerance as follows 1 Launch the Variogram Cloud dialog 2 Entercoal ash asthe Data Setto be analyzed 3 Selectcoal as the Variable to be analyzed and select x and y aS Location 1 and Location2 respectively 4 Change the Azimuth Tolerance from the omnidirectional value of 90 degrees to a narrow 01 5 Save the resulting object ascoal vgcl oud 6 PressOK A two page Graph sheet appears containing a box plot and a scatter plot of the variogram cloud The variogram cloud shows a scatter of high value points for the full range of distance values There is a method in S SPATIALSTATS that can be used to identify points in a variogram cloud to invoke this method enter the following command in the Commands Window while the Variogram Cloud is the current active plot this is required gt identify coal vgcl oud Identifying the high values in the variogram cloud shows that they all are paired with observation 50 which was determined to be an outlier indeed We will remove observation 50 from furthe
27. MathSoft S SPATIALSTATS Version 1 5 Supplement March 2000 Data Andyas Products Divison M athSoft Inc Seattle Washington Proprietary Notice Copyright Notice M athSoft Inc owns both this software program and its documentation Both the program and documentation are copyrighted with all rights reserved by M athSoft The correct bibliographical reference for this document is as follows S SPATIALSTATS Version 1 5 Supplement D ata Analysis Products Division M athSoft Seattle WA Printed in the United States Copyright O 2000 M athSoft Inc All rights reserved CONTENTS Chapter 1 Introduction Spatial D ata Geostatistical Data Lattice Data Spatial Point Patterns Chapter 2 Getting Started Loading the M odule The Spatial M enu Getting Help Chapter 3 Geostatistical Data Exploratory Data Analysis Example The Coal Ash Data V ariogram Cloud Geometric Anisotropy Empirical V ariogram M odel V ariogram Ordinary and Universal K riging Block Kriging Chapter 4 Lattice Data Exploratory Analysis Spatial Neighbors Spatial Correlations Spatial Regression o N DD UI WNN N 10 13 15 17 19 21 27 31 32 35 38 40 Contents Chapter 5 Spatial Point Patterns Exploratory Analysis Spatial Randomness Intensity Appendix A Data and Function Reference 43 44 48 52 55 WELCOME TO S SPATIALSTATS 1 5 Congratulations on acquiring Version 1 5 of the S SPATIALSTATS
28. NCES Cressie Noel 1993 Statistics For Spatial Data Revised Edition Wiley New York SEE ALSO variogram plot vgram fit model variogram nlminb 83 Appendix Data and Function Reference EXAMPLES vg iron lt variogram residuals loc easting northing data iron ore vfit iron lt variogram fit vg iron param c range 8 7 sill 3 5 nugget 4 8 fun spher vgram plot vg iron plot vfit iron add T 84
29. T PATTERNS A mapped spatial point pattern is a collection of points located within a bounded region of space The points can denote locations of naturally occurring phenomena such as earthquakes or plants or social events such as the locations of small towns or the occurrences of a particular disease The data locations or points might be randomly located tending to cluster in groups or follow a regular and predictable pattern A typical data analysis of a point pattern focuses on the question of whether the point locations are completely spatially random CSR or whether we should make an attempt to model an apparent lack of spatial randomness Formal checks for CSR and modeling techniques for spatial point patterns are described in chapter 6 of the S SPATIALSTATS User s M anual In this section we describe ways to use the GU I of S PLUS and S SPATIALSTATS to visualize spatial point patterns and to assess the hypothesis of CSR A data set containing the mapped locations of maple and hickory trees in a 19 6 acre square plot in Lansing Woods Clinton County Michigan will be used for the examples in this section Diggle 1983 The data have been scaled so that they reside on the unit square although this is not necessary for their analysis See the User s M anual for a more complete description of the data set This section describes the following dialogs Spatial Randomness e Intensity 1 Diggle Peter J 1983 Statis
30. The trend is apparent from the bottom plot We will try to model this trend as a second order polynomial in x as part of a Universal K riging fit in the following section 24 To pe rform universal kriging Ordinary and Universal Kriging Choose Spatial gt Universal Kriging from the main menu The dialog below appears Universal Kriging Model Predict Data Data Set coal ash Variable eoa gt Subset Rows with 50 IV Omit Rows with Missing Values Spatial Location Location 1 fx Location 2 y T Correct for Geometric Anisotropy Plot Trend Surface Trend Terms Variogram Inputs F Use Values from a Variogram Fit Variogram Function Range Silk Nugget Results Save As IV Print Results Cancel Apply KE 2of 2 Asan example krige the coal data 1 2 3 Select coal as Location 1 and Lo observation 50 cation 2 Select x and x 2 as trend terms 6 570 oto o osr ooo coal krige Help Launch the Universal Kriging dialog Enter coal ash as the Data Set of interest in the Variable field and x and y respectively Enter 50 in the Subset Rows with field to remove Select a Spherical Variogram Function with Range of 6 570 Sill of 0 109 and Nugget of 0 917 the values fitted using the Empirical Variogram dialog 25 Chapter 3 Geostatistical Data 26 7 Save to an object named coal krige 8 Move to the Predict tab and check the boxes
31. a measure of the fit of yf to y with weights n It is used as a measure of fit of fun to the data in object The default is the sum of squared residuals sum y yf 2 plot it a logical value if TRUE a plot of the variogram and its fitted model is displayed VALUE invisibly returns a named list of the final parameters used This list has the last value of the objective function as an attribute DETAILS This function can be used to fit a variogram or covariogram model by eye The value of objec tive fun is displayed on the plot A weighted least squares objective function for variograms Cressie 1993 p 97 is objective fun lt function y yh n sum n y yh 1 2 REFERENCES Cressie Noel 1993 Statistics For Spatial Data Revised Edition Wiley New York SEE ALSO correlogram plot variogram variogram EXAMPLES vg iron lt variogram residuals loc easting northing data iron ore model variogram vg iron spher vgram range 8 7 sill 3 5 nugget 4 8 69 Appendix Data and Function Reference plot spatial neighbor Plotaspatial neighbor Object plot spatial neighbor DESCRIPTION Plot an object of class spatial neighbor with lines connecting points that are neighbors USAGE plot spatial neighbor x xcoord ycoord line col 1 line type 1 line width 1 matrix id 1 add F arrows F size arrow 0 1 scaled T REQUIRED ARGUMENTS x an object of class spatial neighbor xcoord a numeric vector containing the
32. absolute value for the l2 1 and norm respectively For two vectors x and y these are defined as l 2 lx yil I L yi Iz max Ix y l max dist if max dist is given argument k is ignored and all of the neighbors within distance max dist of each row in x are found drop self a logical value if TRUE then rows with distances equal to 0 and index1 index2 self neighbors are dropped from the returned object This definition retains coincident points as neighbors although their distance apart is zero If quadtree is not supplied k 1 and drop self T a warning is printed since this results in nothing being returned and the value of k is set to 2 VALUE a matrix with three named columns 59 Appendix Data and Function Reference index1 if x is a matrix the row in x for this nearest neighbor If x is not a matrix the value 1 index2 the row in the matrix from which the quad tree was formed for this nearest neighbor If the quad tree was formed from a matrix y then x index1 i and y index2 i are neighbors distances the corresponding nearest neighbor distances DETAILS An efficient recursive algorithm is used to find all nearest neighbors First the quad tree is traversed to find the leaf with medians nearest the point for which neighbors are desired Then all observations in the leaf are searched to find nearest neighbors Finally if necessary adjoining leaves are searched for nearest neighbors
33. access to the S SPATIALSTATS 1 5 dialogs To launch a dialog simply select the desired menu item File Edit View Data Statistics Gatlin Sana 10 ae S ag Yeiogem Cloud Geometric Anisotropy Empirical Variogram Model Variogram Ordinary Kriging Universal Kriging Spatial Neighbors Spatial Correlations Spatial Regression Spatial Randomness Intensity The menu is separated in three sections according to the type of spatial data that the corresponding methodology applies to The first six items correspond to the analysis of geostatistical data the next three to data observed on a lattice and the final two apply primarily to spatial point pattern data Chapter 2 Getting Started GETTING HELP Help is available in a variety of ways Use the Help gt S SPATIALSTATS Help menu item to open the S SPATIALSTATS help file The Help button on a dialog will display help for that dialog The command line function help will provide help on a specified function This supplement is also available online in a PDF file viewable using Adobe Acrobat Use the Help gt Online Manuals menu item to access it GEOSTATISTICAL DATA Geostatistical data also termed random field data consist of measurements taken at fixed locations Variogram estimation and kriging are commonly used with geostatistical data These methods were originally introduced as geostatistical methods for use in
34. ap T SEE ALSO spatial neighbor spatial subset spatial weights EXAMPLES sids nhbr2 lt check islands sids neighbor remap T find neighbor Find the Nearest Neighbors of a Point find neighbor DESCRIPTION Find the k nearest neighbors of a vector x in a matrix of data contained in an object of class quad tree USAGE find neighbor x quadtree quad tree x k 1 metric euclidean max dist NULL drop self F REQUIRED ARGUMENTS x a vector or matrix containing the multidimensional point s at which the nearest neighbors are de sired The vector must have the same number of elements as the number of columns in the numeric matrix used to construct quadtree If a matrix is used the matrix must have the same number of columns as the numeric matrix used to construct quadt ree and nearest neighbors are found for each row in the matrix OPTIONAL ARGUMENTS quadtree an object of class quad t ree containing the sorted matrix of data for which a nearest neighbor search is desired Defaults to quad t ree x if x is a matrix but it is required when x is a vector k the number of nearest neighbors to be found If the data x is the same data that was used to construct the quad tree object then k 1 results in each element having itself as its own nearest neighbor metric acharacter string giving the metric to be used when finding nearest neighbors Partial matching is al lowed Possible values are euclidean city block and maximum
35. are presented and analyzed in Haining 1990 60 Appendix Data and Function Reference REFERENCES Haining R 1990 Spatial Data Analysis in the Social and Environmental Sciences Cambridge Uni versity Press Cambridge SEE ALSO Glasgow SMR Glasgow SMR Standardized Mortality Rates for Glasgow Glasgow SMR SUMMARY The Glasgow SMR data frame contains standardized mortality rates for 87 community medicine areas in Glasgow Scotland for 1980 1982 DATA DESCRIPTION This data frame contains the following columns AllDeaths the standardized mortality rate SMR for all deaths Accidents the SMR for death by accidents Cancer the SMR for deaths due to cancer Respiratory the SMR for deaths due to respiratory disease accidents Heart the SMR for deaths due to ischaemic heart disease Cerebrovascular the SMR for deaths due to cerebrovascular disease Population the population in 1000 s Easting the x coordinate of the community medicine area CMA relative to an arbitrary origin where the x ax is is parallel to the latitude Northing the y coordinate of the CMA relative to an arbitrary origin where the y axis is parallel to the longitude DETAILS The standardized mortality rate for a community medicine area is the observed deaths due to that cause divided by the expected number of deaths given the age and sex combination in that area multiplied by 100 SOURCE The data are presented and analyzed in Haining 1990 REFERENCES
36. as x If TRUE then the corresponding point is inside the given polygon and so on BUG if a ray from a point to an edge intersects a horizontal edge i e is collinear with it the C program will return TRUE even if such point is not in the polygon SEE ALSO poly grid poly area EXAMPLES 100 points on a unit square x lt runif 100 y lt runif 100 A square polygon in the center prenter lt list x c 25 25 75 75 y c 25 75 75 25 Appendix Data and Function Reference pin lt points in poly x y pcenter Plot the unit square and the center square plot x y type n polygon pcenter density 0 col 2 Plot only the points in the center square points x pin y pin col 3 poly grid Generate a Grid Inside a Given Polygonal Boundary poly grid DESCRIPTION USAGE Generates a grid of points and then clips them to lie within a given boundary poly grid boundary nx ny size REQUIRED ARGUMENTS boundary a list with components named x and y or a matrix with 2 columns representing the vertices of a convex polygon Endpoint need not be repeated nx integer representing the number of cells in the horizontal direction ny integer representing the number of cells in the vertical direction OPTIONAL ARGUMENTS size numeric vector containing the size of each cell If it has length one then the cells will be squared with the same side sizes If it has length two then the cells will ha
37. atially autocorrelated there may be a need for spatial modeling A test for spatial autocorrelation can be performed as an exploratory technique to decide whether spatial modeling should be used The null hypothesis is of no correlation and the alternative hypothesis is specifically defined by a weighted neighbor matrix The result is therefore sensitive to the choice of neighbors and weights so it may be desirable to run the autocorrelation under several different scenarios The calculation of spatial autocorrelation assumes constant mean and variance If the process contains trend or non constant variance the results should be used with caution The Spatial Correlations dialog computes spatial autocorrelation and related estimates of variation To compute spatial correlations Choose Spatial gt Spatial Correlations from the main menu The dialog below appears Spatial Correlations x Data Spatial Structure Data Set Neighbor Object sids nhbr30 Variables Options Statistic moran Sampling Type nonfree X Num Permute fi 00 Results Save s sids corl IV Print Results Cancel Apply FE current Help Example Calculate Spatial Correlations for the sids data If you haven t already used the data to create the spatial neighbor object sids nhbr30 follow the steps in the previous section before proceeding Spatial Corrdations The occurrence of SIDS is not likely to have constant variance si
38. bles for example sqrt count log age 1 or I1 2 x The I is required since the operator has a special meaning on the right side of a formula OPTIONAL ARGUMENTS data an optional data frame in which to find the objects mentioned in formula subset expression saying which subset of the rows of the data should be used in the fit This can be a logical vector which is replicated to have length equal to the number of observations or a numeric vector in dicating which observation numbers are to be included or a character vector of the row names to be in cluded na action a function to filter missing data This is applied to the model frame after any subset argument has been used The default with na fail is to create an error if any missing values are found A possi ble alternative is na omit which deletes observations that contain one or more missing values lag a numeric value the width of the lags If missing lag is set tomaxdist nlag nlag an integer the maximum number of lags to calculate tol lag anumeric value the distance tolerance maxdist the maximum distance to include in the returned output The default is half the maximum distance in the transformed data angle a vector of direction angles in degrees clockwise from North to consider as directions of anisotropy ratio a vector of ratios of anisotropy These should all be greater than 1 minpairs the minimum number of pairs of points minimum value for np that must be
39. bor O bject To enter spatial weights consult the help file for the S SPATIALSTATS function slm and enter the argument wei ghts 1 sids births in the Parameters text box 8 Enter sids sl ml in the Save As field 9 PressOK A summary of the spatial regression is displayed in a Report window It includes the actual call to the S SPATIALSTATS function sI m the coefficients of the regression their variance covariance matrix and other parameters of the spatial relationships and covariance matrix structure In particular the coefficients for this model indicate a highly significant effect of non white births on rates of SIDS in North Carolina Coefficients Value Std Error t value Pr gt t Intercept 1 6456 0 2385 6 8990 0 00 nwbirths ft 0 0345 0 0066 5 2068 0 00 41 Chapter 4 LatticeD ata 42 Diagnostic plots on the residuals should follow this analysis to assess the adequacy of the model fitted You can save the residuals by indicating so in the Results tab of the Spatial Regression dialog Alternatively you may extract them from the fitted model using the residuals method as follows gt sids slml resid lt residuals sids sim1 gt summary sids slml resid Min lst Qu Median Mean 3rd Qu Max 106 18 79 7 01 3 61 26 27 77 8 gt qqnorm sids siml resid For example the sequence of commands above would yield a quantile quantile normal plot of the residuals to assess the normality assumption SPATIAL POIN
40. ch to compute Khat Default is 100 The distances for which Khat will be estimated are calculated as seq 0 maxdist ndist both maxdist and ndist will change if not reasonable for the given object boundary points defining the boundary polygon for the spatial point pattern This version accepts only rectan gles for which boundary should be given as a list with named components x and y denoting the corners of the rectangular region For example for the unit square the boundary could be given as bbox x c 0 1 y c 0 1 the bounding box of two diagonally opposed points Defaults to a rectangle covering the range of points plot it logical flag should the resulting K estimates be plotted Default is TRUE VALUE a list containing components values a two column matrix The first column named dist contains the distances at which Khat was com puted and the second column named Khat contains the values of K dist ndist number of distances returned This could be smaller than its input value if the extent of the distances is too large mindist minimum distance between any pair of points maxdev maximum deviation from K t t See DETAILS SIDE EFFECTS if plot it TRUE a plot of the value of K t against distance will be produced on the current graph ics device DETAILS Khat computes Ripley s 1976 estimate of K t for a spatial point pattern K t 4 E number of events lt distance t of an arbitrary event where is the
41. coal as the z Column from the drop down list available and 4 Click the Vary Symbols tab Select z Column on the Vary Size By field and change the Minimum and Maximum Heights to 0 05 and 0 20 respectively so as not to overwhelm the plot with symbols that are too large You may also want to change the Symbol Style on the Symbol tab to a solid circle for better visualization of the significance of their size To explore how the different values of coal percentage vary over the sampling region you may use the Label Point tool from the Graph Tools palette and move through the points clicking on them Point 50 seems to be an outlier as exposed in the User s M anual To superimpose contours of coal ash percentage in the samples 1 Select the 3 columns on the open data window x y and coal in that order 11 Chapter 3 Geostatistical Data 12 2 Then select the graph region in the plot above and Shift click the Contours button on the Plots2D palette Contour lines varying with percentage of coal will be added to the plot of the sampling locations These contours are calculated internally in S PLUS using Akima s fitting method Akima 1978 See the help file for the S PLUS interp for more detail A few cosmetic changes can be made to the resulting plot For example use the Gridding Hist tab of the Contour Plot properties dialog to clear the Extrapolation option and the Labels tab to add labels to the contours set Label Frequ
42. corresponds to complete inhibition at distances up to radius The user should exercise caution when determining the value of radius for if it is too big in relation to the area defined by boundary the algorithm will run out of possible area to place the subsequent disc and the generation of the desired process may be impossible or very slow The option cluster generates a Poisson cluster process This is defined by generating a parent Poisson process with intensity cpar and a daughter process of clusters with radii determined by the value of radius WARNING If radius is too large it may be impossible or nearly impossible to generate the number of requested points The call may hang in some extreme cases REFERENCES Diggle Peter J 1983 Statistical Analysis of Spatial Point Patterns Academic Press London Ripley Brian D 1981 Spatial Statistics John Wiley amp Sons New York Ripley Brian D 1976 The second order analysis of stationary point processes Journal of Applied Probability 13 255 266 SEE ALSO runif rnorm rpois rbinom EXAMPLES A completely random process in the unit square rand lt make pattern 100 plot make pattern 100 process Strauss rad 0 1 c 0 5 plot make pattern 500 proc cluster rad 20 c 10 boundary list x c 0 200 y c 0 200 A nonhomogeneous Poisson pattern with a linear trend in x over a 10 x 10 square lxy lt function x y 1 5 x xy lt make pattern
43. e default arguments S is taken as I minus the sum over i of rho i A i Here I is an identity matrix rho i is a scalar and A i is the i th weight matrix in neighbor If transpose is TRUE then the transpose of this matrix is used for A rho a scalar or vector of constants used in defining the matrix s see argument transpose product let B I minus the sum of rho i A i as described in argument transpose When product is FALSE S B When product is TRUE S is t B B weights if provided the inverse weights are included along the diagonal matrix w and incorporated into the model for s as follows Let R be I minus the sum of rho i A i Then product transpose S F F IRW F T t R 8 8 W T F t R W 3 R 78 region id Appendix Data and Function Reference T T R amp W t R a vector with length equal to the number of regions in the spatial lattice If variables row id and col id of argument neighbor are not integer valued variables with sequential values from 1 to the number or regions in the lattice then argument region id must be specified and is used to obtain a sequential coding of the lattice regions absThreshold the pivot threshold between zero and 1 Values near result in complete pivoting while values near zero result in a strict Markowitz solution In general you should choose a value as close to zero as roundoff error will permit A value of 0 001 has been recommended by Kundert
44. e included or a character vector of the row names to be in cluded na action a function to filter missing data This is applied to the model frame after any subset argument has been used The default with na fail is to create an error if any missing values are found A possi ble alternative is na omit which deletes observations that contain one or more missing values azimuth the clockwise direction angle in degrees from North South Only pairs of points in this direction plus or minus tol azimuth will be included in the output tol azimuth the tolerance angle in degrees tol azimuth greater than or equal to 90 implies the of use all direc tions maxdist the maximum distance to consider The default is half the maximum observed distance bandwidth the maximum perpendicular distance to consider FUN a function of two variables that is to be computed The default function is the contribution to the clas sical empirical variogram for the pair z i z 3 VALUE an object of class vgram cloud that inherits from data frame The columns are distance the distance between the two points gamma the value of FUN for the z i index z jindex iindex the index into the original data for the first value of the pair jindex the index into the original data for the second value of the pair The return object has an attribute call with an image of the call that produced the object DETAILS Methods for class vgram cloud include boxplot plot a
45. e latter are known as islands in S SPATIALSTATS We can use the row names of the data frame to determine which neighbors are special gt row names sids 21 1 Chowan gt row names sids c 28 48 1 Dare Hyde gt Chowan has the most neighbors while Dare and Hyde counties are isolated To plot a spatial neighbor object A neighbor object can be plotted from the Command line in version 1 5 of S SPATIALSTATS by issuing a command such as gt plot sids neighbor xc sids easting yc sids northing scaled T 33 Chapter 4 LatticeD ata The following figure is produced 300 I 200 ii o S 200 1 100 200 300 400 We see roughly the shape of the state of North Carolina as the county seats are joined by line segments to indicate their neighbor relationship 34 Spatial N eghbors SPATIAL NEIGHBORS Lattice modeling is the spatial analogue to time series modeling A time series is modeled by predicting the outcome for each time based on its dependence on the preceding observation or set of observations A spatial process is modeled by predicting the outcome for each region based partially on its dependence on nearby or neighboring regions Choosing a neighborhood structure is the first step in the analysis of lattice data The result determines the covariance structure used for the spatial component of a more general linear regression model Neighbors may be defined as
46. e the random placement of clusters and their number See the DETAILS section for more in formation VALUE an object of class spp whose n points are distributed according to process If process pois son results on a process with zero points the returned value will be a classless matrix with zero rows and a warning will be issued DETAILS The binomial process option generates a spatially random pattern of n points within the given boundary This is in essence a homogeneous Poisson process conditional on the given number of points n 67 Appendix Data and Function Reference 68 The poisson process option generates a Poisson process with intensity lambda This argument is required for this option If lambda is a function the Poisson process is generated by a rejection sam pling algorithm Diggle 1983 a homogeneous Poisson process with intensity maxlambda is generat ed over the region and then points are retained with probability Lambda x y maxlambda The SSI process generates a random pattern where no two points are within the inhibition distance determined by its parameter radius This process is equivalent to sequentially laying down discs of radius radius which will not overlap The Strauss process accepts each randomly generated point with probability cpar s where s is the number of existing points within radius radius of the potential new point The parameter cpar must be in 0 1 for this process where cpar 0
47. e the wrong method will be called SEE ALSO spatial neighbor spatial neighbor object plot scaled plot par segments EXAMPLES Plot the sids neighbor object using the easting and northing it values from sids as the coordinates plot sids neighbor xc sidsSeasting yc sids northing scaled T Create a second order spatial neighbor object on a 10 x 10 grid ng10 lt neighbor grid 10 10 neighbor type second order sn10 lt spatial neighbor ng10 Generate a 10 x 10 set of coordinates xy lt expand grid x 1 10 y 1 10 Plot the spatial neighbor object plot sn10 xc xy x yc xySy Create and plot spatial neighbor object for the bramble canes it nearest neighbors nb lt find neighbor bramble k 2 drop self T sn lt spatial neighbor nb plot sn xc bramble x yc brambleSy plot vgram fit Plot Results from variogram fit plot vgram fit DESCRIPTION Plot a vgram fit object ustually the result from a call to variogram fit USAGE plot vgram fit x line col 1 line type 1 line width 1 add T npoints 100 REQUIRED ARGUMENTS x an object of class Vgram fit OPTIONAL ARGUMENTS line col a numeric value indicating the color for the variogram fit line See the col parameter in the par help file line type a numeric value indicating the line type for the variogram fit line See the 1ty parameter in the par help file line width a numeric value indicating the line width for the variogram fit line See the 1wd paramete
48. eighbor matrix is given as each of the matrix elements are then considered to be neigh bor weights nregion integer stating the total number of regions or spatial units If not given this value is computed from the number of unique elements in row id and col id as the maximum of all the regions given therein max c row id col id symmetric logical flag should the neighbor matrix be considered symmetric If TRUE the spatial weights matrix is computed by assuming that if the i th neighbor pair c row id i col id i has neighbor weight given by w weights i then so does the matrix element c col id i row id i Only half of the weights need be specified in this case If TRUE routine spat ial condense is called to re 75 Appendix Data and Function Reference 76 matrix id VALUE DETAILS SEE ALSO EXAMPLES move redundant values When neighbor matrix is given its symmetry is determined within the function otherwise it defaults to FALSE integer vector of length equal to the total number of spatial neighbors This can be used to differentiate various types of neighbors For example spatial regression models may differentiate between north south neighbors as compared to east west neighbors The values of vector mat rix id should then in dicate the neighbor types If missing a single neighbor type is assumed with one neighbor matrix an object of class spatial neighbor This object inherits from class data frame and de
49. ency to 1 and perhaps change the font to make them more prominent In the figure below the number of contours was also increased and a title inserted Coal Measurements and Sample Locations 22 7 Tag y e eco ooh The outlier 176 is quite apparent and it is driving the shape of the resulting contours 1 Akima H 1978 A Method of Bivariate Interpolation and Smooth Sur face Fitting for Irregularly Distributed Data Points ACM Transactions on Mathematical Software 4 148 164 Variogram Cloud VARIOGRAM CLOUD The variogram cloud is a diagnostic tool that can be used to look for potential outliers or trends and to assess variability with increasing distance Anomalies and non homogeneous areas can be detected by looking at short distances that yield high dissimilarities To plot a variogram cloud Choose Spatial gt Variogram Cloud from the main menu The dialog below appears Variogram Cloud x Data Options Data Spatial Location Data Set coalash gt Location 1 koo Variable oa Location 2 poo x Subset Rows with oO a Correct for Geometric Anisotropy IV Omit Rows with Missing Values pe Direction Say Azimuth o 00 Plots Azimuth Tolerance 01 Scatter Plot M Box Plot Default is Omnidirectional Results Save As coal yacloud F Show Variogram Cloud Results OK Cancel Apply p gt 3of 8 Help Example Use the coal ash data in the S SPATIA
50. es in the Lansing Woods Values of G hat are computed for every neighbor distance in the point process by default The grid of origins in the Fhat calculation is determined by the square root of the total number of points in the given point process For more specifics on the calculations consult the individual S SPATIALSTATS help files for Fhat and G hat Visual judgement of G hat is based on the fact that if there is clustering in the data we would expect to see an excess of short distance neighbors while if there is regularity in the data then there would be an excess of long distance neighbors The interpretation of the Fhat plot is opposite that of the Ghat plot An excess of high distance values is interpreted as clustering As before we could compare this statistic to simulations from a CSR process for a visual interpretation When edge effects need to be considered we can assess the hypothesis of CSR using Monte Carlo techniques For example we can simulate the EDF of nearest neighbor distances from several realizations of aC SR process on A the region containing the original point pattern The average of the simulations provides a reference line and the maximum and minimum provide a simulation envelope The Spatial Randomness dialog provides the ability to draw simulation envelopes for both the K hat and L hat statistics To compute a simulation envelope for an estimate of Lhat 1 Launch Spatial Randomness dialog 2 Ente
51. essing Apply to assess each change 7 PressOK when satisfied 8 Repeat steps 4 7 for the other symbol 44 E xploratory Analysis 9 Chose Insert gt Legend from the main menu and insert a legend Move the legend by dragging it and position as desired on or outside the plot 10 Chose Insert gt Titles gt Main from the main menu and insert a title on top of the plot W hen plotting spatial data such as these it is preferable to have both axis scaled the same way for geometric accuracy that is a scale that conforms to the actual observation locations To scale the axis 1 Right click on the plot region of the scatter plot not on a data point 2 Select Position Size from the middle of the resulting context menu 3 Change the Aspect Ratio from Auto to 1 or set to Proportional Units 4 Click OK 45 Chapter 5 Spatial Point Patterns Insert a title and a legend by choosing Insert from the main menu The resulting plot would look as the one below perhaps with different symbols depending on your choice Lansing Woods Tree Locations a hickory 1 07 EN wT aa gl Ate A a v a mpe BR Y maple a A a a 0 8 af oa b a al a va pe a 0 6 Daten aX Soya Y a ysy A bat v SARA A aa 0 4 7 p s re z a w ANER Fen Fa Vv 027 ieee Fv oem V 0 0 AE Pam Soy Be VY ayy r V Rav No spatial pattern is immediately obvious as the Lansing Woods data is very dense when the 2 species a
52. h covfun spher cov range 4 31 sill 0 14 nugget 0 89 predictions over default 30 x 30 grid pcoal lt predict kcoal plot prediction surface wireframe fit x y data pcoal screen list z 300 x 60 y 0 drape T Lhat Ripley s K Function for a Spatial Point Pattern Object Lhat DESCRIPTION Calculates L t sqrt K t pi where K t is Ripley s K function for a spatial point pattern and L t is linear for a completely random point process USAGE Lhat object maxdist lt lt see below gt gt ndist 100 boundary bbox object plot it T REQUIRED ARGUMENTS 65 Appendix Data and Function Reference object an object of class spp representing a spatial point pattern or a data frame or matrix with first two columns containing locations of a point pattern OPTIONAL ARGUMENTS maxdist ndist boundary numeric value indicating the maximum distance at which Lhat should be estimated Defaults to half the length of a diagonal of the sample s bounding box desired number of default distances at which to compute Lhat Default is 100 The distances for which Lhat will be estimated are calculated as seq 0 maxdist ndist both maxdist and ndist will change if not reasonable for the given object points defining the boundary polygon for the spatial point pattern This version accepts only rectan gles for which boundary should be given as a list with named components x and y denoting the corners of
53. he Data Set of interest 4 Select easting and northing as Variables 1 and 2 respectively 5 Specify Max Dist of 30 keeping Euclidean as the metric and 1 as the number of neighbors to consider 6 Entersids nhbr30 in the Save In field 36 Spatial N eghbors 7 PressOK A Data Window opens containing the spatial neighbor object We can use this object to compute spatial correlations and perform spatial regression Several sources are considered when using the Spatial Neighbors dialog depending on how the neighborhood information is stored in S PLUS or on an ascii file to be read in These are Nearest Nhbrs to be used when you have point locations Row and Col ID to enter data that is already paired up by neighbors with row and column identifiers for each neighbor pair Weight Matrix if the square matrix containing the neighbor weights is available for input e Create Grid to generate regular lattices Read File to browse an ASCII file of varying record length and a set of neighbors per row Click the Help button on the dialog for more specifics on each of these options and the corresponding S SPATIALSTATS function After using this dialog for your data make sure that the results are saved and then explore their structure with both the summary and plot methods illustrated above for objects of class spatial neighbor 37 Chapter 4 LatticeD ata SPATIAL CORRELATIONS 38 If a process is sp
54. ical variogram function The first argument should be distance The remaining arguments are considered parameters that can be changed to update the fit of fun to object lower either a single numeric value or a vector of length equal to the number of parameters giving lower bounds for the parameter values If it is a single value then all parameters have that as their lower bound See the help page for nlminb for more information upper either a single numeric value or a vector of length equal to the number of parameters giving upper bounds for the parameter values If it is a single value then all parameters have that as their upper bound See the help page for nlminb for more information VALUE an object of class vgram fit with components parameters anamed vector with the fitted values for the parameters objective the final value of the objective function funName the fun argument as a character string distRange anumeric vector containing the minimum and maximum distance values from vob DETAILS If fun is one of exp vgram gauss vgram linear vgram power vgram or spher vgram and param is not supplied the function sets special initial starting values for param Otherwise if param is not supplied it is set to a vector of ones The weighted least squares objective function used in the fitting process Cressie 1993 p 97 is objective fun lt function y yh n sum n y yh 1 2 The nlminb function is used for the optimization REFERE
55. ing into account an autoregressive or moving average covariance model reflecting the interactions with neighbors The Spatial Regression dialog fits a linear model with spatial dependence using generalized least squares regression To fit a spatial regression model Choose Spatial gt Spatial Regression from the main menu The dialog below appears Spatial Regression X Model Results Data Spatial Structure Data Set sids Cov Type CAR M Weights X Neighbor Object sids neighbor X Subset Rows with 4 Parameters fights 1 sids births IV Omit Rows with Missing Values Save Model Object Save As sids sim1 Variables Dependent Independent Formula sid ft nwbirths ft Create Formula Cancel Apply kj gt lof 2 Help Spatial Regression Example Fit a Spatial Regression model to the sids data frame See the S SPATIALSTATS User s Manual for a detailed explanation on the choice of regression variables and Covariance M odel This example is equivalent to the example run on page 131 of the M anual 1 Launch the Spatial Regression dialog 2 Entersids asthe Data Set to be modeled 3 Select thesid ft and nwbirths ft columns as Dependent and Independent variables respectively 4 Enter 4 in the Subset Rows with field to remove observation 4 as it was identified as an outlier in section 3 3 of the User s M anual Set the Cov Type to CAR Enter si ds neighbor asthe Neigh
56. ing plot each time along with the objective value printed on the plot to assess how well the specified theoretical variogram matches the empirical variogram 3 A Range of 6 gives a local minima in the objective value After we have selected this Range we may wish to try other values of Sill and Nugget to further reduce the objective value 4 Trying various values suggests that an empirical variance function with range of 6 5 sill of 1 9 and nugget of 1 1 matches the empirical variogram pretty well Enter these values and press Apply 5 PressOK or Cancel to dismiss the dialog Try fitting the variogram without removing observation 50 and see how much influence this value has on the final variogram model You will need to fit an empirical variogram first and then proceed to the Model Variogram dialog The results saved in the object named as stated in the Save As field can be given to one of the kriging dialogs to fit a kriging surface to the spatial process of interest 20 Ordinary and Universal Kriging ORDINARY AND UNIVERSAL KRIGING Kriging is a linear interpolation method that allows predictions of unknown values of a random field from observations at known locations Kriging incorporates a model of the covariance of the random function when calculating predictions of the unknown values S SPATIALSTATS provides two Kriging dialogs to support both ordinary and universal kriging Ordinary kriging uses a random function
57. intensity of the spatial point pattern The theoretical K function for a Poisson completely spatially random process is K t t so L t VK p n is equal to t the distances The default plots K t versus t See function Lhat for estimation of L t REFERENCES Ripley Brian D 1976 The second order analysis of stationary point processes Journal of Applied 63 Appendix Data and Function Reference Probability 13 255 266 SEE ALSO Kenv Lhat EXAMPLES lansing spp lt as spp lansing lansing khat lt Khat lansing spp Khat wheat abline 0 1 krige Ordinary and Universal Kriging krige DESCRIPTION Performs ordinary or universal kriging for two dimensional spatial data The function pre dict krige can then be called to compute interpolation surfaces and prediction errors USAGE krige formula data sys parent subset na action na fail covfun nc 10000 REQUIRED ARGUMENTS formula a formula describing the kriging variable and the spatial location variables and optionally a polynomial trend surface Its simplest form is z T loc x y where z is the kriging variable and x and y are the spatial locations that is z i is observed at the lo cation x i y i The right hand side must contain a call to the function loc A polynomial trend surface is of the form Zz Loel xpy ok eye th TQ vy 22 The polynomial must be in the same variables as the first two arguments used in the loc function A
58. ionary point processes Journal of Applied Probability 13 255 266 SEE ALSO Lenv Khat EXAMPLES 66 lansing spp lt as spp lansing lansing khat lt Lhat lansing spp Lhat wheat abline 0 1 Appendix Data and Function Reference make pattern Generate a Spatial Point Process make pattern DESCRIPTION Generates points in two dimensional space given their desired spatial distribution USAGE make pattern n process binomial object boundary bbox x c 0 1 y c 0 1 lambda maxlambda radius cpar REQUIRED ARGUMENTS n integer denoting the desired number of points in the resulting object OPTIONAL ARGUMENTS process a character string with one of five possible processes for the spatial arrangement of the resulting pat tern This must be one of binomial poisson cluster Strauss or SSI See the DETAILS section for each definition Defaults to binomial for a completely spatially random pro cess conditioned to n points within boundary Partial matching is allowed object a spatial point pattern object An object of class spp When this is given the resulting pattern has the same n and its boundary is that same as the bounding box of object boundary points defining the boundary polygon for the spatial point pattern This version accepts only rectan gles for which boundary should be given as a list with named components x and y denoting the corners of the rectangular region For example for the un
59. it square the boundary could be given as bbox x c 0 1 y c 0 1 the bounding box of two diagonally opposed points Defaults to bbox object if object is given or to the unit square otherwise lambda the intensity when process poisson If lambda is a numerical value then make pattern simu lates a two dimensional homogeneous Poisson process with that constant intensity lambda can also be a function with two arguments that defines the intensity over the region n if given will be ignored if this argument is provided maxlambda if lambda is a function then this should be the maximum value of the function over the region If this is not supplied a nonlinear optimization will be run using nlminb to find the maximum Supplying this value will speed up the simulation and avoid any possible problems with the nonlinear optimiza tion maxlambda is used only if Lambda is a function radius the inhibition distance This is needed for process Strauss SSI and cluster Options Strauss and SSI will NOT generate points closer than radius For this reason this parameter needs to be reasonably small The exception is when process cluster in which case it should contain the desired size of the clusters See DETAILS section for more information cpar the inhibition parameter needed when process Strauss This parameter is also required if pro cess cluster In that case it represents the intensity of the parent Poisson process which will determin
60. kriging simulate non homogeneous Poisson processes and how to create summaries and plots of spatial neighbor objects You will also learn to use the new GU I to access the analytical tools available in previous versions and receive guidance on conducting an analysis of spatial data using the full functionality of the S PLUS for Windows interface This supplemental guide has been organized according to the new menus and dialogs in the GUI of version 1 5 which are in turn organized according to the types of spatial data that can be analyzed using S SPATIALSTATS Geostatistical Lattice and Point Pattern D ata This chapter provides an introduction to these three types of spatial data The S SPATIALSTATS User s Manual contains detailed discussions of each type including mathematical descriptions and assumptions of the diverse methodologies used for their analysis and consequent statistical inference Chapter 1 Introduction SPATIAL DATA Geostatistical Data Lattice Data Spatial data consist of measurements or observations taken at specific locations or within specific regions In addition to values for various attributes of interest spatial data sets also include the locations or relative positions of the data values Locations may be point or areal referenced For example point referenced data are observations recorded at specific fixed locations and might be referenced by latitude and longitude Areal referenced data are observation
61. ly LECZE Help Example Use the coal data once again 1 2 Enter coal ash as the Data Set of interest Selectcoal in the Variable field and x and y respectively as Location 1 and Location 2 Enter 50 in the Subset Rows with field to remove observation 50 Enter c 0 90 as the Angles of anisotropy to explore that is the East West and North South directions 15 Chapter 3 Geostatistical Data 16 5 PressOK A multipanel plot appears in a Graph sheet with several directional variograms for all combinations of four ratio values for each of the 2 directions entered The plot follows gamma distance There are no apparent changes for differing ratio values between rows but the variograms on the left do look different from those on the right The Geometric Anisotropy dialog also provides an Options tab where the user can specify parameters to control the estimation of the variogram values for each combination of angle and ratio values See the dialog s help file for detailed information or section 4 1 4 of the S SPATIALSTATS User s M anual Empirical Variogram EMPIRICAL VARIOGRAM The empirical variogram provides a description of how the data are related correlated with distance The distances are binned and the corresponding variogram values averaged for each bin thereby producing a smoother version of the variog
62. mining applications In recent years these methods have been applied to many disciplines including meteorology forestry agriculture cartography climatology and fisheries This section describes the following dialogs e Variogram Cloud Geometric Anisotropy e Empirical Variogram Model Variogram e Ordinary Kriging e Universal Kriging Chapter 3 Geostatistical Data EXPLORATORY DATA ANALYSIS Example The Coal Ash Data 10 In this section we will give specific examples of EDA for geostatistical data data collected on a continuous spatial surface see chapter 1 of the S SPATIALSTATS User s M anual for a more precise definition Thecoal ash data frame is used in an example of EDA for data collected on an equally spaced grid of locations The coal ash data will then be used to illustrate the use of the S SPATIALSTATS 1 5 dialogs to analyze geostatistical spatial data The coal ash data come from the Pittsburgh coal seam on the Robena Mine Property in Greene County Pennsylvania Cressie 1993 The data frame contains 208 coal ash core samples collected on a grid given by x and y planar coordinates To plot the grid locations 1 Open a view of the data on a Data Window You can do this by selecting Data Select Data from the main menu and then entering coal ash as the Name of the existing data set desired The resulting dialog follows Select Data Of x Source Existing Data Existing Data Name eoalasH z
63. mmands in the Commands Window gt coal no50 lt coal ash 50 Remove 50th row 2 Choose Data gt Select Data from the main menu and open thecoal no50 data frame 3 Select columnsx and coal 4 From the Plots2D palette choose a scatter plot with al oess fit through it by pressing the corresponding button a scatter plot with an L on it 5 Click on the graphsheet and choose Insert gt Annotation gt Reference Line gt Horizontal and add a horizontal dashed line at the mean of the observations that is set Position at 9 740725 to the plot of the observations against x 6 Go back to the Data Window and selecty andcoal thistime 7 Click on the graphsheet with the observations vs x plot and SHIFT click on the scatter plot with al oess fit on the Plots2D palette The plot of coal against y will be superimposed on the other plot 23 Chapter 3 Geostatistical Data 8 Separate the plots by clicking on the plot region not on a data point and selecting Multipanel from the resulting menu 9 Select By Plot as the Panel Type and set the Layout to bea 1 column by 2 rows plot arrangement After inserting a title you see the figure below Coal Ash against Location a a A a A ge S a amp amp A A aaah ea ar Bg eG i me a wv Bet eee Re Ata eG saiheisa r er enn a a A a S E Be hi oy SENRSR ae in to eee a A aR 2 SAR oe at Cfg e222 8 8 a A 4 aa a4 a a a 4 ey a dye coal 0 5 10 15 20 25
64. model of spatial correlation to calculate a weighted linear combination of available samples for prediction of a nearby unsampled location Weights are chosen to ensure that the average error for the model is zero and that the modeled error variance is minimized Universal kriging is an adaptation of ordinary kriging that accommodates trend The trend is modeled as a polynomial function of spatial location Universal kriging can be used to both produce local estimates in the presence of trend and to estimate the underlying trend itself Universal kriging with a constant mean is equivalent to ordinary kriging The Ordinary and Universal Kriging dialogs provide several options for prediction Block Kriging predictions the average over a rectangular area are possible using the dialogs as well as Point predictions either on a grid or at new sampling points Several different plots can be requested to help visualize both the predictions and their standard errors 21 Chapter 3 Geostatistical Data To perform ordinary kriging Choose Spatial gt Ordinary Kriging from the main menu The dialog below appears Ordinary Kriging Bile Es Model Predict Plat Data Variogram Inputs Data Set coal ash IV Use Values from a Variogram Fit Variable coal Variogram Fit coal vgmdl Subset Rows with 50 Variogram IV Omit Rows with Missing Values Function Spherical z Spatial Location Range 6 5698581811728 Location 1
65. nce counties with low birth rates will have more variance The sid ft column contains rates standardized using the Freeman Tukey square root transformation We will look at the spatial correlation of the transformed variable 1 2 6 Launch the Spatial Correlations dialog Selectsids asthe Data Set Select si ds ft from the Variables list Notice that multiple selections are allowed Select sids nhbr30 asthe Neighbor O bject Specify Statistic of moran Sampling Type of free and Num Permute of 100 PressOK A summary of the spatial correlation is displayed in a Report window The small Normal p value and permutation p value suggest that spatial autocorrelation is present for this variable xkk Spatial Correlations Spatial Correlation Estimat Statistic moran Sampling fre Correlation 0 259 Variance 0 00478 Std Error 0 0691 Normal statistic 3 89 Normal p valu 2 sided 9 927e Null Hypothesis No spatial autocorrelati Summary of the permutation correlations Min 1st Qu Median Mean 3rd Qu Me 0 1432 0 08002 0 01972 0 01936 0 03252 0 15 permutation p value 39 Chapter 4 LatticeData SPATIAL REGRESSION 40 To model spatial lattices we look at two levels of variation targe scale change in the mean due to spatial location or other explanatory variables and small scale variation due to interactions with neighbors The change in mean is modeled as a linear model tak
66. nction Reference Assume we have no information about the strength of the spatial association All weights are 1 nghb lt spatial neighbor row id row index col id col index summary nghb Another way to create the same spatial neighbor object nmat lt matrix c 0 1 1 0 Ly Oa Oy Gr Op Oy Le 0 0 0 0 ncol 4 byrow T nghb2 lt spatial neighbor neighbor matrix nmat spatial neighbor object Class spatial neighbor spatial neighbor object DESCRIPTION Class of objects used to define neighbor relationships for spatial data on a regular or irregular lattice GENERATION This class of objects is constructed using the function spatial neighbor Alternatively the func tions read neighbor or neighbor grid may be used In general the user must construct these ob jects whenever estimates of spatial correlation and spatial regression are desired An object of class spatial neighbor contains all the information required to determine which spatial units on a region of interest are neighbors as well as the strength of their relationship METHODS The class spatial neighbor has associated methods print spatial neighbor plot spa tial neighbor and summary spatial neighbor INHERITANCE Class spatial neighbor inherits from class data frame STRUCTURE The spatial neighbor object is in essence a data frame with additional attributes Each row of the data frame denotes a pair of neighboring spatial units
67. nd NOT the region identi fiers which could be character strings or some such These are used to expand the full contiguity ma trix so we should have representation for all indices 1 through nregion though it is possible to have islands in between Use the function check islands to check for these islands and remap their in dexing if that is desirable It is possible to specify two or more types of neighbor relationships For example the user may want to model a spatial relationship depending upon the angle of the line connecting neighbor centers i e considering directional relationships For this example let Type 1 neighbors be north south neighbors and let Type 2 neighbors be east west neighbors neighbors along a diagonal could be modeled with weights proportional to 707 the sine of 45 degrees for instance Consider the elements of row id col id and weights corresponding to a distinct value k of the vector matrix id The spatial neighbor matrix can be expressed as a matrix A k such that A k row id col id weights and all other elements are zero Consider a parameter vector rho of length g many spatial covariance matrices used in spatial regression models can be expressed as a weighted linear combination of the contiguity matrices A k rho k A k for values of k varying 1n lig check islands plot spatial neighbor summary spatial neighbor row index lt c 1 1 2 2 3 col index lt c 2 3 1 3 4 Appendix Data and Fu
68. nd identify If all directions and distances are included the return object will have n n 1 2 rows where n is the number of observations This can get very large even for relatively small n The argument maxdist can be used to limit the size Typically values beyond half the maximum distance in the data are not used in estimating the variogram function REFERENCES Cressie Noel 1993 Statistics For Spatial Data Revised Edition Wiley New York SEE ALSO boxplot vgram cloud identify vgram cloud plot vgram cloud variogram EXAMPLES vl lt variogram cloud coal x y data coal ash plot v1 82 Appendix Data and Function Reference boxplot v1 variogram fit Fit a Variogram Model variogram fit DESCRIPTION Fits a theoretical variogram model to an empirical variogram object using a local minimizer for smooth non linear functions subject to bounded parameters USAGE variogram fit vobj param fun spher vgram lower rep 0 n param upper Inf REQUIRED ARGUMENTS vobj an object that inherits from class variogram representing an empirical variogram Usually the re sult of the variogram function OPTIONAL ARGUMENTS param a named vector with initial values for the parameters to fit Usually these are the nugget sill and range or a subset of these If missing the function will try to determine the parameter names and initial values based on the arguments to the function specified in fun fun a theoret
69. proc poisson boundary bbox x c 0 10 y c 0 10 lambda lxy plot xy Appendix Data and Function Reference model variogram Display a Variogram Object and Theoretical Model model variogram DESCRIPTION Plots an empirical variogram object and displays the fit of a theoretical variogram model on that plot Optionally allows interactive parameter updates to the theoretical model and displays the new fit USAGE model variogram object fun ask T objective fun lt lt see below gt gt plot it T REQUIRED ARGUMENTS object an object that inherits from class variogram this includes classes covariogram and correl ogram The azimuth column should have only one level fun a theoretical variogram function or covariogram or correlogram function depending on the class of object Its first argument should be distance Its remaining arguments are considered parameters that can be changed to update the fit of fun to object OPTIONAL ARGUMENTS additional arguments to fun that do not have default values must be specified here by full name ask a logical value if TRUE a command line menu is displayed allowing the user to change the values of the parameters to fun After changing a value the plot is updated If FALSE the data in object is plotted the value of fun evaluated at object distance is added to the graph and the function re turns objective fun a function with three arguments y yf and n that gives
70. r variogram modeling The variogram cloud provides a diagnostic tool to look for potential outliers or trends and to assess variability with increasing distance It provides the distribution of the variance between all pairs of points at all possible distances and as a consequence it may yield extremely dense point clouds that may be difficult to interpret To reach a point when we can model the variability in the data a smoother version of the variogram is available through the Empirical Variogram dialog G eometric Anisotropy GEOMETRIC ANISOTROPY Anisotropy is present when the spatial autocorrelation of a process changes with direction Unlike a variogram from an isotropic process the variogram from an anisotropic process is not purely a function of distance but is a function of both distance and direction The anisotropy plot is useful for exploring whether the process the data comes from is isotropic or whether the shape of the variogram changes with direction To create an anisotropy plot Choose Spatial gt Geometric Anisotropy from the main menu The dialog below appears Geometric Anisotropy working FEE Data Options Data Spatial Location Data Set coalash Location 1 TE Variable oa ts Location 2 poo H Subset Rows with oOo Angles oa IV Omit Rows with Missing Values Ratios fi 25 1 5 1 75 2 Plots Results IV Plot Variogram Save s IV Add Smooth Line IF Show Variogram Results i Cancel App
71. r in the par help file add a logical value if TRUE no initial plot is drawn only the variogram fitted line is added to the current plot npoints a numeric value the number of to evalute the variogram function at 71 Appendix Data and Function Reference 72 Graphical parameters may also be supplied as arguments to this function see par SIDE EFFECTS a plot is produced on the current graphics device or lines are added to the current plot if add T DETAILS The function specified by x funName must exist It is evaluated at npoints between 0 and x dis tRange 2 This function is a method for the generic function plot for class vgram fit It can be invoked by calling plot for an object of the appropriate class or directly by calling plot vgram fit regardless of the class of the object SEE ALSO variogram fit variogram EXAMPLES vg iron lt variogram residuals loc easting northing data iron ore vfit iron lt variogram fit vg iron param c range 8 7 sill 3 5 nugget 4 8 fun spher vgram plot vg iron plot vfit iron add T points in poly Find Points Inside a Given Polygon points in poly DESCRIPTION Determine whether points are inside a polygon USAGE points in poly x y polygon REQUIRED ARGUMENTS x the X coordinates of the points y the Y coordinates of the points Must be the same length as x nen polygon a list with named components x and y VALUE a logical vector the same length
72. r lansing as the Data Set of interest or press the Roll Back button and skip to step 5 Selectx and y as the Location 1 and 2 variables respectively 4 Type species maple in the Subset Rows with field This will subset those rows of the data frame that correspond to the maples only 5 Check Lhat plot The K hat Lhat O ptions group is enabled Lhat Spatial Randomness 6 Check Construct Simulation Envelope and specify 50 simulations to estimate the envelope 7 Select poisson asthe process to simulate 8 Set the lambda parameter of the Poisson to 10 by entering ambda 10 in the Sim Parameters field 9 PressOK Warning The number of simulations does not need to be large and in fact if a large number of simulations is requested S PLUS may take a long time to complete the simulations The picture below appears 0 6 7 0 4 4 0 2 7 0 0 7 T T T T T 0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 Distance The second order properties of spatial point processes describe how the interaction or spatial dependence between points varies through space These properties are usually described by the second order intensity of the spatial point pattern An alternative description of the second order properties is defined by the K function defined in section 6 3 2 of the S SPATIALSTATS User s M anual 51 Chapter 5 Spatial Point Patterns INTENSITY The intensity of a point pattern is the mean number of points
73. ram which leads to easier modeling You can control the degree of smoothing by adjusting the size of the bins or lags the number of points in each bin and the distance of reliability half the maximum distance over the field of data by default for the variogram Several directional variograms can be returned and as usual a correction for Geometric Anisotropy can be made to the data before processing To plot an empirical variogram Choose Spatial gt Empirical Variogram from the main menu The dialog below appears Empirical ariogram BEI Data Options Data Spatial Location Data Set coal ash Location 1 x Variable coal Location 2 ly Subset Rows with 50 I Corect for Geometric Anisotropy M Omit Rows with Missing Values ia Direction a Azimuth beo Plots Azimuth Tolerance 01 M Plot Variogram Measure Results Type variogram 7 Save As coal vans I Show Variogram Results OK Cancel Apply E gt 10f 5 Help Example Continue analyzing thecoal ash data in S SPATIALSTATS 1 Launch the Empirical Variogram dialog 17 Chapter 3 Geostatistical Data 2 Selectcoal in the Variable field and x and y respectively as Location 1 and Location 2 3 Enter 50 in the Subset Rows with field to remove observation 50 4 Change the Azimuth Tolerance to 01 5 Savetheresultascoal vg ns 6 PressOK A plot of the empirical variogram appears in a Graph sheet Note tha
74. re taken together The data is an example of a bivariate point pattern We can plot the species separately and see if any patterns come to light To separate the plot into 2 panels by species 1 Right click on the scatter plot region again 2 This time select Multipanel from the middle of the resulting context menu 3 From the Panel Type drop down select By Plot Set of Columns to 2 in the Layout group on the same page 46 E xploratory Analysis 4 Click OK The figure below appears Reposition the legend to uncover the axes Lansing Woods Tree Locations a hickory v maple 0 2 0 5 0 8 These plots show that there may be interaction between the two tree species It may be that the presence of one species inhibits the presence of the other 47 Chapter 5 Spatial Point Patterns SPATIAL RANDOMNESS 48 Typical assumptions of interest for point pattern data are 1 The intensity of the point pattern does not vary over the boundary region 2 There are no interactions among the points points neither inhibit nor encourage each other A spatial point pattern with these properties is said to be Completdy Spatially Random See Chapter 6 of the S SPATIALSTATS user s M anual for a more rigorous definition and further examples The Fhat and Ghat statistics are useful for assessing the first assumption constant intensity The Khat and Lhat statistics are useful for assessing the second a
75. rectional and setting arrows TRUE just results in a messy graph size arrows the size of the arrowhead width in inches See the arrows help file for details scaled a logical value if TRUE then scaled plot is used to set up the plot coordinates instead of plot This produces an equally scaled plot which is often useful when xcoord and ycoord are geographic loca tions Graphical parameters may also be supplied as arguments to this function see par SIDE EFFECTS a plot is produced on the current graphics device or lines are added to the current plot if add T DETAILS The coordinate system for the plot is drawn based on the values in xcoord ycoord The graphical parameters specified in are used to draw this initial graph If scaled TRUE the scale ratio parameter to scaled plot can also be passed in the arguments The lines are added through a call to segments or arrows The graphical parameters line col line type and line width are used in the call to segments or arrows This function is a method for the generic function plot for class spatial neighbor It can be in voked by calling plot for an object of the appropriate class or directly by calling plot spa 70 Appendix Data and Function Reference tial neighbor regardless of the class of the object BUGS With S PLUS 5 1 if you are calling this function as a plot method i e plot you must specify the xcoord and ycoord arguments by name not by position otherwis
76. s specific to aregion for example the number of burglaries occurring in census tracts where each census tract is a region In both cases spatial locations may be regular or irregular point locations may fall on a regularly spaced grid or may be irregular with varying distances between points areal locations can comprise equally sized contiguous blocks that might occur in an agriculture field study or may be of variable size and shape such as the city limits within a county Spatial data may be continuous such as the measurements of ore content from a core sample or discrete such as the number of measles cases reported by county Further the locations may come from a spatial continuum such as the point locations within a mining field or a discrete set such as the counties within a state S SPATIALSTATS provides tools for analyzing three specific classes of spatial data geostatistical data lattice data and spatial point patterns G eostatistical data also termed random fidd data are measurements taken at fixed locations The locations are generally spatially continuous Examples of continuous geostatistical data include mineral concentrations measured at test sites within a mine rainfall recorded at weather stations concentrations of pollutants at monitoring stations and soil permeabilities at sampling locations within a watershed An example of discrete geostatistical data is count data such as the number of scallops at a series
77. single value giving the mean weight value for this matrix a vector of indices that are not present in ob ject row id for the i th matrix This will be printed as none by the print method if there are no missing row indices and it is not printed at all if object is a symmetric spatial neighbor matrix since all missing row indices will be islands see below a vector of indices that are not present in ob ject col id for the i th matrix This will be printed as none by the print method if there are no missing column indices and it is not printed at all if ob ject is a symmetric spatial neighbor matrix since all missing column indices will be islands see be low the indices for regions that have no neighbors These indices do not appear in either the ob ject col id or object row id for the i th matrix This will be printed as none by the print method if there are no islands This function is a method for the generic function summary for class spatial neighbor It can be invoked by calling summary for an object of the appropriate class or directly by calling summa ry spatial neighbor regardless of the class of the object spatial neighbor check islands summary Ssids neighbor Create two symmetric spatial neighbor matrices with one island in the second matrix ri se OS Typ 2 3475 Vy lp 27 575 CE SS 25 34 37475041 2737 87 876 mat lt E E 1 1 Typ dy 252727272 sn lt spatial neighbor ri ci symm T matrix mat
78. ss spatial neighbor is a sparse matrix representation of a square matrix or a number of square matrices The function plot spatial neighbor will show a graphical view of the spatial neighbor object and summary spatial neighbor will compute summary statistics on the object The functions spatial multiply and spatial cg solve can be used to form products of the form rho i N i x and rho i N i 1 x for neighbor weight matrices N i vector of constants or parameters rho i and arbitrary vectors x should that be needed to form a neighbor or contiguity matrix as a weighted linear combination of others SEE ALSO spatial neighbor plot spatial neighbor summary spatial neighbor read neigh bor neighbor grid spatial multiply spatial cg solve spatial weights spatial solve Solve Sb x spatial solve DESCRIPTION Solves Sb x for b where S is a sparse matrix obtained from an object of class spatial neigh bor USAGE spatial solve neighbor x transpose F rho 0 product F weights NULL region id NULL absThreshold 0 relThreshold 0 diagPivoting 0 shareMemory F REQUIRED ARGUMENTS neighbor an object of class spatial neighbor containing the sparse matrix representation of the spatial neighbor matrix or matrices see function spat ial neighbor x the right hand side for which a solution is desired Alternatively x can be a matrix In this case a so lution is obtained for each column in x OPTIONAL ARGUMENTS transpose with th
79. ssumption second order intensity which does not depend on absolute location The Spatial Randomness dialog provides plots and saved values for Fhat G hat K hat and L hat Spatial Randomness To calculate measures of spatial randomness Choose Spatial gt Spatial Randomness from the main menu The dialog below appears Spatial Randomness e x Data Khat Lhat Options Data Set lansing hd Location 1 koo n fic Location 2 vb 4 F Subset Rows with species maple pooo M Omit Rows with Missing Values binomi zf Plots a I Ehat Saved Results M Ghat Save In ss I Khat I Fhat I Lhat l Ghat l Khat M ihat Example Explore the Spatial Randomness of the lansing data frame in S SPATIALSTATS 1 Launch Spatial Randomness dialog 2 Enter ansing as the Data Set of interest 3 Selectx andy asthe Location 1 and 2 variables respectively 4 Type species maple in the Subset Rows with field This will subset those rows of the data frame that correspond to the maples only Note that another approach would be to choose Data gt Subset from the main menu to create a data set of just the maples Check Fhat and G hat plots 6 PressOK 49 Chapter 5 Spatial Point Patterns 50 A Graph sheet opens displaying the Fhat and Ghat plots for the maples T hese are plots of the Empirical Distribution Function EDF of the origin to point Fhat and the point to point Ghat nearest neighbor distances for the mapl
80. st have data locations and neighborhood information The locations for the SIDS data are stored in easting and northing Neighborhood information is typically stored in aneighbor matrix where two regionsi and j are neighbors if the ij th element of the neighbor matrix S SPATIALSTATS neighbor information is stored in an object of class spatial neighbor a sparse matrix representation of the neighbor matrix The S PLUS object sids neighbor already contains the neighbor information for the SIDS data To summarize a spatial neighbor We can summarize the neighborhood information calling the summary method for a spatial neighbor as follows gt summary sids neighbor Matrix was NOT defined as symmetric Number of Regions 100 Average Number of Connections 4 020408 Average Weight 0 1306507 Least Number of Connections 1 for Regions with Indices 1 10 16 67 Maximum Number of Connections 8 for Regions with Indices 1 Cressie Noel A C 1993 Statistics for Spatial Data Revised Edition John Wiley and Sons New York 32 Exploratory Analysis 1 21 Missing Row Indices 1 28 48 Missing Column Indices 1 28 48 Indices of Regions with No Connections islands 1 28 48 gt The resulting summary describes the neighborhood as being defined for 100 regions with varying neighbor weights Each county has about 4 neighbors on the average with one county having 8 neighbors and 2 having none Th
81. t restricting our computation to only points in a North South direction is not the same as Correcting for Geometric Anisotropy W hen using the variogram values for those pairs that are related in a North South direction only we are assessing one way the variance covariance of the process changes without making any corrections to it We are trying to understand variability only in the N S direction This makes some sense in applications that relate to physical data where gradients are quite possible Later on we apply our knowledge about this directional variation to the fitting of a kriging surface to the data using Universal K riging techniques 18 M odel Variogram MODEL VARIOGRAM In order to perform kriging it is necessary to specify a theoretical variogram function for the process The Model Variogram dialog is useful for examining the goodness of fit of various theoretical variograms to the observed empirical variogram Typically this dialog will be used repeatedly to determine an appropriate variogram function and parameter values To fit a theoretical variogram Choose Spatial Model Variogram from the main menu The dialog below appears Model Yariogram FEE Variogram Object coal vg ns Plots IV Plot Variogram Results Va lograr Parameters Spl erca bg S ave AS coal g I vomd Function IV Fit Parameters Range 6 5698581811728 Sill 0 1093983263090 Nugget 0 9174357347351 Cancel Apply ce c
82. ted region and locations of earthquake epicenters A marked spatial point pattern includes values of additional related variables at each location The additional variables are often called mark variables and may be used to further refine the analysis of point patterns The Lansing Woods data set introduced in the S SPATIALSTATS User s manual contains a marked spatial point pattern in addition to the locations the tree species were also recorded Chapter 1 Introduction GETTING STARTED This chapter describes how to get started with the S SPATIALSTATS graphical user interface 1 Load the module 2 Examine the Spatial menu 3 Find help on S SPATIALSTATS Chapter 2 Getting Started LOADING THE MODULE The first step in using S SPATIALSTATS 1 5 is to load the module Loading the module will make the spatial analysis functions available create the Spatial menu and load the S SPATIALSTATS 1 5 dialogs To load the module Choose File Load Module from the main menu The dialog below appears Load Module Ox Module Action Load Module Description IV Attach at top of search list Cancel Apply KE current Help To load the S SPATIALSTATS 1 5 module select spatial as the Module and press O K A new menu selection Spatial will appear on the main S PLUS menu bar Select this menu item to access the dialogs to analyze your spatial data The Spatial M enu THE SPATIAL MENU The Spatial menu provides
83. tical Analysis of Spatial Point Patterns Aca demic Press Inc New York 43 Chapter 5 Spatial Point Patterns EXPLORATORY ANALYSIS The exploratory analysis of a spatial point pattern begins with a map of the observations To view a spatial point pattern 1 First open a view of the data on a Data Window You can do this by selecting Data Select Data from the main menu and then entering ansing as the Name of the existing data set desired The resulting dialog follows Select Data ME x Source Existing Data Existing Data Name lansing C New Data h New Data C Import File a I Show Dialog on Startup Cancel Apply KE current Help You could also use the command line directive gt guiOpenView Name lansing classname data frame 2 Proceed by selecting all 3 columns of the data frame in the window starting with the first column column x 3 From the Plots2D palette choose a scatter plot by pressing the first button on the top left hand side corner A scatter plot of the tree locations appears In this scatter plot the points will be plotted with a different symbol for each species You may change the symbol color and shape independently for each species to suit your taste and help you differentiate the 2 species better 4 Position the cursor on a data point and right click 5 Select Symbol from the middle of the resulting context menu 6 Change the symbol s Style and Color as preferred pr
84. uld be expected by random fluctuations This might be due to the deficit of maple trees in the north corners of the plot which might be explained by interaction with hickory trees Appendix Data and Function Reference APPENDIX A DATA AND FUNCTION REFERENCE The functions and data sets described in this appendix are included with S SPATIALSTATS T he information in this appendix is also found in the online help For more information on accessing the online help see Chapter 2 Getting Started 55 Appendix Data and Function Reference 56 Appendix Data and Function Reference anisotropy plot Explore Corrections For Geometric Anisotropy anisotropy plot DESCRIPTION Computes corrections for geometric anisotropy for two dimensional spatial data and plots variograms based on the corrections USAGE anisotropy plot formula formula data data sys parent subset na action lag lt lt see below gt gt nlag 20 tol lag lag 2 maxdist lt lt see below gt gt angle c 0 45 90 135 ratio seq 1 25 2 length 4 minpairs 6 method classical smooth T plot it T panel panel xyplot REQUIRED ARGUMENTS formula formula defining the response and the predictors In general its form is z x y The z variable is a numeric response Variables x and y are the locations All variables in the formula must be vectors of equal length with no missing values NAs The formula may also contain expres sions for the varia
85. uld the envelope be added to an already existing plot of Khat or Lhat for Lenv De faults to TRUE other parameters as needed by the requested process VALUE invisibly returns a list with 4 numeric vectors each representing dist the distances at which all values were computed lower the minimum of all resulting Khat or Lhat for Lenv for the simulations upper the maximum of all resulting Khat or Lhat for Lenv for the simulations average the average of all resulting Khat or Lhat for Lenv for the simulations SIDE EFFECTS if add TRUE an envelope is added to an existing plot of Khat SEE ALSO Khat Lhat make pattern EXAMPLES Khat bramble Kenv bramble nsims 50 Lhat Lenv lansing nsims 50 lansing 62 Appendix Data and Function Reference Khat Ripley s K Function for a Spatial Point Pattern Object Khat DESCRIPTION Calculates K t Ripley s K function for a spatial point pattern USAGE Khat object maxdist lt lt see below gt gt ndist 100 boundary bbox object plot it T REQUIRED ARGUMENTS object an object of class spp representing a spatial point pattern or a data frame or matrix with first two columns containing locations of a point pattern OPTIONAL ARGUMENTS maxdist numeric value indicating the maximum distance at which Khat should be estimated Defaults to half the length of a diagonal of the sample s bounding box ndist desired number of default distances at whi
86. umcircle circle that passes through all three vertices of any triangle in the triangulation EXAMPLES triangulate scallops c lat long variogram cloud Calculate Variogram Cloud variogram cloud DESCRIPTION Calculates all pairwise differences in a random field data set USAGE variogram cloud formula data lt lt see below gt gt subset lt lt see below gt gt na action lt lt see below gt gt azimuth 0 tol azimuth 90 maxdist lt lt see below gt gt bandwidth 1e 307 FUN function zi zj zi zj 2 2 REQUIRED ARGUMENTS 81 Appendix Data and Function Reference formula formula defining the response and the predictors In general its form is Z sty The z variable is a numeric response Variables x and y are the locations All variables in the formula must be vectors of equal length with no missing values NAs The formula may also contain expres sions for the variables e g sqrt count or log age 1 The right hand side may also be a call to the loc function e g 1oc x y The loc function can be used to correct for geometric anisotropy see the loc help file OPTIONAL ARGUMENTS data an optional data frame in which to find the objects mentioned in formula subset expression saying which subset of the rows of the data should be used in the fit This can be a logical vector which is replicated to have length equal to the number of observations or a numeric vector in dicating which observation numbers are to b
87. urrent Example Use the coal ash data in S SPATIALSTATS Note that for the purposes of this example we will fit a theoretical variogram to the North South empirical variogram we estimated above In practice we want to first remove trend and explore the data further as is done in the User s M anual 1 Launch the Model Variogram dialog 2 Select the name of a fitted empirical variogram object from the drop down list in the Variogram Object field Enter the object saved after using the Empirical Variogram dialog coal vg ns 19 Chapter 3 Geostatistical Data 3 Selecta function to fit to the variogram say a Spherical 4 At this point you may check Fit Parameters and have the variogram parameters fitted automatically using the S SPATIALSTATS function variogram fit or enter your own Check the Fit Parameters check box 5 The parameters values are filled in automatically Enter a name in the Save As field to save the resulting variogram model Enter coal vgmdl 6 Press Apply You could also fit the variogram by trying different values of the parameters and looking at the O bjective Function When doing this make sure that the Fit Parameters box is not checked 1 The empirical variogram plot suggests that a Nugget of around 0 7 and a Sill of around 1 1 are appropriate Enter these parameter values 2 Now try various values of Range by entering them one ata time and pressing Apply after each input Look at the result
88. ve width size 1 and height size 2 VALUE DETAILS SEE ALSO EXAMPLES a two column matrix containing the coordinates of the resulting grid A rectangular nx by ny grid is overlaid on the polygon defined by boundary and then those points that fall outside are dropped If size is given then the values nx and ny are redundant and if given will be ignored points in poly plot as spp bramble bramble chull lt bramble chull bramble polygon bramble chull den 0 points poly grid bramble chull size c 1 1 col 2 73 Appendix Data and Function Reference predict krige Point and Block Kriging Prediction predict krige DESCRIPTION Computes point or block kriging predictions and standard errors at locations in newdata using an ob ject returned by krige USAGE predict krige object newdata se fit T grid lt lt see below gt gt blocksize c 1 1 nxy c 1 1 REQUIRED ARGUMENTS object an object of class krige as returned by the function krige OPTIONAL ARGUMENTS newdata a data frame or list containing the spatial locations for the predictions The names must match the names of the locations used in the call to krige see attr object call se fit a logical value if TRUE the standard errors of the predictions are returned Currently the standard er rors are always computed internally This se fit only determines if the returned data frame includes the se column grid a list containing t
89. wo vectors the names of the vectors must match the names of the locations used in the call to krige The vectors are each of length 3 and specify the minimum maximum and number of locations in that spatial coordinate respectively A grid is then computing using expand grid The default value is to use the range of the original location data for the minimum and maximum and 30 points This argument is ignored if newdata is supplied blocksize for block kriging a numeric vector of length 2 specifying the size of the block in x first value and y second value direction The locations specified by newdata or grid are at the center of the blocks nxy for block kriging a numeric vector of length 2 specifying the number of discretization points inside the block If both values are set to 1 the default then point kriging predictions are computed VALUE a data frame where the first two columns are the locations of the prediction along with fit the predicted values se fit the standard error of the prediction Only included if se fit TRUE DETAILS This function is a method for the generic function predict for class krige It can be invoked by calling predict for an object of the appropriate class or directly by calling predict krige regard less of the class of the object REFERENCES Ripley Brian D 1981 Spatial Statistics Wiley New York SEE ALSO krige loc EXAMPLES krige the Coal Ash data kcoal lt krige coal loc x y x
90. x Sill 0 1093983263090 Location 2 y Nugget 0 31 74357347351 l Corect for Geometric Anisotropy Results aot Save As coal ordKrige IV Print Results Cancel Apply KE current Help Example Let us krige the coal data 1 Launch the Ordinary Kriging dialog 2 Entercoal ash as the Data Set of interest 3 Selectcoal in the Variable field and x and y respectively as Location 1 and Location 2 4 Enter 50 in the Subset Rows with field to remove observation 50 5 Check the Use Values from a Variogram Fit box 22 9 Ordinary and Universal Kriging Select coal vgmdl from the drop down list in the Variogram Fit field This is the variogram fit object that we obtained using the Model Variogram dialog The fields for the Variogram parameters automatically fill Save to an object named coal ordKri ge Move to the Plot tab Specify Surface Plots for both the predictions and their standard errors PressOK You see a summary of the fitted object coal ordKri ge in a Report window and the plot on different pages of a Graph sheet The exploratory plots of the data presented in the S SPATIALSTATS User s M anual indicated the presence of a strong gradient in these data This gradient might be apparent if we plot the observations against location To plot the coal observations against location 1 Create a new data set by removing observation 50 from the coal ash data frame by entering these co

Download Pdf Manuals

Related Search

Related Contents

DOSAGE ET MODE D`EMPLOI : UTILISATEURS : DOMAINE D  Cavex Avalloy  1 - Alpine Europe  PDFファイル    Sif 7/73 fra.  Inertial+ User Manual  Rasenmäher R43 BETRIEBSANLEITUNG Tondeuse à gazon R43  Bedienungsanleitung  FXUQ-A_FUQ-C_OM_3P170549-16W  

Copyright © All rights reserved. Failed to retrieve file