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Programita - Thorsten Wiegand

1. Figure 6 The res results file fig2A res The first lines contain the information on the set tings of the analysis the following part contains a table with the results of the analysis The first gives the spatial scale r of the point pattern analysis units of cells the second and third column provide a summary of the Monte Carlo significance test of the null model data at scale r below the confidence limits r inside the confidence limits and above the confi dence limits second column for univariate analysis third column for bivariate analysis col umns 4 5 6 results of univariate analysis column 4 univariate O statistic or L function of the data column 5 lower confidence limit column 6 upper confidence limit and columns 7 8 9 results of bivariate analysis column 7 univariate O statistic or or L function of the data column 8 lower confidence limit column 9 upper confidence limit 12 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Temporary data files During the analysis Programita creates a number of temporary data files which are overwritten by a new analysis Knowing these files you may use the information they contain The files tempp 1 dat and tempp2 dat tThe file tempp1 dat contains a matrix representation of pattern 1 The first line contains information on the dimen sions of the grid 1 number of lines 1 number of columns The follo
2. Oi E ere ne size pat 2 If you e g select 100 t ea gorithm Peter assigns the first encountered 100 adjacent cells Fa 9 celneighborhood to a given object but does consider further adja F 12 cell neighborhood ok cent cells as a separate object A a a A Second there are three different neighbourhoods used to define adjacency to a given cell e the 4 cell neighbourhood includes only the four immediate north east south west neighbours e the 8 cell neighbourhood includes the eight immediate neighbours in cluding the four diagonal neighbours e the 12 cell neighbourhood includes the eight immediate neighbours and additionally the four next north east south west neighbours The 4 and 8 cell neighbourhood assigns only cells which touch directly to ob jects whereas the 12 cell neighbourhood allows you to have objects with small gaps 28 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Separation of joined objects In some cases discrete objects of a given pattern may touch each other In this case the automatic algorithm implemented in Programita will join the two ob jects to one single object One possibility to avoid this problem is to create a Select a null model new category for touching objects and to enable simulations 19 m 99 7 999 the check box Use category for defining ob Pattem1 and 2random jects f Pattern 1 fis pattern 2 random 7 Random labeling 7 _ Cluster proce
3. 4 a pO lier After termination of the simulations of the null model a graph appears showing the confidence limits of your null model right There is a slight clustering at scales r WO dl ard Jes THORSTEN WIEGAND 4 Univariate O ring statistic W M Oll r 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Spatial scale r cells Save results Heterogeneous Poisson null model HP If a univariate point pattern is not homogeneous i1 e the first order intensity of the pattern is not approximately constant the null model of CSR is not suitable for exploration of second order characteristics and a null model accounting for first order effects has to be used to reveal true second order effects A common assumption is that small scale departures from homogeneity CSR are due to interactions among the points whereas larger scale departures are due to exogenous factors Unfortunately large scale heterogeneity critically influ ences the second order characteristics at smaller scales and appropriate null models need to be used to reveal the true second order characteristics see for example the discussion on virtual aggregation in Wiegand and Moloney 2004 The heterogeneous Poisson process is the most simple alternative to CSR if the point pattern shows first order effects The constant intensity of the homogene ous Poisson process is replaced by a function A x y that varies with location x y
4. Masking space limitation Competition for space is an important ingredient of null models for plants with finite size We may encounter situations where plants of the focal species cannot inhabit some areas of the ground e g if it is already occupied by other species Programita allows considering restrictions in the accessible space in an easy way All non accessible cells are summarized as a third pattern called mask and are excluded from the study region The mask may be cells already indicated by code 1 dat or code 9999 asc as cells outside the study area but can be any other code number e g a third species The mask can be enabled by se lecting Irregularly shaped study region in the settings menu Give modus of analysis Note that in the case of an irregularly shaped study region the edge correction method 1 Plants are not allowed to fall outside is automatically enabled the two other two options do not make sense PF i oe id a b 7 ME _ Save map Save index Figure N6 Masking Data file m_ter_22 dat Two strips in the left and in the bottom of the plot are excluded code 11 and shrubs of other species as well code 22 Give code numbers for patterns if mi Patter f1 i 1 1 Pattern 2 2 2 2 blask 22 114 a fa 34 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Circle approximation Programita offers an option to analyze the effect of the irregular shape by ap
5. Your pattern appears on the left and the O ring function of your data appears on the right To determine Monte Carlo confidence limits for CSR enable the check box Calculate confidence limit on the upper left A window with settings for null models appears Select a null model simulations js Wo 39 f 999 Pattern 1 and 2 random f Pattern 1 fis pattem 2 random Random labeling Cluster process Toroidal shift pattem 2 moves Real shape J Pat1 fw Pat 2 al Circle Point Shape fram file If Only one point per cell Onl one point per pattern Heterogeneous Poisson Hard core Save null models W Use category for defining abject Select Pattern 1 fix pattern 2 random You can change the number of replicate simulations of the null model in the box simulations Press Calculate index Programita now performs the simu lations of the null model and shows you the pattern of the null model on the right Note that only the green pattern pattern 2 is randomized whereas the red pattern pattern 1 does not change 52 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA oor T oe hii i TFP i b pi ot w J j m d yn a f ay ia i _ 2 a 3 4 F a feel i n ta tt a i te 4 7 A ei o Fe eee i E E F y sg E a pe s 7 in o A A i seven 4 4 1 Save inder 5 After termination of the simulations of the null model a graph appears showing the confidence limits
6. points lt r from an arbitrary point of the pattern Al where means the number of and E is the expectation operator The K function yields under compete spatial randomness CSR K r nr which is the area of a circle with radius r To remove the scale dependence of K r under CSR and to stabilize the variance a square root transformation of K r called L function is used instead Lin Oy A2 The commonly used estimator for K r was proposed by Ripley 1976 and Ripley 1981 It is based on all distances dij between the ith and jth point of the pattern and is given by A3 where n is the number of points of the pattern in a study region of area A J is a counter variable Z d 1 if dj lt r and dj 0 otherwise and wij is a weighting factor to account for edge effects The weight wj is the proportion of the area of a circle centred at the ith point with radius dij that lies within the study region for reviews on edge correction see e g Haase 1995 or Goreaud amp P lissier 1999 This edge correction is based on the assumption that the region surrounding the study region has a point density and distribution pattern similar to the nearby areas within the boundary The bivariate quantity 42K12 r follows in a straight forward manner and has the 20 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA intuitive interpretation of the expected number of type 2 points within distance r of an arbitrary type 1
7. Random labeling 5 Cluster process Toroidal shift pattern 2 moves Real shape Patifw Pat Circle Point Shape fram file wW Only one point per cell T Onl one point per pattern Hard core Save null models W Use category for defining object Select Pattern 1 fix pattern 2 random and enable the check box Pat 2 to consider objects of finite size and irregular shape for pattern 2 You do not need to enable Pat 1 since pattern 1 is unchanged in this null model and does therefore not require definition of objects 3 Enable the check box Heterogeneous Poisson in the null model window A small settings window for the heterogene ous Poisson process appears Settings for hetero Poisson B a ap give radius A of circle Br Intensity Function from file Test only for pattern 1 f Test only for pattern 2 Test for joint pattern 1 and 2 W Show distribution Kernel m THORSTEN WIEGAND 33 insert the radius R of the moving window 20 and spec ify which data should be used for construction of the first order intensity With a bivariate pattern you have different possibilities You can use pattern 1 Test only for pattern 1 pattern 2 Test only for pattern 2 or pattern 1 and 2 Test for joined pattern 1 and 2 for construction of the first order intensity used for distributing the objects of pattern 2 Which option is most appropriate depends on your biologic
8. but the occurrence of any point remains independent of that of any other P lissier amp Goreaud 2001 Wiegand amp Moloney 2004 The grid based approach allows for straightforward generalization of a hetero geneous Poisson null model for objects of finite size if the heterogeneity occurs for scales well above the size of the plants The intensity function A x y is esti mated using a kernel estimate e g Bailey amp Gatrell 1995 Diggle 2003 Wiegand amp Moloney 2004 Programita includes two different kernel functions for estimating the intensity function vi x y a moving window estimate the default and an estimate using the Epanecnikov kernel To use the Epanecnikov kernel enable the check box kernel 42 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA I Intensity Function from file Test only for pattern 1 Test only for pattern 2 Test for joint pattern 1 and 2 Show distribution Kernel The first method implemented in Programita uses a moving window estimate of the non constant first order intensity A x y Points C R E Areal C R HP1 where C x y R is a circular moving window with radius R that is centered in cell x y the operator Points2 X counts the points of pattern 2 in a region X and the operator Area X determines the area of the region X As edge correction the number of points in an incomplete circle is divided by the proportion of the area of the circle that lies wit
9. dure is repeated until a location is found where both rules are met 32 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Test of possible bias through edge correction Programita allows you to assess the magnitude of possible bias in the distribution of objects relative to the border of the study area If you enable the check box Output data on bias in the Option window for edge correction FigureN4 the results file will contain at the end the line DistDistanceToBorder which gives the total number of cells with x coordinate x that were occupied by an object randomized during any simulation of the null model DistDistanceToBorder 0 1 2 0 63 139 203 268 301 310 322 330 369 403 416 22x Figure N5 Distribution of objects of finite size rela tive to the borders of the study area Shown is the total number of cells in 99 simulations of the null model shown in figure N3 which were occupied by a cell of pattern 1 Filled 450 400 350 A Taft SH ath Q A OO me o wt Oy 1 ew deo a i goi Y b gt Q Q si 250 200 e Method 1 o Method 2 150 Mean 100 Inner mean circles method Plants are not allowed to fall outside open circles method Tor oidal correction Horizon Number of cells distributed 50 0 l 21 4 61 81 101 121 tal expected number of x coordinate cells Figure N5 shows that with method 1 Plants are not allowed to fall outside there are less objects of
10. pattern 2 ran bes a dom without overlap to objects of pattern 1 phx 4 Bottom right randomization of objects of aki x pattern 2 with overlap to objects of pattern 1 HF 5 d s Overlap is indicated by orange color Note 217 FF y that only a few objects overlap i 7 P gt i z d f ngi E P z 7 F 4 T i mae i d a m q came index fi THORSTEN WIEGAND 2 Construction of objects from categorical map A categorical map such as figure N2 top left contains intuitively discernable discrete objects however in order to randomize the position of these objects for a given null model Programita must first define objects This is done with a percolation algorithm which summarizes all adjacent cells of the same type 1 e type or type 2 as one object If pattern 1 of your categorical map comprises objects of finite size and irregular shape enable in the window for the selection of the null model figure N1 the check box Patl If pattern 2 of your categorical map comprises objects of fi nite size and irregular shape enable additionally check box Pat2 After ena bling Patl and or Pat2 a small window Patch determination opens which allows you to select specific settings for the definition of the objects First to avoid very large objects you can define I Sa ne a maximal size of the objects given in number d eee mas patch size pat 1 100 of cells ae patch size he l
11. proximating objects with irregular shape by circular objects of the same size To this end enable the check box Circle in the window for specifying the null model below the real shape check boxes E l b Ps pi E ezi Select a null model j Pa P simulations 19 C 99 C 999 eo w P Pattern 1 and 2 random d Pattern 1 fix pattern 2 random 9 T C Random labeling hs tt a i Cluster process r r 1 Toroidal shift pattern 2 moves I E p Real shape Pat1 Pat2 i a r ra E bi E M Circle Point Shape from file j a 4 SIG z J Only one point per cell F ra Only one point per pattern 7 Heterogeneous Poisson i b Hard core z R 1 Save null models i S p S d Use category for defining object E 4 1 avemap T E _ Figure N7 Circle approximation Left data with objects of irregular shape right null model with approximation of objects by circle of same size Note that the circles are sometimes not perfectly circular be cause they have to be approximated by the underlying grid Point approximation Programita offers an option to compare the analysis with objects of finite size and irregular shape to the common point approximation Programita approxi mates the objects by a point with coordinates being the average x and y coor dinates of the cells belonging to the object To this end enable the check box Point in the window for speci
12. you can mask if required additionally cells which are part of the original study region For example if you study vegetation maps with category 0 bare ground category 1 grass tufts size of one cell and category 2 shrubs size of several cells you may mask the area occupied by shrubs for studying the spatial pattern of the grass tufts If you do not exclude the area occupied by shrubs which cover perhaps 10 or so of the study region a simple null model that randomizes the locations of the grass tufts CSR will distribute tufts at locations where they cannot occur in the field This may introduce a bias in the analysis The possibility to use up to 1000 categories is a convenient feature because you can use the same data for different analyses Be sure that a given code number does not appear in different categories A given cell can either be pattern 1 pat tern 2 empty or mask Format of the dat matrix data file The dat matrix data file is a space or tab delimited ASCI file The first line contains information on the dimensions of the grid 1 number of lines 1 num ber of columns The following lines are the data matrix with the different code numbers In contrast to the ArcView ASCII matrix format you need to insert line breaks Note that the visualization of Programita corresponds to the trans posed matrix figure 9 Give code numbers for patterns 140 1 20 Fatemi j1 f qi H o o a a a d OF O ai 8 aa 8D OD
13. Since you have only one pattern enable pat 1 the in tensity function is used for randomization of pattern 1 and press ok The intensity function appears on the right Click OK in the message window x This is the first order density that will be used in your null model Press Calculate index Programita now performs the simu lations of the heterogeneous Poisson null model and shows you the pattern of the Monte Carlo null model on the FLOht 48 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Toroidal shift Independence of bivariate patterns The null model testing for independence of two point patterns assumes that two independent stochastic processes created the two component patterns e g Goreaud amp Pe lissier 2003 Departure from independence indicates that the two patterns display attraction or repulsion regardless of the univariate pattern of either group by itself Thus the separate second order structures of the patterns in their observed form has to be preserved the dependence between the two pat terns however has to be broken Goreaud amp Pe lissier 2003 This can be achieved by simulations that involve random shifts of the whole of one component pattern relative to the other The same idea applies for plants of finite size and irregular shape pattern 1 remains fixed whereas pattern 2 is ran domly shifted as a whole across the study area using a torus toroidal shift Note that this
14. Spatial scale r cells Since the departures from random labeling occurred for small scales r lt 8 which are within the radius of the shrubs they may be due to slight differences in the size distribution of pattern 1 and pattern 2 Indeed Mulinum shrubs code 12 are at scales r 1 7 dm larger than Senecio shrubs code 13 Mulinum spinosum Senecio filaginaides 1 3 2 3 3 4 4 5 3 6 6 7 7 8 8 9 9 10 gt 10 2 2 9 3 4 4 5 5 6 6 7 7 8 Diameter dm Diameter dm This illustrates that the interpretation of the results of random labeling of objects with finite site and ir regular shape has to be done with care THORSTEN WIEGAND 6l References Bailey T C and Gatrell A C 1995 Interactive spatial data analysis Longman Scientific amp Technical Besag J 1977 Contribution to the discussion of Dr Ripley s paper Journal of the Royal Sta tistical Society B 39 193 195 Diggle P J 2003 Statistical analysis of point patterns Second edition Arnold London Dixon P M 2002 Ripley s K function Encyclopedia of Environmetrics 3 1796 1803 Goreaud F and Pelissier R 1999 On explicit formulas of edge effect correction for Ripley s K function Journal of Vegetation Science 10 433 438 Goreaud F and R P lissier 2004 Avoiding misinterpretation of biotic interactions with the intertype K 2 function population independence vs random labeling hypotheses Journal of Vegetation Science 14 681 6
15. THORSTEN WIEGAND 17 The Settings Menu for Point Pattern Analysis What do you want to do 7 Point pattern analysis 7 Homogeneity test 2 Calculate confidence limits Input data file Screen size 1644_DBH 5 da a 800 600 f 1024 x 768 a dat a_U dat a 1 dat How are your data organized List only dat f Matrix dat or asc Give modus of analysis 2 Combine replicates f Analyze all data in rectangle Irregular shaped study region Which method will you use f Ring wiegand Molonep 3 Circle Ripley mm ring width change set maximal radius max ir to default Select modus of data f Data are given as matris J Data are given as listin grid 7 O List with coordinates no grid 7 Give code numbers for patterns 77 Pr E7 iN E Pattern 1 Pattern 2 Figure 10 The settings menu for Point pattern analysis If you select in the menu What do you want to do Point pattern analysis Programita allows you to select different types of analysis input data input data formats etc If you do not use a res settings file that stores the settings from a previous analysis you need to carefully select all settings manually from the settings menu before performing any analy SIS Programita calculates Ripley s L function Circle in Which method will you use and Wiegand Moloney s O ring statistic Ring in Which method will you use in a grid based impl
16. and Senecio filaginoides white Each cell is 10cm x 10cm Considering finite size and irregular shape One of the limitations of point pattern analysis in plant ecology is that the plants are idealized as points The point approximation is valid where the size of the plants is small in comparison with the spatial scales investigated but may ob scure the real spatial relationships at smaller scales the relationship in which ecologists are mostly interested when interactions among plants are studied Programita contains an extension of the grid based approach to point pattern analysis Wiegand amp Moloney 2004 to deal with objects of finite size and ir regular shape The basic idea is to represent plants in a study area by means of a categorical raster map with a cell size smaller than the size of the plants A plant is represented by one or several adjacent grid cells depending on its size and shape in a map representing categories such as bare ground cover of species 1 cover of species 2 and so on figure F1 The simple grid based estimators of the bivariate functions Kj2 12 and g12 r eqn A9 A10 and All can be easily extended to allow analysis of categorical raster maps such as Figure F1 The size of the smallest plants or other criteria for a minimal required resolution is used to define an appropriate size for the cells To calculate the second order statistics of categorical maps with the esti mators given in equations A
17. data file select Matrix in How are your data organized select Analyze all data in rectangle in Give modus of analysis select the code numbers for pattern 1 and pattern 2 in Give code numbers for patterns write 12 in all windows reserved for pattern 1 the species Mulinum spinosum and write 33 in all windows reserved for pattern 2 the code 33 does not occur in Figure2 asc therefore you define a univariate pattern Give code numbers for patterns Pattern 1 12 12 1212 mw Pattern 2 33 33 33 33 m select Ring Wiegand Moloney in Which method will you use if you like to use the O ring statistic and Circle Ripley if you like to use the L function select an appropriate ring width dr in the box ring width Usually a ring width of one cell is appropriate however ift the intensity A of points in the study region is too low the graph of the O ring statistic will be jagged and selection of a larger ring width dr is appropriate click the button change in set maximal radius rmax to define the maximal scale r of the analysis and insert 30 A too large scale rwx will slow down Programita select Data are given as matrix map in Select modus of data Click button Calculate index Your pattern appears on the left and the O ring function of your data appears on the right To determine Monte Carlo confidence limits for CSR enable the check box Calculate confidence limit on the upper left A window with settings for null mo
18. in ArcView raster format dat files example data files and temporary files int files plug in files for heterogeneous Poisson rep files data files to show results of previous analyses res files results and settings files The manual of Programita Manual FiniteSize Programita2006 pdf and a HTM version of the manual Manual FiniteSize Programita2006 zip are pro vided separately You can use the HTM version as help because it contains many textmarks and internal links for easy navigation through the document THORSTEN WIEGAND 7 Screen size Programita was designed for a screen of 1024 x 768 pixels but it can be run as well using an 800 x 600 screen If you execute Programita in the 1024 x 768 pixel mode it must look like the segment shown in figure 1 Sometimes win dows within Programita are truncated and one cannot see all of some buttons or headers In this case it is as 1f the window is too small to handle them To avoid this problem check the default letter size in the settings of your computer Your computer may scale the letters but not the window sizes and as a consequence the windows appear too small Programita for point pattern analysis pon _ 3 x Point pattern analysis with Ripley s L and the O ring statistic What do you want to do Point pattern analysis Homogeneity test Cluster detection I Combine replicates I Calculate confidence limits Input data file Screen size 1644 DBH 5da af
19. null model for univariate point patterns 1s complete spatial randomness CSR where any point of the pattern has an equal probability of occurring at any position in the study area and the position of a point is independent of the position of any other point 1 e there is no interac tion Plants of finite size and irregular shape are in analogue to CSR distributed randomly as described above This null model operates as a dividing hypothesis to detect further regularity or aggregation in univariate patterns CSR and rectangular study region Example F_CSR_1 res CSR is the basic null model for univariate patterns and most settings for CSR will apply equally for other univariate null models Therefore we explain all steps of the analysis for CSR in detail but skip some of these details in the de scription of the other univariate null models 36 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Input data file Figure2 asc How are your data organized Matrix Give modus of analysis Analyze all data in rectangle Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low select larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 12 12 12 12 shrub species Mulinum spinosum Pattern 2 33 33 33 33 code does not occur highlight the data file Figure2 asc in the window Input
20. null model uses the torus method to treat plants that fall partly out side the study rectangle For this reason no objects need to be defined and ran domized 1 e you do not need to enable real shape Pat1 and or Pat2 and the torus method requires that objects of different patterns are allowed to over lap The latter may introduce a bias towards repulsion at small scales if the ob jects are known not to overlap As an alternative a univariate null model could be introduced to describe the second order structure of pattern 2 and if appropri ate an overlap of plants could be prohibited Toroidal shift Example F_In_1 res This example applies a heterogeneous Poison null model to the previous exam ple Input data file m_te22 dat How are your data organized Matrix Give modus of analysis Analyze all data in rectangle Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low select larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 1 1 1 1 shrub species 1 Pattern 2 2 22 22 22 other shrub species 1 Click button Calculate index Your pattern appears on the left and the O ring function of your data appears on the right THORSTEN WIEGAND 49 To determine Monte Carlo confidence limits for CSR enable the check box Calculate confidence limit on the up
21. of replicates Pattern 1 and 2 random the Monte Carlo simulation of null models dif Piem iik paitem 3 random 9 fers slightly from the point mode Under the Random labeling mode Matrix the null model does not allow to Fuster process Toroidal shift pattern 2 moves have the same category two times in a given Real shape Pat1 Pat2 cell However if you enable the checkbox Shape from file m i E Wo Only one point per cell 7 Only one point per pattern in the null model Aa can Goce lm window figure 7 Programita allows having a T Heterogeneous Poisson mixed category where type 1 and type 2 are Bei Bie Save null models together in one cell Use category for defining abject Figure 7 The null model window for the mode Matrix THORSTEN WIEGAND 15 Preparation of data under matrix mode The input data are a matrix that can have the following code numbers e 0 1 2 999 if the cell is inside the study region e or 9999 if the cell is outside the study region mask Programita reads two different data formats in the matrix mode l a space or tab delimited ASCH file with the dat extension with line breaks 2 the ASCII format of ArcView a asc file without line breaks The head of the asc file must look like this ncols 144 nrows 45 xllcorner 1 ylicorner 1 cellsize 1 nodata_ value 9999 ncols gives the number of columns nrows the number of rows xllcorn
22. results of previous analyses ccccccccccccceececeeeeseeesseseeeeeeeeeeeeeeeeeaaaaas 10 DAV SUMS TESUILS O8 ENC andy GIS arse a a N E A 11 Temporary data eSenior a a O a Ea AA T 12 The input data files dat and asc data files ccccccccccccccessseeeeeceeeeeseeeeeees 14 Preparation of data under matrix mode ccccceeseecccccceceeeeeeeecccseeeeeeeees 15 Format of the adat maix data Te ennea e E E S 16 The Settings Menu for Point Pattern Analysis cccccesesseeeseeeeeeeeeeeees 17 Matie OA lashed cesta eine disks ses teaser T os I7 Irregularly shaped study reQion cccccccsssssssssssseeeeeeccceeeeeeeeeeeeaeaeeeseeeeeeees 18 Irregularly shaped study region for matrix data cceceseseeeeeeeeeeees 18 Maximum scales r and ring Width 7 0 ccccccecccccccceccccceeeeeeeeeeeaaaaseeeeeeeeeess 18 Background of second order Statistics ccccccccsssssssscccccsssssssssssscccccsssssssscsees 19 SECON Order Stas ICG aaa r iwateusiaatedvdveulaa una TO eee awa a 19 Definition of the bivariate K and L functions cc cccccccceeeeeeeeeeeeeeeeees 19 Definition of the g function and of the O ring Statistic eeeeeeees 20 Grid based estimators of second order statistics cccccccccceeeeeeeeeeeeeeeeeeeees 21 Grid based estimator of K and L function cccccccccccccceeeeeeaaeeseeeeeeeees 21 Grid based estimator of g and O function ccccccccceccccccceeeeesaeeeeeeesseeees 22 PCIE CHON
23. two temporal files temp patchNr1 dat and temppatchNr2 dat which are a matrix representation of pattern 1 and pattern 2 respectively but every object is coded with a number numbered according to the order in the list of objects given in temppatch dat and temp patch2 dat The beginning of temppatch1 dat is 4 9 5 2 4 6 23 17311 4 30 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA The corresponding part of temppatchNr1 dat is 1 13 130 0 0 0000 0 0 0000000000000 0 0 0000000000000 0 0 0000000000000 0 0 gogogo OMOMOANHADUAWNE A A nn nn nnn nnnnnoa YUU UU UU d d d g U GU OU Q DN Nn nn nn nnn nnnnnoa YV Y Y d GU UU 0 U GU GU GU GU U Q AA nn nN nnn nnnnnn oa YUU UU d d d 0 d g g GU GU U Q N A Nn nnn nn nnnn NAN V U YV Y U UYU U d gd dU dU U Q Nn A Nn Nn nn nn nnnn oa V U YV U U d U d gd U dU Q Nn nn fh Nn nN nnn nnnnn oa YV UU UYU YUU d U g g U U U Q Xi eZ Ae DIO TE g G ae e Note that the matrix counts the x axis from left to right and the y axis from top to bottom Therefore the minimal coordinates of object 1 are 5 2 and of ob ject 2 MS Randomization of the position of objects Several null models require randomisation of the position of the objects In con ventional point pattern analysis this is easy randomisation of points involves only assignation of random coordinates However a plant of fin
24. 10 and All a point pattern is formally constructed from the map Fig F2 24 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA LYO g 7 E b Ha mo j e ta i au E i a oP i t e HTHH a y P 4 a 1 k i 4 ls a i Be a a ee iS T a a zx r je m at ca k i i a ta aes al fe h a Figure F2 Left categorical map with a bivariate pattern red shrubs of species 1 green shrubs of species 2 Right Point pattern formally constructed from the categorical map shown on the left Note that the grid lines indicate 5 x 5 blocks of cells and not the cell size A given cell obtains one type 1 point if it is covered by one of the categories assigned to pattern e g all shrub species and one type 2 point if it is covered by one of the categories assigned to pattern 2 e g all grass species but is classed as empty if it is not covered by either pattern 1 or 2 but is located within the study area and is assigned to the mask if the cell is outside the study area Although this approach corresponds with conventional point pattern analysis the explicit consideration of real world structures i e objects of finite size and ir regular shape prevents an analytical treatment This extension is therefore a simulation based approach for testing specific ad hoc hypotheses about the spatial dependencies of objects in a particular system Randomization of objects finit
25. 16 18 20 22 24 26 28 30 Spatial scale r cells Here for comparison the results without overlap Note the stark differences in the confidence limits at smaller scales r lt 10 Bivariate O ring statistic W M 012 r ba 0 2 4 6 3 10 12 14 16 18 20 22 24 26 28 30 Spatial scale r cells 54 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Antecedent condition Example F_Ant_3 res This example applies a null model with antecedent condition to example F Ant l res but uses a heterogeneous Poisson null model for distribution of pattern 2 instead of CSR Input data file m_te22 dat How are your data organized Matrix Give modus of analysis Analyze all data in rectangle Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low select larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 1 1 1 1 shrub species 1 Pattern 2 22 2 2 2 other shrubs 1 Click button Calculate index Your pattern appears on the left and the O ring function of your data appears on the right 2 To determine Monte Carlo confidence limits for CSR en able the check box Calculate confidence limit on the upper left A window with settings for null models ap pears Select a null model simulations s 9 0 995 Fattem 1 and 2 random 7 f Pattern 1 fis pattern 2 random
26. 92 Haase P 1995 Spatial pattern analysis in ecology based on Ripley s K function Introduction and methods of edge correction Journal of Vegetation Science 6 575 582 Naves J T Wiegand E Revilla and M Delibes 2003 Endangered species balancing be tween natural and human constrains the case of brown bears Ursus arctos in northern Spain Conservation Biology 17 1276 1289 Ripley B D 1981 Spatial statistics Wiley Schadt S E Revilla T Wiegand F Knauer P Kaczensky U Breitenmoser L Bufka J Cerveny P Koubek T Huber C Stanisa and L Trepl 2002 Assessing the suitability of central European landscapes for the reintroduction of Eurasian lynx Journal of Applied Ecology 39 189 203 Stoyan D and Stoyan H 1994 Fractals Random Shapes and Point Fields Methods of geo metrical statistics John Wiley amp Sons Upton G and Fingleton B 1985 Spatial data analysis by example volume 1 point pattern and quantitative data John Wiley amp Sons Wiegand T and K A Moloney 2004 Rings circles and null models for point pattern analysis in ecology Oikos 104 209 229 Wiegand T Kissling W D Cipriotti P A and Aguiar M R 2006 Extending point pattern analysis to objects of finite size and irregular shape Journal of Ecology
27. C 800 600 a dat 1024 x 768 a_O dat 2 a_1 dat How are your data organized List only dat 2 Matrix dat or asc Give modus of analysis 2 Analyze all data in rectangle Ineqularly shaped study region Which method will you use Ring Wiegand Moloney Circle Ripley fi ring width change set maximal radius rmax default set to default Select modus of data Data are given as matrix 4 Data are given as listin grid List with coordinates no grid i Close Stop Load Settings for Example Replicates Give code numbers for patterns Remove buttons Pattern 1 Fe 7 H 7 Change patterns Patten2 11 11 111 i Series Christine Figure 1 Correct display of the Programita interface under the 1024 x 768 pixel mode 8 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA A quick start Execute Programita Execute and adjust Programita to your screen size Two options gt teen size are given a screen of 800 x 600 pixels and a larger screen of s 4 79 1024 x 768 pixels which is the default Load a settings file to redo an analysis There is a convenient way to quickly start with Calculate Index Close Stop Programita and to learn the settings You can read Load Settings for Example Replicates a file a res file that contains all setting of a pre Select a results file Pee vious analysis and redo this analysis For example you can rep
28. CSR enable the check box Calculate confidence limit on the upper left A window with settings for null models appears 40 11 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Select a null model simulations 99 Wo 99 f 999 Pattern 1 and 2 random Pattern 1 fis pattern 2 random Random labeling Cluster process Toroidal shift pattern 2 moves Real shape fw Paip Pat2 Shape trom file I Only one pont per cell T Onl one point per pattern Heterogeneous Poisson Hard core Save null models Use category for defining object Select Pattern 1 and 2 random You can change the number of replicate simulations of the null model in the box simulations Enable check box Pati to activate the real shape modus where Programita recognizes objects of several adjacent cells A windows opens Patch determination mas patch size pat 1 400 mas patch size pat 2 400 4 cell neighborhood g cell neighborhood 12 cell neighborhood cok Select in window Patch determination the maximal size of a shrub of pattern 1 select 400 as very large value and the type of neighborhood and confirm with ok Press Calculate index Now Programita performs the simu lations of the CSR null model and shows you the pattern of the Monte Carlo null models The objects do not overlap the black holes made by the other species and are only distributed inside the smaller study region
29. Extending Point Pattern Analysis for Objects of Finite Size and Irregular Shape using the Programita software A supplementary user manual with an collection of examples using the Programita software wr 2 x er e Hi i AA in k j eal m eb jE er or 4 4 J i P 7 e T b o F pa A E a 4 a REECE CE EEE EEE z Draft version 27 of July 2006 written by Thorsten Wiegand Department of Ecological Modelling UFZ Centre for Environmental Research PF 500136 04301 Leipzig Germany Tel 49 341 235 2479 Fax 49 341 235 3500 email thorsten wiegand ufz de SUPPLEMENTARY USER MANUAL FOR PROGRAMITA THORSTEN WIEGAND 3 Contents PPO CPO siusessecceves eaaceaectaivavs caaetaxseccaves i ccaseatea ta uas conse tenncedaueescsastieoeteeeessoseeenseuses 5 MANOS UAC shes fate E sas iaia succinic pacains iad wannesasstes deaein E E 5 Before startne aed 0 24 4701 7112 Reem eee eee Peek ee ene 5 Hardware TCQUIT CECI arrini n a Ena T A E ETO 5 Terms of use and copyright agreement ccccceeesseeeecccceceeeeeeeeeeaeaaeeeseeees 6 VANS CALA OM i AA E E E A E E E A AEE E A 6 E 1 2 ST A EAEE E ee E A E E ea en A 7 TAS 2 E OMe Mee nan Ne Cane mee Mery rer ie on Te SNe aa 8 ROC UUS FO Oi INI ir E E E E E EE 8 Load a settings file to redo an analysis ccc eeseesesseeeeccceceeeeeeeeeeeaasaeeeeeeees 8 W at happens onthe SChCEN l inercei n Sereciase eens E 9 Show
30. IOR NE Widia resien aae E ETT ISh 22 Considering finite size and irregular shap ccccsssssssscsccccccccsssssssseecees 23 Randomization of objects finite size and real shape ccccsssseeeeeeeeeeeseees 24 Overlap DEIW CCM pla Sereni ana A 25 Construction of objects from categorical map ccccccceeeeeecceeeeeeeeeeeeees 21 Separation of Joned OD CCS reisean i E E E NE 28 Manual separation of joined objects cc cccccccccccceeceeeeeeeeeeeeeeeeeeeseeeeeeeess 28 Randomization of the position Of objects cccccccccccecceeeeeeeeeeeeeeeeeeeeeees 30 Edge correction 1f objects fall partly outside the study rectangle 31 Test of possible bias through edge COrrectiOn cccccceececeeeeeeeeeeeeeeeees 32 Maskine Space MIMMAON arerin a inca eae 33 Circle approxima ON eia ni TE A A i 34 POG appProx NaNO en a 34 Null models for objects with finite size and irregular shape scccssseees 35 CSR Randomising plant position for univariate patterns 00000eonennnns00ss0 35 4 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA CSR and rectangular study region Example F_CSR_1 res eee 35 CSR in irregularly shaped study region with mask Example F CSR 2 iets Heterogeneous Poisson null model HP ceeeeesesseeeeecceceeeeeeeeeeeeeaeeeseeees Heterogeneous Poisson Example F HP 1 res ccc ecccccceeeeeeeeeeees 3 Heterogeneous Poisson null model plug in inten
31. O89 80 OBO AAA Patten 2 2 2 2 o oo o o 8 a oo OF 8 8 80 8 ai 8 8 8 Gli Al BD oO 2 0 0 0 2 0 OF 2 O oO 2 0 1 i 1i 1 8 0 0 D Mask 1 E o 2 o 0 2 oOo 0O 2 O Q i 1i i 1 1 i 1 D i D i i ih a 1 1 1 1 1 1 1 1 1 ig 0 2 1 rs 1 i i ih 1 1 1 1 1 1 1 1 0 ij i 1 1 1 1 F 1 i ih i ih 1 1 1 1 1 ih Li 0 i i 1 1 1 1 1 1 1 i i ih ih ih i i i ih 2 0 i 1 1 1 1 1 1 i i ih ih 2 ih ih ih 0 0 i i Li Li 1 1 1 i i i i ih ih ih i i 2 i 2 0 1 i Fa Li i i i ii i 1 Fid i ii i Li pi pi oi Li oO 1 i Li i T i T 1 1 i ra ih ii i i ii i 0 i i i i iji i i 1 1 i ih ii o i i pi i 0 i 0 Oo z iji i i Li T T i ij 2 oh gi o 1 i 0 Oo Oo ii F i Li ij iji i i i i i i 0 D 1 1 o 0 oO 1 1 i 1 1 ij ij i Li i ij 0 I 1 1 1 o 0 1 1 ii al 1 1 ij iji i i 1 1 i Oo 1I 1 1 1 i 0 Oo 1 1 i 1 1 ij iji i ih io i Oo 1 1 1 1 l 2 0 2 1 1 1 1 Oo i i Lt i i 1 1 1 1 1 1 o 0 Oo 2 1 1 1 1 i i L iH Oo 1 1 1 1 1 1 1 Figure 9 Example of a dat input data file for the matrix mode Shown are the file small matrix dat left and the visualization in Programita right Red cells of pattern 1 code 1 green cells of pattern 2 code 2 grey empty cells code 0 black mask with cells outside the study region code 1 The first line contains information on the grid 1 number of lines 1 number of columns Note that the visualization of Programita corresponds to the transposed matrix
32. Y USER MANUAL FOR PROGRAMITA Select in window Patch determination the maximal size of a shrub of pattern 1 select 400 as very large value and the type of neighborhood and confirm with ok Enable the check box Heterogeneous Poisson in the null model window A small settings window for the heterogene ous Poisson process appears Settings for hetero Poisson 2 E give radius F of circle Ar Intensity Function from file f Test only for pattern 1 Test only for pattern 2 Test for joint pattern 1 and 2 If Show distribution D Kernel if you want to use the Epanecnikov kernel enable the check box kernel insert the radius R 20 of the moving win dow or the bandwidth R if you use the kernel estimator and specify which data should be used for construction of the first order intensity Since you have only one pattern enable Test only for pattern 1 Press Calculate index Now Programita calculates the moving window estimate of the first order intensity Max Min No data On the left the pattern data on the right the estimated first order intensity The small black gaps are plants of other species which are excluded from the study area This is the intensity estimate using the Epane nikov kernel with bandwidth R 20 Note that it provides a somewhat smoother estimate of the intensity since cells further away are weighted lower but the general pat tern does not change The bandwi
33. al hypothe Sis In this example select Test only for pattern 2 Press Calculate index Now Programita calculates the moving window estimate of the first order intensity F hi gt b 4 b i Max Min On the left Programita shows the pattern data on the right the estimated first order intensity The blue holes are cells occupied by pattern 1 A message window ap pears ee x f This is the first order density that will be used in your null model Save intensity Abbrechen If you want to save the intensity function to use it in other analyses as plug in file select OK otherwise se lect Abbrechen If you save the file the name of the plug in file will be int name R r int where name is the name of the data file here m ter 22 and r is the se lected value for the radius of the moving window or band width here 20 Programita now performs the simulations of the heteroge neous Poisson null model and shows you the pattern of the Monte Carlo null model on the right After termination of the simulations of the null model a graph appears showing the confidence limits of your null model right 56 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Bivariate O ring statistic W M 01r 0 2 4 6 8 10 12 l4 16 18 2 22 24 26 29 30 Spatial scale r cells Antecedent condition Example F_Ant_4 res To show the influence of the selection of the pattern for definition of the first order intensity
34. an isolate specific distance classes More sophisticated estimators of the pair correlation function use kernel functions such as the Epane nikow kernel with a bandwidth A instead of rings with width w see Stoyan amp Stoyan 1994 and references therein scale r 1 Figure A2 Numerical implementation of the L function and the O ring statistic for an irregu larly shaped study region Points of pattern 2 are represented by closed circles the focal point i of pattern 1 as open circle within the red cell Note that we approximate circles and rings with the underlying grid structure Study region grey and white cells area outside the study region black cells Left For numerical implementation of Ripley s bivariate L function we count the number of points of pattern 2 inside the part of the circles around point i of pattern 1 which falls inside the study region 1 e the gray shaded area and the number of cells within this area Right For implementation of the bivariate O function we count the number of points of pat tern 2 inside the part of the ring around point i of pattern 1 which falls inside the study region i e the gray shaded area and the number of cells within this area THORSTEN WIEGAND 23 Figure F1 Categorical map of a 27 4m x 13m study plot in the semiarid grass shrub steppe in Patagonia Argentinia showing individuals of the dominant shrub species Adesmia campestris black Mulinum spinosum dark grey
35. aximal size of a shrub of pattern 1 select 400 as very large value and the type of neighborhood and confirm with ok Select in the window options for edge correction the rules for randomization of the objects Press Calculate index Now Programita performs the simu lations of the CSR null model and shows you the pattern of the Monte Carlo null models After termination of the Simulations of the null model a graph appears showing the O ring function of your data left and the confidence limits of your null model right 38 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Y g be n a i z Univariate O ring statistic W M k a a a a gt 012 3 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Spatial scale r cells eones e Bivariate O ring statistic W M p t s fi amp i d 4 L 4 P 4 a oat L 3 Ld 0 i i I I I I I i i I U I i i i i I i i i 1 4 z oe a 0123 45 67 8 9 1011 12 13 1415 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 P i Th Spatial scale r cells 16 To save the results of the analysis press the button Save results which appears below the graph with the re sults of the univariate analysis and insert a name for the result file The results file will be saved as ASCII file with a res extension in the same directory where programita exe is located It contains the settings your analysis and the results of the
36. confidence limits To show the results of a previous analysis apply the button Replicates figure 2 and a window with a list of results files appears figure 4 Highlight fig 2A rep press select file and Calculate joined statistic The result of the analysis will appear figure 5 Univariate O ring statistic W M Select the result files of single replicates Ei E d daten bd fig2A rep C O1l r 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Spatial scale r cells Save results Bivariate O ring statistic W M O12 r Name of results file hame Only files for random labeling Joined statistic for random labeling 3 Remove file Calculate joined statistic Close 02 4 6 8 10 12 14 16 18 20 22 24 2 28 W Spatial scale r cells Figure 4 A list with result files which were Figure 5 The results of the previous analysis previously saved fig2A rep THORSTEN WIEGAND 11 Save the results of the analysis To save the results of the analysis press the button save results that appears below the graph with the results of the univariate analysis figure 3b and insert a name for the result file The results file will be saved as ASCII file name res in the same directory where the exe file of Programita 1s located The results file figure 6 contains the settings of this analysis and the results of the univariate and the bivariate point pattern analysis The results file name res can be used i
37. dels appears Se lect Pattern 1 and 2 random 11 12 L3 14 15 THORSTEN WIEGAND 37 Select a null model simulations sa i 99 999 f Pattern 1 and 2 random Pattern 7 fix pattem 2 random C Random labeling Cluster process Toroidal shift pattern 2 moves Real shape fw Paip Pat2 Shape from file WwW Only one point per cell Onl one point per pattern Heterogeneous Poisson Hard core Save null models Use category for defining abject You can change the number of replicate simulations of the null model in the box simulations If you want to use the 5th lowest and highest values of 99 or 999 Monte Carlos simulations as confidence limits enable the check box Oo or O99 Enable check box Pati to activate the real shape modus where Programita recognizes objects of several adjacent cells Two windows open Patch determination appear on the left over the map and Options for edge correction right of the window for specification of the null model Patch determination Options for edge conection mas patch size pat 1 400 Randomized objects will appear less mas patch size pat 2 400 frequent close to the border Plants not allowed to fall outside 4 cell neighborhood Toroidal correction Oy Scolinsigibot eed Smaller study aeg 4 12 cell neighborhood cok I Output date on bias Select in window Patch determination the m
38. distances around r are relatively less frequent than they would be under CSR If this is the case for small values of r the pattern shows regularity THORSTEN WIEGAND 21 Grid based estimators of second order statistics The study area is divided into a grid of cells The size of the cells should be de fined as the minimal resolution necessary to respond to the scientific question to be answered and 1s limited by the measurement uncertainty of point coordinates Wiegand amp Moloney 2004 The calculation of point to point distances neces sary for estimation of second order statistics is then based on distances between cells and counting cells and points in cells Grid based estimator of K and L function Wiegand amp Moloney 2004 proposed a simple grid based estimator of the bivariate K function for arbitrarily shaped study regions which is based on the mean number of type 2 points found in complete or incomplete circles of ra dius r around all type 1 points k Pj2 r divided by the area of these circles A I 1 Points C Ar A K r ar A g meo A9 a S Area C r Ny k where 42 n2 A is the intensity of pattern 2 ni is the number of type 1 points 1 1 2 in the study region C 7 is the circle with radius r centred on the Ath type 1 point the operator Points2 X counts the points of the pattern 2 in a region X of area A and the operator Area X determines the total number of cells of the r
39. dth R has a much lar ger impact on the intensity function than the specific selection of the kernel Max Min Wo data If you want to save the intensity function to use it in other analyses as plug in file see next example select OK otherwise select Abbrechen If you save the file the name of the plug in file will be int name R r int THORSTEN WIEGAND 45 where name is the name of the data file here m ter 22 and r is the selected value for the radius of the moving window or bandwidth here 20 f This is the first order density that will be used in your null model Save intensity i l Abbrechen Programita now performs the simulations of the heterogene ous Poisson null model and shows you the pattern of the Monte Carlo null model on the right The objects do not overlap the black holes made by the other species and are only distributed inside the smaller study region but more frequently in the pink and red areas of the intensity map wR T b b p a bu Save map Save index After termination of the simulations of the null model a graph appears showing the confidence limits of your null model right The slight clustering at scales r 10 ll and 12 visible after applying the CSR null model disap pears with the heterogeneous Poisson null model Univariate O ring statistic W M Ollir Spatial scale r cells 46 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Het
40. e correction applied for inner parts of the plot Figure N5 Soe e BORO 4 J Output data on bias Toroidal correction The second method avoids the negative bias produced by the first method by treating the rectangular study plot encompassing the study region as a torus 1 e the part of a plant outside the rectangle is made to appear at the corresponding opposite border However breaking relatively large plants into two smaller plants produces a slight positive bias Smaller study area A third method uses the torus correction but calculates the second order statistics only inside an inner rectangle excluding cells close to the border guard area You can select the wide of the guard area figure N4 For a guard area wider than the diameter of the largest plants the biases of the first and the second method disappear but this may reduce the size of the study rectangle considerably Therefore the guard area may be selected wider than the diameter of most plants but still small enough to yield a large enough study rec tangle To randomise the plants of a given pattern under the above rules Programita performs repeated trials for each individual plant If the provisionally distributed plant which was randomly mirrored and shifted overlaps with an already dis tributed plant of the same pattern or the other pattern if appropriate or falls partly outside an irregularly shaped study area this trial is rejected The proce
41. e size and real shape The key for successful application of Programita is the selection of an appro priate null model that responds to the specific biological question asked The approach for dealing with objects of finite size and irregular shape allows for testing specific null models adapted to the biology of the study system and pro vides a considerable degree of added realism However remember that the null model constitutes a point of reference against which the data are compared Therefore the null model must not necessarily describe all particularities of the real system THORSTEN WIEGAND 25 Specific null hypotheses on the considered system can be tested through analy sis of the spatial pattern of plants with real shape and finite size For example are plants of a given species randomly but without overlaps distributed over the study area are the spatial patterns of two species independent or do plants of species 2 suffer competition or facilitation from plants of species 1 In a later section we present several important benchmark null models to address these questions Although conceptually analogous to the most simple null models in point pat tern analysis 1 e CSR heterogeneous Poisson independence random label ling and antecedent condition Wiegand amp Moloney 2004 consideration of the finite size and irregular shape of plants requires specification of more biological detail Specifically we need to introduce r
42. eat all analysis show in the figures 2A Pe C in Wiegand et al 2006 out_name res Figure 2 Load an example settings file To load a settings file apply the button Load Settings for Example figure 2 and a list with files containing settings of old analysis will appear figure 2 Select a res file for example fig2A res press ok and then the button Calculate Index Now Programita performs the analysis of figure 2A in Wiegand et al 2006 THORSTEN WIEGAND 9 What happens on the screen After loading the settings file fig2A res Programita will automatically select all settings for the data and analysis mode and all settings for the null model that was used in the example fig2A res Two plots will appear on the left appears a plot showing the original point pat tern being analyzed figure 3a and on the right appear the patterns of the Monte Carlo simulations of the null model used for constructing the confidence limits After terminating the simulations of the null model the figure with the simulated patterns of the null model disappears and instead a figure with the result of the analysis appears on the right figure 3b Figure 3a Left the point pattern analyzed in 4 i 4 a nm gt ee S fig2A in Wiegand et al 2006 Right One r hk realization of the Monte Carlo null model that ae l conserves the shape of the shrubs a random p a aie vi oF E 6 pattern CSR u
43. egion X The circles are incomplete if the focal point has a distance smaller than r to the border of the study region Note that this estimator does not scale the number of points in incomplete circles to the expected number within complete circles as is commonly done in point pattern analysis for reviews on edge correction see e g Haase 1995 or Goreaud amp P lissier 1999 The grid based estimator is therefore not affected by the problem that the weights may become unbounded if r becomes larger The grid based estimator of the Z function using equations A2 and A9 yields A F r r r an l La r r y rCj n AO 1 A10 22 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Grid based estimator of g and O function An analogous grid based estimator of the bivariate pair correlation function 212 r is given by D Points Rik r L t 2 A11 ny Zz gt Areal R r 1 k 1 where R r is the ring with radius r and width w centred on the kth type 1 point Wiegand amp Moloney 2004 For the univariate case this estimator was e g used in Condit et al 2000 Selection of ring width The estimator equation A11 involves a technical decision on the width w of the rings The use of rings that are too narrow will produce jagged plots as not enough points will fall into the different distance classes On the other hand if the rings are too wide the pair correlation function will lose the advantage that it c
44. ementation for a given data file selected in Input data file In the window How are your data organized you can select between two types of input data 1 data which are given as a list of points and 2 categorical data which are organized as a matrix Matrix data If your data are a matrix categorical data you need to select Matrix in How are your data organized Data are given as matrix map in Select modus of data and specify in the window Give code numbers for pattern which code numbers of your data matrix make up pattern 1 pattern 2 and the mask The mask defines the area outside the study region if your study region is irregularly shaped 18 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Irregularly shaped study region You can consider any arbitrarily shaped study Give modus of analysis region supported by the grid structure If you se Analyze all data in rectangle lect in the window Give modus of analysis the option 0 ttt Irregularly shaped study region some cells of the rectangular grid are not considered during the Monte Carlo simulation of the null models and cells outside the study region are not counted for the numerical implementation of the Z function and the O ring statistic In contrast if you select Analyze all data in rectangle the study region is the rectangle defined by your grid and all cells of the rectangle count and all cells are considered for simulation of the Monte Carlo Null mode
45. er the smallest x coordinate and yllcorner the smallest y coordinate The cell size must be 1 and the value for no data the mask must be 9999 The matrix mode allows you to use a data ma l Give code numbers for patterns trix with different code numbers however Pattern f3 f4 I5 5 calculation of Wiegand Moloney s O ring sta Faten f 2 tistic and Ripley s L function Programita re Mask qaqa Wal quires a reduction of the original code numbers ee to the four categories Figure 8 Transformation of the original code numbers of the data e the cell is of type 1 pattern 1 matrix to the three categories pattern e the cell is of type 2 pattern 2 1 pattern 2 and mask outside the study region All other categories e the cell is outside the study region which are not set are automatically mask defined as empty cells e the cell is empty If you enable the Matrix or Data are given as matrix option the window Give code number for patterns figure 8 appears and ask you to group your code numbers into the final categories pattern 1 pattern 2 and mask All other cells with code numbers not defines as pattern 1 pattern 2 or mask are defined automati cally as empty cells You can combine up to four code numbers but not 1 to define pattern 1 and pattern 2 and up to four categories including 1 to define the area outside the study region 16 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Thus
46. erogeneous Poisson null model plug in intensity functions Example F_HP_2 res The heterogeneous Poisson null model uses the larger scale intensity of the pat tern for construction of the first order intensity function to reveal the small scale second order characteristics of the pattern In some cases however you may have additional information on variables that determine the environmental heterogeneity e g elevation or soil type In this case Programita allows you to use a plug in file containing the intensity function The plug in a file has the extension int and the following design 1 150 1 50 15573 0000 1 1 1 0 857860 1 21 0 799896 1 31 0 749270 130 129 1 0 074526 130 150 1 Oe T2002 The first line contains as usual the information on the grid size there are 150 cells in x direction and cells in y direction the file contains information on a total of 15573 cells if the study area has irregular shape cells outside the study area do not appear in the list and the grid size is MOOO this information is for the point modus in matrix modus it is always 1 since you cannot change the cell size The first column gives the x coordinate the second column the y coordinate the third column is always 1 and the fourth column gives the value of the intensity function at the given cell Note that the intensity function must be scaled to a maximum value of 1 and should not have negative values Be sure that the dimens
47. f plants of pattern 2 occur more frequently in the neighbourhood of plants of pattern 1 than expected under a random distribution of pattern 2 or expected by the larger scale intensity of pattern 2 Conversely competition 1s indicated if they occur less frequently The appropriate bivariate null model for this question keeps the plants of pattern 1 fixed and distributes the plants of pattern 2 randomly over the area of the study area not occupied by plants of pattern 1 called antecedent condition Wiegand amp Moloney 2004 Note that the null models antecedent condition and independence are not the same because antecedent condition does not preserve the observed second order structure of pattern 2 and thus makes a spe cific assumption on the second order structures of pattern 2 If plants of pattern 2 show large scale variations in their intensity one may distribute them in ac cordance to a heterogeneous Poisson null model which keeps the larger scale intensity of the pattern determined as kernel estimate eq HP1 HP2 and HP3 but destroys all second order structures at smaller scales With a bivariate pattern you have different possibilities to randomize pattern 2 if you select a heterogeneous Poisson null model you can use pattern 1 pattern 2 or the joined pattern 1 and 2 for construction of the first order intensity used for distributing the objects of pattern 2 Which option is appropriate depends on your biological h
48. ferent aspects of random labelling e univariate random labeling uses g1 7 as test statistic e conventional bivariate random labeling uses g 2 r as test statistic Goreaud amp Pe lissier 2003 e the differences g r 21 which tests if pattern 2 is more clustered than pattern 1 e the difference gj2 r g r which evaluates if there are relatively more type 2 points than type points in rings circles of radius r around type points Random labelling for objects of finite size can be performed in analogy to ran dom labelling of point patterns here one needs to randomly assign case labels to n of the n m objects of the joined pattern However the finite size of objects may introduce violation of the assumption of random thinning if the number of plants is small and the size class distribution of the two patterns 1s not approximately the same Therefore care is required by interpreting the results of random labelling for objects of finite size and irregu lar shape Departures from random labelling may stem from differences in the number of objects in pattern 1 and pattern 2 and the size structure of the two patterns 58 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Random labeling Example RL_1 res In this example we hypothesize that the labels pattern 1 and pattern 2 of objects of a bivariate pattern are randomly distributed and apply the random labeling null model To test the random labeling hypothesis m
49. fying the null model below the real shape check boxes Select a null model fi g Give number of replicates C Random labeling Cluster process Toroidal shift pattern 2 moves Real shape V Pat1 Pat2 Circle Point Shape from file J Only one point per cell Only one point per pattern Heterogeneous Poisson Hard core Save null models Use category for defining object Pattern 1 and 2 random z Pattern 1 fix pattern 2 random 3 THORSTEN WIEGAND q Save map 1 35 Figure N7 Point approximation Left data with objects of irregular shape right Point approximation of data shown on the left Null models for objects with finite size and ir regular shape Although the null model for finite size and irregular shape are conceptually analogous to the most simple null models in point pattern analysis 1 e CSR heterogeneous Poisson independence random labelling and antecedent condi tion Wiegand amp Moloney 2004 consideration of the finite size and irregular shape of plants requires specification of more biological detail In the last section we provided all technical details for definition of the null models in the follow ing we will present several examples for null models which use the feature of categorical maps and objects of finite size and irregular shape CSR Randomising plant position for univariate patterns The most simple and most widely used
50. gramita is intended to foster analysis of point pat terns in ecology by providing ecologists a tool that contains null models and pro cedures not supported by most statistical packages but which are essential for a throughout analysis of point patterns The Programita software is not a commer cial venture and may be downloaded and used free of charge for purposes of scientific research and teaching Any commercial application of the program requires the previous permission by the author Publications must acknowledge use of the Programita and cite Wiegand and Moloney 2004 which describes the basic implementation and the procedures used by Programita and Wiegand et al 2006 for analyses of finite size and irregular shape Installation There is no setup procedure installation of Programita requires only the extrac tion of all files from the zip file Progamita FiniteSize zip Make sure that you also access the PDF Manual FiniteSize Programita2006 pdf and HTM ver sions Manual FiniteSize Programita2006 zip of the supplement user manual Place the files into a directory of your choice extracting the zip file will place all files into the sub directory Programita Note that you must place all files in the same directory for simplicity Programita does not use a path variable The zip file contains the following files and file types ProgramitaJulio2006 exe the executable of Programita version 26 of July 2006 asc files example data file
51. he width of the object in y direction ymax ymin the fifth column the number of cells of the objects and the following columns give the pairs of coordinates belonging to the object For example the beginning of the temppatch1 dat is given by 126 1 185 2 2 6 1 186 1 187 2 186 2 187 2 185 3 186 and indicates that there are a total of 126 objects and the minimal x and y coordinates of the first object are 1 and 185 respectively The width in x and y direction of the object is 2 cells xmax 3 xmin 1 ymax 187 ymin 185 and the object comprises the six cells 1 186 1 187 2 186 2 187 2 185 and 3 186 14 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA The input data files dat and asc data files Programita performs point pattern analysis for two different situations First it calculates the O ring statistic and the L function for point patterns which are basically given as a list of points with and without a predefined grid This corre sponds to conventional point pattern analysis In a second mode Programita performs point pattern analysis for categorical maps which can represent objects of finite size and irregular shape In this case the data input is a matrix with categories that can range from 0 to 999 In the following we discuss the data input for categorical maps Select a null model Because Programita works in the matrix mode with a category and not with number of points 12 _ ve number
52. hin the study region This is a simple kernel esti mate with the kernel function O lt d lt R e d HP2 0 d gt R where d is the distance between the focal cell and the counted cell and R the bandwidth The second method uses an Epane nikov kernel recommended by Stoyan and Stoyan 1994 2 Er O lt d lt R 4h R e R HP3 0 d gt R All points x surrounding a given location x were counted within distance R but weighted with the kernel er d where d is the distance between locations x and xi In general the Epanecnikov kernel produces a smoother intensity function than the mowing window estimate An appropriate bandwidth R needs to be lar ger than the typical size of the plants but smaller than the range over which the environmental gradient may vary For implementation of the heterogeneous Poisson process a given plant is provi sionally placed at a random location of the study region but this plant is only retained with a probability proportionally to AB x y averaged over all cells of the plant This procedure is repeated until n points are distributed THORSTEN WIEGAND 43 Heterogeneous Poisson Example F_HP_1 res This example applies a heterogeneous Poison null model to the previous exam ple Input data file m_ter_22 dat How are your data organized Matrix Give modus of analysis Irregularly shaped study region Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low se
53. ion of point pattern analysis to deal with objects of finite size and irregular shape e g plants The objects are approximated by using the underlying grid and may occupy several adjacent grid cells depending on their size and shape Null models correspond to that of point pattern analysis but need to be modified to account for the finite size and irregular shape of plants The procedures used by Programita for performing analyses which consider the finite size and irregular shape of objects are described in detail in Wiegand et al 2006 This document supplements the manual of Programita focusing on analy ses of objects with finite size and irregular shape Before starting Programita Hardware requirements Programita 1s a free unsupported software developed in Borland Delphi4 under a WindowsXP environment Programita is executable under 32 bit operating systems such as Windows98 Windows 2000 Windows XP or WindowsNT Running Programita requires little hard drive space For example for grid sizes lt 200 x200 cells Programita and temporally created files occupy lt 10M How ever analysis of larger grid sizes may be slow for small working memory and low computer speed 6 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Terms of use and copyright agreement The Programita software 1s produced by Thorsten Wiegand in his spare time He is affiliated at the Dept Ecological Modelling UFZ Centre for Environmental Research Leipzig Halle Pro
54. ions of the study rectangle and of the plug in file match exactly This example is the same as the previous example but a plug in file is used to provide the intensity Steps 1 5 are the same as in example F HP 1 res Input data file m_ter_22 dat How are your data organized Matrix Give modus of analysis Irregularly shaped study region Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low select larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 1 1 1 1 shrub species 1 Pattern 2 2 2 2 2 code does not occur Mask 22 11 1 1 22 other shrubs 11 outside strips THORSTEN WIEGAND 47 Enable the check box Heterogeneous Poisson in the null model window A small settings window for the heterogene ous Poisson process appears Settings for hetero Poisson H 37 E give radius F of circle Br Intensity Function from file f Test only for pattern 1 Test only for pattern 2 Test for joint pattern 1 and 2 If Show distribution Kernel to plug in a file containing the intensity function enable the check box Intensity function from file and a window with a list of all int plug in files appears Select a file with the intensity function ing pat 1 i pat pat and 2 close ok elect the iile you Wand to Use Int m Cer 22 R 20 11m
55. ite size may oc cupy several adjacent cells and its shape needs to be preserved Programita achieves this by rotating the objects by 0 90 180 or 270 degrees each of the four variants being equally probably and by randomly shifting the rotated plant as a whole Figure N3 E Figure N3 Randomization of position of objects Left data pattern right randomization of position of original objects of the data pattern Arrows indicate for some objects their rotated and moved counterparts of the null model THORSTEN WIEGAND 31 Edge correction if objects fall partly outside the study rectangle The finite size of the plants requires edge correction since randomly displaced plants may fall partly outside the arbitrarily selected study rectangle This would reduce in the null model the proportion 4 of occupied cells and produces a posi tive bias towards aggregation Programita includes three methods to mitigate this effect figure N4 Plants are not allowed to fall outside Ran Dpue Ua Saar eT can domized plants are not allowed to fall partly out E EE ie enoaiass requent close to the border side the study rectangle This produces a negative A Ciantenoballawecita cailantede bias towards regularity since fewer plants of the Toroidal conection null model are distributed close to the border Thus the intensity of cells simulated by the null Figure N4 Option window for model is smaller at the border and larger in the edg
56. lect larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 1 1 1 1 shrub species 1 Pattern 2 2 2 2 2 code does not occur Mask 22 11 1 1 22 other shrubs 11 outside strips 1 Click button Calculate index Your pattern appears on the left and the O ring function of your data appears on the right 2 To determine Monte Carlo confidence limits for CSR enable the check box Calculate confidence limit on the upper left A window with settings for null models appears Select a null model simulations E Wo 393 f 999 Patten 1 and 2 random Pattern 1 fix pattem 2 random Random labeling Cluster process Toroidal shift pattern 2 moves Real shape J Paip Paz Shape from file W Only one point per cell Only one point per pattem Heterogeneous Poisson Hard core Save null models Use category for defining abject Select Pattern 1 and 2 random 3 You can change the number of replicate simulations of the null model in the box simulations 4 Enable check box Pati to activate the real shape modus where Programita recognizes objects of several adjacent cells A windows opens Patch determination mas patch size pat 1 400 mas patch size pat 2 400 O 4 cell neighborhood f 8 cell neighborhood 12 cell neighborhood ok 44 SUPPLEMENTAR
57. ls Irregularly shaped study region for matrix data If your data are organized as matrix you can Give code numbers for patterns define a mask cells outside the study region Se EE i P ALETA Ib with the category 1 but additionally you can fake GT E use any code number of your data matrix as Ce mask Maximum scales r and ring width dr The analysis is performed for spatial scale r 1 i ring width max The default value of the maximal scale ee set maximal radius max Fmax 18 half of the dimension of the smaller side a to default of the grid however 7max can be changed with the button set maximal radius rmax If you select the O ring statistic you can change the ring width dr in the box ring width The default ring width dr is one cell however if the rings are too narrow Programita will produce jagged plots for O r as not enough points will fall into the different distance classes In this case you may select a larger ring width THORSTEN WIEGAND 19 Background of second order statistics Second order statistics Definition of the bivariate K and L functions For stationary and isotropic point processes all second order characteristics can be expressed by means of the intensity 4 and K r Ripley s K function The quantity AK r has the intuitive interpretation of the expected number of further points within distance r of an arbitrary point of the process which is not counted Ripley 1981 A K r E
58. n the same way as fig2A res in the previous section to load the setting and to re peat the analysis Pointpattern analysis of file D Programita Figure2 asc Method Wiegand Moloney ring with 99 replicates for confidence limits ring width 3 Test Model 12random Il1shape X 400 400 8 smaller 4 E 8 the null assumed homogeneous pattern s Analysis modus standard only one point per cell allowed All cells within the rectangle were considered for calculating the indices number cells of pattern 1 1015 number cells of pattern 2 0 the rectangular area contains 130 274 35620 cells diml dim2 pattern 1 was coded with numbers 11 11 11 11 pattern 2 was coded with numbers 2 2 2 2 the mask was coded with numbers 1 1 1 trr Oll r Bite Bitr 012 r HI E124 r r 0 3999665 0 3856460 0 4191957 0 0000000 0 0000000 0 0000000 r r 0 3003174 0 2881745 0 3228708 0 0000000 0 0000000 0 0000000 rr 0 1851738 0 1748624 0 2109540 0 0000000 0 0000000 0 0000000 rr 0 0968253 0 0864759 0 1244389 0 0000000 0 0000000 0 0000000 rr 0 0579106 0 0485426 0 0833683 0 0000000 0 0000000 0 0000000 rr 0 0391140 0 0313220 0 0652772 0 0000000 0 0000000 0 0000000 rr 0 0329553 0 0220457 0 0598031 0 0000000 0 0000000 0 0000000 rr 0 0361094 0 0185628 0 0582927 0 0000000 0 0000000 0 0000000 rr 0 0425961 0 0186002 0 0557966 0 0000000 0 0000000 0 0000000 rr 0 0484125 0 0196974 0 0514856 0 0000000 0 0000000 0 0000000
59. nd order structure than the joined pattern of case and control although there is a non significant tendency to be more clustered at scales r lt 10 Interestingly ex changing case and controls 1 e enabling g21 yields a different results Univariate random labeling with g function W M 012 3 4 5 67 8 9 1011 12 1314 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Spatial scale r cells Save results for scales r l 7 pattern 2 18 Significantly less clustered than the joined pattern 10 Using the test statistic gl2 gll shows that cells of pattern 2 surround cells of pattern 1 as cells of pattern ie 60 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Bivariate random labeling with g function W M gi2 gll 012 3 4 5 67 8 9 1011 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Spatial scale r cells 11 However the test with the test inverse statistic g21 g22 shows that cells of pattern 1 surround cells of pat tern 2 at scales r 1 7 less than cells of pattern 2 Bivariate random labeling with g function W M 0 Se 0123 4 5 6 7 8 9 1011 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Spatial scale r cells 12 Finally the test statistic g22 gll shows that pattern 2 is at scales r 1 7 less clustered than pattern 1 Bivariate random labeling with g function W M 22 gll 012 3 4 5 67 8 9 1011 12 1314 15 16 1718 19 20 21 22 23 24 25 26 27 28 29 30
60. of your null model right There is a slight tendency to aggregation at small scales r 1 3 since the objects in the data map left map touch more that the randomized objects right map Bivariate O ring statistic W M Or 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Spatial scale r cells THORSTEN WIEGAND 53 Antecedent condition Example F_Ant_2 res This example repeats analysis of the previous example F Ant 1 res but allows overlap between objects of pattern 1 and pattern 2 Comparison between exam ples F Ant 1 res and F Ant 2 res demonstrates the difference such small difference in settings of the null model can make Therefore you must carefully select the null model in accordance to the scientific question and knowledge on hand To allow overlap between pattern 1 and pattern 2 enable the check box Only one point per pattern in the null model window Select a null model simulations so IV 99 999 Pattern 1 and 2 random 7 f Pattern 1 fix pattern 2 random 7 Random labeling 7 Cluster process Toroidal shift pattern 2 moves Real shape fw Pad Pat 2 d Circle Point Shape from file Wo Only one point per cell W Only one point per pattem Heterogeneous Poisson Hard core Save null models W Use category for defining abject The departure from the null model observed in the previ ous example disappears Bivariate O ring statistic W M 02 4 6 8 10 12 14
61. omization of the objects of finite size and irregular shape The first two columns are the minimal x and y coordinates of the object the third column gives the width of the object in x direction xmax xmin the fourth column gives the width of the object in y direction ymax ymin the THORSTEN WIEGAND 29 fifth column the number of cells of the objects and the following columns give the pairs of coordinates belonging to the object For example the beginning of the temppatch1 dat may be given by 1 1 385 2 2 6 1 186 1 187 2 186 2 187 2 185 3 186 This indicates that there is 1 object and the minimal x and y coordinates of the object are 1 and 185 respectively The width in x and y direction of the object is 2 cells xmax 3 xmin 1 ymax 187 ymin 185 and the object comprises six cells 1 186 1 187 2 186 2 187 2 185 and 3 186 This is the objects with its coordinates 185 186 187 Z X X X X X X W N You can now split this object manually into two objects object 1 with coordinates 1 186 1 187 2 187 and object 2 with coordinates 2 186 2 185 3 186 185 186 187 185 186 187 i X X 2 X 2 x x 3 3 xX To split them you need to modify the file accordingly 1 e adding one objects 2 changing the minimal x and y coordinates the width the number of cells belonging to the object and To help you with this task Programita creates
62. ore comprehensibly we use several test statistics Input data file Figure2 asc How are your data organized Matrix Give modus of analysis Analyze all data in rectangle Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low select larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 13 13 13 13 shrub species 1 JUI A wN e Pattern 2 12 12 12 12 shrub species 2 highlight the data file Figure2 asc in window Input data file select Matrix in How are your data organized select Analyze all data in rectangle in Give modus of analysis select Data are given as matrix map in Select modus of data select the code numbers for pattern 1 and pattern 2 in Give code numbers for patterns write 12 in all windows reserved for pattern 1 the first shrub species and write 13 in all windows reserved for pattern 2 the second shrub spe cies Give code numbers for patterns Pattern 1 12 12 1212 H gt Fatem2 13 13 1313 m enable the check box Calculate confidence limit on the upper left A window with settings for null models ap pears Enable the check boxes Pat 1 and Pat 2 and se lect Random labeling Select a null model simulations a9 iw 99 999 Pattern 1 and 2 random Pattern 1 fix pattem 2 random f Random labeling Cluster
63. pattern 1 close to the border filled circles x coordinates 1 2 3 and 4 and 127 128 129 and 130 as expected horizontal line because objects partly overlapping the border of the study area are rejected However more than 4 cells away from the border the objects are randomly distributed Consequently the inner mean gray horizontal is slightly elevated This is be cause hardly an object has a diameter larger than 4 cells figure N3 The bias for cells closer than 4 cells from the border disappears for the method 2 Toroidal correction open circles in figure N5 Note that the size of the objects and the condition that they are not allowed to overlap leads to fluctuations in the density The third method of edge correction uses the second method Toroidal correc tion but measures the second order properties only inside an inner rectangle excluding cells close to the border guard area Because the objects are ran domly distributed this leads to a slightly fluctuating number of cells belonging to pattern 1 of the null model inside the inner rectangle To check the degree of fluctuation which may become large if the inner rectangle is small the results file contains at the end a line Anzpat1 and Anzpat2 which gives the number THORSTEN WIEGAND 33 of cells of pattern 1 and pattern 2 respectively in each replicate simulation of the null model Anzpatl 364 350 427 406 407 342 371 349 362 375 383 373 374 376 397 378
64. per left A window with settings for null models appears Select a null model simulations E O e 1 999 C Pattem 1 and 2 random 7 Pattern 1 fix pattem 2 random 7 Random labeling 7 Cluster process Toroidal shift pattern 2 moves Real shape Paip Paz Shape from file wW Onl one point per cell If Only one point per pattern Heterogeneous Poisson Hard core Save null models Select Toroidal shift You can change the number of replicate simulations of the null model in the box simulations Press Calculate index Programita now performs the simu lations of the Toroidal shift null model and shows you the pattern of the null model on the right Note that the green pattern is shifted as indicated by the black arrows and the objects may overlap indicated by orange color p a F F a qr Mw ar 2c i T i E n 7 T F a in j Eoi a i 4 Save index f i After termination of the simulations of the null model a graph appears showing the confidence limits of your null model right 50 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA Antecedent condition Randomising only pattern 2 The investigation of some processes such as investigating facilitation or compe tition 1s less concerned with independence but try to disprove instead that plants of one pattern are randomly distributed in the neighbourhood of the plants of the other pattern Facilitation is indicated i
65. point Definition of the g function and of the O ring statistic The pair correlation function g r Stoyan amp Stoyan 1994 is related to the K function by 1 dK r g r ip oe A4 Stoyan and Penttinen 2000 provide a heuristic definition g r is related to the probability p r that each of two infinitesimally small discs dx and dy which are a distance r away contain a point of the process P r 2 g r dxdy AS Rearrangement of equation A4 and dK r K r dr K r yields Ag r 4 K r dr K r 2ar dr A6 where 2ar dr is the area of a ring with radius r and infinitesimal width dr Thus for isotropic patterns the quantity AK r AK r dr 2ndr Ag r may be inter preted in analogy to equation Al as the expected number of points in a ring with radius r and width dr centred at an arbitrary point of the process A g r 2nrdr E r lt points lt r dr from an arbitrary point of the pattern A7 where means the number of and E is the expectation operator The O ring statistic O r Ag r A8 is thus the conditional intensity of points at distance r from an arbitrary point of the pattern For a homogeneous Poisson process CSR g r 1 Values g r gt 1 indicate that interpoint distances around r are relatively more frequent than they would be under CSR If this is the case for small values of r typically there is cluster ing Conversely values g r lt 1 indicate that interpoint
66. process Toroidal shift pattern 2 moves Real shape fw Pati fy Pate Circle Paint Shape fram 4 Ww Only one point per cell 7 D Only one point per pattem Heterogeneous Poisson Hard core Save null models Use category for defining abject click Calculate index Programita now performs the Simulations of the random labeling null model and shows you the pattern of the null model on the right Note that the cells occupied by pattern 1 or pattern 2 are exactly the same but the color pattern 1 or pattern 2 changes THORSTEN WIEGAND 59 i r a F T F i T F 7 wee We ee a E E at a a a 445 mr k d 445 ae e lt i e po E ho a k k d E a 1 k al d T ae en oe a eee i E p T oe ee ee E 8 After termination of the simulations a window appears Select one option close ak giz 3 e ga 7 f gl2 gl1 f gel g22 f gel gll 7 gl2 q22 ge2 gl1 7 qll g22 f gl2 gel 7 gel agil2 9 Enable first g12 to test for univariate random labeling Programita shows the univariate gll function instead of the O ring statistic together with the confidence limits for the univariate random labeling null model Univariate random labeling with g function W M 012 3 4 5 67 8 9 1011 12 131415 16 1718 19 20 21 22 23 24 25 26 27 28 29 30 Spatial scale r cells Save results The results show that the cases pattern 1 have the same seco
67. rger continuous areas are non accessible e g example in Wiegand amp Moloney 2004 Input data file m_ter_22 dat How are your data organized Matrix Give modus of analysis Irregularly shaped study region Which method will you use Ring or Circle Ring width 1 if intensity A of cells is too low select larger ring widths Set maximal radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 1 1 1 1 shrub species 1 Pattern 2 2 2 2 2 code does not occur Mask 2211 1 1 22 other shrubs 11 outside strips 1 Highlight the data file m ter 22 dat in the window Input data file 2 select Matrix in How are your data organized 3 Enable the check box Irregularly shaped study region 4 Select the code numbers for pattern 1 pattern 2 and the mask in Give code numbers for patterns write 1 in all boxes re served for pattern 1 shrub species 1 and write 2 in all boxes reserved for pattern 2 the code 2 does not occur in m ter 22 dat therefore you define a univariate pattern write in the boxes reserved for the mask 22 other shrub species and 11 the outside strips Give code numbers for patterns Pattern f f GN BH Pattern 2 2 2 HE E Mak eman m 5 Click button Calculate index Your pattern appears on the left and the O ring function of your data appears on the rrght 6 To determine Monte Carlo confidence limits for
68. sed to construct the confidence iii a e limits f s 4 4 g Z t r z a a 4 5 d l w 5 P i ft aa A Ei A bd Li g sz H A 3 z 4 Ps oe A as ETE a a i He a n s ee 1 i d A i P i E amp a t m Save map 5 Save index Figure3b After termination of the a Monte Carlo simulations of the null model a figure with the result ap pears on the right The figure shows Wiegand Moloney s O ring statistic E or Ripley s L function together 0 12 3 4 5 6 7 8 9 201i 12 1314 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 with the confidence limits for the MK onn ee Bivariate O ring statistic W M specific null model chosen The top figure shows the results of the uni variate point pattern analysis the bottom figure shows the results of the bivariate analysis if a second type i23 4678 9 1011 12 1314 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 of points was specified In fig2A ns only one type of points was used Therefore there appears no result for the bivariate analysis O12 r 10 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA What do you want to do Show results of previous analyses F Point pattern analysis 2 Homogeneity test Programita offers a convenient possibility to show the results of previous analyses This works only if the option Combine replicates was enabled when doing Combine replicates 2 the ori ginal analysis Calculate
69. sity functions Example TB S a eps 92 ees isee coe haa eee acta tea i EE ect asus E cian Cee Sipenaiten T 46 Toroidal shift Independence of bivariate patterns ccceeeeeeeesceeeeseeeeeeeeeee 48 Toroidal shit Example F Ma bies reisan a ESEE eee 48 Antecedent condition Randomising only pattern 2 00 0 0 cccceeeeeeeeeeeeees 50 Antecedent condition Example F Ant 1 reS ce cccccccccesssseeeeeeeeeeees 51 Antecedent condition Example F Ant 2 reS cccccceeeseescesceseessesseeseeees 53 Antecedent condition Example F Ant 3 reS cccceeeeescecescessesseeseeseeees 54 Antecedent condition Example F Ant 4 res ccccecececeeesseeeeeeeseeee 56 Randont WAG WING neen a E stad meeaseeesenc teas a 57 Random labeling Example RL 1 7ES Janeane a bia 58 RETEN OS aen E T TO O E E 6l THORSTEN WIEGAND 5 Programita Abstract The Programita software allows you to perform univariate and bivariate point pattern analysis with Ripley s Z function the pair correlation function and the O ring statistic Programita contains standard and non standard procedures for most practical applications Procedures for non standard situations include the possibility to perform point pattern analyses for arbitrarily shaped study regions and Programita offers a wide range of non standard null models such as hetero geneous Poisson null models or cluster null models The grid based implementation of Programita allows for a straight forward ex tens
70. ss In this case Programita will base the definition of Toroidal shift pattem 2 moves Real shape 7 Pat1 Pat 2 the objects not only on cells occupied by pattern 1 Fp fees alias za ata a D I Circle Point or pattern 2 but additionally on the category of Pia er ea the cells in your original map In this case an ob m ea pelea i Heterogeneous Poisson ject can only comprise cells of the same category Eo Save null models Another simple possibility to avoid the problem of joined objects is dividing the cells into 4 smaller cells and leaving empty cells between adjacent but distinct objects Finally you may manually separate objects Manual separation of joined objects To manually separate objects run first one analysis where adjacent but separate objects may be joined by Programita Use the temporal files temppatch1 dat and temppatch2 dat which are automatically created by Programita and change the name of temppatch1 dat to name pal and temppatch2 dat to name pa2 for name use an appropriate name The subfix pal and pa2 are reserved for manually modified files containing the information on the objects of pattern 1 and pattern 2 respectively These files are the object representation of your data of pattern 1 and pattern 2 The first line in the files gives the number of objects of pattern 1 or pattern 2 the next lines contain the information on the objects each object is a line necessary for rand
71. ta organized Data are given as matrix map in Select modus of data and after specifying in the window Give code numbers 26 SUPPLEMENTARY USER MANUAL FOR PROGRAMITA for pattern which code numbers of your data matrix make up pattern 1 pattern 2 and the mask apply the check box Calculate confidence limits A window for selection of the null model appears Figure N1 appears The check box Only one point per cell is always enabled in the matrix Select a null model mode To allow overlap of objects of pattern 1 Fsmistons is 0 93 0 939 f Pattern 1 and 2 random 4 and pattern 2 enable the check box Only one Pattem fix pattem 2 random gt Random labeling gt Cluster process Toroidal shift pattern 2 moves point per pattern Figure N2 top right shows a randomization of Ce tee ae Ht SPE Mom mie f Circle Point pattern 2 without overlap to pattern 1 null model A anene anire Pattern 1 fix pattern 2 random and figure N2 Only one point per pattem bottom shows randomization of pattern 2 with the e i same null model but allowing overlap to pattern 7 Save null models Figure N1 Window for selection of null model Figure N2 Rules for overlap between objects T 5 a a in null models Top left data pattern without F a overlap pattern 1 red pattern 2 green Top 3 e Re right randomization of objects of pattern 2 T E T d Null model Pattern 1 fix
72. ulations of the null model for de fining 95 99 confidence limits e g Stoyan and Stoyan 1994 The individual O and 2K function tThe individual O r of a cell x y of pattern 1 gives the density of pattern 1 cells in a ring with radius r around the focal cell x y The individual AK r of a cell x y of pattern 1 gives the num ber of pattern cells in a circle with radius r around the focal cell x y THORSTEN WIEGAND 13 The file temp indO dat gives the individual O 7 or AK r for each point of the pattern The file is comma delimited and the first two columns give the x and y coordinates of the cell of pattern 1 The next columns give the individual uni variate O r or AK i1 r O11 r or K11 r for the cell x y at scale r and the individual bivariate O12 r or AK 12 r O12 r or K12 r for the cell x y at scale r The end of the file gives the settings file figure 5 used to generate the data The files temppatchI dat and temppatch2 dat tlhese files are the object representation of your data of pattern 1 and pattern 2 The first line gives the number of objects of pattern 1 or pattern 2 the next lines contain the informa tion on the objects necessary for randomization of the objects of finite size and irregular shape The first two columns are the minimal x and y coordinates of the object the third column gives the width of the object in x direction xmax xmin the fourth column gives t
73. ules and options to decide on the overlap between plants explain how to construct objects from a categorical map how to randomize the position of the objects and how to take edge effects into account which may arise if randomized objects partly overlap the limits of the study region How this is implemented in Programita is explained in the following Overlap between plants An important difference between conventional point pattern analysis and analy sis of categorical maps is that points cannot overlap except where they occupy exactly the same location but for randomisation of the position of plants of fi nite size rules are needed to determine if they are allowed to overlap or not Since a category and not a number of points is assigned to each cell overlap of two plants of the same pattern is not allowed However to avoid that several small plants may occupy the same grid cell the size of the grid cell may be re duced Depending on the null hypothesis we may or may not allow overlap of plants of two different patterns This difference is important for data collection and mapping as for the application of null models For example allowing overlap of the two component patterns of a bivariate pattern is relevant for a null model to approximate for instance third dimension effects which may occur if smaller plants such as grass tufts grow inside larger unpalatable shrubs After selecting Matrix in How are your da
74. univariate and the bivariate point pattern analysis 17 Programita uses as default the lowest and highest O r the different simulations of the null model as confidence limits However it automatically produces two temporally files Uni confidence env Bi confidence env that contain the O r for all simulations of the null model The col umns of these files are the scales r 1 Imax and the lines are the different simulations of the null model temporary files are overwritten if you start a new analy Sis THORSTEN WIEGAND 39 CSR in irregularly shaped study region with mask Example F_CSR_2 res This example shows application of the mask Competition for space is an impor tant ingredient of null models for plants with finite size One may encounter situations where plants of the focal species cannot inhabit some areas of the ground e g if it is already occupied by other species Programita facilitates an elegant extension of the null models CSR and antecedent condition to consider restrictions in the accessible space All non accessible cells are summarized as a third pattern called mask and are excluded from the study region Masking is especially important for the null model antecedent condition Not considering the plants of the third pattern will reduce the intensity of pattern 2 which then appears to be aggregated in respect to pattern 1 For univariate analy sis 1 e CSR this effect is only important if la
75. we repeat the previous example F Ant 3 res but use the data of pattern 1 and pattern 2 for construction if the first order intensity Compare the difference of the first order intensity to the previous example However since both pattern 1 and pattern 2 follow CSR there is no difference in the re sult to the previous example Bivariate O ring statistic W M Ol2 r 3 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Spatial scale r cells i THORSTEN WIEGAND 57 Random labelling Random labelling is the appropriate null model for a null hypothesis corre sponding to the absence of interaction between the two types of points if the points of both patterns were created by one process e g trees but a subse quent process created the labels pattern 1 surviving trees and pattern 2 dead trees Goreaud amp Pelissier 2003 The null model of random labelling then assumes that the process which assigned label to plants was a random proc ess In a wider sense random labelling is an appropriate null model to test the null hypothesis that the labels do not show any spatial structure An important ingredient of testing for random labelling is the invariance of g and K functions under random thinning 1 e randomly removing points does not change the second order properties of uni or bivariate patterns Thus 211 222 r 212 r 221 r This property suggests different test statistics to test dif
76. wing lines are the data matrix with the pattern The numbers are not code numbers as in the matrix data format but give a 1 if the cell is occupied by category of pattern 1 and a 0 if the cell is not occupied by the category of pattern 1 The file tempp1 dat does not contain information on an irregularly shaped study re gion The file tempp2 dat is the analogue to tempp1 dat for pattern 2 The file temphab dat f you analyze an irregularly shaped study region Programita creates a matrix representation of your study region analogously to the files tempp1 dat and tempp2 dat The file tempp12 dat The file tempp12 dat is the list in grid representa tion of your data containing the information on cells of pattern 1 pattern 2 and the study region The files Bi_confidence env and Uni_confidence env Programita uses the lowest and highest O r or L r of the different simulations of the null model as default confidence limits However it automatically produces two temporally files Uni_confidence env Bi_confidence env that contain the O r or L r for all simulations of the null model The columns of these files are the scales r 1 Fmax and the lines are the different simulations of the null model You may use this information to construct confidence limits with different definitions Note that Programita offers the possibility to use also the 5th highest and 5th lowest O r or L r out of 99 or 999 replicate sim
77. ypothesis Test only for pattern 1 f Test only for pattern 2 Test for joint pattern 1 and 2 If you want to investigate if there are small scale second order effects indicating competition or facilitation between plants of two different species you may use the first order intensity of pattern 2 for randomization of plants of pattern 2 However if you hypothesis is that the distribution of pattern 2 may depend on pattern e g if pattern 2 are seedlings and pattern 1 the parent trees you may use the first order intensity of pattern 1 for randomization of plants of pattern 2 Finally if your hypothesis is that the two patterns follow the same environ mental heterogeneity you may use the first order intensity of the joined pattern for randomization of plants of pattern 2 THORSTEN WIEGAND 51 Antecedent condition Example F_Ant_1 res This example applies a null model under antecedent condition to the previous example Input data file m_te22 dat How are your data organized Matrix Give modus of analysis Analyze all data in rectangle Which method will you use Ring or Circle Ring width Set maxim 1 if intensity A of cells is too low select larger ring widths al radius rmax 30 too large scales slow down Programita Select modus of data Data are given as matrix map Give code numbers for pattern Pattern 1 1 1 1 1 shrub species 1 Pattern 2 22 2 2 2 other shrub species Click button Calculate index

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