PGS user manual version 0.2-0
Contents
1. Similarly rectangular and quincunx lattices of point patterns can be created by using the specific generator functions PPRectLat2 and PPQcxLat2 For planar lattices of quadrats or segments use the generators QHexLat2 QRectLat2 SHexLat2 and SRectLat2 The function LLat2 creates planar lattices of lines For a complete documentation of these functions see the on line help A plot function is implemented for planar lattices of figures The com mand line gt plot plat xlim c 0 3 ylim c 0 3 pch cex 0 3 yields the graphic shown in Figure 3 1 Figure 3 1 Plot of the object plat representing a lattice of point patterns 3 3 Specific generators of some particular 3D lat tices of figures The package provides three specific generators of 3D lattices of horizon tal point patterns rectangular body center rectangular and face centered rectangular lattices of horizontal point patterns For instance the command line gt p3lat PPRectLat3 dx 1 dy 1 dz 1 n 4 creates a 3D cubic lattice of point patterns with 4 points lying in a horizontal square of side length equal to 0 2 default value Body centered and face centered lattices of horizontal point patterns can be created by using the generators PPBCRectLat3 and PPFCRectLat3 Complete documentations of the generators are available in the on line help 3 4 Generator FigLat The generator function FigLat can be used to create user defined lattices of figures The lattice2. G Kendall On the number of lattice points inside a random oval Quarterly Journal of Mathematics Oxford Second Series 19 1 26 1948 D G Kendall and R Rankin On the number of points of a given lattice in a random hypershere Quarterly Journal of Mathematics Oxford Second Series 4 178 189 1953 K Ki u and M Mora Asymptotics for geometric spectral densities and a stochastic approach of the lattice point problem Mathematicae Notae 2004 K Ki u and M Mora Precision of stereological planar predictors Journal of Microscopy 222 3 201 211 June 2006 K Ki u and M Mora Advances on the precision of several stereolog ical volume estimators In V C et al editor Stereology and Image Analysis Ecs10 Proceedings of the 10th European Congress of ISS volume 4 of The MIRIAM Project Series Bologna Italy 2009 ESCU LAPIO Pub Co B Mat rn Estimating area by dot counts In J Lanke and G Lindgren editors Contributions to Probability and Statistics in Honour of Gunnar Blom pages 243 257 Department of Mathemati cal Statistics University of Lund 1985 B Mat rn Precision of area estimation a numerical study Journal of Microscopy 153 3 269 284 1989 G Matheron Les variables r gionalis es et leur estimation Masson 1965 G Matheron The theory of regionalized variables and its applica tions Technical report Centre de morphologie math matique Ecole des Mines de Paris 19
3. Specific generators of some particular planar lattices of figures The package provides specific generator functions for some common planar lattices of figures e The figure can be a finite point pattern a quadrat a segment or a line e The 2D vector lattices are rectangular hexagonal or quincunx For instance the command line gt plat PPHexLat2 delta 1 n 4 creates an hexagonal lattice of point patterns with 4 points lying in a square of side length equal to 0 2 default value see arguments hp and vp below for changing the default value The argument delta determines the distance between homologous points in two neighbour point patterns The second argument n gives the number of points in each point pattern The object plat is printed as follows gt print plat 2D lattice of point patterns Vector Lattice An object of class VecLat dimspace 2 dimsupp 2 determinant 0 8660254 generating matrix 1 2 1 1 0 5000000 2 0 0 8660254 determinant 0 8660254 Figure PointPattern coord 1 2 3 4 dd 0 0 2 0 2 0 0 2 0 0 0 0 2 0 2 The function print displays the generating matrix of the hexagonal vector lattice with its determinant and the column matrix coord containing the Cartesian coordinates of the points Other arguments can be passed to the function PPHexLat2 hp and vp determine the size of the point pattern bounding rectangle and h3 defines the orientation of the point pattern
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5. N Comprehensive R Archive Network Therefore it can be downloaded and installed using e g install packages or package installation and management tools available in R GUI s see R general documentation Note that this package is distributed under the CeCILL free license which is rather similar to the GPL license In order to load pgs package in a R session just type gt library pgs in the R command window The documentation of pgs is available in the on line help of R Chapter 3 Lattice of figures The sampling devices considered here are lattice of figures A figure is a simple geometric test set such as a single point a finite point pattern a quadrat a segment a line or a plane By translations the figure is repeated in space at regularly spaced locations The translation vectors form a vector lattice 3 1 Classes of geometric objects In pgs package geometric entities are represented by objects The package operates on three classes of geometric objects e Simple figures are represented by objects of the class Figure e The vector lattices are represented by objects of the class VecLat e The lattices of figures are represented by objects of the class FigLat Note that the classes have been designed for lattices and figures in spaces with arbitrary dimensions not only in 2D and 3D spaces Most users will handle only FigLat objects generated using predefined generating functions see sections 3 2 and 3 3 3 2
6. PGS user manual version 0 2 0 Ki n Ki u and Marianne Mora December 11 2013 Contents 1 Introduction 2 2 Installation of pgs package 4 Ql Prerequisites 2s vic 54 550 e 25590 23 LQ Giya ie UY ek Be Sew 4 2 2 Obtaining and installing pgs 4 3 Lattice of figures 6 3 1 Classes of geometric objects 2 020048 6 3 2 Specific generators of some particular planar lattices of figures 6 3 3 Specific generators of some particular 3D lattices of figures 8 3 4 Generator FigLat 0 2000002 eae 8 4 Planar area prediction 10 4 1 MSE approximation formulae in the isotropic case 10 4 2 Practical design of a sampling scheme 12 4 3 Variance and MSE estimation 13 4 3 1 MSE estimation in the isotropic case 14 4 3 2 MSE estimation under mild anisotropy 15 Chapter 1 Introduction Using stereological methods it is possible to estimate geometric parameters planar area volume surface area number etc of spatial structures 2D or 3D from partial observations Partial observations consist of intersections of the spatial structure under study and figures test sets such as points quadrats lines planes In practice the figures are often systematically distributed in the containing space the whole sampling device forms a lattice of figures Such sampling devices are commonly used both because they are easy to implement and because of the
7. his framework The feature to be predicted is the total area of the structure The term prediction is preferred to estimation because the area is a random variable instead of a fixed parameter The structure under study is sampled by a lattice of figures see examples in Figure 4 1 The area predictor is the content of the intersection of the structure with the lattice of figures divided by the sampling intensity mean content of the lattice of figures seen in a region with unit area For instance if the sampling device is a lattice of points the area predictor is equal to the number of lattice points hitting the structure multiplied by the area of a tile of the lattice If the sampling device is a lattice of quadrats the area predictor is the area of the structure observed in the quadrats divided by the fraction of the plane covered by the quadrats If the lattice of figures is uniformly randomly translated the area pre dictor is unbiased the conditional mean prediction given the structure is equal to the area The approximations for the prediction MSE as given in are obtained under the assumption that the normals to the boundary of the structure are isotropically distributed Note that if the lattice of figures is isotropically randomly rotated the MSE approximations hold even for anisotropic boundaries 4 1 MSE approximation formulae in the isotropic case Under the assumption of isotropy the MSE approximation formulae i
8. ions Journal of Microscopy 122 143 157 1981 M Garcia Finana and L M Cruz Orive Explanation of apparent paradoxes in Cavalieri sampling Acta Stereologica 17 297 302 1998 M Garcia Finana and L M Cruz Orive Fractional trend of the vari ance in cavalieri sampling Acta Stereologica 19 71 79 2000 M Garcia Finana and L M Cruz Orive New approximations for the efficiency of cavalieri sampling Journal of Microscopy 199 224 238 2000 X Gual Arnau and L M Cruz Orive Consistency in systematic sam pling Advances in Applied Probability 28 982 992 1996 X Gual Arnau and L M Cruz Orive Variance prediction under sys tematic sampling with geometric probes Advances in Applied Proba bility 28 982 992 1998 X Gual Arnau and L M Cruz Orive Systematic sampling on the circle and on the sphere Advances in Applied Probability 32 3 628 647 2000 17 12 13 14 15 17 18 19 20 21 22 23 H J G Gundersen and E B Jensen The efficiency of systematic sampling in stereology and its prediction Journal of Microscopy 147 3 229 263 1987 H J G Gundersen E B V Jensen K Ki u and J Nielsen The efficiency of systematic sampling in stereology reconsidered Journal of Microscopy 193 3 199 211 1999 A M Kellerer Exact formulae for the precision of systematic sampling Journal of Microscopy 153 3 285 300 1989 D
9. ir statistical efficiency see e g 12 Provided that the lattice of figures is uniformly randomly translated most stereological estimators turn out to be unbiased However assessing the precision of stereological estimators is usually not straightfoward A naive approach consists of considering the sampling figures as non correlated sta tistical units However as shown by Gundersen amp Jensen 12 this approach tends to overestimate the estimation variability An alternative approach is based on a theory developped by Kendall 15 16 and Matheron 22 23 This approach provides asymptotic approximations of mean square errors MSE The approximations converge when the sampling density grows Stereological applications of Kendall Matheron theory have been developped in several papers 12 2 13 20 21 7 8 11 4 6 9 10 3 1 5 14 General MSE formulae for planar area and volume estimation predic tion have been derived 17 18 These formulae can be used for a large set of sampling devices point lattices point pattern lattices quadrat lattices series of parallel lines strips Due to their very simple form the formu lae can be used to compare sampling devices independently of the spatial structure under study Hence they can be very useful in a priori sampling design Also they can be used for MSE estimation from data see e g 19 The R package pgs provides tools for computing MSE approximations and estimates for area and v
10. lattice of figures For the extension to be valid the data provided to the function area mse est must be invariant under the stretching This is the case for example for lattices of point patterns and quadrats The function area mse est first estimates the stretching in order to provide the mean perimeter and the MSE estimates For the lattice of point patterns and the data given in ppldata one gets gt area mse est ppldata mse only FALSE iso FALSE B est 1 40 47671 15 deformation 1 2 1 0 9363625 0 6710058 2 0 5056472 0 7056111 mse est 1 0 468871 The argument iso is set equal to FALSE to specify that a stretching has to be applied The component list deformation returned by area mse est defines the stretching matrix 16 Bibliography L M Cruz Orive Estimating volumes from systematic hyperplane sections Journal of Applied Probability 22 518 530 1985 L M Cruz Orive On the precision of systematic sampling a review of Matheron s transitive methods Journal of Microscopy 153 3 315 333 1989 L M Cruz Orive Systematic sampling in stereology In Proceedings of the 49th Session of the International Statistical Institute volume 2 pages 451 468 1993 L M Cruz Orive Precision of Cavalieri sections and slices with local errors Journal of Microscopy 193 182 198 1999 L M Cruz Orive and A O Myking Stereological estimation of vol ume ratios by systematic sect
11. nctions read table or read csv The sampling grid parameters and the collected data are passed as ar guments to the function FigLatData gt ppldata FigLatData pplat pp counts 14 Both the mean perimeter and MSE estimates can be computed with the function area mse est gt area mse est ppldata mse only FALSE iso TRUE B est 1 41 47922 mse est 1 0 5150249 e The first argument passed to the function is the lattice of figures and the array containing the data The data array is reduced to a matrix as in the example when a single structure is sampled e The second argument is set equal to TRUE the default value is FALSE to specify that estimates must be computed under the as sumption of isotropy e The third argument is set equal to TRUE default value if both the mean perimeter and MSE estimates are required Otherwise only the MSE estimate is provided The function area mse est has an extra argument diff2use defining the data covariations to be used for the MSE estimation See the on line help for more details 4 3 2 MSE estimation under mild anisotropy The MSE estimation method can be extended to the case where the bound ary can be made isotropic by a specific area preserving deformation stretch ing the structure along a given direction compressing it in the orthogonal direction For sake of simplicity this transformation is called stretching The stretching is applied to both the structure and the
12. nted by plat see Figure 3 1 The point counts are given by gt counts c 5 3 1 4 1 3 4 2 4 5 The area predictions are computed as follows gt areaPred counts 4 plat vlat det gt areaPred 1 1 4433757 0 8660254 0 2886751 1 1547005 0 2886751 0 8660254 1 1547005 8 0 5773503 1 1547005 1 4433757 The prediction variance is gt var areaPred 1 0 1814815 The prediction variance is the sum of the area variance and of the area prediction MSE The MSE formula as given in requires parameters de pending on the lattice of figures and the mean perimeter of the structure under study Let us consider the case where the shape parameter B vA may be approximated by 5 in view of the nomogram given in 12 The mean perimeter is approximated by gt B 5 sqrt mean areaPred gt B 1 4 805623 The MSE is computed using the function area mse gt area mse plat B 1 0 1068132 The area variance may be computed as the difference between the pre diction variance and the prediction MSE gt var areaPred area mse plat B 1 0 07466825 13 Above the MSE has been estimated based on a visual estimate of the shape parameter B V A It should be noticed that some figures like segments lines and quadrats provide data for perimeter estimation For instance consider line sampling The total intercept length provides the area estimate while the total number of intercepts provides the perimeter estimate In s
13. nvolve two terms e one term depending only on sampling parameters and involving the multidimensional Epstein zeta function 10 F o s eS TEZA SF TAT AA TAN U T TU UU TT TT TT Gam 4 HR gst HH oO 0 00 0 0 We et Ht sO WUN BH n n Fs d Oe FP Q an a A St Te DLSA a n A t d HSN BH n j BIB k eis HH D OH n n ale 4 b 4 HOR ROR c n Figure 4 1 Examples of lattices of figures used for area prediction A lattice of figures is superimposed on a planar structure shown in grey The total area of the planar structure is predicted from point counts length or area measurements a Hexagonal lattice of points b lattice of point patterns c lattice of quadrats d lattice of segments e lattice of infinite lines f lattice of infinite strips 11 e one term depending on the boundary of the structure under study the mean perimeter The MSE approximation is computed using the function area mse gt area mse plat B 1 L 3 1 0 02222672 e The first argument plat is the sampling test system e The second argument B is the mean perimeter of the structure to be provided e The third argument is an integer used as a criterion for stopping sum mation of the Epstein ze
14. of figures plat defined above can also be obtained by the command line gt plat FigLat 2 HexLat2 sqrt 3 2 PP2 n 4 h 1 5 e The first argument passed to FigLat is the dimension of the space where the structure lies 2 for planar structures e The second argument passed to the function FigLat is a VecLat object representing the vector lattice Such a vector lattice can be created by the general generator VecLat The unique argument of the function VecLat is the generating matrix of the vector lattice Planar hexagonal lattices may be generated in a simpler way by the specific generator HexLat2 The argument of the function is the deter minant of the generating matrix that is the area of a fundamental tile of the hexagonal lattice Similarly the generator RectLat2 is used for planar rectangular lattices The third argument defines the figure here a point pattern represented by a PointPattern object The generator PointPattern generates such objects it takes as a unique argument a column matrix containing the Cartesian coordinates of the points The specific function PP2 creates some particular planar point pat terns The 3D generator PP3 converts these planar point patterns into horizontal 3D point patterns Chapter 4 Planar area prediction The spatial structure of interest is considered as a random compact planar set Note that the case where the structure is deterministic may be consid ered as a special case of t
15. olume predictors Notice that this document does not aim to be a self contained introduc tion to methods for assessing the precision of stereological estimators The reader is referred to e g 18 for a presentation of those methods Note that the current version 0 2 0 of the package is still at an experi mental stage of development Chapter 2 Installation of pgs package 2 1 Prerequisites R is a free statistical software which consists of many packages see http waw R project org R is available for many platforms PC Windows PC Linux Mac OsX It is assumed here that R is already installed We have not tested the pgs package under old versions of R Any fairly recent version should fit The pgs package depends on three other R pack ages e The package methods is a standard package of R It should be already installed in most cases If not it can be downloaded and installed from the Web site of R e The package R2Cuba performs numerical integration It can be down loaded and installed from the Web site of R e The package gsi is a front end to the GNU Scientific Library It can be downloaded and installed from the Web site of R Linux users should install first the Gnu Scientific Library http www gnu org software gs1 Most installation procedures are dependency aware and should install these required packages if not yet installed 2 2 Obtaining and installing pgs The package pgs is available on CRA
16. ta function 4 2 Practical design of a sampling scheme Let us consider the case where the figure is fixed and the vector lattice is defined up to a scaling parameter The function latscale computes the scaling parameter u such that the prediction coefficient of error is equal to a given value gt u latscale PPHexLat2 n 4 A 1 shape 5 CE n 0 05 upper 2 gt u 1 0 3487114 e The first argument passed to the function is the lattice of figures The associated vector lattice is supposed to be unit i e its determi nant must be equal to 1 The lattice of figures plat created in Sec tion 3 2 does not fulfill this condition determinant equal to 0 866 The vector lattice associated with the lattice of figures defined by PPHexLat2 n 4 is unit e The second argument A is a rough estimate of the mean area e The third argument is a rough estimate of the shape parameter B yA It may be determined approximately using the nomogram provided in 12 e The fourth parameter is the nominal coefficient of error e The fifth parameter is an upper bound for the interval where u is to be searched Hence in order to get area predictions with a coefficient of error equal to 5 one should use an hexagonal lattice where the spacing between homol ogous points is equal to 0 349 12 4 3 Variance and MSE estimation Let us consider the case where a series of say 10 structures have been sam pled The sampling lattice of figures is represe
17. uch a case a MSE estimate is obtained by providing the perimeter estimate as the argument B of the function area mse Below we focus on MSE estimation based uniquely on data used for area prediction 4 3 1 MSE estimation in the isotropic case It is supposed that individual data for each sampling figure are available The mean perimeter estimation is based on short range data covariations empirical covariogram near the origin and depends on the sampling fig ure geometry Then the MSE can be estimated by providing that mean perimeter estimate to the function area mse All estimate computations can be done with the function area mse est As an example consider the case where a single planar structure is sam pled by the rectangular lattice of point patterns pplat defined by gt pplat PPRectLat2 1 1 5 0 4 The first and second arguments are the horizontal and vertical spacings of the vector lattice The third and fourth arguments define the point pattern 5 points lying in a square of side length equal to 0 4 4 points lying at the square corners and one middle point The data to be used are contained in the matrix pp counts displayed as 00000000550 00453000451 00000002554 00000055555 10000555025 50001530000 31015300000 04455000000 00001000000 Each entry of the matrix pp counts contains the individual data of a sampling point pattern The matrix may be filled from input files either text or csv files use respectively the fu
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