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Fuzzy classification Mapping

This is report I've written after my stage in Wageningen Winland Staring Center in Holland. It presents technique of mapping results of fuzzy classification - i.e. where each point belongs to several classes simultaneously with certain membership value

Data, representing vegetation information, are values of coverage of several species. This data are given in percents (see appendix A). Usage of absolute coverage values is not good for classification because influence of dominant species is much more then influence of rare species, but rare species are often indicators of some significant ecological conditions.

The simplest way to avoid this is use diagonal metric during fuzzy classification. But, unfortunately diagonal metric with this data causes collapse of clusterization algorithm. Same results were obtained when ¹/_{max.
coverage}was used as weights for species. It has almost same effect as diagonal transformation, because all data is greater or equal than zero and there are significant share of zeros in data matrix.

Logarithmic transformation can also be used to pay more attention to rare species. Some sort of this transformation was used for original hard classification (table 2). But it was not really logarithmic, because value 0 was assigned to absent species, while they must have infinite values in really logarithmic transformation. This nearly excludes presence/absence differences from classification. But this difference is significant for vegetation classification, especially for rare indicator species. Due to this real logarithmic transformation is better. But clustering algorithm cannot operate with infinities, produced by this transformation. Therefore infinite values was replaced by large negative numbers. We think that negative number with absolute value two times larger than maximal possible positive value is enough for represent infinity in case of cluster analysis. Due to this decimal logarithm was used and absence was represented by -4 (log(100)=2). In this case too much attention is payed to presence/absence species and there are no classes with large average coverage of frequent species.

Another approach is voluntary assign extra weight for indicator species. in our case 19 species of 67 was selected as especially significant, due to their significance to existing classifications of vegetation or for indication of soil acid-alkaline conditions (table 1)

table 1. Species with extra weight table 2. Balls, assigned to coverages

 

1.Carex Hostiana Coverage Ball 

2.Carex Nigra 0 0 

3.Carex Panicea  1 

4.Carex Pulicaris  2 2 

5.Cirsium Dissectum  3-4 3 

6.Dactylorhiza Maculata  4-7 4 

7.Danthonia Decumbens  8-13 5 

8.Erica Tetralix  14-30 6 

9.Eriophorum Angustifolium  31-50 7 

10.Eupatorium Cannabinum 51-70 8 

11.Festuca Ovina 71-100 9 

12.Galium Uliginosum 

13.Juncus Acutiflorus .

14.Luzula Multiflora

15.Lysimachia Vulgaris

16.Parnassia Palustris

17.Peucedanum Palustre

18.Ranunculus Flammula

19.Valeriana Dioica



Program MacFuzzy (A.W.Ward et al,1992) was used for fuzzy classification of data. We used fuzzy k-means without extra grades, because attempts to use extra grade algorithm cause calculation errors. Value 1.5 was chosen for fuzzy exponent after several iterations.

We had computed three kinds of classification - classification of untransformed data, classification of data, transformed as described above and classification with extra (multiplied by 2) weights of species shown in table 1, but without transformation of coverages.

Last of them seems to resemble with original classification better, and it was chosen for mapping.

We had made classifications for each number of classes from 5 to 9, because there are too few data points (57) to estimate larger number of classes and original hard classification has 9 classes.

Analyzing statistical measures of this classifications, computed by MacFuzzy program (fig 1), we decide that classification with 7 classes is best.

fig 1 Statistics of classifications with different number of classes.

Membership interpolation

Main constraint, which must be applied to membership interpolation is that sum of memberships in all classes must be equal to one. This is always true for observation points, due to fuzzy classification algorithm. Some interpolation methods as inverse square distance method, ensures that it would be true for all interpolation points. But methods, which take into account statistical properties of data, as kriging, cannot ensure this constraint. There are several methods to deal with this problem. One of them is symmetric log-ratio transformation, described in (McBratney,De Grujter, Brus,1992) . After this transformation distribution of transformed memberships is nearly normal and after backward transformation sum of memberships is always one. This transformation was used in our case.

This transformation can be defined by equations:

 mⁱ

tⁱ=ln 

(mⁱ)^1/k

(straight transformation)

and

exp(tⁱ)

mⁱ=

exp(tⁱ)

(inverse transformation)

Transformed memberships tⁱ was interpolated by ordinary kriging on grid with 1m cell (235x72) using 8 neighboring points. Spherical model was used for variograms. This model was chosen after fitting of several models (exponential, Gaussian and spherical, with estimating nugget and with assuming zero nugget) as best. Parameters of estimated variograms for classification with extra weights and no transformation of coverages is shown in table 3. Kriging was performed by program, written by authors on Turbo Pascal language, using source codes provided by Peter Finke.

table 3 Parameters of estimated variograms



Class Range Sill Nugget

(m) (%²) (%²)



7a 7.81 4.948 0

7b 8.45 4.077 0

7c 25.25 6.004 0

7d 18.92 6.144 0

7e 8.39 3.191 0

7f 15.45 7.041 0

7g 12.87 5.272 0



fig 2 Example of variogram fitting

Mapping of fuzzy classes

Conventional mapping methods do not allow to represent more than 2-3 quantitative parameters on one map. But in fuzzy set applications in geography, we need to represent memberships in several classes. Common technic is representation of class number with maximal membership.

But if we create such maps, even if them are accompanied by maps of uncertainty, i.e. maps of max. membership values, most of fuzzy approach advantages are lost. We still have sharp borders between classes and we see only one class in each point of map, when we have information of membership of each point in all classes.

Now we present another approach. It allow to create really fuzzy maps. Main concept of this approach is based on the point, that grids of interpolated memberships are not very fine, due to observation accuracy and kriging computation time. We usually use grids with not more than several hundred cells in each direction. But resolution of modern output devices is several times higher and we can represent every cell of our grid by square, which contains many dots.

And this dots can have different colors. Because sum of memberships is always equal to one, percent of dots with particular color can correctly represent membership in respective class, as shown on fig.3.

Program, which creates such maps was written on Turbo Pascal.

fig 3. Representation of grid cell in fuzzy mapping algorithm.

Results overview

Results of fuzzy classification are shown in appendices B and C. Appendix B shows a class centroid table. This table contain coverages of species for hypothetic "central" points of seven classes. Even for species shown in table 1 these are unweighted coverage values, because results was back-transformed. This data can be interpreted as description of vegetation in points with "clear" classes.

Appendix C shows memberships of each of 57 observation points in these classes.

Fig 4: Fuzzy map of vegetation classes

Appendix A Observation point coverages

101 102 103 104 105 106 107 108 109 110 111 112 113 114

Agrostis Canina 1 1 1 5 1 3 1 1 1 0 1 0 1 1

Agrostis Stolonifera 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Alnus Glutinosa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Amelanchier Lamarckii 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Angelica Sylvestris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Anthoxanthum Odoratum 0 0 0 0 0 0 1 0 0 0 0 0 0 0

Betula Pubescens 0 1 1 0 0 0 0 0 0 1 0 0 0 0

Musci 0 0 20 20 20 5 0 3 3 30 20 30 20 5

Calamagrostis Canescens 1 1 20 20 1 1 1 1 1 0 0 0 0 1

Cardamine Pratensis 0 0 0 1 0 0 0 0 0 0 0 0 0 0

Carex Flacca 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Hostiana 5 20 0 0 20 7 30 5 20 5 1 1 0 12

Carex Lasiocarpa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Nigra 1 0 1 1 1 1 0 0 0 1 1 1 20 0

Carex Oederi 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Panicea 3 10 3 0 20 5 30 5 3 1 1 30 40 3

Carex Pulicaris 12 20 1 0 1 1 5 7 20 7 5 1 1 5

Centaurea Jacea 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Cirsium X Spurium 0 0 0 0 0 1 0 0 0 0 1 0 0 0

Cirsium Dissectum 50 1 30 1 40 50 1 1 1 0 0 1 1 1

Cirsium Palustre 1 1 0 2 0 0 1 5 1 1 1 5 5 0

Dactylorhiza Incarnata 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Dactylorhiza Maculata 0 1 1 1 3 3 1 1 1 1 7 3 3 0

Danthonia Decumbens 1 3 0 0 1 0 3 1 30 12 20 7 1 0

Deschampsia Flexuosa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Equisetum Palustre 0 0 1 0 0 0 0 0 0 0 0 0 0 0

Erica Tetralix 0 1 0 0 0 0 0 1 1 1 1 1 1 1

Eriophorum Angustifolium 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Eupatorium Cannabinum 0 0 1 1 0 1 1 0 0 0 0 0 0 0

Festuca Ovina 20 30 0 0 1 1 0 30 3 50 50 30 1 0

Festuca Rubra 0 1 0 0 0 0 1 0 0 0 0 0 0 0

Filipendula Ulmaria 1 0 0 0 0 0 0 5 2 1 0 0 0 0

Frangula Alnus 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Galium Palustre 0 0 1 1 0 0 0 0 0 0 0 0 0 0

Galium Uliginosum 0 0 1 1 0 1 1 1 1 1 0 1 1 0

Gentiana Pneumonanthe 0 1 0 0 0 0 0 0 0 0 1 0 0 0

Hieracium 0 0 0 0 0 0 0 0 0 1 0 0 0 0

Holcus Lanatus 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Hydrocotyle Vulgaris 1 1 0 1 0 0 10 0 0 0 0 0 0 0

Hypochaeris Glabra 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Juncus Acutiflorus 0 0 10 20 5 7 0 1 0 0 5 1 3 0

Juncus Conglomeratus 0 0 0 0 0 0 0 1 1 1 0 1 1 1

Juncus Squarrosus 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lonicera Periclymenum 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lotus Uliginosus 0 0 0 0 0 0 0 0 0 0 1 1 0 0

Luzula Multiflora 1 1 0 1 0 0 0 1 1 1 1 1 1 0

Lysimachia Vulgaris 1 0 0 12 0 0 7 0 0 0 0 3 3 1

Lythrum Salicaria 0 0 1 1 0 1 0 1 1 0 0 1 0 1

Mentha Aquatica 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Molinia Caerulea 3 1 1 3 5 10 7 1 1 1 1 3 3 50

Parnassia Palustris 0 0 0 0 0 1 0 0 1 0 0 0 0 0

Peucedanum Palustre 3 5 3 5 1 1 7 5 5 1 1 5 2 3

Phragmites Australis 0 1 0 1 0 1 1 1 1 0 0 1 0 1

Pinus Sylvestris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Potentilla Erecta 3 5 1 0 1 1 5 5 5 3 5 10 12 1

Potentilla Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Quercus Robur 1 0 0 0 0 0 0 0 0 0 0 0 0 0

Ranunculus Acris 0 0 0 1 0 0 0 0 0 0 0 0 0 0

Ranunculus Flammula 0 1 0 1 0 0 0 0 0 0 0 1 0 0

Salix Aurita 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Salix Cinerea 0 0 0 0 0 0 0 0 0 0 1 0 0 0

Salix Repens 0 0 0 0 0 0 0 0 0 0 1 0 0 0

Sphagnum 0 0 1 60 1 1 0 80 1 0 7 20 50 5

Succisa Pratensis 0 0 0 0 3 1 0 0 0 7 0 1 0 0

Taraxacum Officinale 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Valeriana Dioica 0 1 0 0 0 0 0 0 0 1 0 0 0 0

Viola Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

115 116 117 118 119 120 121 122 123 124 125 126 127 128

Agrostis Canina 1 0 1 3 1 3 1 3 10 0 10 3 1 3

Agrostis Stolonifera 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Alnus Glutinosa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Amelanchier Lamarckii 0 0 0 0 0 0 0 1 0 0 0 0 0 0

Angelica Sylvestris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Anthoxanthum Odoratum 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Betula Pubescens 0 0 1 0 0 0 0 0 0 0 0 0 0 0

Musci 60 20 10 10 80 90 30 10 5 1 1 10 5 10

Calamagrostis Canescens 1 0 0 1 0 10 1 0 0 0 0 5 5 1

Cardamine Pratensis 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Flacca 0 0 0 0 0 0 1 0 12 1 0 0 0 0

Carex Hostiana 20 5 3 1 1 0 20 3 12 20 0 0 1 5

Carex Lasiocarpa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Nigra 0 0 1 1 0 1 1 0 0 0 0 1 0 1

Carex Oederi 1 1 0 0 0 0 3 0 0 0 0 0 0 0

Carex Panicea 1 3 1 5 12 0 3 1 1 40 5 3 60 3

Carex Pulicaris 3 1 1 0 0 0 1 3 1 1 0 1 1 3

Centaurea Jacea 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Cirsium X Spurium 0 0 0 0 0 0 0 0 1 1 0 1 1 0

Cirsium Dissectum 5 70 1 1 0 20 10 40 7 10 1 3 1 10

Cirsium Palustre 2 0 2 2 12 0 1 1 2 0 5 5 1 1

Dactylorhiza Incarnata 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Dactylorhiza Maculata 1 1 3 1 1 0 1 3 5 0 5 1 1 5

Danthonia Decumbens 1 1 30 7 5 0 20 1 0 10 0 1 0 3

Deschampsia Flexuosa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Equisetum Palustre 0 0 0 0 0 0 0 0 0 0 0 1 0 0

Erica Tetralix 1 0 1 1 10 0 0 1 1 0 0 0 0 0

Eriophorum Angustifolium 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Eupatorium Cannabinum 0 0 0 0 0 0 0 0 0 0 0 1 1 1

Festuca Ovina 0 0 0 0 12 0 0 12 10 1 1 0 0 0

Festuca Rubra 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Filipendula Ulmaria 0 0 0 0 0 1 0 0 0 0 0 0 0 0

Frangula Alnus 0 0 0 0 1 0 0 1 1 0 0 0 0 0

Galium Palustre 0 0 0 0 0 1 1 0 0 0 0 0 0 0

Galium Uliginosum 1 0 1 0 1 0 1 0 0 0 1 1 1 1

Gentiana Pneumonanthe 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Hieracium 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Holcus Lanatus 0 0 0 0 1 0 0 0 0 0 1 1 0 0

Hydrocotyle Vulgaris 0 0 0 1 0 0 5 0 0 0 0 0 0 1

Hypochaeris Glabra 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Juncus Acutiflorus 1 3 5 7 3 7 7 5 1 5 10 10 7 5

Juncus Conglomeratus 5 1 1 3 0 7 3 1 1 1 0 0 0 0

Juncus Squarrosus 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lonicera Periclymenum 0 0 0 0 0 0 0 0 0 0 0 1 0 0

Lotus Uliginosus 0 0 0 0 1 1 0 0 0 0 0 0 1 0

Luzula Multiflora 0 0 1 1 1 0 0 1 3 0 1 1 1 0

Lysimachia Vulgaris 1 3 1 7 0 0 0 0 1 1 10 10 10 10

Lythrum Salicaria 1 1 0 0 0 1 0 0 0 0 0 1 0 0

Mentha Aquatica 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Molinia Caerulea 30 5 3 3 12 1 3 12 12 3 3 1 3 3

Parnassia Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Peucedanum Palustre 7 1 1 1 0 0 1 1 1 3 1 2 1 1

Phragmites Australis 1 0 0 0 0 0 0 0 0 0 0 1 1 1

Pinus Sylvestris 0 0 0 0 0 0 0 0 0 0 0 0 0 1

Potentilla Erecta 1 1 7 10 12 1 1 1 1 1 5 1 1 3

Potentilla Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Quercus Robur 0 0 0 0 0 0 0 0 0 0 1 0 0 0

Ranunculus Acris 0 0 0 0 0 0 0 0 0 1 0 0 0 0

Ranunculus Flammula 0 0 0 0 1 5 1 0 0 1 0 0 1 1

Salix Aurita 0 0 1 0 0 0 0 0 0 0 1 0 1 1

Salix Cinerea 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Salix Repens 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Sphagnum 10 50 70 80 0 0 40 70 70 5 90 80 10 50

Succisa Pratensis 0 0 0 0 1 0 0 0 0 1 0 0 0 0

Taraxacum Officinale 0 0 0 0 0 1 0 0 0 0 0 0 0 0

Valeriana Dioica 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Viola Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

129 130 131 132 133 134 135 201 202 203 204 205 206 207

Agrostis Canina 20 0 0 3 0 1 1 1 20 3 10 20 0 1

Agrostis Stolonifera 0 0 0 0 0 0 0 0 0 0 1 1 0 0

Alnus Glutinosa 1 0 1 0 0 0 0 0 0 0 0 0 0 0

Amelanchier Lamarckii 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Angelica Sylvestris 0 0 0 0 0 1 0 0 0 0 0 0 0 0

Anthoxanthum Odoratum 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Betula Pubescens 0 1 0 1 0 1 0 0 0 1 0 0 0 1

Musci 1 40 10 40 5 1 5 10 0 5 50 0 0 0

Calamagrostis Canescens 1 5 0 0 0 1 3 0 0 1 0 0 0 0

Cardamine Pratensis 0 0 0 0 0 0 1 0 0 0 0 0 0 0

Carex Flacca 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Hostiana 30 0 20 0 20 5 0 0 0 0 0 0 0 0

Carex Lasiocarpa 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Nigra 0 0 0 1 0 0 0 3 1 20 1 1 10 1

Carex Oederi 1 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Panicea 1 0 10 1 5 10 40 1 0 0 0 1 0 0

Carex Pulicaris 1 40 20 0 5 5 3 0 0 0 0 0 0 0

Centaurea Jacea 1 0 0 0 0 0 0 0 0 0 0 0 0 0

Cirsium X Spurium 0 1 1 0 0 0 0 0 0 0 0 0 0 0

Cirsium Dissectum 3 1 1 1 1 1 10 0 0 0 0 0 0 0

Cirsium Palustre 0 1 5 3 0 1 0 0 0 0 0 0 0 0

Dactylorhiza Incarnata 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Dactylorhiza Maculata 0 1 1 1 1 1 0 0 0 0 0 0 0 0

Danthonia Decumbens 5 0 30 1 60 30 0 5 1 12 0 7 3 3

Deschampsia Flexuosa 0 0 0 0 0 0 0 0 0 0 20 0 0 0

Equisetum Palustre 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Erica Tetralix 0 1 0 50 0 0 0 3 0 0 0 1 5 40

Eriophorum Angustifolium 0 0 0 0 0 0 0 1 0 1 0 0 3 1

Eupatorium Cannabinum 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Festuca Ovina 0 1 0 1 3 30 0 1 0 0 0 0 0 1

Festuca Rubra 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Filipendula Ulmaria 0 0 0 0 0 5 20 0 0 0 0 0 0 0

Frangula Alnus 0 0 0 0 1 1 0 0 0 0 0 0 0 1

Galium Palustre 1 0 0 0 0 0 1 0 0 0 0 0 0 0

Galium Uliginosum 1 1 0 0 1 0 0 0 0 0 0 0 0 0

Gentiana Pneumonanthe 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Hieracium 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Holcus Lanatus 0 0 1 0 0 0 0 0 0 0 0 0 0 0

Hydrocotyle Vulgaris 0 0 0 0 1 0 1 0 0 0 0 0 0 0

Hypochaeris Glabra 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Juncus Acutiflorus 0 1 3 1 0 0 0 1 3 5 0 3 0 0

Juncus Conglomeratus 3 1 0 0 0 0 0 1 1 0 0 1 0 1

Juncus Squarrosus 0 0 0 1 0 0 0 0 0 0 0 0 0 0

Lonicera Periclymenum 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lotus Uliginosus 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Luzula Multiflora 0 1 1 0 0 1 0 0 0 0 0 1 0 0

Lysimachia Vulgaris 1 3 1 1 0 0 5 1 3 1 0 1 0 0

Lythrum Salicaria 0 0 0 0 1 0 0 0 0 0 0 0 0 0

Mentha Aquatica 1 0 0 0 0 0 1 0 0 0 0 0 0 0

Molinia Caerulea 5 20 5 20 3 1 5 2 1 5 30 7 30 1

Parnassia Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Peucedanum Palustre 1 1 1 0 1 10 7 0 1 1 0 1 0 0

Phragmites Australis 1 1 1 0 1 1 1 1 1 1 1 3 1 0

Pinus Sylvestris 0 0 0 0 0 0 0 0 0 1 0 0 1 1

Potentilla Erecta 1 3 1 5 3 5 3 3 1 1 0 3 3 20

Potentilla Palustris 0 0 0 0 0 0 5 0 0 0 0 0 0 0

Quercus Robur 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Ranunculus Acris 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Ranunculus Flammula 1 0 0 0 1 0 1 0 0 0 0 0 0 0

Salix Aurita 0 0 0 0 0 1 0 1 5 0 0 1 0 2

Salix Cinerea 0 0 1 0 0 0 0 0 0 0 0 0 0 0

Salix Repens 0 0 0 0 0 0 0 0 0 10 0 0 0 0

Sphagnum 5 0 40 0 0 0 0 90 90 20 0 90 70 90

Succisa Pratensis 0 0 0 0 0 0 0 1 0 0 0 0 0 0

Taraxacum Officinale 0 1 0 0 0 0 0 0 0 0 0 0 0 0

Valeriana Dioica 0 0 0 0 0 3 0 0 0 0 0 0 0 0

Viola Palustris 0 0 0 0 0 0 0 20 1 0 0 0 5 1

208 209 210 211 212 213 214 215 216 217 218 219 220 221 230

Agrostis Canina 1 1 3 3 20 20 20 0 1 1 10 40 40 3 1

Agrostis Stolonifera 0 0 0 0 0 0 0 5 0 1 1 1 0 0 0

Alnus Glutinosa 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Amelanchier Lamarckii 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Angelica Sylvestris 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Anthoxanthum Odoratum 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Betula Pubescens 1 1 0 0 0 0 1 0 1 0 0 0 1 0 0

Musci 1 0 0 0 5 0 1 0 0 0 40 0 3 3 1

Calamagrostis Canescens 0 0 0 0 0 0 0 5 0 0 0 1 0 0 0

Cardamine Pratensis 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Flacca 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Hostiana 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0

Carex Lasiocarpa 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0

Carex Nigra 5 1 17 1 1 1 1 1 20 0 1 3 5 0 3

Carex Oederi 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Carex Panicea 0 0 0 0 0 0 0 0 0 10 3 10 1 20 0

Carex Pulicaris 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

Centaurea Jacea 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Cirsium X Spurium 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0

Cirsium Dissectum 0 0 0 0 0 0 1 0 0 1 1 1 0 1 0

Cirsium Palustre 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0

Dactylorhiza Incarnata 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Dactylorhiza Maculata 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Danthonia Decumbens 5 40 1 1 1 0 0 0 10 40 20 0 10 10 30

Deschampsia Flexuosa 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Equisetum Palustre 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Erica Tetralix 1 5 5 3 0 0 0 0 0 0 0 0 1 0 5

Eriophorum Angustifolium 1 0 1 3 1 0 1 0 1 1 0 1 1 3 1

Eupatorium Cannabinum 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Festuca Ovina 1 1 0 1 0 0 0 0 0 0 3 0 0 0 0

Festuca Rubra 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Filipendula Ulmaria 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Frangula Alnus 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

Galium Palustre 0 0 0 0 0 0 1 3 0 0 0 1 0 1 0

Galium Uliginosum 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Gentiana Pneumonanthe 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Hieracium 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Holcus Lanatus 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Hydrocotyle Vulgaris 0 0 0 5 0 0 1 0 0 1 10 0 0 1 0

Hypochaeris Glabra 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Juncus Acutiflorus 0 0 0 1 1 1 1 1 0 1 0 0 0 1 0

Juncus Conglomeratus 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0

Juncus Squarrosus 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lonicera Periclymenum 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lotus Uliginosus 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Luzula Multiflora 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Lysimachia Vulgaris 0 0 0 3 20 20 30 10 3 10 1 30 2 30 1

Lythrum Salicaria 0 0 0 0 0 0 1 3 1 0 1 1 0 0 0

Mentha Aquatica 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Molinia Caerulea 3 22 22 3 1 0 1 0 3 1 30 3 20 3 3

Parnassia Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Peucedanum Palustre 0 0 0 10 1 1 2 1 1 1 3 1 1 1 1

Phragmites Australis 1 1 1 1 1 3 3 10 1 1 1 1 1 1 1

Pinus Sylvestris 1 0 1 0 1 0 0 0 1 1 1 1 0 1 0

Potentilla Erecta 5 3 3 3 1 0 1 0 1 0 5 1 1 1 10

Potentilla Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Quercus Robur 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

Ranunculus Acris 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Ranunculus Flammula 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0

Salix Aurita 0 0 1 5 0 0 0 0 0 1 0 0 0 0 0

Salix Cinerea 0 1 0 0 0 0 0 0 0 0 0 0 1 1 1

Salix Repens 50 0 0 0 1 0 0 0 30 20 0 0 0 0 0

Sphagnum 70 80 40 90 80 0 30 0 50 5 0 0 20 3 50

Succisa Pratensis 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0

Taraxacum Officinale 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0

Valeriana Dioica 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Viola Palustris 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1

Appendix B. Class centre coverages

7a 7b 7c 7d 7e 7f 7g

Agrostis Canina 0.98 2.56 0.86 1.57 22.50 6.33 4.83

Agrostis Stolonifera 0.01 0.37 0.01 0.00 0.34 0.07 0.39

Alnus Glutinosa 0.01 0.11 0.01 0.00 0.03 0.00 0.02

Amelanchier Lamarckii 0.00 0.00 0.00 0.11 0.00 0.00 0.00

Angelica Sylvestris 0.00 0.00 0.22 0.00 0.00 0.00 0.00

Anthoxanthum Odoratum 0.13 0.01 0.01 0.00 0.00 0.00 0.14

Betula Pubescens 0.02 0.25 0.63 0.11 0.37 0.21 0.11

Musci 6.78 7.83 15.39 11.90 2.63 4.88 51.16

Calamagrostis Canescens 1.91 1.24 0.54 2.62 0.64 1.20 1.90

Cardamine Pratensis 0.22 0.01 0.00 0.00 0.01 0.03 0.00

Carex Flacca 0.22 0.16 0.01 0.01 0.02 0.61 0.02

Carex Hostiana 8.84 8.15 6.44 7.09 1.49 1.33 5.25

Carex Lasiocarpa 0.00 0.00 0.00 0.00 0.24 0.12 0.00

Carex Nigra 1.41 2.56 0.31 0.36 0.95 1.15 0.38

Carex Oederi 0.01 0.25 0.01 0.19 0.04 0.01 0.21

Carex Panicea 40.01 3.54 7.16 5.91 4.13 1.59 3.33

Carex Pulicaris 2.24 4.68 7.84 3.53 0.64 0.58 3.08

Centaurea Jacea 0.00 0.04 0.00 0.00 0.03 0.00 0.01

Cirsium X Spurium 0.43 0.11 0.24 0.23 0.21 0.13 0.05

Cirsium Dissectum 5.27 2.34 0.81 47.69 1.01 1.32 4.12

Cirsium Palustre 1.20 0.83 1.44 0.35 0.56 1.36 2.29

Dactylorhiza Incarnata 0.00 0.05 0.00 0.00 0.01 0.00 0.03

Dactylorhiza Maculata 0.85 0.67 2.50 1.77 0.10 1.14 0.50

Danthonia Decumbens 1.91 8.60 7.86 0.45 1.67 3.62 3.04

Deschampsia Flexuosa 0.00 0.02 0.00 0.00 0.02 0.00 3.54

Equisetum Palustre 0.00 0.01 0.00 0.10 0.00 0.07 0.01

Erica Tetralix 0.60 1.99 1.27 0.40 0.95 2.74 3.87

Eriophorum Angustifolium 0.07 0.44 0.02 0.01 0.78 0.64 0.04

Eupatorium Cannabinum 0.33 0.09 0.01 0.33 0.02 0.13 0.07

Festuca Ovina 1.24 1.18 36.98 5.72 0.32 1.71 2.72

Festuca Rubra 0.13 0.01 0.17 0.00 0.01 0.00 0.15

Filipendula Ulmaria 4.34 0.20 1.42 0.21 0.03 0.13 0.15

Frangula Alnus 0.01 0.09 0.23 0.11 0.02 0.14 0.16

Galium Palustre 0.25 0.32 0.02 0.11 0.60 0.04 0.18

Galium Uliginosum 0.25 0.18 0.18 0.17 0.04 0.15 0.21

Gentiana Pneumonanthe 0.00 0.00 0.39 0.00 0.00 0.00 0.00

Hieracium 0.00 0.00 0.22 0.00 0.00 0.00 0.00

Holcus Lanatus 0.01 0.08 0.01 0.00 0.00 0.15 0.15

Hydrocotyle Vulgaris 1.50 0.66 0.20 0.22 0.37 0.52 1.78

Hypochaeris Glabra 0.00 0.00 0.00 0.00 0.00 0.00 0.14

Juncus Acutiflorus 3.08 2.20 1.34 4.49 0.85 3.82 1.81

Juncus Conglomeratus 0.42 0.55 0.38 0.35 0.16 0.68 1.93

Juncus Squarrosus 0.01 0.01 0.01 0.00 0.01 0.00 0.04

Lonicera Periclymenum 0.00 0.00 0.00 0.00 0.00 0.07 0.00

Lotus Uliginosus 0.23 0.07 0.32 0.01 0.01 0.00 0.27

Luzula Multiflora 0.38 0.24 0.96 0.32 0.04 0.57 0.22

Lysimachia Vulgaris 5.42 2.98 0.45 0.85 20.05 4.40 0.99

Lythrum Salicaria 0.05 0.44 0.13 0.52 0.52 0.16 0.54

Mentha Aquatica 0.22 0.08 0.00 0.00 0.03 0.02 0.01

Molinia Caerulea 4.29 7.68 1.81 6.21 4.58 6.18 20.70

Parnassia Palustris 0.00 0.07 0.01 0.22 0.00 0.00 0.01

Peucedanum Palustre 1.87 0.73 2.02 0.80 0.65 0.86 1.01

Phragmites Australis 0.64 1.40 0.54 0.25 2.09 0.84 0.74

Pinus Sylvestris 0.03 0.39 0.01 0.01 0.31 0.23 0.19

Potentilla Erecta 3.77 2.60 4.89 1.50 1.05 4.02 3.55

Potentilla Palustris 1.08 0.00 0.00 0.00 0.00 0.00 0.00

Quercus Robur 0.00 0.00 0.00 0.20 0.00 0.07 0.17

Ranunculus Acris 0.22 0.01 0.00 0.00 0.01 0.03 0.00

Ranunculus Flammula 0.68 0.31 0.27 0.03 0.09 0.06 0.75

Salix Aurita 0.22 0.22 0.39 0.01 0.04 1.07 0.02

Salix Cinerea 0.03 0.16 0.23 0.00 0.19 0.08 0.02

Salix Repens 0.16 4.62 0.35 0.05 0.51 2.61 0.27

Sphagnum 11.60 25.65 5.86 18.13 12.68 77.41 4.77

Succisa Pratensis 0.25 0.01 1.66 0.69 0.01 0.15 0.16

Taraxacum Officinale 0.01 0.02 0.01 0.01 0.22 0.00 0.16

Valeriana Dioica 0.00 0.01 1.04 0.00 0.00 0.00 0.00

Viola Palustris 0.03 0.11 0.01 0.00 0.08 1.85 0.01

Appendix C. Memberships in observation points

7a 7b 7c 7d 7e 7f 7g

91101 0.012 0.020 0.021 0.909 0.012 0.009 0.017

91102 0.037 0.084 0.768 0.017 0.030 0.019 0.044

91103 0.041 0.129 0.033 0.571 0.054 0.033 0.139

91104 0.026 0.190 0.023 0.019 0.089 0.584 0.068

91105 0.067 0.051 0.021 0.771 0.024 0.016 0.050

91106 0.002 0.003 0.001 0.989 0.002 0.001 0.002

91107 0.655 0.131 0.041 0.023 0.059 0.027 0.062

91108 0.040 0.152 0.190 0.031 0.051 0.483 0.053

91109 0.054 0.594 0.091 0.026 0.071 0.041 0.124

91110 0.005 0.011 0.957 0.004 0.006 0.005 0.013

91111 0.005 0.012 0.956 0.004 0.006 0.006 0.011

91112 0.211 0.093 0.489 0.024 0.045 0.043 0.094

91113 0.699 0.088 0.028 0.019 0.041 0.078 0.048

91114 0.052 0.452 0.046 0.026 0.092 0.051 0.280

91115 0.014 0.071 0.016 0.011 0.019 0.015 0.855

91116 0.017 0.026 0.013 0.881 0.017 0.024 0.022

91117 0.006 0.100 0.006 0.004 0.011 0.861 0.012

91118 0.001 0.006 0.001 0.000 0.002 0.989 0.001

91119 0.035 0.062 0.059 0.016 0.030 0.021 0.777

91120 0.037 0.088 0.044 0.080 0.051 0.037 0.663

91121 0.031 0.600 0.028 0.033 0.039 0.110 0.159

91122 0.029 0.089 0.037 0.612 0.038 0.146 0.049

91123 0.013 0.130 0.020 0.014 0.033 0.764 0.027

91124 0.956 0.015 0.006 0.005 0.006 0.004 0.008

91125 0.004 0.017 0.003 0.002 0.010 0.961 0.004

91126 0.002 0.017 0.002 0.002 0.006 0.967 0.004

91127 0.908 0.022 0.013 0.009 0.019 0.012 0.016

91128 0.017 0.430 0.013 0.018 0.060 0.421 0.041

91129 0.064 0.376 0.060 0.040 0.248 0.065 0.148

91130 0.067 0.232 0.110 0.053 0.104 0.068 0.366

91131 0.062 0.640 0.045 0.021 0.048 0.105 0.080

91132 0.084 0.193 0.099 0.065 0.128 0.098 0.334

91133 0.070 0.535 0.100 0.032 0.079 0.050 0.135

91134 0.009 0.027 0.933 0.004 0.009 0.005 0.012

91135 0.951 0.014 0.006 0.006 0.010 0.005 0.010

91201 0.003 0.021 0.003 0.002 0.006 0.959 0.006

91202 0.005 0.030 0.005 0.004 0.022 0.925 0.009

91203 0.010 0.845 0.012 0.005 0.043 0.039 0.046

91204 0.007 0.040 0.009 0.005 0.023 0.009 0.906

91205 0.005 0.029 0.005 0.004 0.018 0.932 0.008

91206 0.008 0.095 0.008 0.006 0.018 0.846 0.019

91207 0.053 0.170 0.054 0.044 0.081 0.518 0.080

91208 0.026 0.197 0.027 0.020 0.051 0.634 0.044

91209 0.012 0.121 0.014 0.009 0.023 0.796 0.023

91210 0.017 0.590 0.017 0.010 0.057 0.250 0.059

91211 0.001 0.009 0.001 0.001 0.003 0.982 0.002

91212 0.014 0.073 0.012 0.010 0.126 0.742 0.023

91213 0.001 0.006 0.001 0.001 0.986 0.002 0.003

91214 0.005 0.026 0.004 0.003 0.933 0.019 0.010

91215 0.034 0.528 0.036 0.016 0.229 0.041 0.114

91216 0.020 0.435 0.020 0.012 0.057 0.414 0.042

91217 0.070 0.598 0.044 0.018 0.139 0.046 0.085

91218 0.008 0.072 0.013 0.005 0.027 0.009 0.866

91219 0.021 0.033 0.013 0.009 0.881 0.018 0.024

91220 0.031 0.133 0.032 0.021 0.619 0.079 0.086

91221 0.197 0.167 0.040 0.023 0.449 0.046 0.077

91230 0.012 0.512 0.014 0.007 0.033 0.392 0.030