1. Community and gradient analysis: Matrix approaches in macroecology The world comes in fragments

2. Basic metrics of food webs S = 19 species L max = 19*18/2 = 171 possible links between two species L = 35 realized links between two species Connectance: C = 35/171 Ch = 100 total length of all food chains Li = 40 is the total number of chains ChL = 100/40 = 2.5 is the average chain length L/S = 35/19 = 1.8 is the mean number of links per species A pitcher plant (Nepenthes albomarginata) food web Nepenthes albomarginata

3. Food web metrics translated into matrix metrics N = 28 Fill = 28/80=0.35 D m =28/10=2.8 D n =28/8=3.5

4. Metrics of species associations in biogeographic matrices The C-score as a metric of negative associations The Clumping-score as a metric of positive associations Checkerboards The Togetherness-score as a metric of niche overlap

5. The additive nature of the C-score C Mixed = CS – C Turn - C Segr. Numbers of checkerboards for entries within the area AT are a measure of spatial species turnover. Numbers of checkerboards for entries within the area ATC are a measure of turnover independent species segregation. The rank correlation of matrix entries is a metric of spatial turnover. 1 1 1 2 1 3 2 1 2 2 ……. 7 10 8 10 R 2 = 0.347 R 2 is a more liberal metric than C turn. The correlation of ordination scores is also a metric of turnover but even less selective.

6. Range size coherence There are 17 embedded absences. The number of embedded absences is a measure of species range size coherence. Coherent range size Scattered range size The metric depends strongly on the ordering of rows and columns

7. The measurement of nestedness The distance concept of nestedness. Sort the matrix rows and olumns according to some gradient. Define an isocline that divides the matrix into a perfectly filled and an empty part. The normalized squared sum of relative distances of unexpected absences and unexpected presences is now a metric of nestednessis.

8. Nestedness based on Overlap and Decreasing Fill (NODF) NODF is a gap based metric and more conservative than temperature.

9. The disorder measure of Brualdi and Sanderson Ho many cells must be filled or emptied to achieve a perfectly ordered matrix. The Brualdi Sanderson measure is a count of this number Discrepancy is a gap counting metric.

10. How to measure species aggregation? Compartmented matrix Nearest neighbor metricsd ij Join count statistics Nearest neighbour is a presence – absence metric Join count operates on presence – absence and abundance matrices A sum of cell entries around a focal cell multiplied by the entry of the focal cell Other metrics proposed: Morisita Simpson Soerensen Block variance Ordination score variance Marginal variances NND has weak power at higher matrix fill These metrics have very low power a moderate to small matrix size and high or low matrix fill.

11. Abundance based metrics The C-score extension The metric CA is a count of the number of abundance checkerboards in the matrix. Other 2x2 submatrices catch matrix properties that have not well defined ecological meaning.

12. Nestedness in abundance matrices The metric is a sum of all pairs in the matrix (first sorted accoding to species richness then sorted according to weights), where the weight in the row/column of lower species richness is smaller than the weight in the row/column of higher species richness

13. Whole matrix SegregatedAggregated Aggregated nested Data typePA null models A null models Independent of matrix sortingC-scoreClumping scoreCS/ClumpingPAAll NestPairs A All Togetherness A and PAAll Species only SegregatedAggregated Data typePA null models A null models Simpson dissimilaritySimpson similarity PANo fixed - fixed Soerensen dissimilarity PANo fixed - fixed Morisita A and PANo fixed - fixedAll Chao A and PAAll Other joint occurrence/absence metrics PANo fixed - fixed Dependent of matrix sortingWhole matrix SegregatedAggregated Aggregated nested Data typePA null models A null models NND PAAll Block A and PAAll Join-coint A and PAAll Other distance based metrics A and PAAll NODFA and PAAll BRPAAll TPAAll Species only SegregatedAggregated Data typePA null models A null models Embedded absences PAAll For seriationr2r2 PAAll C Turn PAAll C Segr PAAll Morisita PAAll A complete table of methods for co-occurrence analysis

14. Pattern detection in large matrices These programs use cluster analysis and ordination to sort the matrix according to numbers of occurrences. Didstance metrics are then used to identify compartments. They generate hypotheses about matrix structure. They do not fully allow for statistical inference. WAND: ecological network analysis Pajek: software for social network analysis KliqueFinder: software for compartment analysis