1. Ecological modelling

2. Our program 1.Matrix algebra I 2.Matrix algebra II 3.General additive models 4.Jackknifing and bootstrapping 5.Population models 6.Richness patterns in communities 7.Nestedness analysis 8.Indicator species analysis

3. Additional sources K. Kaw. 2002. Introduction to matrix algebra http://www.autarkaw.com/books/matrixalgebra/index.html http://www.ems.bbk.ac.uk/faculty/phdStudents/efthyvoulou/Kaw.pdf Introduction to matrix algebra and linear models: http://nitro.biosci.arizona.edu/courses/EEB581- 2006/handouts/LinearI.pdf Matrix cook book http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf /imm3274.pdf Matrix http://en.wikipedia.org/wiki/Matrix_the ory Matrix algebra and regression http://www.stat.tugraz.at/courses/files/s05.pdf http://en.wikipedia.org/wiki/Matrix_(mathematics) http://matwww.ee.tut.fi/Kost/MatrixAlgebra-toc.html A first course in linear algebra (free online textbook) http://linear.ups.edu/download.html

4. Mathe online http://www.mathe- online.at/mathint.html

5. A vector can be interpreted as a file of data A matrix is a collection of vectors and can be interpreted as a data base The red matrix contain three column vectors Handling biological data is most easily done with a matrix approach. An Excel worksheet is a matrix.

6. A general structure of databases The first subscript denotes rows, the second columns. n and m define the dimension of a matrix. A has m rows and n columns. Two matrices are equal if they have the same dimension and all corresponding values are identical. Column vector Row vector

7. Solving systems of linear equations Takakazu Shinsuke Seki (1642-1708) Determinants to solve linear equations Gottfried Wilhelm Leibniz (1646-1716) Determinants to solve linear equations Arthur Cayley (1821-1895) Formal matrix algebra The Nine Chapters on the Mathematical Art. (1000BC-100AD). Systems of linear equations, Gaussian elimination Matrix approaches Johann Carl Friedrich Gauss (1777 – 1855) Gaussian elimination, inverse Olga Taussky-Todd (1906-1995) Finite value matrices

8. In biology and statistics are square matrices A n,n of particular importance The symmetric matrix is a matrix where A n,m = A m,n. Lower and upper triangular matrices Some elementary types of matrices The diagonal matrix is a square and symmetrical. Unit matrix I is a matrix with one row and one column. It is a scalar (ordinary number).

9. Matrix operations Addition and Subtraction Addition and subtraction are only defined for matrices with identical dimensions S-product

10. The inner or dot or scalar product Assume you have production data (in tons) of winter wheat (15 t), summer wheat (20 t), and barley (30 t). In the next year weather condition reduced the winter wheat production by 20%, the summer wheat production by 10% and the barley production by 30%. How many tons do you get the next year? (15*0.8 + 20* 0.9 + 30 * 0.7) t = 51 t. The dot product is only defined for matrices, where the number of columns in the first matrix equals the number of rows in the second matrix.

11. We add another year and ask how many cereals we get if the second year is good and gives 10 % more of winter wheat, 20 % more of summer wheat and 25 % more of barley. For both years we start counting with the original data and get a vector with one row that is the result of a two step process

12. The number of columns in the first matrix must equal the number of rows in the second matrix.

13. Transpose A’ ot A T

14. Matrix add in for Excel: www.digilander.libero.it/foxes/SoftwareDownload.htm

15. always exists and gives a symmetric matrix only if A is square and symmetric Some properties of the transpose If A is orthogonal A’A is diagonal, but AA’ need not to be diagonal

16. Specieswroswronwilterswisosmillip Pterostichus nigrita (Paykull)026153018392 Platynus assimilis (Paykull)0010090117 Amara brunea (Gyllenhal)1100194001 Agonum lugens (Duftshmid)11220000 Loricera pilicornis (Fabricius)00100030 Pterostichus vernalis (Panzer)112120170 Amara plebeja (Gyllenhal)00001204 Badister unipustulatus Bonelli00004103 Lasoitrechus discus (Fabricius)00010010 Poecilus cupreus (Linnaeus)00000200 Amara aulica (Panzer)01000000 Anisodatylus binotatus (Fabricius)00000020 Bembidion articulatum (Panzer)00000010 Clivina collaris (Herbst)00000020 Ground beetles on Mazurian lake islands (Mamry) Photo Marek Ostrowski Carabus auratus Carabus problematicus

17. Specieswroswronwilterswisosmillip Pterostichus nigrita (Paykull)026153018392 Platynus assimilis (Paykull)0010090117 Amara brunea (Gyllenhal)1100194001 Agonum lugens (Duftshmid)11220000 Loricera pilicornis (Fabricius)00100030 Pterostichus vernalis (Panzer)112120170 Amara plebeja (Gyllenhal)00001204 Badister unipustulatus Bonelli00004103 Lasoitrechus discus (Fabricius)00010010 Poecilus cupreus (Linnaeus)00000200 Amara aulica (Panzer)01000000 Anisodatylus binotatus (Fabricius)00000020 Bembidion articulatum (Panzer)00000010 Clivina collaris (Herbst)00000020 Panagaeus cruxmajor (Linnaeus)024001051 Poecilus versicolor (Sturm)00000002 Pterostichus gracilis Dejean)00000000 Stenolophus mixtus00010000 Pseudoophonus rufipes (De Geer)001300532 Harpalus latus (Linnaeus)00000302 Agonum duftshmidi Shmidt00100000 Harpalus solitaris Dejean00001010 Species associations

20. S Panagaeus cruxmajor (Linnaeus) Poecilus versicolor (Sturm) Pterostichus gracilis Dejean) Stenolophus mixtus Pseudoopho nus rufipes (De Geer) Harpalus latus (Linnaeus) Agonum duftshmidi Shmidt Harpalus solitaris Dejean wros00000000 wron240000000 wil000013010 ter00010000 swi10000001 sos00005300 mil50003001 lip12002200 Specieswroswronwilterswisosmillip Pterostichus nigrita (Paykull)026153018392 Platynus assimilis (Paykull)0010090117 Amara brunea (Gyllenhal)1100194001 Agonum lugens (Duftshmid)11220000 Loricera pilicornis (Fabricius)00100030 Pterostichus vernalis (Panzer)112120170 Amara plebeja (Gyllenhal)00001204 Badister unipustulatus Bonelli00004103 Lasoitrechus discus (Fabricius)00010010 Poecilus cupreus (Linnaeus)00000200 Amara aulica (Panzer)01000000 Anisodatylus binotatus (Fabricius)00000020 Bembidion articulatum (Panzer)00000010 Clivina collaris (Herbst)00000020 Species Panagaeus cruxmajor (Linnaeus) Poecilus versicolor (Sturm) Pterostichus gracilis Dejean) Stenolophus mixtus Pseudoopho nus rufipes (De Geer) Harpalus latus (Linnaeus) Agonum duftshmidi Shmidt Harpalus solitaris Dejean Pterostichus nigrita (Paykull)24540531004586139 Platynus assimilis (Paykull)1172340029226110 Amara brunea (Gyllenhal)44200202122019 Agonum lugens (Duftshmid)2400226020 Loricera pilicornis (Fabricius)1500022013 Pterostichus vernalis (Panzer)590022993217 Amara plebeja (Gyllenhal)5800181401 Badister unipustulatus Bonelli760011904 Lasoitrechus discus (Fabricius)50013001 Poecilus cupreus (Linnaeus)000010600 Amara aulica (Panzer)240000000 Anisodatylus binotatus (Fabricius)100006002 Bembidion articulatum (Panzer)50003001 Clivina collaris (Herbst)100006002

21. Probabilities of co-occurrence

22. The entries of the matrix give the sum of probabilities that two species meet on any of the islands.

23. Assume you are studying a contagious disease. You identified as small group of 4 persons infected by the disease. These 4 persons contacted in a given time with another group of 5 persons. The latter 5 persons had contact with other persons, say with 6, and so on. How often did a person of group C indirectly contact with a person of group A? A 1 2 3 4 B12345B12345 B 1 2 3 4 5 C123456C123456 A 1 2 3 4 C123456C123456 We eliminate group B and leave the first and last group. No. 1 of group C indirectly contacted with all members of group A. No. 2 of group A indirectly contacted with all six persons of group C.

24. Instead of contact we use probabilities of being infected. Person 1 of group C has the highest probability of being infected.