Transcript Slajd 1

Ecological modelling
A
B
C
Our program
1.
2.
3.
4.
5.
6.
7.
8.
Matrix algebra I
Matrix algebra II
General additive models
Jackknifing and bootstrapping
Population models
Richness patterns in communities
Nestedness analysis
Indicator species analysis
Additional sources
http://en.wikipedia.org/wiki/Matrix_(mathematics)
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/EEB5812006/handouts/LinearI.pdf
http://matwww.ee.tut.fi/Kost/MatrixAlgebra-toc.html
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
A first course in linear algebra (free online textbook)
http://linear.ups.edu/download.html
Matrix algebra and regression
http://www.stat.tugraz.at/courses/files/s05.pdf
Mathe online
http://www.matheonline.at/mathint.html
Species
Taxon
Guild
Nanoptilium kunzei (Heer, 1841)
Acrotrichis dispar (Matthews, 1865)
Ptiliidae
Ptiliidae
Necrophagous
Necrophagous
Mean
length
(mm)
0.60
0.65
Acrotrichis silvatica Rosskothen, 1935
Ptiliidae
Necrophagous
Acrotrichis rugulosa Rosskothen, 1935
Ptiliidae
Acrotrichis grandicollis (Mannerheim, 1844)
Acrotrichis fratercula (Matthews, 1878)
Ptiliidae
Ptiliidae
Site 1
Site 2
Site 3
Site 4
0
13
0
0
0
4
0
7
0.80
16
0
2
0
Necrophagous
0.90
0
0
1
0
Necrophagous
0.95
1
0
0
1
Necrophagous
1.00
0
1
0
0
1
0
0
0
13
0
0
8
3
0
5
23
0
5
0
2
5
0
4
0
0
5
6
0
6
9
2
0
0
1
0
0
Carcinops pumilio (Erichson, 1834)
Histeridae
Predator
2.15
Saprinus aeneus (Fabricius, 1775)
Histeridae
Histeridae
Histeridae
Staphylinidae
Histeridae
Histeridae
Histeridae
Predator
Predator
Predator
Predator
Predator
Predator
Predator
3.00
Gnathoncus nannetensis (Marseul, 1862)
Margarinotus carbonarius (Hoffmann, 1803)
Rugilus erichsonii (Fauvel, 1867)
Margarinotus ventralis (Marseul, 1854)
Saprinus planiusculus Motschulsky, 1849
Margarinotus merdarius (Hoffmann, 1803)
3.10
3.60
3.75
4.00
4.45
4.50
A vector can be
interpreted as a
file of data
Handling biological data is most easily done with a matrix approach.
An Excel worksheet is a matrix.
A matrix is a
collection of
vectors and can
be interpreted
as a data base
The red matrix
contain three
column vectors
A general structure of databases
 a11

A
a
 m1
 a1 
 
a2 

V
 a3 
 
 a4 
a1n 


a mn 
The first subscript denotes rows,
the second columns.
n and m define the dimension of a matrix.
A has m rows and n columns.
V   a1 a 2 a3 a 4 
Row
vector
Column
vector
 a11 a12

V   a 21 a 22
a
 31 a 32
a13 

a 23 
a 33 
 a11 a12

V   a 21 a 22
a
 31 a 32
a13 

a 23 
a 33 
Two matrices are equal if they have the same dimension and all corresponding
values are identical.
Solving systems of linear equations
The Nine Chapters on the
Mathematical Art.
(1000BC-100AD).
Systems of linear equations,
Gaussian elimination
Takakazu Shinsuke Seki
(1642-1708)
Determinants to solve
linear equations
Gottfried Wilhelm
Leibniz
(1646-1716)
Determinants to
solve linear
equations
Matrix approaches
Johann Carl Friedrich Gauss
(1777 – 1855)
Gaussian elimination, inverse
Arthur Cayley
(1821-1895)
Formal matrix
algebra
Olga Taussky-Todd
(1906-1995)
Finite value matrices
Some elementary types of matrices
In biology and statistics are square matrices An,n
of particular importance
1

3
A
5

4

2
4
6
3
3
5
7
2
4

6
8

1 
The symmetric matrix is a matrix where
An,m = A m,n.
1

2
A
3

4

2
4
5
6
3
5
7
8
4

6
8

1 
Lower and upper triangular matrices
1

2
A
3

4

0
4
5
6
0

0
0

1 
0
0
7
8
1

0
A
0

0

2
4
0
0
3
5
7
0
4

6
8

1 
The diagonal matrix is a square and symmetrical.
1

0
A
0

0

0
4
0
0
0
0
7
0
Λ  3 is a matrix with one row and one column.
It is a scalar (ordinary number).
0
1


0
0
A

0
0



0
1

0
1
0
0
0
0
1
0
0

0
0

1 
Unit matrix I
Matrix operations
Addition and Subtraction
1

2
A
3

3
 a11  b11 ... ... a1m  b1m 
2 3   2 4 0   2 8 1   5 14 4 


 
 
 

...
...
...
...


2 4   1 2 0   7 5 5  10 9 9 
AB 



...
... ...
... 
5 7   6 9 1   0 0 1   9 14 9 


 
 
 

 a  b ... ... a  b 
1 0 1 1 4 5 6 1  9 8 5
nm
nm 
 n1 n1
Addition and subtraction are only defined for matrices with identical dimensions
S-product
1

2
A
3

3
2
2
5
1
3 1
 
4  2

7 3
 
0  3
 b11

 ...
B  
...

 b
 n1
2
2
5
1
3 1
 
4  2

7 3
 
0 3
2
2
5
1
... ... b1m 

... ... ... 
 B
... ... ... 

... ... bnm 
3 3 6
 
4 6 6

7   9 15
 
0 9 3
9
1


12 
2
 3
3
21


0
3
2
2
5
1
3   3 1
 
4 32

7   3 3
 
0   33
32
32
3 5
3 1
A  B   B  A  1B  A
AB  BA
A  (B  C)  (A  B)  C
 A  A
 (A  B)   A   B
A(   )  A  A
3 3 

34
37 

30
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.
 0.8 


P  15 20 30    0.9   15*0.8  20*0.9  30*0.7  51
 0.7 


A  B   a1
 b1  n
 
... a n    ...    a i bi  scalar
 b  i 1
 n
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.
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
 0.8 1.1 


P  15 20 30    0.9 1.2   15*0.8  20*0.9  30*0.7 15*1.1  20*1.2  30*1.25    51 78 
 0.7 1.25 


 a11 ... a1m   b11

 
A  B   ... ... ...    ...
 a ... a   a
nm   m1
 n1
 m
a1i bi1 ...
... b1k   
i 1


... ...    ...
...

... a mk   m
  a ni bi1 ...
 i 1
A  B  B A
(A  B)  C  A  (B  C)  A  B  C
(A  B)  C  A  C  B  C

a
b

1i ik 
A B ... A1Bk 
i 1
  1 1

...    ... ...
... 
 

m
A
B
...
A
B
m
1
m
k


a ni bik 

i 1

m
1 
 2 4  

  2
3 5 3
 
 2 4   2 4 6   2  2  4  5 2  4  4  6 2  6  4  7   24 32 40 





3
5
5
6
7
3

2

5

5
3

4

5

6
3

6

5

7

 
 
  31 42 53 
 2 4

   2 4 6
3
5


The number of columns in the first matrix must equal the number of rows in the
second matrix.
AijB jkCkl Dlm...Z yz  Ciz
AijB jk  Cik
A
2x3
1
1
2
2
1
2
4
3
3
3
2
4
3
2
3
1
B
3x2
AB
2x2
17
9
18
12
5 2x4
5
106
66
87
51
86
48
2x3
ABCD
1153
687
1943
1167
1011
597
C
2x4
ABC
4
1
D
4x3
1
3
1
4
6
2
3
5
3
2
4
1
175
105
Transpose A’ ot AT
1
2
3 4



2
.
171828

1
8
9


  3.14159 3.56 4 3 


2
1
2
3
4

 

 1
4 3 2 1
 2 3 4 5  3

 4

T
 1 2.171828  3.141459


2

1
3
.
56




3
8
4


4

9
3


1
  29 30

2 
  21 20
3 
  39 40
4 
AB
 a11
T
 a11 ... ... a1n  

  ...
...
...
...
...

 
a
  ...
...
...
a
mn 
 m1
a
 1n
1

 2 1 3 4  2

 * 
1
2
3
4

 3
4

(A  B)T  BT  AT
2
3
2
1
... am1 

... ... 
... ... 

... amn 
4

3   29 21 39


4   30 20 40

5 
BT  A T
Matrix add in for Excel:
www.digilander.libero.it/foxes/SoftwareDownload.htm
Some properties of the transpose
A ' A  AA '
A ' A  AA '
always exists and gives a
symmetric matrix
only if A is square
and symmetric
If A is orthogonal A’A is diagonal, but AA’
need not to be diagonal
A
A'
Orthogonal matrix
3
-1
2
2
1
-1
-1
0.5
2
3
-1
-1
2
2
0.5
1
-1
2
AA'
11
3.5
2
3.5
8.25
1
2
1
6
A'A
14
0
0
0
6
0
0
0
5.25
Ground beetles on Mazurian lake islands (Mamry)
Carabus problematicus
Carabus auratus
Photo Marek
Ostrowski
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus
(Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
wros
0
0
1
1
0
1
0
0
0
0
0
wron
2
0
1
1
0
1
0
0
0
0
1
wil
61
1
0
2
1
21
0
0
0
0
0
ter
53
0
0
2
0
2
0
0
1
0
0
swi
0
0
19
0
0
0
1
4
0
0
0
sos
18
9
40
0
0
1
2
1
0
2
0
mil
39
0
0
0
3
7
0
0
1
0
0
lip
2
117
1
0
0
0
4
3
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
1
2
0
0
0
Species associations
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus (Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
wros
0
0
1
1
0
1
0
0
0
0
0
0
0
0
wron
2
0
1
1
0
1
0
0
0
0
1
0
0
0
wil
61
1
0
2
1
21
0
0
0
0
0
0
0
0
ter
53
0
0
2
0
2
0
0
1
0
0
0
0
0
swi
0
0
19
0
0
0
1
4
0
0
0
0
0
0
sos
18
9
40
0
0
1
2
1
0
2
0
0
0
0
mil
39
0
0
0
3
7
0
0
1
0
0
2
1
2
lip
2
117
1
0
0
0
4
3
0
0
0
0
0
0
Panagaeus cruxmajor (Linnaeus)
Poecilus versicolor (Sturm)
Pterostichus gracilis Dejean)
Stenolophus mixtus
Pseudoophonus rufipes (De Geer)
Harpalus latus (Linnaeus)
Agonum duftshmidi Shmidt
Harpalus solitaris Dejean
0
0
0
0
0
0
0
0
24
0
0
0
0
0
0
0
0
0
0
0
13
0
1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
5
3
0
0
5
0
0
0
3
0
0
1
1
2
0
0
2
2
0
0
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus (Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
wros
0
0
1
1
0
1
0
0
0
0
0
0
0
0
wron
2
0
1
1
0
1
0
0
0
0
1
0
0
0
wil
61
1
0
2
1
21
0
0
0
0
0
0
0
0
ter
53
0
0
2
0
2
0
0
1
0
0
0
0
0
swi
0
0
19
0
0
0
1
4
0
0
0
0
0
0
sos
18
9
40
0
0
1
2
1
0
2
0
0
0
0
mil
39
0
0
0
3
7
0
0
1
0
0
2
1
2
lip
2
117
1
0
0
0
4
3
0
0
0
0
0
0
S
wros
wron
wil
ter
swi
sos
mil
lip
Panagaeus
cruxmajor
(Linnaeus)
0
24
0
0
1
0
5
1
Poecilus
versicolor
(Sturm)
0
0
0
0
0
0
0
2
Pterostichus
gracilis
Dejean)
0
0
0
0
0
0
0
0
Stenolophus
mixtus
0
0
0
1
0
0
0
0
Pseudoopho
nus rufipes
(De Geer)
0
0
13
0
0
5
3
2
Harpalus
latus
(Linnaeus)
0
0
0
0
0
3
0
2
Agonum
duftshmidi
Shmidt
0
0
1
0
0
0
0
0
Harpalus
solitaris
Dejean
0
0
0
0
1
0
1
0
Species
Pterostichus nigrita (Paykull)
Platynus assimilis (Paykull)
Amara brunea (Gyllenhal)
Agonum lugens (Duftshmid)
Loricera pilicornis (Fabricius)
Pterostichus vernalis (Panzer)
Amara plebeja (Gyllenhal)
Badister unipustulatus Bonelli
Lasoitrechus discus (Fabricius)
Poecilus cupreus (Linnaeus)
Amara aulica (Panzer)
Anisodatylus binotatus (Fabricius)
Bembidion articulatum (Panzer)
Clivina collaris (Herbst)
Panagaeus
cruxmajor
(Linnaeus)
245
117
44
24
15
59
5
7
5
0
24
10
5
10
Poecilus
versicolor
(Sturm)
4
234
2
0
0
0
8
6
0
0
0
0
0
0
Pterostichus
gracilis
Dejean)
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Stenolophus
mixtus
53
0
0
2
0
2
0
0
1
0
0
0
0
0
Pseudoopho
nus rufipes
(De Geer)
1004
292
202
26
22
299
18
11
3
10
0
6
3
6
Harpalus
latus
(Linnaeus)
58
261
122
0
0
3
14
9
0
6
0
0
0
0
Agonum
duftshmidi
Shmidt
61
1
0
2
1
21
0
0
0
0
0
0
0
0
Harpalus
solitaris
Dejean
39
0
19
0
3
7
1
4
1
0
0
2
1
2
Probabilities of co-occurrence
Species
wros
Pterostichus nigrita (Paykull)
0
Platynus assimilis (Paykull)
0
Amara brunea (Gyllenhal)
0.59
Agonum lugens (Duftshmid)
0.02
Loricera pilicornis (Fabricius)
0
Pterostichus vernalis (Panzer)
0.1
Amara plebeja (Gyllenhal)
0
Badister unipustulatus Bonelli
0
Lasoitrechus discus (Fabricius)
0
Poecilus cupreus (Linnaeus)
0
Amara aulica (Panzer)
0
Anisodatylus binotatus (Fabricius)
0
Bembidion articulatum (Panzer)
0
Clivina collaris (Herbst)
0
Species
wros
Panagaeus cruxmajor (Linnaeus)
0
Poecilus versicolor (Sturm)
0
Pterostichus gracilis Dejean)
0
Stenolophus mixtus
0
Pseudoophonus rufipes (De Geer)
0
Harpalus latus (Linnaeus)
0
Agonum duftshmidi Shmidt
0
Harpalus solitaris Dejean
0
wron wil
0.79 0.01
0 0.83
0.97
0
0.06 0.18
0 0.08
0.59 0.88
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
wron wil
0
0
0
0
0
0
0
0
0 0.22
0
0
0 0.17
0
0
ter
0.14
0
0
0.74
0
0.87
0
0
0.02
0
0
0
0
0
ter
0
0
0
0.11
0
0
0
0
swi sos
0 0.15
0 0.53
0.11 0.02
0
0
0
0
0 0.4
0.19 0.09
0.4 0.03
0
0
0 0.42
0
0
0
0
0
0
0
0
swi sos
0
0
0
0
0
0
0
0
0 0.83
0 0.29
0
0
0
0
mil
0.37
0
0
0
0.97
1
0
0
0
0
0
0.8
0.72
0.65
mil
0
0
0
0
0.98
0
0
0
lip
0.14
0.86
0.47
0
0
0
0.86
0.58
0
0
0
0
0
0
lip
0.7
0.38
0
0
0.66
0.64
0
0
kor hel
0
0
0.76 0.59
0 0.87
0
0
0
0
0
0
0
0
0
0
0
0
0.12
0
0
0
0
0
0
0
0
0
kor hel
0
0
0
0
0
0
0
0
0.58 0.04
0.35
0
0
0
0
0
guc
0.45
0.62
0.54
0
0
0
0
0
0
0
0
0
0
0
guc
0
0
0
0
0.32
0
0
0
gil
0.51
0.2
0
1
0.27
0
0.19
0.34
0
0
0
0
0
0
gil
0
0
0.38
0
0.51
0.18
0
0.81
ful
0.56
0.03
0.39
0
0.56
0
0
0
0
0
0
0
0
0
ful
0
0
0
0
0.19
0.15
0
0.85
dab
0.01
0.85
0.47
0.37
0.89
0
0.37
0
0
0
0
0
0
0
dab
0
0
0
0
0.62
0
0
0
3pog
0.28
0.37
0
0
0
0
0
0
0
0
0.59
0
0
0
3pog
0
0
0
0
0.17
0.17
0.22
0
2pog
0.74
0.83
0
0
0.46
0
0
0
0
0
0
0
0
0
2pog
0
0
0
0
0.54
0.25
0
0
1pog
0.18
0
0
0.89
0
0
0
0
0
0
0
0
0
0
1pog
0
0
0
0
0.53
0
0.17
0
R  P1P2
T
Species
Panagaeus cruxmajor
Poecilus versicolor
(Linnaeus)
Pterostichus
(Sturm)
Stenolophus
gracilis Dejean)
Pseudoophonus
mixtus Harpalus
rufipes
latus
Agonum
(De
(Linnaeus)
Geer)
duftshmidi
HarpalusShmidt
solitaris Dejea
Pterostichus 0.034035
nigrita (Paykull)
0.036504 0.213372 0.347488 4.640972 2.625121 0.682791 0.328616
Platynus assimilis
0.35977
(Paykull)
0.385866 0.047028
0 2.791692 2.228522 0.572894 0.16836
Amara brunea0.22055
(Gyllenhal)
0.236548
0
0 2.735377 1.078382
0 0.005715
Agonum lugens (Duftshmid)
0
0 0.149993 0.477588 2.060951 0.613648 0.432521 0.206119
Loricera pilicornis (Fabricius)
0
0 0.062924
0 2.527301 1.257689 0.254665 0.235746
Pterostichus vernalis0(Panzer) 0
0 0.287552 1.234233 0.455731 0.020836
0
Amara plebeja
0.126953
(Gyllenhal)
0.136162 0.145696
0 1.244267 0.955163
0 0.200214
Badister unipustulatus
0.209502 Bonelli
0.224699 0.059252
0 0.895534 1.127406
0 0.081424
Lasoitrechus discus (Fabricius)
0
0
0 0.158252 0.18667
0
0
0
Poecilus cupreus (Linnaeus)
0
0
0
0 0.804519 0.570117
0
0
Amara aulica (Panzer)
0
0
0
0 0.015829 0.016889 0.004067
0
Anisodatylus binotatus
0 (Fabricius)
0
0
0 0.006664
0
0
0
Bembidion articulatum
0 (Panzer) 0
0
0 0.333915
0
0
0
Clivina collaris (Herbst)
0
0
0
0 0.082518
0
0
0
The entries of the matrix give the sum of probabilities that two species meet on any of the islands.
15
R( Pterostichus  Panagaeus)   PPterosticus PPanagaeus
i 1
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?
B
C
A
B
1 2 3 4 5
1 2 3 4
1 0 0 0 1 1
1 0 1 1 1




0
1
0
0
0

 2
0 1 0 0 2
We eliminate


0 0 0 1 1 3
A  1 0 0 1 3
group B and leave


B



the first and last
0 0 0 1 1 4
0 0 0 1 4
 0 1 0 0 0
group.
0 1 0 0
5



 5
 0 1 0 0 0 6


No. 1 of group C
C
A
indirectly
1 2 3 4
contacted with all
1 0 0 0 1
1 1 1 1 1
members of group

 1 0 1 1 

 0 1 0 0 2
A.
0 1 0 0 0 
0
1
0
0
 
0 0 0 1 1 
 3
No. 2 of group A
0
1
0
1
  1 0 0 1  

C  BA  
indirectly
 0 1 0 1 4
0 0 0 1 1 
contacted with all
0 1 0 0 0  0 0 0 1 0 1 0 0 5
six persons of

  0 1 0 0 

 
0 1 0 0 0 

group C.


0 1 0 0 6
Instead of contact we use probabilities of being infected.
A
B
1
2
3
4
5
C
1
2
3
4
5
6
1
0.3
0
0.3
0
0
1
0.3
0
0
0
0
0
2
0
0.3
0
0
0.3
3
0.2
0
0
0
0
4
0.2
0
1
1
0
2
0
0.3
0
0
0.3
0.3
B
3
0
0
0
0
0
0
4
0
0
0.1
0.1
0
0
5
0.2
0
0.2
0.2
0
0
3
0.06
0
0
0
0
0
4
0.06
0
0.1
0.1
0
0
Sum
0.27
0.09
0.16
0.16
0.09
0.09
A
C
1
2
3
4
5
6
1
0.09
0
0
0
0
0
2
0.06
0.09
0.06
0.06
0.09
0.09
C  BA
Person 1 of group C has the highest
probability of being infected.