• Species correlated - respond to same environmental gradients
• Respond to multiple gradients
• Represent high dimensional patterns in reduced dimensional space

##     Sp1 Sp2 Sp3 Sp4 Sp5
## Q1    5  10  19   9   6
## Q2    5   9  19   9   6
## Q3    6  11  20  10   5
## Q4    5   9  20  10   5
## Q5    3   8  19   9   5
## Q6    4   9  19  10   5
## Q7    3   9  21  11   3
## Q8    4  10  19   9   5
## Q9    6  10  20  11   4
## Q10   5  10  19   9   6

• Variables
  
• Objects

##     Sp1 Sp2 Sp3 Sp4 Sp5
## Q1    5  10  19   9   6
## Q2    5   9  19   9   6
## Q3    6  11  20  10   5
## Q4    5   9  20  10   5
## Q5    3   8  19   9   5
## Q6    4   9  19  10   5
## Q7    3   9  21  11   3
## Q8    4  10  19   9   5
## Q9    6  10  20  11   4
## Q10   5  10  19   9   6

GOALS

• Data reduction
• Represent multivariate patterns in reduced ordination space

##     Sp1 Sp2 Sp3 Sp4 Sp5
## Q1    5  10  19   9   6
## Q2    5   9  19   9   6
## Q3    6  11  20  10   5
## Q4    5   9  20  10   5
## Q5    3   8  19   9   5
## Q6    4   9  19  10   5
## Q7    3   9  21  11   3
## Q8    4  10  19   9   5
## Q9    6  10  20  11   4
## Q10   5  10  19   9   6

• R-mode
• Q-mode

Presence/absence

library(vegan)
decostand(data[,-1], method="pa")

##        Sp1 Sp2 Sp3 Sp4 Sp5 Sp6 Sp7 Sp8 Sp9 Sp10
## Site1    1   0   0   1   1   0   0   0   0    0
## Site2    0   0   0   1   1   0   1   1   0    0
## Site3    0   0   0   1   1   0   1   1   0    0
## Site4    0   0   0   0   0   0   0   1   0    1
## Site5    0   0   1   0   0   0   0   1   1    1
## Site6    0   1   1   0   0   0   0   0   1    0
## Site7    0   0   1   0   0   1   0   0   1    0
## Site8    0   0   0   0   1   0   1   1   0    0
## Site9    0   0   0   1   1   0   1   0   0    0
## Site10   0   0   0   1   1   0   0   0   0    0

• Maximum per column

• Maximum per column

library(vegan)
decostand(data[,-1], method="max")

##        Sp1 Sp2       Sp3        Sp4       Sp5 Sp6       Sp7   Sp8       Sp9
## Site1    1   0 0.0000000 0.90277778 0.1063830   0 0.0000000 0.000 0.0000000
## Site2    0   0 0.0000000 0.34722222 0.8297872   0 1.0000000 0.575 0.0000000
## Site3    0   0 0.0000000 0.08333333 0.8936170   0 1.0000000 0.775 0.0000000
## Site4    0   0 0.0000000 0.00000000 0.0000000   0 0.0000000 1.000 0.0000000
## Site5    0   0 0.2857143 0.00000000 0.0000000   0 0.0000000 0.850 0.8181818
## Site6    0   1 0.5714286 0.00000000 0.0000000   0 0.0000000 0.000 1.0000000
## Site7    0   0 1.0000000 0.00000000 0.0000000   1 0.0000000 0.000 0.9090909
## Site8    0   0 0.0000000 0.00000000 0.2765957   0 1.0000000 0.925 0.0000000
## Site9    0   0 0.0000000 0.83333333 1.0000000   0 0.6666667 0.000 0.0000000
## Site10   0   0 0.0000000 1.00000000 0.7234043   0 0.0000000 0.000 0.0000000
##             Sp10
## Site1  0.0000000
## Site2  0.0000000
## Site3  0.0000000
## Site4  1.0000000
## Site5  0.8571429
## Site6  0.0000000
## Site7  0.0000000
## Site8  0.0000000
## Site9  0.0000000
## Site10 0.0000000

• Wisconsin double standardization

  Sites	Sp1	Sp2	Sp3	Sp4	Sp5	Sp6	Sp7	Sp8	Sp9	Sp10
  Site1	5	0	0	65	5	0	0	0	0	0
  Site2	0	0	0	25	39	0	6	23	0	0
  Site3	0	0	0	6	42	0	6	31	0	0
  Site4	0	0	0	0	0	0	0	40	0	14
  Site5	0	0	6	0	0	0	0	34	18	12
  Site6	0	29	12	0	0	0	0	0	22	0
  Site7	0	0	21	0	0	5	0	0	20	0
  Site8	0	0	0	0	13	0	6	37	0	0
  Site9	0	0	0	60	47	0	4	0	0	0
  Site10	0	0	0	72	34	0	0	0	0	0

• standardize columns (species) by maximum
• standardize rows (sites) by totals

• Wisconsin double standardization

library(vegan)
wisconsin(data[,-1])

##              Sp1       Sp2       Sp3        Sp4        Sp5     Sp6       Sp7
## Site1  0.4977203 0.0000000 0.0000000 0.44933078 0.05294896 0.00000 0.0000000
## Site2  0.0000000 0.0000000 0.0000000 0.12617043 0.30152049 0.00000 0.3633708
## Site3  0.0000000 0.0000000 0.0000000 0.03028155 0.32472135 0.00000 0.3633786
## Site4  0.0000000 0.0000000 0.0000000 0.00000000 0.00000000 0.00000 0.0000000
## Site5  0.0000000 0.0000000 0.1016401 0.00000000 0.00000000 0.00000 0.0000000
## Site6  0.0000000 0.3888889 0.2222222 0.00000000 0.00000000 0.00000 0.0000000
## Site7  0.0000000 0.0000000 0.3437500 0.00000000 0.00000000 0.34375 0.0000000
## Site8  0.0000000 0.0000000 0.0000000 0.00000000 0.12563421 0.00000 0.4542160
## Site9  0.0000000 0.0000000 0.0000000 0.33333333 0.40000000 0.00000 0.2666667
## Site10 0.0000000 0.0000000 0.0000000 0.58024691 0.41975309 0.00000 0.0000000
##              Sp8       Sp9      Sp10
## Site1  0.0000000 0.0000000 0.0000000
## Site2  0.2089382 0.0000000 0.0000000
## Site3  0.2816185 0.0000000 0.0000000
## Site4  0.5000000 0.0000000 0.5000000
## Site5  0.3023793 0.2910603 0.3049203
## Site6  0.0000000 0.3888889 0.0000000
## Site7  0.0000000 0.3125000 0.0000000
## Site8  0.4201498 0.0000000 0.0000000
## Site9  0.0000000 0.0000000 0.0000000
## Site10 0.0000000 0.0000000 0.0000000

• Range

• Range

library(vegan)
enviro1 <- within(enviro, 
  Substrate <- as.numeric(Substrate))
decostand(enviro1[,-1], method="range")

##            pH      Slope  Pressure  Altitude Substrate
## 1  0.00000000 0.32743363 0.0000000 0.0000000         0
## 2  0.26086957 0.76991150 0.3802817 0.5948478         1
## 3  0.30434783 0.71681416 0.4366197 0.6370023         1
## 4  0.39130435 0.61061947 0.6619718 0.8852459         1
## 5  0.47826087 0.46902655 0.8309859 0.9496487         1
## 6  0.60869565 0.69911504 0.9859155 1.0000000         0
## 7  0.60869565 0.00000000 1.0000000 0.9976581         0
## 8  0.39130435 1.00000000 0.6338028 0.8571429         1
## 9  1.00000000 0.68141593 0.3098592 0.4192037         0
## 10 0.04347826 0.08849558 0.2816901 0.4145199         0

• Center

• Center

library(vegan)
enviro1 <- within(enviro, 
  Substrate <- as.numeric(Substrate))
apply(enviro1[,-1],2,scale, scale=FALSE)

##          pH Slope Pressure Altitude Substrate
##  [1,] -0.94 -2.36    -39.2   -576.9      -0.5
##  [2,] -0.34  2.64    -12.2    -68.9       0.5
##  [3,] -0.24  2.04     -8.2    -32.9       0.5
##  [4,] -0.04  0.84      7.8    179.1       0.5
##  [5,]  0.16 -0.76     19.8    234.1       0.5
##  [6,]  0.46  1.84     30.8    277.1      -0.5
##  [7,]  0.46 -6.06     31.8    275.1      -0.5
##  [8,] -0.04  5.24      5.8    155.1       0.5
##  [9,]  1.36  1.64    -17.2   -218.9      -0.5
## [10,] -0.84 -5.06    -19.2   -222.9      -0.5

• Standardize - scale

• Standardize - scale

library(vegan)
enviro1 <- within(enviro, 
   Substrate <- as.numeric(Substrate))
apply(enviro1[,-1],2,scale)

#OR
library(vegan)
decostand(enviro1[,-1], method="standardize")

##             pH      Slope   Pressure   Altitude  Substrate
## 1  -1.39885358 -0.6635569 -1.6862750 -2.0691070 -0.9486833
## 2  -0.50596832  0.7422840 -0.5248101 -0.2471164  0.9486833
## 3  -0.35715411  0.5735831 -0.3527412 -0.1179990  0.9486833
## 4  -0.05952568  0.2361813  0.3355343  0.6423593  0.9486833
## 5   0.23810274 -0.2136878  0.8517409  0.8396220  0.9486833
## 6   0.68454537  0.5173495  1.3249303  0.9938457 -0.9486833
## 7   0.68454537 -1.7038792  1.3679476  0.9866725 -0.9486833
## 8  -0.05952568  1.4733213  0.2494999  0.5562810  0.9486833
## 9   2.02387326  0.4611158 -0.7398962 -0.7851058 -0.9486833
## 10 -1.25003937 -1.4227110 -0.8259306 -0.7994522 -0.9486833