Chapter 6 Matrix algebra, very briefly
Matrix algebra is well-suited for ecology, because most (if not all) data sets we work with are in a matrix format.
6.1 Data sets are matrices
Ecological data tables are obtained as object-observations or sampling units, and are often recorded as this:
| Objects | \(y_1\) | \(y_2\) | \(\dots\) | \(y_n\) |
|---|---|---|---|---|
| \(x_1\) | \(y_{1,1}\) | \(y_{1,2}\) | \(\dots\) | \(y_{1,n}\) |
| \(x_2\) | \(y_{2,1}\) | \(y_{2,2}\) | \(\dots\) | \(y_{2,n}\) |
| \(\vdots\) | \(\vdots\) | \(\vdots\) | \(\ddots\) | \(\vdots\) |
| \(x_m\) | \(y_{m,1}\) | \(y_{m,2}\) | \(\dots\) | \(y_{m,n}\) |
where \(x_m\) is the sampling unit \(m\); and \(y_n\) is the ecological descripor that can be, for example, species present in a sampling unit, locality, or a chemical variable.
The same ecological data table can be represented in matrix notation like this:
\[Y = [y_{m,n}] = \begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ y_{m,1} & y_{m,2} & \cdots & y_{m,n} \end{bmatrix}\]
where lowercase letters indicate elements, and the subscript letters indicate the position of these elements in the matrix (and in the table!).
Moreover, any subset of a matrix can be recognized.
We can subset a row matrix, as below:
\[\begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ \end{bmatrix}\]
We can also subset a column matrix, as below:
\[\begin{bmatrix} y_{1,1} \\ y_{2,2} \\ \vdots \\ y_{m,2} \end{bmatrix}\]
6.2 Association matrices
Two important matrices can be derived from the ecological data matrix: the association matrix among objects and the association matrix among descriptors.
Using the data from our matrix \(Y\),
\[ Y = \begin{array}{cc} \begin{array}{ccc} x_1 \rightarrow\\ x_2 \rightarrow\\ \vdots \\ x_m \rightarrow\\ \end{array} & \begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ y_{m,1} & y_{m,2} & \cdots & y_{m,n} \end{bmatrix} \end{array} \]
one can examine the relationship between the first two objects:
\[x_1 \rightarrow \begin{bmatrix} y_{1,1} & y_{1,2} & \cdots & y_{1,n} \\ \end{bmatrix} \]
\[x_2 \rightarrow \begin{bmatrix} y_{2,1} & y_{2,2} & \cdots & y_{2,n} \\ \end{bmatrix} \]
and obtain \(a_{1,2}\).
We can populate the association matrix \(A_{n,n}\) with the relationships between all objects from \(Y\):
\[A_{n,n} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n,1} & a_{n,2} & \cdots & a_{n,n} \end{bmatrix}\]
Because \(A_{n,n}\) has the same number of rows and columns, it is denoted a square matrix.
Therefore, \(A_{n,n}\) has \(n^2\) elements.
We can also obtain the relationship between the first two descriptors of \(Y\), \(y_1\) and \(y_2\):
\[\begin{bmatrix} y_{1,2} \\ y_{2,2} \\ \vdots \\ y_{m,2} \end{bmatrix}\]
\[\begin{bmatrix} y_{1,1} \\ y_{2,1} \\ \vdots \\ y_{m,1} \end{bmatrix}\]
and store it in \(a_{1,2}\).
We can populate the association matrix \(A_{m,m}\) with the relationships between all descriptors from \(Y\):
\[A_{m,m} = \begin{bmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,m} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,m} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,m} \end{bmatrix}\]
This \(A_{m,m}\) is a square matrix, and it has \(m^2\) elements.
These matrices, \(A_{n,n}\) and \(A_{m,m}\), are the basis of Q-mode and R-mode analyses in ecology.
R-mode constitutes of analyzing the association between descriptors or species, while Q-mode analyzes the association between OTUs, objects or sites.