Basic concepts in ordination

What is ordination? Finding a concise and useful summary of the patterns within multivariate data. An arrangement of units in a uni- or multidimensional order. There are two forms of ordination: Direct ordinations - organizing observations along some known gradient( e.g. space, time, or elevation) Indirect ordinations – detect trends within data without the user needing to define end-points or gradients within their data. Indirect ordinations are powerful tools for probing and exploring multivariate data.

D Direct ordinations Plant community succession in time on sand dunes at the southern end of Lake Michigan (Olsen , 1958).

D Direct ordinations The distribution of plant communities away from a stream edge in a floodplain forest (Hughes and Cass, 1997).

D Indirect ordinations Indirect ordinations produce a set of gradients which are inherent in the data, called ordination axes. An approach to producing a meaningful summary of the patterns underlying multivariate data. Data space – an abstract high-dimensional mathematical space where the number of dimensions equals the number of variables being plotted against each other. Each variable is considered an axis and each axis is oriented at 90 degrees to all previous axes. Samples which consists of similar measurements will be close to each other in data space and samples which have very different composition will be widely separated.

D Indirect ordinations All variables are of equal importance in ordination techniques. Multivariate data is difficult to visualize because it contains too many dimensions to allow for easy plotting of all possible graphs. Ordination techniques take as input an object in high-dimensional data space, and produce as output an object in a new lower-dimensional data space. Both the raw data space and the new data space consist of axes meeting at 90 degrees.

D Indirect ordinations All ordination techniques involve calculating new variables, called ordination or axis scores. The axes produced by an ordination will be in descending order of importance, with the first axis being the most informative, the second axis the second most informative, etc. The most useful ordination diagram plots the first ordination axis against the second. Ordination techniques do not directly provide probability levels, an ordination cannot be said to be statistical significant.

Indirect ordinations Plot of bivariate data: data points with similar values for both properties (one distinct cluster). property 2 data points with similar values for property 1 (1 or 2 clusters). property 1

Indirect ordinations Ordination diagram: D Ordination diagrams look like a bivariate graph with axes labeled with an axis number (ordination axis) not with the name of a measured variables. Ordination axis derives from the raw data by the techniques which was invoked. Each point defines the properties of the entire row of values collected for each observation. Variables Observations Axis 2 Axis 1

Indirect ordinations Bi-plots: D Ordination of properties of the columns of the matrix (variables) are overlain on top of the main ordination diagram in order to highlight relationships between the two sets of information. The second set of information is shown as arrows that run from (0,0) to the coordinate in question. Variables Cluster with high values of v5 and low values of v6. v1 v2 v3 v4 v5 v6 Observations Axis 2 v5 v6 Axis 1

Indirect ordinations The choice of variables: No division of the data into “dependent” and “independent” variables. Example of selection of variables: study of soil (pH, elemental contents, cation exchange capacity ..) morphometric study (available body dimensions) biological community ( all available species) Each variable should be checked for normality and outliers.

D Indirect ordinations Log-transform the data (log(x+1) for data containing zeros) - makes it more likely to pick up underlying trends. Decide if variables that are valid but appear to be of little importance should be excluded. Sensitivity to the inclusion of scarce species: Robust (Bray-Curtis ordination, Principal Component Analysis) Overemphasize the importance of rare species (correspondent analysis and DCA).