| Unitary associations |
| Typical application | Assumptions | Data needed |
| Quantitative biostratigraphical correlation | None | Presence/absence matrix with horizons in rows, taxa in columns |
Unitary Associations analysis (Guex 1991) is a method for biostratigraphical correlation (see Angiolini & Bucher 1999 for an example application). The data input consists of a presence/absence matrix with samples in rows and taxa in columns. Samples belonging to the same section (locality) must be assigned the same color, and ordered stratigraphically within each section such that the lowermost sample enters in the lowest row. Colors can be re-used in data sets with large numbers of sections (see alveolinid.dat for an example).
Overview of the method
The method of Unitary Associations is logical, but rather complicated, consisting of a number of steps. For details, see Guex 1991. The implementation in PAST does not include all the features found in the standard program, called BioGraph (Savary & Guex 1999), and advanced users are referred to that package. The basic idea is to generate a number of assemblage zones (similar to 'Oppel zones') which are optimal in the sense that they give maximal stratigraphic resolution with a minimum of superpositional contradictions. One example of such a contradiction would be a section containing a species A above a species B, while assemblage 1 (containing species A) is placed below assemblage 2 (containing species B). PAST (and BioGraph) carries out the following steps:
1. Residual maximal horizons
The method makes the range-through assumption, meaning that taxa are considered to have been present in all levels between the first and last appearance in any section. Then, any samples with a set of taxa that is contained in some other sample are discarded. The remaining samples are called residual maximal horizons. The idea behind this throwing away of data is that the absent taxa in the discarded samples may simply not have been found even though they originally existed. Absences are therefore not as informative as presences.
2. Superposition and co-occurrence of taxa
Next, all pairs (A,B) of taxa are inspected for their superpositional relationship: A below B, B below A, A together with B, or unknown. If A occurs below B in one locality and B below A in another, they are considered to be co-occurring although they have never actually been found together.
The superpositions and co-occurrences of taxa can be viewed in the biostratigraphic graph. In this graph, taxa are coded as numbers. Co-occurrences between pairs of taxa are shown as solid blue lines. Superpositions are shown as dashed red lines, with long dashes from the above-occurring taxon and short dashes from the below-occurring taxon.
3. Maximal cliques
Maximal cliques are groups of co-occurring taxa not contained in any larger group of co-occurring taxa. The maximal cliques are candidates for the status of unitary associations, but will be further processed below. In PAST, maximal cliques receive a number and are also named after a maximal horizon in the original data set which is identical to, or contained in (marked with asterisk), the maximal clique.
4. Superposition of maximal cliques
The superpositional relationships between maximal cliques are decided by inspecting the superpositional relationships between their constituent taxa, as computed in step 2. Contradictions (some taxa in clique A occur below some taxa in clique B, and vice versa) are resolved by a 'majority vote'. The contradictions between cliques can be viewed in PAST.
The superpositions and co-occurrences of cliques can be viewed in the maximal clique graph. In this graph, cliques are coded as numbers. Co-occurrences between pairs of cliques are shown as solid blue lines. Superpositions are shown as dashed red lines, with long dashes from the above-occurring clique and short dashes from the below-occurring clique. Also, cycles between maximal cliques (see below) can be viewed as green lines.
5. Resolving cycles
It will sometimes be the case that maximal cliques are now ordered in cycles: A is below B, which is below C, which is below A again. This is clearly contradictory. The 'weakest link' (superpositional relationship supported by fewest taxa) in such cycles is destroyed.
6. Reduction to unique path
At this stage, we should ideally have a single path (chain) of superpositional relationships between maximal cliques, from bottom to top. This is however often not the case, for example if A and B are below C, which is below D, or if we have isolated paths without any relationships (A below B and C below D). To produce a single path, it is necessary to merge cliques according to special rules.
7. Post-processing of maximal cliques
Finally, a number of minor manipulations are carried out to 'polish' the result: Generation of the 'consecutive ones' property, reinsertion of residual virtual co-occurrences and superpositions, and compaction to remove any generated non-maximal cliques. For details on these procedures, see Guex 1991. At last, we now have the Unitary Associations, which can be viewed in PAST.
8. Correlation using the Unitary Associations
The original samples are now correlated using the unitary associations. A sample may contain taxa which uniquely places it in a unitary association, or it may lack key taxa which could differentiate between two or more unitary associations, in which case only a range can be given. These correlations can be viewed in PAST.
9. Reproducibility matrix
Some unitary associations may be identified in only one or a few sections, in which case one may consider to merge unitary associations to improve the geographical reproducibility (PAST does not carry out this procedure automatically in the present version). The reproducibility matrix should be inspected to identify such unitary associations.