Fig 1.
A Gaussian response curve (left) and a U-shaped response curve (right).
The left panel also indicates the parameters μk (optimum), ak (maximum) and tk (tolerance).
Table 1.
The parameters used for the bell-shaped response functions for species k = 1, …, 12.
For all species, the scaling parameters sk are set to (ηk+ζk) so as to make the maxima comparable.
Table 2.
The parameters used for the U-shaped response functions for species k = 13, …, 20. For all species, the scaling parameters sk are set to so as to make the maxima comparable.
Fig 2.
The bell-shaped response functions of the first twelve species (left panel) and the U-shaped response functions of the next eight species (right panel).
Fig 3.
Results of the simulation study.
(a) the averages of the estimated environmental gradients as a function of the penalty parameter δ; the intervals shown on top are proportional to the total variance of the estimates. (b) the average number of bell-shaped response functions as a function of the penalty parameter δ. (c) for each of the 20 species the graph shows the evolution of the ’s as δ changes. (d) for each of the 20 species the graph shows the evolution of the Sum of Squared Errors (SSE) of the fits of the response functions for the penalty parameter moving from δ = 0 (symbol: +) to δ = −1 (symbol: O); the dots represent the intermediate results with larger dots representing smaller penalisation.
Fig 4.
The relative changes of average LLR (left) and average SSE (right) as a function of the penalty parameter δ.
Fig 5.
Results for the case study in the first dimension.
Estimated coefficients of environmental gradient (a) and the average number of bell-shaped response functions (b) as a function of penalty parameter δ. Relative changes of average LLR (c) and average SSE (d) as a function of penalty parameter δ.
Fig 6.
Cross-validated (10-fold) results for the case study in the first dimension.
Relative change of the average LLR (a) and average SSE (b) as a function of the penalty parameter δ.
Fig 7.
Results for the case study in the second dimension.
Estimated coefficients of environmental gradient (a) and the average number of bell-shaped response functions (b) as a function of penalty parameter δ. Relative changes of average LLR (c) and average SSE (d) as a function of penalty parameter δ.
Fig 8.
Cross-validated (10-fold) results for the second dimension.
Relative change of the average LLR (a) and average SSE (b) as a function of the penalty parameter δ.
Table 3.
Comparison of the estimated environmental gradients and the model fits from three ordination methods applied to the Antarctic lakes data.
Dimension 1 and Dimension 2 refer to models fitted with the environmental scores on dimensions 1 and 2, respectively. MSE gives the mean squared error calculated only among Bell-shaped species, MSE* stands for the mean squared error calculated from all species.
Fig 9.
Ordination diagram of the BECOA analysis of the Antarctic lake data, with penalisation parameter δ being -1.7 for the first dimension and = 0.7 for the second dimension.
Numbers represent lakes. The points represent the species optima, with symbols indicating the shape of the corresponding species response function when δ = 0: p1, U-shaped in 1st and 2nd dimension; p2, bell-shaped in 1st dimension; p3, bell-shaped in 2nd dimension; p4, bell-shaped in 1st and 2nd dimension.
Fig 10.
Ordination diagram of the CCA analysis of a subset of Antarctic data.
Fig 11.
Ordination diagram of the FCOA analysis of a subset of Antarctic data.
Fig 12.
Ordination diagram of the BECOA analysis of a subset of the Antarctic lake data, with penalisation parameter δ being -1.7 for the first dimension and = 0.7 for the second dimension.
Numbers represent lakes. The points represent the species optima, with symbols indicating the shape of the corresponding species response function when δ = 0: p1, U-shaped in 1st and 2nd dimension; p2, bell-shaped in 1st dimension; p3, bell-shaped in 2nd dimension; p4, bell-shaped in 1st and 2nd dimension. Species labels are added.