Redundancy analysis

Fig. 1 Results from a redundancy analysis of stream nutrient concentrations vs. catchment characteristics in 31 headwater catchments in Catalonia (Spain). Note how phosphorus (SRP) and ammonium (NH4) appear associated with urban point and diffuse sources (Urban percent urban land use, Inhabs number of inhabitants, eq. inhab inhabitant equivalents), whereas nitrate (NO3) concentration is positively associated with irrigated agriculture (IrrAg) and bovine cattle units (ECU), and negatively with mean catchment slope (slope) and percent forest land (Forest). The other variables are Area log area, P precipitation, NonIrrAg non irrigated agriculture. Data are from the Catalan water agency (http //ww.gencat.cat/ac)...

Reduced rank regression (RRR), also known as redundancy analysis (or PCA on Instrumental Variables), is the combination of multivariate least squares regression and dimension reduction [7]. The idea is that more often than not the dependent K-variables will be correlated. A principal component analysis of Y might indicate that A (A m) PCs may explain Y adequately. Thus, a full set of m... 

Fig. 4. Redundancy analysis ordination diagram based on the antibiotic resistance MPN data. The diamond ( ), the open square (o) and the triangle (A) indicate BG, DDF and DEF, respectively. The value in the parentheses indicates eigenvalue for the axis. The solid and broken arrows indicate significant environmental gradients at p=0.05 and 0.10, respectively.

The ordination of 23 taxa or groups of taxa in relation to the environmental predictors was examined by means of multivariate analyses. Nutrients (nitrates, total phosphate, and ammonia), conductivity, total suspended solids and water flow were retained by the redundancy analysis (RDA) (Fig. 4). The first RDA axis (RDA1 16.6% of the variance) showed a strong correlation between nutrients and suspended solids and (negatively) water flow. Large centric diatoms and Scenedesmus sp.pl. best defined this situation, which was characteristic of the meandering zone, mostly in autumn. Other taxa (Coelastrum microporum,... 

Fig. 4 Redundance analysis (RDA) on phytoplankton communities collected during the surveys of 2005-2006 and their relationship with the environmental variables, such as the Total Phosphorus (PTOT), Water Flow (WATFL), Total Suspended Solids (TSS), ammonia concentration (NH4), nitrate concentration (N03), and conductivity (COND). Acronyms were also used for phytoplankton taxa representation...

Redundancy analysis was able to explain 47% of total variance in flavour in relation to the 62 aroma compounds in the flrst two components PLS explained only 36%. Neither method, however, correlated either raspberry ketone or linalool with important aroma notes, suggesting concentrations of flie im ct compounds are not important in determining varietal character. 

An interesting additional feature of using potentiometric sensors in array mode has been pointed out, which is to profit from the extra information to gain confidence in measurements, that is to perform redundant analysis. This strategy can be the basis for automated fault detection of the sensory elements [48] and also be an aid for more robust calibrations [49]. 

The use of these latent variables provides PLS with a greater capacity than MLR to distill most of the information contained in a large number of descriptors and properties into a smaller number of factors and thereby avoid some of the ravages of the Curse of Dimensionality. It is especially suited to dealing with systems containing many highly correlated descriptors. PLS is related to the methods of principal components analysis (PCA) and maximum redundancy analysis (MRA), in that all three methods augment the raw desaiptors with matrices derived from variance found in the descriptors themselves (PCA), the properties to be modeled (MRA), or a combination of both (PLS). ... 

A technique derived from a principal components approach is the coupling of PCA with redundancy analysis (RDA) (van der Brink et al. 1996). The utility of the technique is that it provides a depiction of the treatment trajectories in an ecological space, and the statistical significance can be examined using a permutation test. One of the proposed benefits of the technique is that it can determine recovery, a dubious distinction in light of the ground work laid in Chapter 2. In common with other PCA techniques, the technique does assume a linear response. 

An ordination diagram has been used by van der Brink et al. (1996) to plot the path of the various treatment groups using the axes generated by the redundancy analysis. This method has the advantage of seeing a number of variables at once and the trajectory of each treatment over the course of the experiment. The plots are still two-dimensional representations, and variability is not pictured. 

Lambert, Z. V, Wildt, A. R., Durand, R. M. (1988). Redundancy analysis An alternative to canonical correlation and multivariate multiple regression in exploring intcrset associations. Psychological Bulletin, 104, 282-289. 

As already stated in the introduction to this chapter, the present material is largely novel, and only a few relevant references are available. The neceessary mathematics is expounded in university textbooks we have applied the theorems given in Dieudonne (1970). The approach to the observability/redundancy analysis as presented here can be characterized as equation-oriented cf. Romagnoli and Stephanopoulos (1980), or Crowe (1989). For another (graph-oriented) approach, see Stanley and Mah (1981a and b), Kretsovalis and Mah (1988), summarized in Mah (1990). 

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