PCA and MDS both do a similar job in the sense that they extract underlying dimensions or components from data, so why are plots of PCA factor loadings often so different to those yielded by MDS?
In MDS you try to project n-dimensional data points to a (usually) 2-dimensional space in a manner that similar points in the n-dimensional space will project to near distances in a plane: you´re projecting a multidimensional space preserving the inter point distances, employing distance and a loss function to analyze the proximities of data points, while in PCA you project a multidimensional space to the directions of maximum variability using covariance/correlation matrix to analyze the correlation between data points and variables.
In MDS you try to project n-dimensional data points to a (usually) 2-dimensional space in a manner that similar points in the n-dimensional space will project to near distances in a plane: you´re projecting a multidimensional space preserving the inter point distances, employing distance and a loss function to analyze the proximities of data points, while in PCA you project a multidimensional space to the directions of maximum variability using covariance/correlation matrix to analyze the correlation between data points and variables.
It is important to have in mind that PCA assumes a linear relationship between the data and the underlying latent variables represented by the principal components. If this relationship is non-linear, for example unimodal, which is the case if you're looking at species abundances and their relationship to underlying environmental gradients. Then PCA will have problems providing a good representation of the distance between observations. In contrast, MDS assumes no relationship at all and only strives to optimise the fit between the dissimilarity between observations and their euclidean distance in the MDS ordination space. Thus, depending on the type of input data, PCA and MDS can produce quite different results.
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