I have PCA data (first two axes) for which I want to determine a way to account for this variation between some "groups" ("group", which can be represented with ordihull polygons).
Is there a way to assess variation that accounts for the percent of variation captured by each PCA axis? For instance, if the first axis captures 60% variation, but the second axis only captures 20%, is there a way to attribute more variation to the first axis than second?
In terms of measuring PCA 'variation', I am leaning toward the "dispersion" metric presented in Laliberte & Legendre 2010 (see attached figure for their explanation), which accounts for the distance of each metric from the polygon centroid. Is there a way to "weigh" this metric by each axis' percent variation contribution? Or is there altogether a better solution to this problem?
Dear Lanna,
Maybe you could share the source of your data. For example spectroscopic data requires different approaches than data from surveys.
I don´t see a reason why you want to introduce the percentage of variation as a weighing factor. The 60 % variation in the first PC just signifies that this is the biggest portion of CORRELATING information. Depending on the data source this does not automatically mean that the first PC should be weighed higher compared to the 2. PC. One example: You have 4 different sensors in a vessel. Three are capturing the temperature and one detects the pH. The first PC might explain 75% of the variation and still just be based on temperature. The second PC might just contain 25%, however this can still be the information you are looking for. Especially in a PCA the first PC must not contain the relevant information (in my data it is often the background noise / scattering) and thus must not necessarily be weighed higher.
Please keep in mind that this is different for PLS (or for classification PLS-DA, sometimes also called DLS) as the PCs are aligned to the trend of additional information you provide on the dataset (Y matrix).
Regards
Marek
you may also focus on individuals values in the first two axes. you can use metrics on these values to discriminate your groups.
By the way Marek is completely right, you first should be aware that PCA can not be applied on any data
Marek's point about the %v.a.f. along each axis not necessarily indicating a sensible weighting is a good one. There is nothing to stop you from calculating the empirical variances of the PCA scores for the points along each axis. If you calculate the total variance and the variance within each group, if you then sum the within-group variances and subtract them from the total variance you find the between group variance. This allows you to calculate an empirical between- to within-group variance ratio along each axis that will give you some indication of the "significance" (using the term loosely) of the variance in each dimension. Depending on the underlying nature of your data these v.r.'s might have approximate F distributions, but I'd be very wary of pushing that sort of assumption too hard - I think this is the point that Marie is making. In any case, if you have an a priori group structure, why not use an ordination method that is specifically designed to take such structure into account - Canonical Variates Analysis (LDA) or non-linear discriminant analysis?
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