I need to use multivariate statistical techniques to extract information about the similarities or dissimilarities between to extract information about the similarities or dissimilarities between sampling sites, identification of water quality variables responsible for spatial variations in groundwater quality, and the influence of possible sources (natural and anthropogenic) on the water quality parameters.
The selection of PC1 Vs PC2 or PC1Vs PC3 depends on the availability of % of variation in the components.
Better select two components having maximum variation.
I am not sure if there exists a "best" criterion. It can be done using some of the heuristic techniques mentioned above which are always biased to the user's judgement. However, there is a vast literature of how to do this automatically using statistical techniques which have nice theoretical and asymptotical properties. A very famous work is "Wax, Mati, and Thomas Kailath. "Detection of signals by information theoretic criteria." IEEE Transactions on Acoustics, Speech, and Signal Processing 33.2 (1985): 387-392.". However, if you do not have enough observations compared to the variables, you can also check some sample-poor techniques like "Nadakuditi, Raj Rao, and Alan Edelman. "Sample eigenvalue based detection of high-dimensional signals in white noise using relatively few samples." IEEE Transactions on Signal Processing 56.7 (2008): 2625-2638.", and "Kritchman, Shira, and Boaz Nadler. "Non-parametric detection of the number of signals: Hypothesis testing and random matrix theory." IEEE Transactions on Signal Processing 57.10 (2009): 3930-3941."
The PC's that contributed the highest percentage to total variation which is 100% are chosen while the rest are ignored.
There are different tests for determining the number of statistically significant PC axes (e.g. Bartletts, Broken Stick). Plotting the eigenvalues you can easily see the asymptote. However, at the end of the day, you should rely more on what you are trying to show with your analyses and not strictly rely on the number of significant axes.
As you know, the axis are orthogonal and therefore often used in follow up analyses, e.g. when comparing a response variable (e.g. diversity) to the PC axes (ENV gradient). So how many or which axes to use is also dependent on what relationships you are interested in studying (by interpreting the loadings you can determine what axes are of interest). I find that I seldom work with more than 3 and often only the 1st and 2nd PC axes as these are the main gradients.
Frequently, if I have many variables in my dataset, I reduce the number of explanatory variables using correlation and PCA and then redo the PCA with the revised list of variables and use the revised PC axes in my analyses.
The Eigenvalues of the components, the Eigenvalues shall be more than 1 and the loadings on the components greater than .4.
The default number is usually 4 components.
Examine the components. The first PC often reflects a global quality that all variables share. For example, I'm looking at measurements of brain blood vessels, and the first component simply reflects how big the brain was. The subsequent components are more interesting, in that they can identify groups of variables within the whole that tend to vary together. But statistical considerations have to go together with a theory in order to look for meaningful (not just "significant") components.
Básicamente cada uno de los componentes principales explica un porcentaje de la varianza (%) en orden decreciente. Las representaciones en ejes cartesianos representan la relaciones entre CP1 Y CP2, CP1 Y CP2, CP2 Y CP3, y asi sucesivamente. Lo ideal es que el análisis incluya un número de componentes que expliquen al menos el 85% de la varianza.
Criteria to chose principal componants is eigenvalue that should be greater than 1, only those componants need to be select.
There is no straightforward "best way" to select the number of components. It varies in use cases. If there is no indication for number of components in the use cases, the I believe Scree Plots are a good tool.
In Scree Plotting, we calculate the "score" of each components. Then we start the component associated with highest score. We keep adding the components one by one in the selected set and calculate the cumulative variances explained by them. Set a threshold, say 90%. Whenever the cumulative variance hits the threshold, we stop adding. Then the number of components in selected set approximates the variance of your data as per your desired level.
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