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40

SVD Singular-value decomposition is at the root of the three kindred techniques. Let $\bf X$ be $r \times c$ table of real values. SVD is $\bf X = U_{r\times r}S_{r\times c}V_{c\times c}'$. We may use just $m$ $[m \le\min(r,c)]$ first latent vectors and roots to obtain $\bf X_{(m)}$ as the best $m$-rank approximation of $\bf X$: $\bf X_{(m)} = U_{r\times m}...


40

There are many different ways to produce a PCA biplot and so there is no unique answer to your question. Here is a short overview. We assume that the data matrix $\mathbf X$ has $n$ data points in rows and is centered (i.e. column means are all zero). For now, we do not assume that it was standardized, i.e. we consider PCA on covariance matrix (not on ...


25

Explanation of a loading plot of PCA or Factor analysis. Loading plot shows variables as points in the space of principal components (or factors). The coordinates of variables are, usually, the loadings. (If you properly combine loading plot with the corresponding scatterplot of data cases in the same components space, that would be biplot.) Let us have 3 ...


16

Consider upvoting @amoeba's and @ttnphns' post. Thank you both for your help and ideas. The following relies on the Iris dataset in R, and specifically the first three variables (columns): Sepal.Length, Sepal.Width, Petal.Length. A biplot combines a loading plot (unstandardized eigenvectors) - in concrete, the first two loadings, and a score plot (rotated ...


13

Warning: R uses the term "loadings" in a confusing way. I explain it below. Consider dataset $\mathbf{X}$ with (centered) variables in columns and $N$ data points in rows. Performing PCA of this dataset amounts to singular value decomposition $\mathbf{X} = \mathbf{U} \mathbf{S} \mathbf{V}^\top$. Columns of $\mathbf{US}$ are principal components (PC "scores")...


13

These terms appear in some books on multivariate statistics. Suppose you have n individuals by p quantitative features data matrix. Then you can plot individuals as points in the space where the axes are the features. That will be classic scatterplot, aka variable space plot. We say, the cloud of individuals span the space defined by the axes-features. You ...


12

Do you mean, e.g., in the plot that the following command returns? biplot(prcomp(USArrests, scale = TRUE)) If yes, then the top and the right axes are meant to be used for interpreting the red arrows (points depicting the variables) in the plot. If you know how the principal component analysis works, and you can read R code, the code below shows you how ...


11

I have a better visualization for the biplot. Please check following figure. In the experiment, I am trying to mapping 3d points into 2d (simulated data set). The trick to understand biplot in 2d is finding the correct angle to see same thing in 3d. All the data points are numbered, you can see the mapping clearly. Here is the code to reproduce the ...


9

Well it appears Kevin Wright should be given most of the credit to try to help explain the confusion (from R-help mail list); The arrows are not pointing in the most-varying direction of the data. The principal components are pointing in the most-varying direction of the data. But you are not plotting the data on the original scale, you are ...


8

It is described in Michael Friendly's American Statistician paper on corrgrams, Preprint PDF here. See section on correlation ordering. Also if you look at the source of the corrgram library you will see some other potential ways to order the data as well. To describe what the code is doing in a nut-shell, the variables in the correlation matrix are ordered ...


7

The dots are the respondents and the colours are the genders. This, you know. The principal axes of your plot represent the first and second PC scores and individuals are plotted on that basis. Somebody in the lower left hand quadrant got low scores on both. PC2 seems to flag "male" and "female" interests. I don't know what PC1 means, but it probably ...


7

Principal components analysis and Linear discriminant analysis outputs; iris data. I will not be drawing biplots because biplots can drawn with various normalizations and therefore may look different. Since I'm not R user I have difficulty to track down how you produced your plots, to repeat them. Instead, I will do PCA and LDA and show the results, in a ...


5

I redid your PCA in SPSS (I'm not R user). It was PCA based on covariances. I confirm your analysis. Eigenvalues (component variances) and the proportion of overall variance explained I 145.7983424 .9834567 II 2.4525573 .0165433 Eigenvectors (cosines of rotation of variables into components) I II X .7235615578 -....


4

X1 and X2 are "redundant" in the sense of linear duplicates of each other if they correlate perfectly ($r=1$). Then the two variable vectors must coincide, be collinear in the space (that space - where variables are drawn as vectors, arrows - is called "subject space"). But from the plot, without knowing the variable correlations, you can't tell if the ...


4

My understanding is that biplots of linear discriminant analyses can be done, it is implemented in fact in R packages ggbiplot and ggord and another function to do it is posted in this StackOverflow thread. Also the book "Biplots in practice" by M. Greenacre has one chapter (chapter 11, see pdf) on it and in Figure 11.5 it shows a biplot of a linear ...


3

PCA tries to project your data onto a new set of dimensions where the variances in your data are captured such that you can classify/cluster them visually or by using a hopefully simple algorithm. The variance plot tells you how much the new set of dimensions capture variances in decreasing order. Biplot is the projection of your data on the first two ...


3

Why are there negative values in the range of the plot? Because the input contingency table of positive values (e.g. frequencies) was standardized by the analysis in a special way relative the central point. How is the center of the plot (i.e., the 0.0 point) found? It is the weighted mean row profile and the weighted mean column profile. It is thus the ...


3

I am going to provide an example but keep in mind that it is not 100% clear that the lda.scalings_ stores the eigenvectors (loadings of the variables). This is something that I am currently trying to verify. Example using iris data and sklearn: import numpy as np import matplotlib.pyplot as plt from sklearn import datasets import pandas as pd from sklearn....


2

I computed $$\small b(r) = \sqrt{\small\text{comp}_1(r)^2 + \text{comp}_2(r)^2} $$ and $$\small \text{err}(r)= \small \text{comp}_{1\&2}(r) - b(r) $$ where $r$ indicates the row-index, so comp1&2 $\approx b$ very likely means the squareroot of the communality up to the two first pc's. In a biplot with the two pc's as axes the values comp1&2 give ...


2

The general advice is Use scaling 1 where you want a biplot focussed on the sites/samples and the (dis)similarity between them in terms of the species (or variables), use scaling2 where you want to best represent the correlations between species (or variables). As these numeric scaling codes are really a reflection of software implementations from the DOS ...


2

Your interpretation is mostly correct. The first PC accounts for most of the variance, and the first eigenvector (principal axis) has all positive coordinates. It probably means that all variables are positively correlated between each other, and the first PC represents this "common factor". The second PC (looks like it has much smaller variance) contrasts ...


2

I know this was asked over a year ago, and ttnphns gave an excellent and in-depth answer, but I thought I'd add a couple of comments for those (like me) that are interested in PCA and LDA for their usefulness in ecological sciences, but have limited statistical background (not statisticians). PCs in PCA are linear combinations of original variables that ...


1

I think you will need to impute values for the missing variables. There are several ways of doing this - if you're lucky the choice won't matter that much, so I'd try a couple. One obvious way is to just assign the column mean of V3 (and other missing variables) to your new observation (as chl implies in his comment); another way would be to create a ...


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