Linked Questions

374
votes
3answers
238k views

Relationship between SVD and PCA. How to use SVD to perform PCA?

Principal component analysis (PCA) is usually explained via an eigen-decomposition of the covariance matrix. However, it can also be performed via singular value decomposition (SVD) of the data matrix ...
75
votes
7answers
203k views

What are principal component scores?

What are principal component scores (PC scores, PCA scores)?
72
votes
5answers
115k views

Loadings vs eigenvectors in PCA: when to use one or another?

In principal component analysis (PCA), we get eigenvectors (unit vectors) and eigenvalues. Now, let us define loadings as $$\text{Loadings} = \text{Eigenvectors} \cdot \sqrt{\text{Eigenvalues}}.$$ I ...
17
votes
2answers
15k views

Weighted principal components analysis

After some searching, I find very little on the incorporation of observation weights/measurement errors into principal components analysis. What I do find tends to rely on iterative approaches to ...
18
votes
1answer
9k views

Positioning the arrows on a PCA biplot

I am looking to implement a biplot for principal component analysis (PCA) in JavaScript. My question is, how do I determine the coordinates of the arrows from the $U,V,D$ output of the singular vector ...
20
votes
2answers
34k views

Methods to compute factor scores, and what is the “score coefficient” matrix in PCA or factor analysis?

As per my understanding, in PCA based on correlations we get factor (= principal component in this instance) loadings which are nothing but the correlations between variables and factors. Now when I ...
18
votes
2answers
13k views

What are the four axes on PCA biplot?

When you construct a biplot for a PCA analysis, you have principal component PC1 scores on the x-axis and PC2 scores on the y-axis. But what are the other two axes to the right and the top of the ...
19
votes
1answer
15k views

Interpreting 2D correspondence analysis plots

I've been searching the internet far and wide... I have yet to find a really good overview of how to interpret 2D correspondence analysis plots. Could someone offer some advice on interpreting the ...
17
votes
1answer
15k views

What is the proper association measure of a variable with a PCA component (on a biplot / loading plot)?

I am using FactoMineR to reduce my data set of measurements to the latent variables. The variable map above is clear for me to interpret, but I am confused when ...
10
votes
1answer
3k views

Is PCA still done via the eigendecomposition of the covariance matrix when dimensionality is larger than the number of observations?

I have a $20\times100$ matrix $X$, containing my $N=20$ samples in the $D=100$-dimensional space. I now wish to code up my own principal component analysis (PCA) in Matlab. I demean $X$ to $X_0$ first....
11
votes
1answer
9k views

Arrows of underlying variables in PCA biplot in R

At the risk of making the question software-specific, and with the excuse of its ubiquity and idiosyncrasies, I want to ask about the function biplot() in R, and, ...
9
votes
1answer
5k views

Data space, variable space, observation space, model space (e.g. in linear regression)

Suppose we have the data matrix $\mathbf{X}$, which is $n$-by-$p$, and the label vector $Y$, which is $n$-by-one. Here, each row of the matrix is an observation, and each column corresponds to a ...
4
votes
1answer
8k views

(Multiple) Correspondence Analysis for count data entered as binary variables

I have a data set, 1014 cases and 55 variables which are binary and is in the form of ...
2
votes
2answers
4k views

How can one interpret the Stata output for Multiple Correspondence Analysis?

As an alternative to conducting exploratory factor analysis on a set of data, with binary responses, I have been suggested to use Multiple Correspondence Analysis (MCA). Following is a curtailed and ...
5
votes
4answers
760 views

When you do PCA (or any dimensionality reduction), what is “the number of dimensions”?

Fundamental question When you do PCA (or any dimensionality reduction), what is "the number of dimensions"? I always thought that the thing you measure (ie, the variable) is the number of dimensions: ...

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