Questions tagged [pca]
Principal component analysis (PCA) is a linear dimensionality reduction technique. It reduces a multivariate dataset to a smaller set of constructed variables preserving as much information (as much variance) as possible. These variables, called principal components, are linear combinations of the input variables.
530
questions
1308
votes
27
answers
883k
views
Making sense of principal component analysis, eigenvectors & eigenvalues
In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.
I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like ...
- 13.4k
562
votes
3
answers
439k
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 ...
- 99.1k
206
votes
7
answers
190k
views
PCA on correlation or covariance?
What are the main differences between performing principal component analysis (PCA) on the correlation matrix and on the covariance matrix? Do they give the same results?
- 2,240
260
votes
15
answers
286k
views
What are the differences between Factor Analysis and Principal Component Analysis?
It seems that a number of the statistical packages that I use wrap these two concepts together. However, I'm wondering if there are different assumptions or data 'formalities' that must be true to use ...
- 6,872
52
votes
4
answers
49k
views
Does the sign of scores or of loadings in PCA or FA have a meaning? May I reverse the sign?
I performed principal component analysis (PCA) with R using two different functions (prcomp and princomp) and observed that the ...
- 977
104
votes
5
answers
204k
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 ...
- 1,339
81
votes
7
answers
137k
views
Using principal component analysis (PCA) for feature selection
I'm new to feature selection and I was wondering how you would use PCA to perform feature selection. Does PCA compute a relative score for each input variable that you can use to filter out ...
- 2,331
42
votes
1
answer
56k
views
Doing principal component analysis or factor analysis on binary data
I have a dataset with a large number of Yes/No responses. Can I use principal components (PCA) or any other data reduction analyses (such as factor analysis) for this type of data? Please advise how I ...
- 421
24
votes
1
answer
51k
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 ...
- 453
84
votes
4
answers
31k
views
How to visualize what canonical correlation analysis does (in comparison to what principal component analysis does)?
Canonical correlation analysis (CCA) is a technique related to principal component analysis (PCA). While it is easy to teach PCA or linear regression using a scatter plot (see a few thousand examples ...
- 943
58
votes
1
answer
39k
views
How does centering the data get rid of the intercept in regression and PCA?
I keep reading about instances where we center the data (e.g., with regularization or PCA) in order to remove the intercept (as mentioned in this question). I know it's simple, but I'm having a hard ...
- 2,305
84
votes
6
answers
31k
views
Is there any good reason to use PCA instead of EFA? Also, can PCA be a substitute for factor analysis?
In some disciplines, PCA (principal component analysis) is systematically used without any justification, and PCA and EFA (exploratory factor analysis) are considered as synonyms.
I therefore ...
- 849
163
votes
5
answers
137k
views
What's the difference between principal component analysis and multidimensional scaling?
How are PCA and classical MDS different? How about MDS versus nonmetric MDS? Is there a time when you would prefer one over the other? How do the interpretations differ?
- 4,253
163
votes
1
answer
179k
views
How to reverse PCA and reconstruct original variables from several principal components?
Principal component analysis (PCA) can be used for dimensionality reduction. After such dimensionality reduction is performed, how can one approximately reconstruct the original variables/features ...
- 99.1k
215
votes
6
answers
245k
views
Can principal component analysis be applied to datasets containing a mix of continuous and categorical variables?
I have a dataset that has both continuous and categorical data. I am analyzing by using PCA and am wondering if it is fine to include the categorical variables as a part of the analysis. My ...
- 2,151
143
votes
6
answers
79k
views
Should one remove highly correlated variables before doing PCA?
I'm reading a paper where author discards several variables due to high correlation to other variables before doing PCA. The total number of variables is around 20.
Does this give any benefits? It ...
- 1,541
126
votes
4
answers
238k
views
PCA and proportion of variance explained
In general, what is meant by saying that the fraction $x$ of the variance in an analysis like PCA is explained by the first principal component? Can someone explain this intuitively but also give a ...
- 3,183
44
votes
1
answer
21k
views
PCA and Correspondence analysis in their relation to Biplot
Biplot is often used to display results of principal component analysis (and of related techniques). It is a dual or overlay scatterplot showing component loadings and component scores simultaneously. ...
- 54.5k
70
votes
8
answers
56k
views
Is PCA followed by a rotation (such as varimax) still PCA?
I have tried to reproduce some research (using PCA) from SPSS in R. In my experience, principal() function from package psych ...
- 3,368
19
votes
2
answers
19k
views
PCA and exploratory Factor Analysis on the same dataset: differences and similarities; factor model vs PCA
I would like to know if it makes any logical sense to perform principal component analysis (PCA) and exploratory factor analysis (EFA) on the same data set. I have heard professionals expressly ...
- 191
38
votes
1
answer
38k
views
Is there Factor analysis or PCA for ordinal or binary data?
I have completed the principal component analysis (PCA), exploratory factor analysis (EFA), and confirmatory factor analysis (CFA), treating data with likert scale (5-level responses: none, a little, ...
- 423
31
votes
5
answers
8k
views
How can top principal components retain the predictive power on a dependent variable (or even lead to better predictions)?
Suppose I am running a regression $Y \sim X$. Why by selecting top $k$ principle components of $X$, does the model retain its predictive power on $Y$?
I understand that from dimensionality-reduction/...
- 2,193
50
votes
3
answers
51k
views
PCA and the train/test split
I have a dataset for which I have multiple sets of binary labels. For each set of labels, I train a classifier, evaluating it by cross-validation. I want to reduce dimensionality using principal ...
- 6,559
41
votes
2
answers
20k
views
How does Factor Analysis explain the covariance while PCA explains the variance?
Here is a quote from Bishop's "Pattern Recognition and Machine Learning" book, section 12.2.4 "Factor analysis":
According to the highlighted part, factor analysis captures the covariance between ...
- 3,275
34
votes
1
answer
43k
views
Best factor extraction methods in factor analysis
SPSS offers several methods of factor extraction:
Principal components (which isn't factor analysis at all)
Unweighted least squares
Generalized least squares
Maximum Likelihood
Principal Axis
Alpha ...
- 13.9k
31
votes
7
answers
39k
views
Testing for linear dependence among the columns of a matrix
I have a correlation matrix of security returns whose determinant is zero. (This is a bit surprising since the sample correlation matrix and the corresponding covariance matrix should theoretically be ...
- 3,051
21
votes
3
answers
7k
views
I'm getting "jumpy" loadings in rollapply PCA in R. Can I fix it?
I have 10 years of daily returns data for 28 different currencies. I wish to extract the first principal component, but rather than operate PCA on the whole 10 years, I want to rollapply a 2 year ...
- 1,029
15
votes
1
answer
24k
views
Steps done in factor analysis compared to steps done in PCA
I know how to perform PCA (principal component analysis), but I would like to know steps that should be used for factor analysis.
To perform PCA, let us consider some matrix $A$, for instance:
...
- 775
31
votes
2
answers
19k
views
Why are there only $n-1$ principal components for $n$ data if the number of dimensions is $\ge n$?
In PCA, when the number of dimensions $d$ is greater than (or even equal to) the number of samples $N$, why is it that you will have at most $N-1$ non-zero eigenvectors? In other words, the rank of ...
- 411
29
votes
2
answers
25k
views
Why PCA of data by means of SVD of the data?
This question is about an efficient way to compute principal components.
Many texts on linear PCA advocate using singular-value decomposition of the casewise data. That is, if we have data $\bf X$ ...
- 54.5k
21
votes
1
answer
30k
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 ...
- 345
43
votes
1
answer
59k
views
What is the intuitive reason behind doing rotations in Factor Analysis/PCA & how to select appropriate rotation?
My Questions
What is the intuitive reason behind doing rotations of factors in factor analysis (or components in PCA)?
My understanding is, if variables are almost equally loaded in the top ...
- 5,463
34
votes
3
answers
36k
views
How to perform orthogonal regression (total least squares) via PCA?
I always use lm() in R to perform linear regression of $y$ on $x$. That function returns a coefficient $\beta$ such that $$y = \beta x.$$
Today I learned about ...
- 2,237
11
votes
3
answers
27k
views
Can the scaling values in a linear discriminant analysis (LDA) be used to plot explanatory variables on the linear discriminants?
Using a biplot of values obtained through principal component analysis, it is possible to explore the explanatory variables that make up each principle component. Is this also possible with Linear ...
- 1,683
35
votes
3
answers
57k
views
PCA on correlation or covariance: does PCA on correlation ever make sense? [closed]
In principal component analysis (PCA), one can choose either the covariance matrix or the correlation matrix to find the components (from their respective eigenvectors). These give different results (...
- 639
87
votes
3
answers
22k
views
What is the intuition behind SVD?
I have read about singular value decomposition (SVD). In almost all textbooks it is mentioned that it factorizes the matrix into three matrices with given specification.
But what is the intuition ...
- 1,239
24
votes
1
answer
14k
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 ...
- 450
40
votes
5
answers
12k
views
Examples of PCA where PCs with low variance are "useful"
Normally in principal component analysis (PCA) the first few PCs are used and the low variance PCs are dropped, as they do not explain much of the variation in the data.
However, are there examples ...
- 403
17
votes
1
answer
21k
views
Questions on PCA: when are PCs independent? why is PCA sensitive to scaling? why are PCs constrained to be orthogonal?
I am trying to understand some descriptions of PCA (the first two are from Wikipedia), emphasis added:
Principal components are guaranteed to be independent only if the data set is jointly normally ...
- 621
81
votes
3
answers
71k
views
What is the difference between ZCA whitening and PCA whitening?
I am confused about ZCA whitening and normal whitening (which is obtained by dividing principal components by the square roots of PCA eigenvalues). As far as I know,
$$\mathbf x_\mathrm{ZCAwhite} = \...
- 12.2k
67
votes
9
answers
28k
views
Advanced statistics books recommendation
There are several threads on this site for book recommendations on introductory statistics and machine learning but I am looking for a text on advanced statistics including, in order of priority: ...
9
votes
1
answer
3k
views
Plotting a discriminant as line on scatterplot
Given a data scatterplot I can plot the data's principal components on it, as axes tiled with points which are principal components scores. You can see an example plot with the cloud (consisting of 2 ...
- 54.5k
82
votes
6
answers
41k
views
What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)?
I've read a lot about PCA, including various tutorials and questions (such as this one, this one, this one, and this one).
The geometric problem that PCA is trying to optimize is clear to me: PCA ...
- 3,450
27
votes
1
answer
15k
views
How LDA, a classification technique, also serves as dimensionality reduction technique like PCA
In this article
, the author links linear discriminant analysis (LDA) to principal component analysis (PCA). With my limited knowledge, I am not able to follow how LDA can be somewhat similar to PCA.
...
- 6,245
13
votes
1
answer
7k
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....
- 2,268
28
votes
1
answer
9k
views
What exactly is called "principal component" in PCA?
Suppose $u$ is the vector that maximizes the variance of the projection of the data with design matrix $X$.
Now, I have seen materials that refer $u$ as the (first) principal component of the data, ...
- 1,813
3
votes
1
answer
3k
views
One-hot-encoding gives untractable amount of classes
I'm performing regression on the price of bycicles based on their brand, model and submodel. These features are hierarchical: one model belongs only to one brand but one brand can have many models. ...
109
votes
2
answers
269k
views
Why do we need to normalize data before principal component analysis (PCA)? [duplicate]
I'm doing principal component analysis on my dataset and my professor told me that I should normalize the data before doing the analysis. Why?
What would happen If I did PCA without normalization?
...
- 5,527
33
votes
3
answers
13k
views
Visualizing a million, PCA edition
Is it possible to visualize the output of Principal Component Analysis in ways that give more insight than just summary tables? Is it possible to do it when the number of observations is large, say ~...
- 5,440
20
votes
2
answers
6k
views
Low variance components in PCA, are they really just noise? Is there any way to test for it?
I'm trying to decide if a component of a PCA shall be retained, or not. There are a gazillion of criteria based on the magnitude of the eigenvalue, described and compared e.g. here or here.
However, ...
- 201