# Questions tagged [principal-component-analysis]

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.

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### Why can $X$ be approximated as $X V V^T$?

I was given the statement If $\vec{x_1}, \dots, \vec{x_n}$ are close to a k-dimensional subspace $\mathbb{V}$ with orthonormal basis $V \in \mathbb{R}^{d \times k}$, then the data matrix can be ...
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### Am I Misunderstanding Principal Component Analysis (PCA)?

Principal Component Analysis (PCA) has routinely caused me to question my understanding of mathematics, particularly linear algebra. Once again, PCA is present and I would like to engage to the ...
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### Questions on robust PCA

I am reading a paper on robust PCA by Candès et al. In section 2.3, the authors said $\| \mathcal P_{\Omega} \mathcal P_T \| < 1$ is equivalent to $\Omega \cap T = \{0\}$. Can someone please ...
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### How to use two SPSS factor scores to generate one social vulnerability index value

I am trying to produce a single value representing social vulnerability based on 6 z-score normalized demographic variables for 55 geographic areas. I have successfully done a factor analysis on the 6 ...
0answers
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### Angle Preserve Dimension Reduction

PCA is known to be one of the most-used Dimension Reduction algorithms. But does it preserve angles between vectors? If no, are there any algorithms that do preserve angles and (what is the same) ...
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### Intuition behind performing principal components analysis on the linear approximation of a time-series from another time-series.

I was reading a recent paper and was trying to understand the novel factor analysis method that they introduce. I am not terrific at linear algebra so I was hoping to get some intuition behind what ...
1answer
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### Approximating large quadratic optimization problems

For some positive-definite matrix $A \in \mathbb{R}^{K \times K}$ I want to solve the quadratic optimization problem $$\max_{x\in [0,1]^K} x^T A x \\ \text{s.t.} \\ \sum_{i=1}^{K}x_{i}=1$$ The problem ...
0answers
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### Finding Orthonormal Basis using SVD and comparing it with Gram-Schmidt shows different result

I was trying to find the orthonormal basis for the column space of the following matrix "A" \begin{pmatrix} -1 & -1 & 2 & 3 \\ -1 & 1 & -3 & -4 \\ 2 & -2 & 5 ...
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### Principal component Eigen values

I am learning principal component : In Below question I am facing problem to calculate the eigen values and eigen vector. Is there any short method to determine that. Consider the following covariance ...
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### Maximizing the variance of projected data

Show that this expression can be written as: I have tried multiple approaches like going through a backward proof by taking individual norms and trying to reconstruct but I am unable to find a clue. ...
1answer
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### Maximizing the Lagrangian with respect to vector - help solve

I need help solving this following equation, a Lagrangian problem that I encountered during my studies in principal component analysis (PCA). One should maximize the variance with respect to the first ...
0answers
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### Probability of PCA parameters?

Let's say I have a PCA model computed using SVG and have kept $L$ parameters, so I have matrix of mean values $\mu$, a matrix with the components $W$ and a vector of standard deviations $s$(?). I can ...
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### Low Loadings in PCA

I am new user of this forum, and I am not sure whether I posted my question appropriate place, sorry for that. I am running PCA for determining gentrification score for census tracts. Actually, I am ...
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### Intuition behind eigen vectors

Why do we use the eigen vectors corresponding to the low eigen values of the laplacian matrix in most of the applications? Do the eigen vectors of the Laplacian matrix represent only frequencies or ...
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### PCA dimensions question

I read this quote in some lectures about PCA: " Also, since n points in a p-dimensional space defines a linear subspace whose dimension is at most n−1, we would find that p−n+ 1 eigenvalues are ...