Questions tagged [rotation]

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Describe a geometrical shape as a piecewise function

Consider a cube filled with random particles. Let's say the particles in cube are rotated around the z-axis through the center of the cube. Here, the rotation is proportional to the height of the cube,...
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Predict score based on rotated PCA

I have done a varimax rotated PCA on my "test" dataset. I would like to use this PCA (i.e. the varimax rotated PCA) on a new dataset and predict the scores. This function ...
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Rotation Matrix from Covariance of 3D point-cloud

I am trying to retreive rotation matrix from a rotated 3D point cloud covariance matrix, using SVD decomposition (as done in SimNet and MVTrans). Here how I computed the covariance matrix from 3D ...
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Clustering and coordinate rotation

Does the coordinate system rotation affect the clustering result? Which approach could be used to eliminate the influence of coordinate system rotation in clustering? Any help would be appreciated!
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Why the correlation between factors are penalized when weight is large negative in Oblimin rotation?

I'm facing difficulties in interpreting the criterion for the Oblimin rotation. In my knowledge, the following criterion shall be minimized in oblimin rotation. $$\sum_{ij} (\sum_{v}{l_i}^2{l_j}^2 - \...
No Ru's user avatar
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Rotation-sensitivity of SVD

Suppose I perform a truncated SVD on a symmetric, PSD matrix $A \in R^{N \times d}$ (lowering the dimensionality from $d$ to $k$). Further suppose that there is a rotation matrix $Q$ such that some of ...
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CNNs Scale/Rotation Invariance

CNNs are translation-invariant due to the pooling layer. How can we make them scale/rotation invariant? I have beginner-level knowledge of Deep Learning so please help me understand.
Murtaza Kazmi's user avatar
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Choosing a rotation method for ESEM

I am trying to decide on which oblique rotation method to use for my ESEM analysis (with MLR estimator). MPLUS provides a number of options (GEOMIN, QUARTIMIN, OBLIMIN, CRAWFER, etc.). I was wondering ...
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How can I generate a completely balanced combination

Attaching a code that generates unique combinations of 8 letters but, it still lacks a condition wherein all letters should have equal counts per column. For my code, basically, letter a to h should ...
Xpeculate's user avatar
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2 answers
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Gaussian distribution: moments, independence and rotation

I have a few questions with respect to the gaussian distribution, its moments and independence. So a gaussian distribution is fully specified by its first two moments, the mean and variance (or ...
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Good article examples of PAF/PCA

Can anyone give me (and probably to the rest of the world) good article examples which uses especially PAF (and/or PCA) factor analysis with oblique and/or orthogonal rotations?
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Which rotation type for principal component regression?

I would like to perform a principal component regression (PCR), but feel a little confused about the rotation type to be used in the principal component analysis (PCA) step. First I perform a PCA to ...
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Is PCA effectively just a rotation of the data, if you were to keep all principal components? [duplicate]

If so, why would you apply another rotation after you already found the "variance-maximizing" - and in that sense optimal - rotation. Wouldn't the second rotation lead to a - again - non-variane-...
PeterPancake's user avatar
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Oblique vs. Orthogonal Rotation for EFA

Will orthogonal relationships show up when using Oblique rotation? Based on the articles I have read on EFA rotation my understanding is that although oblique rotation procedures might be expected ...
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What is the Rotation Matrix in PCA?

I'm trying to implement th Local Coordinate System (LCS) of this paper. It's all clear to me about how it works, but the only thing that I' dont understand is the "rotation" mechanism. Quoting the ...
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What is a rotation matrix and how to implement it?

In Revisiting the VLAD Image Representation the authors introduce Local Coordinate System, i.e. they: we learn off-line (for each visual word) a rotation matrix Qi from training descriptors mapped ...
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$\mathbf{Cov}(x,y)$ under rotation of axes

If $\mathbf{Cov}=0$ let $u=x \cdot cos(\theta)+y \cdot sin(\theta)$ and $v =y \cdot cos(\theta)-x \cdot sin(\theta)$. What will be $\mathbf{Cov}(u,v)$? Will it be $0$?
Sumedha ghosh's user avatar
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How to use varimax rotated PCA to produce raster layers?

I have a raster stack of 19 layers called "raster_bio". The code to do PCA analysis is: ...
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linear regression after rotation

I have a set of 2 dimensional points [x,y], with a barycenter in 0,0 and I'm rotating it. I'm wondering why the linear regression of this set of points is not rotating of the same amplitude. Below ...
Ricky Bobby's user avatar
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Find the rotation between set of points

I have two sets (sourc and target) of points (x,y) that I would like to align. What I did so ...
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Which matrix should be interpreted in factor analysis: pattern matrix or structure matrix?

When doing a factor analysis (by principal axis factoring, for example) or a principal component analysis as factor analysis, and having performed an oblique rotation of the loadings, - which matrix ...
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Is Mahalanobis distance equivalent to the Euclidean one on the PCA-rotated data?

I've been led to believe (see here and here) that Mahalanobis distance is the same as the Euclidean distance on the PCA-rotated data. In other words, taking multivariate normal data $X$, the ...
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How to organise an iterative manual rotation of n component pairs?

I am currently building a q.rotate() function for the qmethod R package for Q Methodology. As is desirable for Q, I'd like users to be able to iteratively rotate ...
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Does the order of rotations matter in rotating PCA loadings (by-hand)?

Suppose I have retained 3 principal components, and I want to rotate their loadings by hand (yeah, that's rare, but it is commonly used in Q Methodology). Does it matter in which order I rotate the ...
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Meaning of negative elements in Principal Component Analysis(PCA) rotated component matrix

Suppose that we have this rotated component matrix from PCA (SPSS output): ...
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How we can calculate squared factor loadings in Principal Component Analysis(PCA)?

This snapshot is from THIS article: What is Squared factor loadings and how we can calculate it. Any link between above snapshot and below snapshot of SPSS output? there isn't any relationship ...
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Isn't an oblique rotation against the whole spirit of principal component analysis?

I am studying principal component analysis (PCA) as a method to deal with multicollinearity. And when studying the rotation method -- which, if my understanding is correct, is the center of the PCA -...
Po Stulat's user avatar
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131 views

Different results based on two target rotation approaches

I am trying to do some data analyses based on target rotation in R using two different approaches. Here is a small simulation in r. ...
user2702330's user avatar
12 votes
1 answer
3k views

How to generate uniformly random orthogonal matrices of positive determinant?

I've got probably a silly question about which, I must confess, I'm confused. Imagine repeated generating of uniformly distributed random orthogonal (orthonormal) matrix of some size $p$. Sometimes ...
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2 votes
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Statistical model for axis angle rotations

I would like to describe a large number of measurements of rotations $\textbf{x}_i$. Each rotation is described by its rotation axis $\textbf{v} =\frac{\textbf{x}}{|\textbf{x}|}$ and the rotation ...
Chopian's user avatar
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Why does my loading matrix following PCA with a varimax rotation contain only ones and zeros? [duplicate]

I'm running a PCA using the R function prcomp. This is the function: d2.pca <- prcomp(sel.d2, center=TRUE, scale.=TRUE) So ...
David VR's user avatar
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Are matrix Fisher r.v.s closed under multiplication?

With appropriate parameters, a matrix Fisher distribution provides a distribution over SO(3) (i.e. over rotations in $R^3$). See this MathOverflow post for a few notes describing the distribution. ...
Bill Bradley's user avatar
3 votes
1 answer
1k views

Multiple linear regression through orthogonal matrices

An example of linear regression could look like: $min \sum_{i=0}^{m}||x_i A - y_i||_2^{2}$, where ${x_i, y_i} \in \mathbb{R}^n$ and $A \in \mathbb{R}^{n\times n}$. I am interested in knowing how do ...
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Varimax Rotated factors are very much NOT uncorrelated?

On the back of an earlier question i am having an issue. Software = SAS JMP Pro 11 Earlier Question My rotated factors (post Principal Components Analysis [PCA]) are not at all uncorrelated. I have ...
Samuel's user avatar
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2 answers
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Predict only the first N principal components in a PCA analysis

I'm using R to analyze a very large dataset. I conduct a PCA on one dataset, PCA <- prcomp(formula = ~., data = train, scale = T, na.action=na.exclude) and ...
mrsoltys's user avatar
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Extracting nonorthogonal sources in ICA/PCA/blind source seperation problem

My problem is essentially a 'blind source separation' problem. I have 3 non-orthogonal sources (or basis functions) and N random linear combinations (mixes) of said sources. My problem is to obtain ...
DankMasterDan's user avatar
2 votes
1 answer
885 views

Why direct oblimin rotation results in greater eigen values?

I came across this in the Wikipedia page about Factor Analysis. Is that true that direct oblimin rotation results in greater eigen values? If that is true, what's the reason behind it and does it ...
Ehsan88's user avatar
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Procrustes Analysis of 3d point cloud without defined landmarks

I am working with several hundred 3d point clouds generated using a 3d scanner and would like to be able to compare their shapes using something like a procrustes analysis. Instead of manually ...
user2364084's user avatar
1 vote
2 answers
235 views

How many PCs should varimax rotation be applied to?

I have a list of 25 air pollutants many of which are strongly correlated. I was hoping to reduce down to a short list of eigenvectors which would each be composed of a small number of the pollutants. ...
Scott's user avatar
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1 answer
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A textbook error w.r.t structure and pattern loadings

I have this picture in Lattin representing structure and pattern loadings in factor analysis. If $Z$ (an observed variable) $=w_1 F_1+w_2 F_2$ (according to factor model), then the pattern loadings of ...
Bravo's user avatar
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To rotate or not to rotate post-PCA and pre-cluster analysis

Questions in respect to rotation post-PCA have been answered before -> its all in the hands of the researcher... Same answer to the question if rotation (orthogonal or not) makes sense before plugging ...
nafets's user avatar
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3 votes
2 answers
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Exploratory factor analysis - promax & factor cross-loadings

I have a question regarding the best practice for dealing with cross-loadings on factors after conducting an exploratory factor analysis using a promax rotation. Just to give a bit of background ...
Madeline's user avatar
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4 votes
1 answer
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Is it acceptable to rotate factors with PCA for binary data?

What issues, if any, might there be in rotating factors in order to obtain factor/component loadings of binary data? Is it acceptable to rotate the factors when doing a traditional PCA? (Assuming I’m ...
Deryl H.'s user avatar
6 votes
1 answer
4k views

Can I somehow compute variance explained by PC after Oblique rotation in PCA?

Let´s say that my PCA analysis extracted 2 components, which explain 80% of the variance before rotation. The components were then rotated using oblique (Direct Oblimin) rotation, so SPSS cannot ...
Noro's user avatar
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3 votes
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Detecting reflection in non-orthogonal rotation

I've known that, in orthogonal rotation, if the rotation matrix has determinant of -1 then reflection is present. Otherwise the determinant is +1 and we have pure rotation. May I extend this "sign-of-...
ttnphns's user avatar
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4 votes
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Rotation matrices and prior invariance for arbitrary dimensions

I have a question about a rotation matrix, which can be represented in 2 dimensions as: $$R_{2}(\theta)=\begin{bmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{bmatrix}$$ For ...
probabilityislogic's user avatar