# Questions tagged [orthogonal]

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### OPLSDA in non linear data [closed]

I am wondering if it would be possible to run OPLSDA on non-parametric data. If yes, do I need to carry previous steps to account for non linearity? If no, what is the solution? - I have seen a ...
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### How to directly know the backward selection model when independent variables are orthogonal?

According to this output, the independent variables are orthogonal. Please tell me, when doing the backward selection, why it can be directly known that it should be reduced to 5th order model?
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### Lasso Least Squares Estimator for Matrix with Orthonormal Columns

$A\in\mathbb{R}^{nxm}$ and $y ∈ \mathbb{R}^n$. Consider the least squares problem: $$\text{minimize}||Ax−y||^2\text{ with respect to }x∈R^m$$ where $x^{LS}$ is the Lasso least square estimator for a ...
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### A Primer on Orthogonal GARCH Model Covariance Matrix

I am trying to replicate Table 3a: Correlation Matrix from this paper (Page 11): http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.201.7226&rep=rep1&type=pdf. (I believe there is a ...
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### Why the first principal component is mostly negative while the second component is mostly positive?

I am running PCA for a fleet management data frame $X$, where each column is a city, each row is a date, there are 50 cities and 500 dates. I run PCA on $A=X^{T}X$. Then the first component $v_{1}$ ...
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### Scatter Plot - Basics [closed]

I am stuck in understanding a basic scatter plot. I am working in two dimensions i.e. there are two variables X & Y. So, the question is that in the scatter plot, what do the two axes mean? ...
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For example, if I have the data $$\begin{array}{l|l|l|l|l|l|l} A & low & & medium & & high & \\ \hline B & standard & new & ... 0answers 72 views ### Using varimax – rotated PCA for clustering via Gaussian Mixture Model? After extracting the Principle Components of my data, I apply Gaussian Mixture Models for clustering. I used a subset of the orthogonal basis of the Principle Components and projected my data onto ... 0answers 113 views ### Orthogonal contrasts for coefficients of regression Suppose that we want to test the following hypothesis H_{0}:b_{1}+b_{3}-2b_{2}=0 where b_{1},b_{2},b_{3} are coefficient derived from a linear regression.We can see that H_{0} is similar to ... 1answer 167 views ### 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 ... 0answers 147 views ### How is multivariate Gaussian distribution is determined by its second moments alone? The following statement is given in Unsupervised Learning chapter of the book Elements of Statistical Learning. Since the multivariate Gaussian distribution is determined by its second moments ... 2answers 2k views ### The linear transformation of the normal gaussian vectors I am facing difficulty in proving the following statement. It is given in a research paper found on Google. I need help in proving this statement! Let X= AS, where A is orthogonal matrix and ... 0answers 60 views ### Statistics: orthogonality vs uncorrelatedness vs independence [duplicate] In this post I would like someone to summarize and relate these 3 concepts of statistics (in the context of stats). 1) I remember that uncorrelated does NOT imply independence (e.g. the case where ... 1answer 273 views ### Does orthogonal and zero mean of two RV X,Y imply that they are uncorrelated? I understand that two uncorrelated RV X,Y are orthogonal if at least one of both is of zero mean. But can you reverse this statement if you expand the preconditions to both RV X,Y being of zero mean? ... 1answer 40 views ### which angle and axis to chose to get a 90 degrees angle between those 2 vectors I am suddenly puzzled by ho to know (when in 3D) with respect to which axis is the vector being rotated when the dot product between then is =0. for example: if i rotate 90degrees (pi/2 radians) along ... 2answers 158 views ### orthogonality in 2D vs higher dim vectors considering that 2 vectors such as x_2=\begin{bmatrix}1 & 1 \end{bmatrix} and y_2=\begin{bmatrix} -1 & 1 \end{bmatrix} are orthogonal in 2D (i.e. their scalar product is 0) however ... 2answers 2k views ### Why are PCA eigenvectors orthogonal but correlated? I've seen some great posts explaining PCA and why under this approach the eigenvectors of a (symmetric) correlation matrix are orthogonal. I also understand the ways to show that such vectors are ... 1answer 1k views ### Orthogonal initialization of weight matrix Searching for the way to initialize the matrix weights as orthogonal (i.e. W*W^T = I and all the eigenvalues are equal either 1 or -1),(I was wrong) I found this ... 0answers 283 views ### Calculating orthogonalized impulse response functions for vector error corrrection models Background: I am working on orthogonal impuls response functions (OIRFs) for vector error correction models (VECMs). Its an exercise to develop understanding. I am given a bivariate VECM:$$ \Delta ...
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What is the sense or background of orthogonal polynomials (regarding using mixed models)? I would like to know why they shall or should be orthogonal. Is it to build independent sample points? On Is ...
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### How to standardize the data matrix before applying SVD for PCA?

I am trying to enhance the contrast in the images I get after scanning a surface using Thermography (Principal Component Thermography ~Rajic, which is basically an application of Principal Component ...
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### Balancing out in an orthogonal design

A definition of orthogonality in the context of statistics is An experimental design is orthogonal if the effects of any factor balance out (sum to zero) across the effects of the other factors. ...
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### Statistically orthogonal - explanation?

I did see the related question here but my question is more related to the actual explanation of the orthogonality itself. So the following design is orthogonal (this is a latin square to be precise):...
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### What is the intercept term in a mixed effects model using orthogonal polynomials to model time?

I'm using a mixed effects model (lmer) in R to model eye-tracking data using orthogonal polynomials (poly(time,3)) for time. The response variable is log(looks to target/looks to competitor). The ...
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### Orthogonality in ANOVA and Regression Analysis

I read the following (Wikipedia) regarding contrast coding of categorical variables: Unlike when used in ANOVA, where it is at the researcher’s discretion whether they choose coefficient values ...
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### Distribution involving orthogonal matrix

If $Y∼N(\mu,I\sigma^2)$ and $Q$ is any orthogonal matrix of appropriate dimension, how do I find the distribution of $QY$?
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### Uses of the Helmert matrix

In chapter 8 of "Matrix Algebra from a Statistician's Perspective", the author describes the construction of an orthogonal matrix, the first row of which is proportional to some row vector of non-...
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### Linear regression: *Why* can you partition sums of squares?

This post refers to a bivariate linear regression model, $Y_i = \beta_0 + \beta_1x_i$ . I have always taken the partitioning of total sum of squares (SSTO) into sum of squares for error (SSE) and sum ...
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### What are multivariate orthogonal polynomials as computed in R?

Orthogonal polynomials in an univariate set of points are polynomials that produce values on that points in a way that its dot product and pairwise correlation are zero. R can produce orthogonal ...
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### Optimization with orthogonal constraints

I am working on computer vision, and have to optimize an objective function involves matrix $X$ and matrix $X$ is an orthogonal matrix. $$maximize \ \ f(X)$$ $$s.t \ \ X^T X=I$$ Where $I$ is the ...
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### How to force the slope=1 in orthogonal regression

I would like to model the functional relationship between two acoustic measurements of traffic noise exposure, e.g. noise during day (x) and noise during night (y), expressed in Decibel values, using ...
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### Planned contrasts - pros & cons of reducing a 2x3 design into 6-level single factor?

I have a 2x3 factorial design, and wish to explore specific hypotheses with planned (orthogonal) contrasts. Some people recommend that in order to specify the contrasts, this design can be replaced ...
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### Dropping columns from an orthogonal design matrix?

Hello: I’m working with a three factor (ANOVA) design that I wish to use in an MCMC chain to estimate the parameters for the main effects and treatment interactions. I wish to run MCMC analyses ...
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### Orthogonal polynomials for regression

Is it possible to define orthogonal polynomials on the interval $[0, +\infty[$ ? Maybe using the Gram-Schmidt process from the monomial basis $(1, x, x^2, ...)$? My problem is that I have some data ...
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### PCA: Cannot Understand one part of Derivation of Principal Comonents

I'm stuck on something in the derivation of the Principal Components. We have random vectors $X$ of dimension $p$. We want to find linear combinations of $X$, $a'X,$ where $a \in \mathbb{R}^{p}$ that ...
I am trying to understand the behavior of distributions over the Unitary group (i.e. the set of square matrices $P$ such that $P^tP = I_d$), or in general distribution over the Stiefel manifold (set ...