Questions tagged [orthogonal]

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The orthogonal rotated data

Suppose the multivariate one-way anova model for the raw data , i.e. $$ \label{Example_model_1} \mathbf{y}_{ij}=\mathbf{\mu}+\mathbf{z}_i+\mathbf{e}_{ij}, ~~ i=1,\ldots,m,~~j=1,\ldots,n_i,~~~~~~~~~~...
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How to interpret coefficients obtained from an ordinal variable in a linear regression?

I'm following this example https://stats.idre.ucla.edu/r/library/r-library-contrast-coding-systems-for-categorical-variables/ I was wondering what is the correct interpretation of the coefficients ...
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Find a vector that satisfies the following: i) it has a given correlation with a second vector and ii) it is orthogonal to a set of vectors

I would like to generate a vector $\vec{u}$ of dimension $n$, so that i) it has a given correlation $r$ with a second vector $\vec{v}$ and ii) it is orthogonal to a set of $m$ vectors $A = \{\vec{w}_1,...
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44 views

Orthogonal contrasts, ANOVA, why are there only as many contrasts there are degrees of freedom?

For example, if I have the data $$ \begin{array}{l|l|l|l|l|l|l} A & low & & medium & & high & \\ \hline B & standard & new & ...
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50 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 ...
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78 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 ...
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118 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 ...
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119 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 ...
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1k 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 $...
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51 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 ...
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186 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? ...
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29 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 ...
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114 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 ...
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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 ...
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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 ...
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243 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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391 views

Advantage of orthogonal polynomials

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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861 views

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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90 views

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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206 views

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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460 views

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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220 views

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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1answer
214 views

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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536 views

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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966 views

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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2k views

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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98 views

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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81 views

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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155 views

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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1answer
544 views

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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110 views

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 ...
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27 views

Orthogonal arrays

This is from a note I found I have one question, could t=3?
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45 views

Marginal distribution of distribution on the Unitary group

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 ...