Questions tagged [orthogonal]

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Alternatives to PCA with orthogonal datasets?

http://blog.audio-tk.com/2008/02/04/dimensionality-reduction-principal-components-analysis/ "It is obvious that PCA does not respect the manifold structure. One has to use 3 dimensions to describe ...
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1answer
18 views

Contrasts in a Completely Randomized Design (Unbalanced)

Four catalysts that may affect the concentration of one component in a three-component liquid mixture are being investigated. Consider a completely randomized experiment, where $n_1 = 5$, $n_2 = 4$, $...
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Genstat warning message an71

When running an anova analysis of an experiment using balanced incomplete blocks using Genstat, I get this warning message: Standardized residuals approximate if not an orthogonal design Please ...
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50 views

Independence of components in PCA

Let's have spatio-temporal dataset ($Y \in \mathbb{R}^{L \times T}$). Where $L$ stands for spatial grid points and $T$ for time. Now let's say that the noise of the system follows a multivatiate ...
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22 views

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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38 views

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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40 views

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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32 views

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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63 views

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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19 views

Variance of Random Vector in the Circular Orthogonal Ensemble

Let $x$ be a (uniformly) randomly chosen column of a random orthogonal matrix (of size $K$ x $K$) distributed according to Haar measure. What is $\mathbb{E}[x]$, $\mathbb{E}[x x^T]$, $Cov(x, x)$, and $...
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25 views

Granger's representation theorem: Johansen's version

In his book 'Likelihood based inference in cointegrated Var', in order to get the expression for the Granger's representation theorem,, Johansen claims that: (1) $$\beta \bot(\alpha' \bot \beta \bot ...
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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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16 views

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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86 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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91 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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134 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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Orthogonality of residuals in linear regression

In multiple linear regression, I came across the statement that both $e$(residual) and predicted $y$ are projections of actual y and $e$ is orthogonal to predicted $y$. I was trying to visualize the ...
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213 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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185 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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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 $...
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66 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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1answer
330 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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1answer
48 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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194 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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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 ...
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1answer
2k 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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294 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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524 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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1k 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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1answer
121 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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1answer
256 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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1answer
718 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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295 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
302 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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754 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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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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1answer
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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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1answer
3k 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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1answer
183 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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104 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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176 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
685 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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1answer
120 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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28 views

Orthogonal arrays

This is from a note I found I have one question, could t=3?
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51 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 ...