# Questions tagged [orthogonal]

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### What is the appropriate design for a discrete choice experiment?

I need to develop a discrete choice experiment in R, which is an unlabeled experiment. However, after delving into the theory behind discrete choice experiments, it remains unclear whether there is a ...
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### Prove two orthogonal contrasts are uncorrelated

I am reading some ANOVA notes and have a problem with proving two orthogonal contrasts are uncorrelated. Here is what I tried: Assume that $\bar X_i, i=1,\dots,n$ are uncorrelated. Suppose we have two ...
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### Partially orthogonal error term in Regression

Suppose I have a model: $Y = X_1 \beta_{1} + X_2\beta_2 + C$, where $C$ is independent to $X_1$ but not $X_2$. If we naively perform linear regression, say by concatenating $X= (X_1 X_2)$ and ...
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### Two-way ANOVA with interaction term / orthogonal design

The two-way ANOVA model with interaction for some continuous variable $y$ can be expressed as $$y = X\mu + \varepsilon,$$ where $X$ is the design matrix (the first column of $X$ contains the constant, ...
100 views

### interpreting polynomial regression output when the regressors are orthogonal (vs. raw)

I want to show an inverted U-shape relationship between two variables: "minutes spent in a room A" and "trustworthiness in others". The hypothesis is that those who have low and ...
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### Are eigenvectors of PCA guaranteed to be orthonormal?

Are eigenvectors (principal components) of PCA orthonormal or only orthogonal ? Or only some of them are orthonormal or they are orthonormal if data were normalized before doing PCA ?
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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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For example, if I have the data $$\begin{array}{l|l|l|l|l|l|l} A & low & & medium & & high & \\ \hline B & standard & new & ... • 938 0 votes 0 answers 246 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 ... 1 vote 0 answers 244 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 ... • 85 5 votes 2 answers 15k views ### 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 ... 0 votes 1 answer 588 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 ... 1 vote 0 answers 455 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 ... • 692 11 votes 2 answers 5k 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 ... • 692 1 vote 0 answers 106 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 ... • 152 0 votes 1 answer 866 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? ... • 235 1 vote 1 answer 532 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 ... • 152 0 votes 2 answers 258 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 ... • 152 11 votes 3 answers 6k 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 ... • 131 1 vote 1 answer 3k 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 ... • 163 2 votes 0 answers 364 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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