Questions tagged [orthogonal]
THIS TAG HAS UNCLEAR/VAGUE SCOPE. Please use more specific tags instead.
62
questions
1
vote
0
answers
34
views
How can I add the constraint that regression parameters are orthogonal to a design matrix in an MCMC algorithm?
Suppose I have a linear regression like
$$y_i=X\boldsymbol{\beta} + \alpha_i + \epsilon_i,$$
where $\epsilon_i\sim N(0,\sigma^2)$ are i.i.d. Further, suppose that I want to add the constraint that $\...
- 1,777
0
votes
0
answers
21
views
In what sense are "all regression predictors in a balanced factorial ANOVA orthogonal"?
This question Why are all regression predictors in a balanced factorial ANOVA orthogonal? asks why all predictors in a balanced ANOVA setting are orthogonal. My question is in what sense are the ...
- 246
2
votes
1
answer
47
views
Is the coefficient of an orthogonal independent variable affected by the near-multicollinearity of two other independent variables?
Suppose there is a regression with three variables $X_1, X_2, X_3$. Say $X_1$ and $X_2$ have near-multicollinearity, but $X_3$ is (nearly) orthogonal to both.
Will $\beta_3$ experience the same ...
3
votes
1
answer
84
views
What estimation methods other than ordinary least squares guarantee the orthogonality of predictions and residuals?
From my question here, it is evident that estimation approaches to linear regression other than ordinary least squares can result in the predictions and residuals lacking orthogonality, despite the ...
- 46.7k
1
vote
0
answers
40
views
Quadratic regression with orthogonal polynomials vs. raw polynomials with QR decomposition
I'm using rstanarm to estimate random slopes for second-order polynomial coefficients. My model has the basic form:
...
- 11
0
votes
1
answer
145
views
Prove OLS consistency
Consider the linear model
$$
Y={\underbrace{X_i}_{K\times 1 }}^\top\beta+U_i
$$
and assume
(0) There is no intercept in the model
(1) $E(X_i U_i)=0_K$ [orthogonality]
(2) $E(X_i X_i^\top)$ has rank $K$...
- 560
1
vote
0
answers
26
views
The p value of my OPLS-DA model was -nan(ind), I don't know, what's mean, and the residual MS was infinite, I need help here?
I obtained for the first time strange cross-validation results of my OPLS-DA model, the CV ANOVA was -nan(ind), SD residual was inf, MS residual was inf also, moreover, F test equal to zero.
The ...
- 11
0
votes
0
answers
22
views
Cross-covariance in context of Andrews plot
As shown in this Cross-Validated post
Close curves on an Andrews plot
I don't understand how, in the accepted answer, the cross-covariance can be defined as,
$$\int_{-\pi}^{\pi}f_xf_ydt$$
Considering ...
2
votes
1
answer
229
views
some thought about independence and orthogonal, please comment on this if it's wrong
It seems that linearly independent is totally different from independent of random variable concept. Non-zero vectors Orthogonality must imply linearly independence.
In Statistics, the relation of ...
- 239
1
vote
0
answers
223
views
decision boundaries of random forests
I was told that decision boundaries of RandomForests can be non-orthogonal. See Figure 7-5 in Geron's book Hands-On Machine Learning with Scikit-Learn & TensorFlow p.g. 187 edition 1. This is not ...
- 155
0
votes
0
answers
24
views
Proving non-correlation with very disperse distributions
I'm fairly new to statistics and came up with a problem.
I have a sample with a variation coefficient CV = 0.517 for variable x, and I want to prove this variable is not correlated with a second ...
- 111
0
votes
0
answers
42
views
One-dimensional subspace clustering
Consider the inner product space $(\mathbb{R}^n, \langle\cdot,\cdot\rangle)$ and suppose that there are one-dimensional orthogonal subspaces $\{V_i\}_{i=1}^n$ such that $\mathbb{R^n} = \oplus_{i=1}^n ...
- 35
2
votes
0
answers
99
views
What does "nearly orthogonal" mean for a matrix?
I am currently working on experimental designs. I have read several times (e.g Kuhfeld 2010) that for the main effects to be identified, the design matrix has to be orthogonal or "nearly ...
0
votes
0
answers
50
views
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 ?
- 547
2
votes
1
answer
47
views
Check orthogonality of batched vectors, of non square matrix
I have a batch of vectors $X$ that have row vectors of size $n$, and batch size of $k$, so $$\begin{bmatrix}
v_{11} & ... & v_{1n} \\
v_{21} & ... & v_{2n} \\
&\;\;\vdots \notag \\
...
- 547
2
votes
2
answers
247
views
Prove two orthogonal contrasts are statistically independent
Linear combination $C=\sum_{i=1}^{n} a_i \bar{X}_i$ is called a (estimated) contrast if $\sum_{i=1}^{n} a_i=0$. Two contrasts are called orthogonal if $\sum_{i=1}^{n} a_i b_i = 0$; simplest example ...
- 556
0
votes
1
answer
143
views
Using Orthogonal Main Effects Plan to select profiles for conjoint analysis
I am trying to create a code in Python to select orthogonal profiles given some attributes and levels.
For eg:
...
- 1
3
votes
0
answers
82
views
Estimating Moving Average Impact Matrix after running VECM
How to estimate moving average impact matrix after running VECM model in R? It is given as $ \hat{\beta_c}( \hat{\alpha_c}$ $\hat{\Gamma}$ $\hat{\beta_c}$)$^{-1}$ $\hat{\alpha_c}'$, where $\alpha$, $\...
- 138
2
votes
1
answer
652
views
Orthogonality and uncorrelated
In linear regression suppose we parition the regressors X (with k variables and n observations) into two sets X1 (with k1 variables) and X2 (with k2 variables) where k1 and k2 sum to k.
I found some ...
- 85
1
vote
1
answer
141
views
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 ...
user271077
2
votes
1
answer
255
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$, $...
- 353
1
vote
0
answers
101
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 ...
- 123
0
votes
2
answers
102
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?
- 1
1
vote
0
answers
151
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 ...
- 65
0
votes
1
answer
432
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}$ ...
- 235
0
votes
1
answer
126
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?
...
2
votes
0
answers
72
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 $...
- 21
1
vote
1
answer
89
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 ...
- 131
0
votes
0
answers
22
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,...
2
votes
1
answer
339
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 & ...
- 908
0
votes
0
answers
221
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
1
vote
0
answers
232
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
12k
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 ...
- 51
0
votes
1
answer
511
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
378
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
4k
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
100
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
781
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?
...
- 205
1
vote
1
answer
443
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
247
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
5k
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
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 ...
- 153
2
votes
0
answers
357
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 ...
- 21
1
vote
0
answers
899
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 ...
- 2,939
0
votes
0
answers
2k
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 ...
- 31
2
votes
1
answer
222
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.
...
- 95
1
vote
1
answer
487
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):...
- 95
3
votes
1
answer
1k
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 ...
- 61
6
votes
0
answers
442
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 ...
- 3,758
1
vote
1
answer
405
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 $QY$?
- 11