All Questions
Tagged with multivariate-analysis covariance
64 questions
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How can I assume covariance (rho) between two psychiatric disorders for multivariate meta-analysis
I want to do a multivariate meta-analysis, as I have several effect sizes per study. For example, I want to investigate the risk (Odds Ratio) of Conduct Disorder (CD) and Oppositional Defiance ...
0
votes
1
answer
71
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Comparing linear slopes between two scatterplot matrices
Below is a reproducible example of code that produces a dataset and plot roughly similar to what I am working on. The dataset is comprised of multiple columns for gene transcript abundance values and ...
- 1
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0
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53
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Multivariate sample covariance
I have a set of $X_1,...,X_n$ samples with covariance $\Sigma_1,...,\Sigma_n$.
The multivariate sample mean is then $$ \left(\sum_{i=1}^n \Sigma_i^{-1} \right)^{-1} \left(\sum_{i=1}^n \Sigma_i^{-1} ...
1
vote
1
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188
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Deriving covariance of joint distributions of MVN [Linear Gaussian systems?]
Let $z$ ∈ R^L be an unknown vector of values, and $y$ ∈ R^D be some noisy measurement of z. We assume these variables are related by the following joint distribution
$p(z) \sim N(z|\mu_{z}, \Sigma_{z})...
- 111
1
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1
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55
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Variance of $X + \alpha^\top Y$ where $X$ is a scalar random variable and $Y$ is a random vector [duplicate]
Consider a scalar random variable $X\in\mathbb{R}$, a vector random variable $Y\in\mathbb{R}^n$ and a constant (non-random) vector $\alpha\in\mathbb{R}^n$. I want to compute
$$
\mathbb{V}[X + \alpha^\...
- 667
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0
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115
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Evaluate relative quality of covariance matrix relative to a set
My ultimate goal is a way to evaluate a group of "m" covariance matrices (all size n*n) so I can pick an arbitrary one and calculate "this one is tighter than the average covariance ...
- 101
1
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0
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22
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No covariance results in Matrixnormal distribution?
Let's say we estimate the matrix $\beta_{N\times N}$ which is the coefficient matrix of a simple multiple linear regression $Y=X\beta+\epsilon$ where $Y, X, \epsilon\in R^{T\times N}$. If we assume:
$$...
- 703
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91
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How to call $\mathrm{Cov}(X,Y,Z)$? [duplicate]
Let
$$\mathrm{Cov}(X,Y,Z) = \mathrm E[(X-\mathrm E(X))(Y-\mathrm E(Y))(Z-\mathrm E(Z))]$$
This can be regarded as a generalization of covariance to three random variables.
I am writing a text where I ...
- 4,552
1
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0
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100
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Multivariate datasets comparison in R
I would like to compare the variance of two multivariate datasets describing the same population between them (e.g. Covariance) but also the specificity of one dataset regarding the total variance of ...
- 111
1
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1
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555
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Combining two covariance matrices -- Multiplying two multi-variate Gaussian PDFs
I want to multiply two Normal probability density functions,
$$ {\displaystyle f_{\mathbf {X} }(x_{1},\ldots ,x_{k})={\frac {\exp \left(-{\frac {1}{2}}({\mathbf {x} }-{\boldsymbol {\mu }})^{\mathrm {T}...
- 273
2
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152
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Best approximation of the Mahalanobis distance by standardized Euclidean distance
I am looking for the best way to approximate the Mahalanobis distance by the standardized Euclidean distance, which would reduce the number of the required multiplications. The easiest way is the ...
- 390
2
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0
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97
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Robust Covariance in Multivariate / Multi-response OLS
Assume we are in the OLS setting with $y = X\beta + \epsilon$. When $y$ is a response vector, and $X$ are covariates, we can get two types of covariance estimates:
The homoskedastic covariance
$cov(\...
- 1,662
1
vote
1
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661
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What is the conditional covariance matrix of $(X_2,X_3)^T$ given $X_1$?
$X=(X_1,X_2,X_3)^T\sim N_3(\mu,\Sigma).$ Suppose $X_1,...,X_{20}$ are i.i.d. observations from $X$. The sample mean vector and the covariance matrix are then defined by
$$ \bar{x} = (1,0,2)^T,\quad S=...
- 314
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0
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1k
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How do you interpret generalized variance?
Per Wiki, generalized variance is the determinant of a covariance matrix: https://en.wikipedia.org/wiki/Generalized_variance
I have heard that if the determinant is small, there is strong correlation ...
- 3,263
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1
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209
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Should allowing multiple DVs to covary in SEM influence beta coefficients?
I am running a replication study to test a model with 8 IVs and 3 DVs (all variables are continuous).
In the initial study, I had a moderate sample (≈ 200), and thus relied upon multivariate multiple ...
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1
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0
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996
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two-way MANCOVA with two covariates in SPSS
I am running a two-way MANCOVA which needs to be adjusted by two covariates. Problem is, I am not entirely sure whether I clarified all assumptions correctly and how to finally deal with two ...
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1
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6k
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Does the covariance of i.i.d. random vectors/multivariate random variables have any zero terms?
If we have i.i.d. random variables, $X$ and $Y$, then $\text{Cov}(X,Y)=0$.
But let's say we have i.i.d. random vectors $\boldsymbol{X}$ and $\boldsymbol{Y}$, where $\boldsymbol{X}=(X_{1},...,X_{p})$ ...
- 484
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0
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2k
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Dealing with non positive definite matrix covariance (possible numeric issue)
I'm generating random number of a multivariate skew normal distribution. Here is my code:
...
- 352
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2
answers
1k
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Why Multivariate Distribution with Covariance Matrix with Rank 1 Works?
So I want to sample from multivariate normal distribution and have this code where mean is 0 and I added covariance matrix with all entries to 1 implying all random variables are equally correlated.
...
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2
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7k
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Covariance between two binomial random variables [closed]
Consider a binomial random variable $X$ with parameter $p$ and another binomial random variable $Y$ with parameter $q$. What is the covariance of $X$ and $Y$?
How well does the proof generalize to $n$...
3
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1
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74
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Does $\text{cov}(a_1' X, a_2' X) = 0$ imply $a_1 \cdot a_2 = 0$?
Let $X$ be a $p$-dimensional random vector with $p$ principal components $y_1, y_2, \dots, y_p$. By definition, a restriction put on the second principal component $y_2 = a_2'X$ is
$$
\text{cov}(y_1, ...
- 1,817
2
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1
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193
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Computing conditional probabilities on multivariate data from covariances
I am struggling to implement some Bayesian algorithm which I hope you may help me with.
I am required to compute all probabilities of the form:
$$P(Z_i\le z_i\;|\;Z_1=z_1, \dots, Z_{i-1}=z_{i-1}) \;\...
- 23
2
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1
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1k
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How to show sample correlation is sample covariance for standardized values?
Given a matrix $X$ and the resulting sample correlation matrix $R$, consider the standardized observations:
$$\frac{(x_{jk} - \bar x)} {\sqrt{S_{kk}}} \quad k=1,2,...,p \quad j=1,2,...,n$$
Show that ...
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6
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1
answer
400
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Are all symmetric matrices with diagonal elements 1 and other values between -1 and 1 correlation matrices?
A question for the statisticians and other math lovers: Are all symmetric matrices with diagonal elements 1 and other values between $-1$ and 1 correlation matrices?
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3
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1
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54
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If one dimension of the data is scaled by a factor, how would it affect the probability of the Gaussian distribution?
I have fitted a maximum likelihood Gaussian distribution $N(\mu, \Sigma)$ on a multidimensional data set $X$. I wonder how would $p(X)$ change if one dimension of $X$ is scaled by a factor?
It's ...
- 16.8k
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1
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304
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Multivariate statistics sampling and standard error problem
I have a group of workers listening to audio from many various audio files and typing it out. I want to compare the average transcription accuracy of these old workers with 2 new workers.
I have a ...
-1
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1
answer
556
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Cleaning Up the Data with Mahalanobis Distance [duplicate]
I have a data set and I want to cleaning up my data set from the ouliers, so I decide to use the Mahalanobis distance to find the outliers. But I have a problem here since my covariance matrix isn't ...
2
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0
answers
204
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What is Cov(X,Y) when X,Y is a F-distribution (a,b) and (c,d) ,in case X,Y Not independent [closed]
I want to know Covariance of random variable between $X$ and $Y$ when $X$ is a F-distribution with degree of freedom $a$ and $b$ and $Y$ is a F-distribution with degree of freedom $c$ and $d$,in case $...
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1
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4k
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sequential/recursive/online calculation of sample covariance matrix
I am solving the next exercise, but I have spent a lot of time and I can´t.
For random vectors, $X_1,X_2,...\in\mathbb{R}^p$ The sample covariance with $\Sigma_1=0$ is given by:
$$\hat\Sigma_n=\frac{1}...
- 1,003
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2
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2k
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Prove that $\mathrm{Cov}(x^TAx,x^TBx) = 2 \mathrm{Tr}(A \Sigma B \Sigma) + 4 \mu^TA \Sigma B \mu$
Suppose $\vec x \sim N(\vec \mu, \Sigma)$ is multivariate normal. I want to see that
$$\mathrm{Cov}(\vec x^TA\vec x,\vec x^TB\vec x) = 2 \mathrm{Tr}(A \Sigma B \Sigma) + 4 \vec \mu^TA \Sigma B \vec \...
- 831
2
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1
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337
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Methods to prove that a guess for the covariance matrix is correct
Suppose we are interested in the covariance matrix $\Sigma$ of a few MLE estimators $\hat \theta_1,\hat \theta_2,\cdots,\hat \theta_n$. For each $j$, $\hat \theta_j$ is normally distributed and ...
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0
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26
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Statistical models for dependency of responses?
What are some ways I can model the dependency or correlation of my responses?
The data:
How many hours does the patient spend doing X?
How many hours does the patient spend doing Y?
How many ...
- 11
1
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0
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309
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Deriving the conditional distributions of a multivariate normal distribution (more than 2) [duplicate]
(This is a follow up on another question Deriving the conditional distributions of a multivariate normal distribution.)
I struggle when I condition several variables on another. My question is how to ...
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3
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3k
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Uncorrelated/orthogonal random vectors
For random variables $X,Y\in\mathbb{R}$ we say that they are orthogonal if $E(XY)=0$ and uncorrelated if $E((X-E(X))(Y-E(Y))=0$. In what follows I assume all random variables to be centered so ...
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1
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377
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solve an argmax problem of two unknowns given covariance matrices
The problem is:
For a pair of random variable $(X,Y)$, where $X \in \mathbb{R}^p$, $Y \in \mathbb{R}^q$, write $\Sigma_{XX}=var(X)$, $\Sigma_{YY}=var(Y)$, $\Sigma_{YX}=cov(Y,X)$.
Define
$(\alpha^*, \...
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3
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1
answer
384
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multivariate hypergeometric distribution
Is there a good textbook or a journal article which describes the properties of the multivariate hypergeometric distribution? I am particularly interested in the covariance structure. Wikipedia gives ...
- 266
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130
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How to deal with covariance between factors in an RDA (is it a big problem)?
I'm trying to figure out which factors are driving differences in my biotic matrix, among other things, relative abundance weighted mean community traits (similar to how Moillot et al 2013 [TREE] ...
- 151
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0
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174
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Can the determinant of the covariance of the union of sets be described as a function of the sum of the determinants of the covariance of subsets
Assume my data can be partitioned into subsets such that all my data $ w= \bigcup_{i \in N} w_i $, where $w_i$ are disjoint and so $ \bigcap_{i \in N} w_i = {\emptyset}$. How can I write the $\det\...
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0
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909
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Why are principal components of the residuals from a multivariate regression correlated with the estimated coefficients?
Say I have some data that follows a general linear model:
$$
Y = XB + E
$$
for which: $Y \in \Re^{n \times m}$, $X \in \Re^{n \times p}$ and $B \in \Re^{p \times m}$
Further, let's assume (1) that ...
- 7,752
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0
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202
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Comparing variances of vectors
For vectors we can use the covariance matrix which contains the variances per variable and the covariances.
Now I want to compare the variances of multiple point clouds.
It is hard to compare the ...
- 121
5
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1
answer
472
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Artificial Data Generation based on Data Distribution
My data looks something this (for example):
...
- 83
5
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2
answers
976
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Example of dependence with zero covariance
This is a constructivist question.
Please provide a bi-variate distribution or density/mass function of two absolutely continuous/discrete (but not mixed-type) random variables, which (may) have ...
- 60.7k
2
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1
answer
226
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Univariate Normal Converted to Multivariate Normal: Covariance Derivation
I am reading the paper available at this link:
https://drive.google.com/file/d/0B2_rKFnvrjMARnU1QjB4anR3RDA/edit?usp=sharing
I am having trouble understanding section 5.1 (page 2741).
Essentially ...
- 595
1
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0
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67
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Is the multivariate Gauss the only pdf incorporating covariances? [closed]
I am wondering whether there is another probability density function known to literature which is similar to the multivariate normal distribution in the respect that the pdf incorporates the ...
- 1,631
1
vote
2
answers
180
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If x = y*y, and you know var(y), var(z), and cov(y,z), do I know cov(x,z)?
If I know that x = y*y, and I know a whole of statistics pertaining to y, such as the variance and its covariance with other variables, can I analytically solve for the variance and covariance of x? ...
2
votes
1
answer
245
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Changing a multivariate Gaussian along a dimension
I have a multivariate Gaussian parameterised by a mean vector $\mu$ and a precision matrix $\Sigma$. Now, I want to set the Gaussian along a given dimension $i$ to a point mass i.e. I set the ...
- 4,730
6
votes
3
answers
5k
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Multivariate logistic distribution
The normal distribution can be generalized into the multivariate normal distribution.
Can the logistic distribution also be generalized into a similar multivariate distribution?
Is there a ...
- 1,631
2
votes
1
answer
182
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Expectation of correlated variables
I'm looking to compare effects $ \delta = \frac{\mu_T - \mu_C}{\sigma}$ for two studies, compared with the same control group. In order to find the covariance of effects for treatment A and treatment ...
- 33
2
votes
1
answer
270
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Would the group means of PC scores differ from the PC scores of group means?
I have $2$ $n\times p$ matrices, where $n$ are the rows (samples), and $p$ the columns (measurements). Each matrix has samples and measurements from different groups. I call these the "raw" data. I'...
- 21
2
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0
answers
129
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Independence of real and imaginary parts of the scalar product of multivariate complex normal vectors
Reading the paper "Distribution of Inner Product of Two Complex Gaussian Vectors and its Application to MPSK Performance" by Mallik R.K. I can not get one of the steps.
The author assumes two complex ...
- 191