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Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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Distribution of the correlation coefficient based on quadratic forms

Let $x,y$ be two independent random correlated vectors following the same multivariate (real or complex) centred normal distribution, and let $A$ be a non-negative linear operator. We can read here, ...
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Specifying interactions, quadratics and ratios in regression model

I would like to model a linear regression with the dependent continuous variable y and the independent continuous variables x and z: The interaction of x and z is expected to increase y, but y is ...
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Cut-off based on an ordinal variable in unbalanced panel data

I am currently looking for an appropriate statistical analysis for my research questions. I have a continuous variable (score) and an ordinal variable (test). Score is quadratically related to Test, i....
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Propensity matching affects significance of polynomial degrees differently

I have a regression as follows: $$ y = \alpha + \mu L + \beta_1 x + \beta_2 x^2 + \varepsilon $$ where L is a dummy, and x is a control variable. Both $x$ and $x^2$ are significant when I run the ...
Babak Fi Foo's user avatar
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Distribution of norm of fixed vector projected onto a Gaussian subspace

Let $\Sigma \in \mathbb{R}^{m \times m}$, $\Theta_0 \in \mathbb{R}^{k \times m}$, $v = \Theta_0 \beta \in \mathbb{R}^k$ with $\| v \| = 1$ and $\Theta \sim \mathcal{N}(\Theta_0, \mathrm{Id}_k \otimes \...
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Ratio of cubic and quadratic form in random variables is approximately normal?

Let be $x_{1},x_{2},x_{3}$ i.i.d. random variables following a normal distribution with $\mu=0$ and $\sigma=1$. I'm intrigued by the following random variable, which is a ratio of a cubic form and a ...
rgvalenciaalbornoz's user avatar
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Moments of sum of squares of independent gaussians $X_i \sim \mathcal{N}(\mu_i,\sigma^2_i)$, or $||X||^2$

Say that we have $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. Is there some formula to calculate analytically the expected value of the sum $S = \sum_i^n X_i^2$?. This is equivalent to computing $\...
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Formulas, approximations, or bounds for $\mathbb{E}\left( \frac{X}{\lVert X \rVert} \right)$, $X\sim N(\mu, \Sigma)$?

In another question, I asked for $\mathbb{E}\left( \frac{X}{\lVert X \rVert} \right)$, in the case where $X \in \mathbb{R}^d \sim N(\mu, I_{d})$. Somebody posted an exact formula based on the symmetry ...
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Formulas or approximations for $\mathbb{E}\left( \frac{X}{\|X\|} \right)$, $X\sim N(\mu, Id)$?

This is a cross-posting of this math SE question. I want to compute or approximate the following expected value with some analytic expression: $\mathbb{E}\left( \frac{X}{||X||} \right)$ , where $X \in ...
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Interpreting linear and quadratic terms with same sign

I am running a second order model with betweenness centrality and closeness centrality as independent variables and cognitive demand as dependent variable. The results shows that betweenness ...
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1 answer
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Covariance of quadratic form and a random vector of type $\mathbf{G}\,\mathbf{y}$

Assume that the $p \times 1$ vector $\mathbf{y}$ has multivariate Normal distribution with $\mathbb{E}[\mathbf{y}] = \boldsymbol{\mu}$ and $\mathrm{V}[\mathbf{y}] = \boldsymbol{\Sigma}$. Let $\mathbf{...
andre's user avatar
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When is an interaction in quadratic regression significant?

I run a regression with a continuous variable $x_2$ and a binary grouping variable $x_1$. We suppose that the effect of $x_2$ is quadratic on the dependent variable $y$ and our question is whether the ...
LulY's user avatar
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How to interpret interactions involving quadratic terms in a GLM?

I'm trying to interpret the output of a GLM from a tutorial that models species abundances as a function of environmental data and species traits. The function ...
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4 votes
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Expectation of two Quadratic form

Assume $\mathbf{h} \in C^{N \times 1}$ is a Gaussian vector with zero mean and Covariance matrix $\mathbf{R}$. Also $\mathbf{A} \in C^{N \times N}$ is a deterministic diagonal matrix. In this case, ...
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Is there a high probability bound of quadratic forms?

I am wondering about the following: For a symmetric matrix $A \in \mathbb{R}^{n \times n}$ and vector $x \in [-1,1]^n$, if $X$ is a random vector in $\mathbb{R}^n$ such that w.h.p. $X_i \not\in [-1,1] ...
swuk's user avatar
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3 votes
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Expected value of Rayleigh quotient, non-centered Gaussian vector

Let $X \sim \mathcal{N}\left(\mu, \Sigma \right)$, and let $A$ be a symmetric matrix. My understanding is that the Rayleigh quotient of vector $X$ is given by: $$R=\frac{X^T A X}{X^T X}$$ I've been ...
dherrera's user avatar
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7 votes
1 answer
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Expected value of the outer product of normalized, non-centered Gaussian vector

I have a multidimensional random variable $X \sim \mathcal{N} \left(\mu, I_d \right)$. Ideally, I would like to know the expected value of the normalized outer product of the latent variable with ...
dherrera's user avatar
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2 votes
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How to interpret quadratic-by-quadratic interaction (X^2*W^2)?

I want to know how to interpret the quadratic-by-quadratic interaction (e.g., X^2*W^2). I have looked for textbooks or scholarly articles on this issue, but all I found is “Statistics methods and ...
user376288's user avatar
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2 answers
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Reconciling OLS as a linear regression model and polynomial regressors [duplicate]

A bit of a naive question. I understand that OLS is used for a linear regression model (for example, Wikipedia page for OLS: OLS is a type of linear least squares method for choosing the unknown ...
Ploit88's user avatar
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2 votes
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What is $E\left[\frac{X^2}{X^2+Y^2}\right]$ if $X$ and $Y$ are normally distributed but not iid.?

I assume that X and Y are normally distributed with individual mean and variance. So far, I have found that an analytic expression exists for $E[X^2+Y^2]$, $E[X^2*Y^2]$ and $E[X^2*(X^2+Y^2)]$, all ...
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How to sample from a distribution with log likelihood a quadratic form

In the context of the article BiRank: Towards Ranking on Bipartite Graphs specially the Bayesian interpretation section 5.2. I want to make inference on the vector non-negative vector $\mathbf{u}\in \...
Girigio's user avatar
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Is there any intuitive meaning to $\beta^T A \beta=1$?

In a statistical paper, I found that $\beta^T A \beta=1$, where $\beta$ and $A$ are a vector and matrix of constants, respectively. In the paper, the author utilizes only the $\beta$ that satisfies $\...
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Deriving ADF unit root test form for the time series with quadratic deterministic trend

I have the following time series process $y_t $ $$\Delta y_t = \delta + \gamma t + \epsilon_t$$ where $e_t$ is white noise process with the variance of $\sigma^2$. I guess that whereas $\Delta y_t$ is ...
1190's user avatar
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1 vote
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How do I fit a constrained logistic regression model via quadratic programming in R?

I trying to find $\pi_{1}, \pi_{2}, \pi_{3}$ for model: $$ Y = \pi_{1}X_{1} + \pi_{2}X_{2} + \pi_{3}X_{3} + \epsilon, $$ with constraints: $\Sigma_{k}\pi_{k}=1$ and $\pi_{k}\geq0$. (All $\pi$ are ...
user366142's user avatar
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Derivation of the vertex value of quadratic term in GLM in R

I have a quadratic term in a GLM and I am interested in the vertex value (+ the standard error and confidence interval of the vertex) of the quadratic term. To my knowledge, there is no automatic ...
Kris's user avatar
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Both quadratic and linear term are insignificant

I am trying to fit a model which has age as a control variable and mental health score as my dependent variable. I centered the age because 0 is not meaningful in my analysis. I tried age, age^2, log(...
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Quadratic Approximation for Log-Likelihood Ratio Processes, Why and How

I'm trying to understand why the quadratic equation can approximate the log likelihood ratio. Is this approximated using Taylor's series or normal distribution equation or anything else?
Ela's user avatar
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Proving $\sum_{i=1}^n(X_i-\overline X_n)^2-\sum_{i=1}^m(X_i-\overline X_m)^2 \sim \chi^2_{n-m}$

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d $N(0,1)$ random variables. For $2\le m<n$, let $S_m^2=\sum_{i=1}^m(X_i-\overline X_m)^2$ and $S_n^2=\sum_{i=1}^n(X_i-\overline X_n)^2$ where $\overline X_m=\...
StubbornAtom's user avatar
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How do I calculate the variance of a Hermitian form?

Suppose $\mathbf{x}\sim\mathcal{CN}\left(\mathbf{0},\mathbf{I}_n\right)$ is a circular complex Gaussian random vector, and $\mathbf{Q}$ is a Hermitian matrix. How do I calculate the variance of the ...
Raymond's user avatar
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Interaction between quadratic term and dummy variable

Suppose I have a linear regression: $Y=\beta_1+\beta_2X+\beta_3X^2+\beta_4D$ where $D$ is a dummy variable that takes value 0 and 1. If I want to examine if the effect of $X$ on $Y$ for $D=0$ and $D=1$...
newt335's user avatar
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State space model equation

I would appreciate your help on the following I have a quadratic equation and need to write it in a state space format according to a model below. My equation is the following below, where T is the ...
HelenA's user avatar
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2 votes
1 answer
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Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution

According to Golam Kibria & Joarder (2006, p.7) available here and Kotz & Nadarajah (2004, p. 19) visible in google, the distribution of $X'\Sigma^{-1}X /p$, for a known correlation matrix $\...
Denis Cousineau's user avatar
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Partial derivative of a Group Lasso

I am looking at the gradient descent method for group lasso questions. Here's what I am currently stuck at. Given the quadratic form of the objective function: $$ f(x) = \frac{1}{2} x^T V x - m^T x + \...
Kevin Choon Liang Yew's user avatar
2 votes
1 answer
337 views

Expectation of the product of two independent random vectors and a positive-definite matrix

I am trying to compute the following: $\mathbb{E}[X^T\Omega^{-1}\epsilon]$, where $X$ is a random matrix, $\epsilon$ is a random vector, $\Omega$ is a real positive-definite matrix, and $\mathbb{E}[X^...
Charles's user avatar
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5 votes
1 answer
143 views

For a general multivariate normally distributed $\boldsymbol{X}$, what is the expectation of $1/(\boldsymbol{X}^T \boldsymbol{X})$

For $\boldsymbol{X} \sim \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\Sigma})$, where $\boldsymbol{\mu} \in \mathbb{R}^N$, $\boldsymbol{\Sigma} \in \mathbb{R}^{N \times N}$ is positive definite, how to ...
Zifeng Zhang's user avatar
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Variance and Covariance of Fixed Effects expressed in Quadratic Form

I know that for a simple vector $x$ of length $n$, the variance of this vector $\sigma_x^2= \frac{1}{n}\sum_{i=1}^{n} x_{i}^{2}-\left(\frac{1}{n}\sum_{i=1}^{n} x_{i}\right)^{2}$ can be written as $x^{\...
Alalalalaki's user avatar
5 votes
1 answer
363 views

Expectation of double quadratic form

I want to compute the following expectation $E(\hat{Y_k}'A\hat{Y_l}\hat{Y_k}'A\hat{Y_l})$ where $A$ is a symmetric non-random matrix and $E(\hat{Y_k}) = Y_k$, $E(\hat{Y_l}) = Y_l$. Additionally, $\hat{...
Schneeflocke's user avatar
1 vote
1 answer
66 views

Manipulation of quadratic form

How is this derivation obtained? It seems like it is possible to do so from first principles for using the series definition for quadratic form, but that seems tedious. Is there a faster way to ...
shem's user avatar
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1 vote
0 answers
62 views

Distribution of quadratic form?

Let $\mathbf{y = X \boldsymbol{\beta} + \epsilon}$, where $\mathbf{X} \in \mathcal{R}^{N\times p}$, $\boldsymbol\beta \in \mathcal{R}^p$ and $\boldsymbol{\epsilon} \sim N(0,\sigma^2 \mathbf{I}_N)$. ...
user61062's user avatar
1 vote
0 answers
255 views

Multivariate Normal Quadratic MGF: Eigendecomposition to Matrix form

If $X \sim \mathcal{N}(\mu, \Sigma)$ is a multivariate normal, then the quadratic $X^TAX$ has moment generating function $$M_{X^TAX}(t)= \frac{1}{\sqrt{\det(I - 2tA\Sigma)}}\exp\left(-\frac{1}{2}\mu'[...
ItsAllPurple's user avatar
1 vote
1 answer
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SVM loss function

I am going through Bishop's book and especially SVM. I am trying to understand the logic behind minimizing the specific loss $argmax_{\mathbf{w}} \frac{1}{2}||\mathbf{w}||^{2}$. On page 327, in 7.3 we ...
Jose Ramon's user avatar
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77 views

Combining two terms into a quadratic form

I have an objective function defined by $ min_{Y_{t}} \hspace{2mm} ||X_{t} - Y_{t}D_{t}^{T}||_{F}^{2} + \lambda_{2}\sum_{i,j} w_{i,j}||\mathbf{y}_{i} - \mathbf{p}_{j}^{t}||_{2}^{2}$ where capital $T$ (...
Upendra01's user avatar
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6 votes
1 answer
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Confidence interval from summary function

Here is a summary data from a texbook ...
Em Ae's user avatar
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covariance of squared projections

Given a vector $x$ of independent mean-zero random variables, and two nonrandom orthogonal unit vectors $u,v$, does $u'v=0$ imply $cov(x'uu'x,x'vv'x)=0$? If so, what is the proof? If not, what happens ...
Hasse1987's user avatar
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1 answer
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Ridge and Quadratic Programming for Portfolio Norm Optimization

Much like this post: Quadratic Programming and Lasso, I'm trying to integrate RIDGE Penalty in a dedicated quadratic solver. In my case, I am working with quadprog from MATLAB. Unlike LASSO where you ...
Samuel Normandeau's user avatar
1 vote
0 answers
376 views

Interpreting moderated quadratic effect in mixed effect models

I study the effect of the same stimulus ($X$), displaced in 4 different conditions (categorical $M$, categories=High, Medium, Low, in addition there is "Control" condition, as a reference ...
user6606453's user avatar
3 votes
0 answers
435 views

Interpretation: Adding quadratic term makes linear term insignificant (OLS regression)

I'm conducting a multiple OLS regression. My main model contains a significant effect (p < .5) of x on y. I want to test in a robustness check whether x is related to y in a curvilinear/quadratic ...
user18075's user avatar
  • 647
1 vote
1 answer
2k views

How to plot quadratic model? [closed]

I have fit a polynomial glm in R with x and x^2 as the predictor of interest. ...
SanMelkote's user avatar
1 vote
0 answers
4k views

How to measure correlation in polynomial regression?

I have two variables that have a quadratic relationship. I can fit such an equation and get the R-squared, but how can I measure the degree to which the two variables are associated? Does a ...
Xe M's user avatar
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3 votes
2 answers
396 views

Prove independence of quadratic and linear form

Suppose $(X,Y)$ is bi-variate normal distribution with correlation $\rho$ and mean $(0,0)^T$ and both variances 1, and $(X_1,Y_1), (X_2,Y_2),...,(X_n,Y_n)$ is a i.i.d sample from the bi-variate normal ...
Statisfun's user avatar
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