Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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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 ...
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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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2 votes
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 ...
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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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Bounding the error of a quadratic form based on distance between two quadratic form matrices

In survey statistics, it is common to estimate sampling variance for a sum using a quadratic form. For example, I want to estimate the number of people in the U.S. who have been diagnosed with a ...
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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] ...
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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 ...
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7 votes
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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 ...
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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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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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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 \...
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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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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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175 views

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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168 views

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=\...
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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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1 answer
355 views

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 $D=1$...
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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 ...
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1 answer
132 views

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
274 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
135 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
1 vote
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78 views

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
317 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
58 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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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
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238 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
1k views

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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73 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 $T$ (...
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6 votes
1 answer
894 views

Confidence interval from summary function

Here is a summary data from a texbook ...
Em Ae's user avatar
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0 answers
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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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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
338 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
408 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
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1 vote
1 answer
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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
3k 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
376 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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2 votes
2 answers
704 views

Concavity of SVM dual formulation

These notes have derived the following dual formulation of the SVM optimisation problem using KKT conditions that I have followed It then states that the objective function is quadratic and concave (...
samlu1999's user avatar
1 vote
1 answer
530 views

R: quadratic term in a beta regression

My hypothesis is that along a gradient of habitat structural complexity 'a', the fish diversity 'y' increases until an optimum level but decreases after that (which kinda I have noticed in the graphic ...
Barbara Quirino's user avatar
3 votes
1 answer
3k views

Distributions of Quadratic form of a normal random variable

I am looking for ways to prove that the moment generating function of $X'AX$ given that $X \sim N(\vec{\mu}, \vec{\Sigma})$ and $A$ is symmetric is defined as: $$M_{X'AX}(\vec{t})= \frac{1}{|I-2tA\...
Xorion 1997's user avatar
0 votes
1 answer
617 views

Panel data OLS regression: plot and quadratic fitting line

I am conducting an OLS regression panel data analysis with package PLM in R. I use the following script to obtain a plot and fitting line of variables D and ...
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