Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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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=\...
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2 votes
1 answer
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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 ...
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1 vote
1 answer
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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 $D=1$...
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0 answers
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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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  • 11
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 $\...
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1 vote
0 answers
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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 + \...
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whether $T_3$ has a chisquare distribution subject to a multiplicative constant

Suppose $X_1,...,X_n$ are random samples from $N(0, \sigma^2)$, and $\bar X_n = n^{-1}\sum_{i=1}^{n}X_i$. Let $Y =$ $Y_1 \choose {Y_2}$, where $Y_1 = X_1 - \bar X_n$, $Y_2 = X_2 - \bar X_n$, be a ...
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2 votes
1 answer
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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^...
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  • 131
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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^{\...
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5 votes
1 answer
170 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{...
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0 votes
0 answers
23 views

limit behaviour of a quadratic form

I'm reading a book on portfolio optimization and risk management, and I wanna clarify what the author wants to say. Let $\mathbf{X}=[X_1,...,X_n]$ be a random vector with mean $\mathbf{\mu}=E\{\mathbf{...
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  • 337
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1 answer
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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 ...
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0 answers
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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 $\boldsymbol{\epsilon} \sim N(0,\sigma^2 \mathbf{I}_N)$. ...
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1 vote
0 answers
132 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'[...
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1 vote
1 answer
657 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 ...
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27 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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7 votes
1 answer
118 views

Confidence interval from summary function

Here is a summary data from a texbook ...
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0 answers
30 views

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 ...
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  • 516
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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 ...
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1 vote
0 answers
103 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 ...
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2 votes
0 answers
178 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 ...
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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. ...
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0 answers
2k 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 ...
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  • 21
3 votes
2 answers
177 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 ...
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  • 635
2 votes
2 answers
279 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 (...
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1 vote
1 answer
270 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 ...
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3 votes
1 answer
1k 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\...
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0 votes
1 answer
179 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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  • 11
1 vote
2 answers
374 views

Assessing model fit of two models by computing t-test on coefficients in R

I am attempting to assess whether my explanatory variable of "startingpos" is best modelled via a quadratic regression model or a linear regression model. One way I can do this is to compute the ...
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3 votes
1 answer
695 views

Fitting a quadratic regression in R

I am trying to fit a quadratic regression model in R. Here is an example of my dataframe: ...
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3 votes
1 answer
205 views

Including a quadratic effect for an ordinal variable in a regression analysis

It's common for many datasets to have ordinal versions of numerical variables, such as age groups (e.g. "Under 20", "20-30", "30-40", etc.) or time groups (e.g. "Less than 15 minutes", "15-30 minutes",...
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  • 303
6 votes
1 answer
175 views

Variance Ratio Formula

I've been trying to minimize/maximize the ratio of quadratic forms given by $$Q(c)=\frac{c^\top \Sigma c}{c^\top \text{diag}(\Sigma) c}$$ where $\Sigma$ denotes a covariance matrix of some $n$-...
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1 vote
0 answers
172 views

Variance of Quadratic form

From the Wikipedia In general, the variance of a quadratic form depends greatly on the distribution of ${\displaystyle \varepsilon }$ . However, if ${\displaystyle \varepsilon }$ does follow a ...
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2 votes
1 answer
236 views

Expectation of Quadratic Form with Two Random Vectors

Assume I have two independent $(N \times 1)$ random vectors, $\epsilon_{1} \sim N(0,\Sigma_1)$ and $\epsilon_{2} \sim N(0,\Sigma_2)$. We could assume $\Sigma_1=\Sigma_2$ for my purposes but a general ...
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0 votes
0 answers
46 views

Self-Study: If $x \sim \mbox N_p(\mu; V)$ and A is a matrix then how to show that $E[(x-\mu)(x-\mu)'A(x-\mu) = 0$?

If we assume the random vector $x$ to be normally distributed with $N_p(\mu; V)$ then $E[(x-\mu)(x-\mu)'(x-\mu)] = 0_p$. If I am not mistaken, this can be shown using the moment generating function of ...
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  • 469
3 votes
1 answer
146 views

Determine statistical difference of slopes of quadratic relationship in a Poisson regression

I'm looking for a statistical or mathematical way to test the difference between two slopes. Others have asked related questions but my problem is quite particular. I'm running a Poisson regression ...
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  • 485
3 votes
2 answers
112 views

Maximization of quotient of quadratic forms in linear regression

I would like to find maximum of the following function: $$I = \max_{a\in \mathbb{R}^p} \frac{(a'\hat{\beta})^2}{S^2a'(X'X)^{-1}a},$$ where $X$ is a design matrix and of course $Y$ is normally ...
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  • 512
2 votes
1 answer
60 views

How to generate data such that an equation needs to hold?

Can I create or generate $\{y_i\}_{i=1}^{4}$ data set such that this equation holds $$ \sum_{i=1}^{4}\sum_{j=1}^{4}m_{ij}y_{i}y_{j}=6 $$ where $$ m=\left[ \begin{array}{cccc} 13 & 12 & 3 &...
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  • 323
1 vote
1 answer
142 views

What is the relationship between quadraric and categorical logistic regression models?

Consider two logistic regression models Y on x, one where x appears in the model as a categorical variable, and one where x appears in the predictor as both a linear term and as a quadratic term. ...
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1 vote
1 answer
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What is the relationship between a quadratic model and categorical model?

Using logistic explore the association between lung reactivity and risk of chronic respiratory disease. The dataset contains information on a combined measure of lung function exposure respcat ...
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  • 67
0 votes
0 answers
26 views

Proof of distribution of $Y^{T}AY$

I am reading a proof that given $Y \sim N(\mu, \Sigma)$, where $\Sigma$ is positive definite, $Y^{T}AY \sim \chi^{2}_{p}(\mu^{T}A\mu)$ iff $A \Sigma A =A$ and $A$ has rank $p$. One of the steps in ...
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  • 101
2 votes
1 answer
1k views

Variance of quadratic form for multivariate normal distribution

This is a homework problem I’m trying to solve but I can’t seem to solve Q1b without using the theorem. I am also given the fact that $$E(y’Ay)=tr(A\Sigma)+\mu’A\mu$$ I’ve tried using the trace-...
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0 votes
1 answer
64 views

Regression: Why does using quadratic expressions work with linear estimators? [duplicate]

My questions is, that I see people using R´s lm() (linear regression model) with Y ~ X^2 e.g. here: Simple non-linear regression problem But I dont see how and ...
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  • 13
0 votes
0 answers
30 views

Modeling approach to showing a parabolic effect is greater than that expected by scale boundaries

I have conducted an experiment where raters rate different nations on a DV. Each observation is a different nation. I calculated a mean and SD of the DV for each nation. The data we are working with ...
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2 votes
0 answers
188 views

Asymptotic normality of quadratic form?

Let $X$ be a $p$-dimensional vector that is asymptotically normal such that $$\sqrt{n}(X - \mu_X) \stackrel{d}\longrightarrow N(0, \Sigma)$$, and let $H$ be a random $p\times p$ symmetric matrix, ...
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  • 21
5 votes
1 answer
4k views

How to include an interaction with a quadratic term? [closed]

I want to predict $y$ with $x_{1}$ and $x_{2}$ and I suppose that $x_{2}$ has a quadratic effect on $y$ and that there is an interaction. How to model that? I've look in previous questions but there ...
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2 votes
1 answer
387 views

Quadratic term of standardized predictor in logistic regression

A random intercept logistic regression is performed to assess the association between $Y$: Disease (Yes/No) and Standardized Predictor($X_1$) adjusting for control variables ($X_2$, $X_3$) based on ...
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  • 603
4 votes
1 answer
493 views

How powerful are second order interactions?

A lot of applications in statistics and machine learning model a phenomenon by second order interactions of variables and get good results. By second order interactions I mean, for a general variable $...
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1 vote
1 answer
129 views

Why do we use quadratic form for random vectors? [closed]

I am studying linear regression. I have studied this in the past, but this is my first time exposing myself to the matrix form of multiple linear regression. My matrix algebra/linear algebra skills ...
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  • 93
2 votes
1 answer
877 views

Quadratic polynomial - how to test correlation between x and y?

I have one dependent variable (fish abundance) and one independet variable (time), both continuous. I would like to test the correlation between them because I expect that the abudance changes over ...
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