# Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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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
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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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1 vote
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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 ...
• 11
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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 $\... 1 vote 0 answers 43 views ### 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: $$f(x) = \frac{1}{2} x^T V x - m^T x + \... 0 votes 0 answers 21 views ### whether T_3 has a chisquare distribution subject to a multiplicative constant Suppose X_1,...,X_n are random samples from N(0, \sigma^2), and \bar X_n = n^{-1}\sum_{i=1}^{n}X_i. Let Y = Y_1 \choose {Y_2}, where Y_1 = X_1 - \bar X_n, Y_2 = X_2 - \bar X_n, be a ... • 21 2 votes 1 answer 64 views ### Expectation of the product of two independent random vectors and a positive-definite matrix I am trying to compute the following: \mathbb{E}[X^T\Omega^{-1}\epsilon], where X is a random matrix, \epsilon is a random vector, \Omega is a real positive-definite matrix, and \mathbb{E}[X^... • 141 1 vote 0 answers 36 views ### Variance and Covariance of Fixed Effects expressed in Quadratic Form I know that for a simple vector x of length n, the variance of this vector \sigma_x^2= \frac{1}{n}\sum_{i=1}^{n} x_{i}^{2}-\left(\frac{1}{n}\sum_{i=1}^{n} x_{i}\right)^{2} can be written as x^{\... 5 votes 1 answer 185 views ### Expectation of double quadratic form I want to compute the following expectation E(\hat{Y_k}'A\hat{Y_l}\hat{Y_k}'A\hat{Y_l}) where A is a symmetric non-random matrix and E(\hat{Y_k}) = Y_k, E(\hat{Y_l}) = Y_l. Additionally, \hat{... 1 vote 1 answer 42 views ### Manipulation of quadratic form How is this derivation obtained? It seems like it is possible to do so from first principles for using the series definition for quadratic form, but that seems tedious. Is there a faster way to ... • 134 1 vote 0 answers 40 views ### Distribution of quadratic form? Let \mathbf{y = X \boldsymbol{\beta} + \epsilon}, where \mathbf{X} \in \mathcal{R}^{N\times p}, \boldsymbol\beta \in \mathcal{R}^p and \boldsymbol{\epsilon} \sim N(0,\sigma^2 \mathbf{I}_N). ... 1 vote 0 answers 144 views ### Multivariate Normal Quadratic MGF: Eigendecomposition to Matrix form If X \sim \mathcal{N}(\mu, \Sigma) is a multivariate normal, then the quadratic X^TAX has moment generating function$$M_{X^TAX}(t)= \frac{1}{\sqrt{\det(I - 2tA\Sigma)}}\exp\left(-\frac{1}{2}\mu'[... 1 vote 1 answer 783 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 ... 0 votes 0 answers 36 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$(... • 1,596 6 votes 1 answer 241 views ### Confidence interval from summary function Here is a summary data from a texbook ... • 361 0 votes 0 answers 31 views ### covariance of squared projections Given a vector$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$? If so, what is the proof? If not, what happens ... • 536 0 votes 1 answer 741 views ### Ridge and Quadratic Programming for Portfolio Norm Optimization Much like this post: Quadratic Programming and Lasso, I'm trying to integrate RIDGE Penalty in a dedicated quadratic solver. In my case, I am working with quadprog from MATLAB. Unlike LASSO where you ... 1 vote 0 answers 147 views ### Interpreting moderated quadratic effect in mixed effect models I study the effect of the same stimulus ($X$), displaced in 4 different conditions (categorical$M$, categories=High, Medium, Low, in addition there is "Control" condition, as a reference ... 3 votes 0 answers 246 views ### Interpretation: Adding quadratic term makes linear term insignificant (OLS regression) I'm conducting a multiple OLS regression. My main model contains a significant effect (p < .5) of x on y. I want to test in a robustness check whether x is related to y in a curvilinear/quadratic ... • 637 1 vote 1 answer 2k views ### How to plot quadratic model? [closed] I have fit a polynomial glm in R with x and x^2 as the predictor of interest. ... • 631 0 votes 0 answers 2k views ### How to measure correlation in polynomial regression? I have two variables that have a quadratic relationship. I can fit such an equation and get the R-squared, but how can I measure the degree to which the two variables are associated? Does a ... • 21 3 votes 2 answers 212 views ### Prove independence of quadratic and linear form Suppose$(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)$is a i.i.d sample from the bi-variate normal ... • 615 2 votes 2 answers 339 views ### Concavity of SVM dual formulation These notes have derived the following dual formulation of the SVM optimisation problem using KKT conditions that I have followed It then states that the objective function is quadratic and concave (... 1 vote 1 answer 326 views ### R: quadratic term in a beta regression My hypothesis is that along a gradient of habitat structural complexity 'a', the fish diversity 'y' increases until an optimum level but decreases after that (which kinda I have noticed in the graphic ... 3 votes 1 answer 1k views ### Distributions of Quadratic form of a normal random variable I am looking for ways to prove that the moment generating function of$X'AX$given that$X \sim N(\vec{\mu}, \vec{\Sigma})$and$A$is symmetric is defined as: $$M_{X'AX}(\vec{t})= \frac{1}{|I-2tA\... 0 votes 1 answer 254 views ### Panel data OLS regression: plot and quadratic fitting line I am conducting an OLS regression panel data analysis with package PLM in R. I use the following script to obtain a plot and fitting line of variables D and ... • 11 1 vote 2 answers 484 views ### Assessing model fit of two models by computing t-test on coefficients in R I am attempting to assess whether my explanatory variable of "startingpos" is best modelled via a quadratic regression model or a linear regression model. One way I can do this is to compute the ... 3 votes 1 answer 780 views ### Fitting a quadratic regression in R I am trying to fit a quadratic regression model in R. Here is an example of my dataframe: ... 3 votes 1 answer 258 views ### Including a quadratic effect for an ordinal variable in a regression analysis It's common for many datasets to have ordinal versions of numerical variables, such as age groups (e.g. "Under 20", "20-30", "30-40", etc.) or time groups (e.g. "Less than 15 minutes", "15-30 minutes",... • 315 6 votes 1 answer 187 views ### Variance Ratio Formula I've been trying to minimize/maximize the ratio of quadratic forms given by$$Q(c)=\frac{c^\top \Sigma c}{c^\top \text{diag}(\Sigma) c}$$where \Sigma denotes a covariance matrix of some n-... • 1,217 1 vote 0 answers 192 views ### Variance of Quadratic form From the Wikipedia In general, the variance of a quadratic form depends greatly on the distribution of \varepsilon } . However, if \varepsilon } does follow a ... • 1,629 3 votes 1 answer 284 views ### Expectation of Quadratic Form with Two Random Vectors Assume I have two independent (N \times 1) random vectors, \epsilon_{1} \sim N(0,\Sigma_1) and \epsilon_{2} \sim N(0,\Sigma_2). We could assume \Sigma_1=\Sigma_2 for my purposes but a general ... 0 votes 0 answers 47 views ### Self-Study: If x \sim \mbox N_p(\mu; V) and A is a matrix then how to show that E[(x-\mu)(x-\mu)'A(x-\mu) = 0? If we assume the random vector x to be normally distributed with N_p(\mu; V) then E[(x-\mu)(x-\mu)'(x-\mu)] = 0_p. If I am not mistaken, this can be shown using the moment generating function of ... • 491 3 votes 1 answer 181 views ### Determine statistical difference of slopes of quadratic relationship in a Poisson regression I'm looking for a statistical or mathematical way to test the difference between two slopes. Others have asked related questions but my problem is quite particular. I'm running a Poisson regression ... • 505 3 votes 2 answers 126 views ### Maximization of quotient of quadratic forms in linear regression I would like to find maximum of the following function:$$I = \max_{a\in \mathbb{R}^p} \frac{(a'\hat{\beta})^2}{S^2a'(X'X)^{-1}a},$$where X is a design matrix and of course Y is normally ... • 542 2 votes 1 answer 73 views ### How to generate data such that an equation needs to hold? Can I create or generate \{y_i\}_{i=1}^{4} data set such that this equation holds$$ \sum_{i=1}^{4}\sum_{j=1}^{4}m_{ij}y_{i}y_{j}=6 $$where$$ m=\left[ \begin{array}{cccc} 13 & 12 & 3 &... • 333 1 vote 1 answer 148 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. ... • 67 1 vote 1 answer 72 views ### 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 ... • 67 0 votes 0 answers 27 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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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-...