A quadratic form is a homogeneous polynomial of order two.

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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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### Covariance of Quadratic Form Proof [duplicate]

I've been able to find result that $Cov(Y^TAY, Y^TBY) = 2 tr(A \Sigma B \Sigma) + 2 \mu^T A \Sigma B \mu$ for symmetric matrices $A$ and $B$ with $Y \sim \mathcal{N}(\mu, \Sigma)$ multiple places ...
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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 ...
1 vote
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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 ...
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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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### 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 ...
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### Confidence bounds for coefficients of a fit of data set obtained with another fit

I fitted an equation to a set of data points. Then I substracted the fit previously obtained to another set of data points. After that, I fitted another equation to this new data (result of the ...
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 ...
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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 ...
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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?
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### how do I enforce a matrix is invertible when optimizing for it with MOSEK?

I am using MOSEK to optimize a quadratic objective with linear constraints (QP). One of the variables is a whole matrix. The matrix becomes non-invertible sometimes, which is not good for my needs. I ...
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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 ...
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### 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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### Confidence interval from summary function

Here is a summary data from a texbook ...
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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 ...
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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 ...
1 vote
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### 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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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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### 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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### 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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### 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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### 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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