Questions tagged [quadratic-form]
A quadratic form is a homogeneous polynomial of order two.
178
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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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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 ...
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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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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 ...
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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 ...
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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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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 ...
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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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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
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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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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:
$$
f(x) = \frac{1}{2} x^T V x - m^T x + \...
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1
answer
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Expectation of the product of two independent random vectors and a positive-definite matrix
I am trying to compute the following: $\mathbb{E}[X^T\Omega^{-1}\epsilon]$, where $X$ is a random matrix, $\epsilon$ is a random vector, $\Omega$ is a real positive-definite matrix, and $\mathbb{E}[X^...
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Variance and Covariance of Fixed Effects expressed in Quadratic Form
I know that for a simple vector $x$ of length $n$, the variance of this vector $\sigma_x^2= \frac{1}{n}\sum_{i=1}^{n} x_{i}^{2}-\left(\frac{1}{n}\sum_{i=1}^{n} x_{i}\right)^{2}$ can be written as $x^{\...
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Expectation of double quadratic form
I want to compute the following expectation $E(\hat{Y_k}'A\hat{Y_l}\hat{Y_k}'A\hat{Y_l})$ where $A$ is a symmetric non-random matrix and $E(\hat{Y_k}) = Y_k$, $E(\hat{Y_l}) = Y_l$. Additionally, $\hat{...
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Manipulation of quadratic form
How is this derivation obtained? It seems like it is possible to do so from first principles for using the series definition for quadratic form, but that seems tedious. Is there a faster way to ...
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Distribution of quadratic form?
Let $\mathbf{y = X \boldsymbol{\beta} + \epsilon}$, where $\mathbf{X} \in \mathcal{R}^{N\times p}$, $\boldsymbol\beta \in \mathcal{R}^p$ and $\boldsymbol{\epsilon} \sim N(0,\sigma^2 \mathbf{I}_N)$. ...
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Multivariate Normal Quadratic MGF: Eigendecomposition to Matrix form
If $X \sim \mathcal{N}(\mu, \Sigma)$ is a multivariate normal, then the quadratic $X^TAX$ has moment generating function
$$M_{X^TAX}(t)= \frac{1}{\sqrt{\det(I - 2tA\Sigma)}}\exp\left(-\frac{1}{2}\mu'[...
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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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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 ...
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0
answers
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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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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
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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
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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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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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1
answer
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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:
$$M_{X'AX}(\vec{t})= \frac{1}{|I-2tA\...
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Panel data OLS regression: plot and quadratic fitting line
I am conducting an OLS regression panel data analysis with package PLM in R.
I use the following script to obtain a plot and fitting line of variables D and ...
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2
answers
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Assessing model fit of two models by computing t-test on coefficients in R
I am attempting to assess whether my explanatory variable of "startingpos" is best modelled via a quadratic regression model or a linear regression model. One way I can do this is to compute the ...
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Fitting a quadratic regression in R
I am trying to fit a quadratic regression model in R. Here is an example of my dataframe:
...
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1
answer
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Including a quadratic effect for an ordinal variable in a regression analysis
It's common for many datasets to have ordinal versions of numerical variables, such as age groups (e.g. "Under 20", "20-30", "30-40", etc.) or time groups (e.g. "Less than 15 minutes", "15-30 minutes",...
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Variance Ratio Formula
I've been trying to minimize/maximize the ratio of quadratic forms given by
$$Q(c)=\frac{c^\top \Sigma c}{c^\top \text{diag}(\Sigma) c}$$
where $\Sigma$ denotes a covariance matrix of some $n$-...
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Variance of Quadratic form
From the Wikipedia
In general, the variance of a quadratic form depends greatly on the
distribution of ${\displaystyle \varepsilon }$ . However, if
${\displaystyle \varepsilon }$ does follow a ...
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1
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Expectation of Quadratic Form with Two Random Vectors
Assume I have two independent $(N \times 1)$ random vectors, $\epsilon_{1} \sim N(0,\Sigma_1)$ and $\epsilon_{2} \sim N(0,\Sigma_2)$.
We could assume $\Sigma_1=\Sigma_2$ for my purposes but a general ...
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Self-Study: If $x \sim \mbox N_p(\mu; V)$ and A is a matrix then how to show that $E[(x-\mu)(x-\mu)'A(x-\mu) = 0$?
If we assume the random vector $x$ to be normally distributed with $N_p(\mu; V)$ then $E[(x-\mu)(x-\mu)'(x-\mu)] = 0_p$. If I am not mistaken, this can be shown using the moment generating function of ...
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Determine statistical difference of slopes of quadratic relationship in a Poisson regression
I'm looking for a statistical or mathematical way to test the difference between two slopes. Others have asked related questions but my problem is quite particular.
I'm running a Poisson regression ...
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2
answers
133
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Maximization of quotient of quadratic forms in linear regression
I would like to find maximum of the following function:
$$I = \max_{a\in \mathbb{R}^p} \frac{(a'\hat{\beta})^2}{S^2a'(X'X)^{-1}a},$$
where $X$ is a design matrix and of course $Y$ is normally ...
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1
answer
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How to generate data such that an equation needs to hold?
Can I create or generate $\{y_i\}_{i=1}^{4}$ data set such that this equation holds
$$
\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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