Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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Online estimation of a quadratic form

I have a functional form $y = x^T Q x + b^T x + c$, with $Q, b, c$ to be estimated, $x \in \mathbb{R}^n$ and $n$ varies around 10-20, depending on the problem. $x$ is sampled from a known Gaussian ...
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51 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. ...
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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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7 views

Interpreting the results of using an ordered variable among independent variables in a model

I am running an ordered probit model since the dependent variable is ordered (using "MASS" package in R). Also, there is an ordered variable among the independent variable. It shows the results as L,...
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When building a multiple linear regression model, is it possible to form models with both linear and non-linear (quadratic) relationships?

Through backward elimination, I have reduced my model from 6 linear factors to 1, accounting for 68% of variance. I have also found that by squaring one of the variables I previously included, that ...
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25 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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45 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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34 views

Distribution of quadratic forms in mixed model

I have a question related to the distribution or asymptotic distribution of quadratic forms that arise in the linear mixed model. Suppose, $$Y=X\beta + H\delta + \epsilon$$ where $\epsilon \sim N_n(0,...
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75 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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35 views

Why does quadratic optimization produce non-stable results

Edit: posing the question in a different way I am running a mean-variance optimization, which is a quadratic optimization problem. I run the optimization 2000 times for different levels of risk (via ...
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21 views

expected value of quadratic form when both X and A are random matrices?

i know how expected value of X'AX is calculated when A is deterministic matrix. but i do not know when A is diagonal but random, how expected value of X'AX is calculated. both of X and A are matrices
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31 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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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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330 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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33 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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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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Inflection Point in Quadratic Model

I have a panel data and I am estimating a qudaratic model with fixed effects. The following model is estimated using stata. ...
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79 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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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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59 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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49 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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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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83 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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45 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 ...
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224 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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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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54 views

Maximal inequality for quadratic forms as functions of matrices?

In general, an $M$-estimator is defined as a maximizer of some objective function $Q_n(\theta)$: $\hat{\theta} = \arg\max_{\theta} Q_n(\theta)$. Suppose that $q(\theta) = E Q_n(\theta)$ is uniquely ...
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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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86 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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632 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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138 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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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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1answer
55 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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1answer
300 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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693 views

Mean and Variance of SSE

First, let \begin{equation} SSE = \overrightarrow{y}'(I - H) \overrightarrow{y} \end{equation} where \begin{equation} \overrightarrow{y} \sim MN(\textbf{X} \overrightarrow{\beta}, \sigma^{2}I) \\ H ...
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Test of concavity for repeated measures data

First question here. I'm trying to figure out what statistical test is appropriate for testing whether a series of data is concave or convex. Specifically, this is coming from a human subjects ...
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319 views

Interpretation of Cox Hazard Model with quadratic term

I am having trouble finding information on how to interpret coxph model hazard ratios with a quadratic term. Some of my variables are continuous count data, whereas others are continuous percentages. ...
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67 views

Quadratic model of data?

Is it possible to fit a quadratic or polynomial model with this type of data? Two inputs, input one is a temperature sensor: Input two is a valve opening on a scale from 0-100: This is a scatter ...
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356 views

Is the Quadratic Approximation of Log-Likelihood Equivalent to the Normal Approximation of the MLE?

Let $X_1, X_2, ..., X_n \sim \text{IID N}(\theta, \sigma^2)$ with $\sigma^2$ known, and let $\hat{\theta}$ be the MLE of the mean. (1) How can I show that in this case, the following is true? $$\...
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353 views

Covariance of Two Quadratic Forms

We're looking for the $\operatorname{Cov}\left[x^T A x, ~x^T B x\right]$ where $x$ is random variable and mean-centered, but not independent and $A$ and $B$ are symmetric matrices. The fundamental ...
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Sums of degenerate quadratic forms

I am searching for an analogue of the fact: let $\Sigma_1 , \Sigma_2> 0$ in $\mathbb R^{m \times m}$ and let $x,c_1, c_2 \in \mathbb R^m$ be arbitrary. Let $\Sigma_3^{-1} = \Sigma_1^{-1} + \Sigma_2^...
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Center or square first when creating a quadratic term with a grand-mean centered variable in a multilevel model

I have a standard 2-level hierarchical linear model that I’m doing (household water use is the dependent variable), and one of the level-1 (household-level) variables (average cost) includes a ...
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Is there any sort of higher-order SVD (quadratic and above) for dimensionality reduction?

X-Posted on math.stackexchange, apologies, though I thought this was equally relevant to both communities. I'm wondering if there exists any higher-order SVD for dimensionality reduction. Note that ...
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Interpreting a Quadratic Term in Binary Logistic Regression

Apologies in advance for my limited stats knowledge. I hope someone can help. I am trying to understand how to interpret the coefficients of both the linear and quadratic term in a binary logistic ...
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579 views

Distribution of quadratic form of multivariate normal with linear term

Suppose that $A$ is a symmetric non-random matrix and $X\sim N(\mu,\Sigma)$ and $b \in R^n$ is a non-random vector. Then what is the distribution of $$X^tAX+b^tX \quad ?$$ The distribution without ...
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Interpret Quadratic Regression with Asinh transformation

I have a regression equation that uses covariates of the following form asinh(y) = b0 + b1 asinh(x) + b2 (asinh(x))^2 + error and am wondering how to intepret the ...
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Accounting for non-linear trend in SOME samples of a linear mixed model using quadratic

I have the following data: Pine Forest Biomass ~ Age | Plot: Each black curve represents whole-plot biomass for each individual plot I want to formally examine ...
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67 views

Insignificant x, but significant x squared

I have estimated the following model to capture increasing/decreasing marginal effect of $x$ on $y$. : $y=\alpha + \beta_1x+ \beta_2x^2 +e$ where $\beta_1$ is statistically insignificant, but $\...
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401 views

Asymptotic normality of a quadratic form

Let $\mathbf{x}$ be a random vector drawn from $P$. Consider a sample $\{ \mathbf{x}_i \}_{i=1}^n \stackrel{i.i.d.}{\sim} P$. Define $\bar{\mathbf{x}}_n := \frac{1}{n} \sum_{i=1}^n \mathbf{x}_i$, and $...