Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

Filter by
Sorted by
Tagged with
0
votes
0answers
10 views

what is the difference between MDA and LDA?and differences between LDA and QDA? I need compelet answer thanks for your help

what is the difference between MDA and LDA?and differences between LDA and QDA? I need compelet answer thanks for your help
0
votes
0answers
34 views

Clarification regarding $(X'BX)(X'AX)$ distribution

Assume $X = (X_1, \ldots, X_n)' \sim \mathcal{N}(0, \Sigma)$ is a random normal vector. I'm looking for the distribution of the following form: $$Z = (X'BX)(X'AX)$$ here $X \in \mathbb{R}^{n \times 1}$...
1
vote
1answer
15 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 ...
0
votes
0answers
32 views

Multivariate normal distribution -testing a quadratic form of mean vector

Suppose $X_{1},X_2,...,X_n$ are i.i.d.observations from a multivariate normal distribution $N(\mu,\Sigma)$ where $\Sigma$ is known. Assume that $a$ and $b$ are given vectors. Use the likelihood ratio ...
0
votes
0answers
27 views

Transformation of dependent normally distributed random variables

If $Y_1,Y_2,...Y_n$ are normally distributed random variables with mean $E(Y_i)=\mu\;,Var(Y_i)=\sigma^2\;and\;Cov(Y_i,Y_j)=s[i,j=1,2,...,n;i\neq j]$ and we take the transformation $Z_i=Y_i^2$, then ...
1
vote
0answers
31 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 $\boldsymbol{\epsilon} \sim N(0,\sigma^2 \mathbf{I}_N)$. ...
0
votes
0answers
19 views

quadratic regression - model assumptions not respected

I've just finished performing a quadratic regression. The results are significant, but when plotting the residuals, I noticed that the assumptions of the model (i.e. homogeneous variances and normal ...
0
votes
0answers
37 views

Convert back standardized parameters for a quadratic fit

I have a model with standardized data: $x' = \frac{x - \bar{x}}{S_x}$ $y' = \frac{y - \bar{y}}{S_y}$ I’m trying to figure out how to convert back the coefficients of a quadratic fit to the original ...
1
vote
0answers
70 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
1answer
280 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
0answers
21 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 $T$ (...
7
votes
1answer
90 views

Confidence interval from summary function

Here is a summary data from a texbook ...
0
votes
0answers
28 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 ...
0
votes
1answer
212 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
0answers
54 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 ...
2
votes
0answers
101 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 ...
0
votes
0answers
25 views

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 ...
1
vote
1answer
668 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. ...
0
votes
0answers
586 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 ...
2
votes
1answer
95 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
1answer
119 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
1answer
568 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
1answer
75 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 ...
1
vote
2answers
177 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
1answer
449 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: ...
1
vote
1answer
97 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",...
6
votes
1answer
160 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
vote
0answers
100 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 ...
2
votes
1answer
131 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
0answers
43 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 ...
3
votes
1answer
102 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 ...
3
votes
1answer
73 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 ...
2
votes
1answer
43 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 &...
1
vote
1answer
115 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. ...
1
vote
1answer
52 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 ...
1
vote
1answer
680 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-...
0
votes
1answer
57 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 ...
0
votes
0answers
26 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 ...
2
votes
0answers
152 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, ...
6
votes
1answer
2k 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 ...
2
votes
1answer
255 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 ...
4
votes
1answer
281 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 $...
1
vote
1answer
94 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 ...
2
votes
1answer
649 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 ...
3
votes
1answer
1k 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 ...
0
votes
0answers
41 views

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 ...
0
votes
1answer
599 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. ...
0
votes
1answer
70 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 ...
4
votes
1answer
614 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? $$\...
2
votes
1answer
590 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 ...