Questions tagged [quadratic-form]

A quadratic form is a homogeneous polynomial of order two.

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Problem on quadratic form [migrated]

Show that $\{x : x^{T}Ax \le 1\}$ is bounded if $A$ is positive definite, where $x^T=[x_1,x_2,...,x_n]$ and $A$ is an $n\times n$ matrix.
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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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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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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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165 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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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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35 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 $...
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1answer
31 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
47 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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296 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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Summing linear and quadratic effects of a variable in polynomial models?

I have a regression model that includes two predictor variables, but each is represented by a linear and quadratic term. $$y \sim a + b_1x + b_2x^2 + b_3y + b_4y^2$$ All the terms are significant, ...
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21 views

Prediction Interval on a Product

As a (perhaps) less offensive twist on this question, suppose that $z_i$ are independent $p$-variate standard normals: $z_i \sim \mathcal{N}\left(0, I\right).$ Let $a$ be an unknown $p$-vector. ...
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11 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 ...
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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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56 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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172 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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119 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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15 views

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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Regressing quadratic form x'Ax onto dot product involving x [closed]

I have the following simple panel relation to estimate: $y_{i,t} = \beta_i x_{i,t} + \epsilon_{i,t}$, where $y$ and $x$ however are linked through: $y_{i,t} = w_i'\Omega_t w_i, \\ x_{i,t} = w_i'f_t,...
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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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1answer
954 views

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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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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1answer
51 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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352 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 $...
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656 views

How to report results of a polynomial regression with a discrete independent variable

I have been recently trying to fit a linear model to my data. The dependent variable is continuous and the independent variable is numeric and discrete. When I first test the assumptions concerning ...
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If I have significant linear and quadratic terms, can I interpret both of these results?

I would have thought there'd be an answer to this question on here already, but I've been unable to find it, so I apologize if this is a repost. If I have a regression model of the form ...
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515 views

Mixed Effects model with quadratic term in fixed, but not in random part

Say one wishes - and believes - to fit a mixed effects model where time ($x_{ij}$ below) is included in a linear and quadratic term in the fixed part, but only linear in the random part (hence random ...
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1answer
1k views

Equivalent to ANOVA for non-linear (quadratic) relationships

I am trying to perform the equivalent of a repeated-measures ANOVA using data that have a non-linear relationship. The independent variable position runs from -20 ...
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598 views

Quadratic form and Chi-squared distribution

It's about the demostration of the quadratic forms and chi-squared distribution. Let's split the problem: We have a $n$ vector with n standardized normal distribution called $z={[z_1,z_2...z_n]}$. ...
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1answer
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Linear and quadratic term interpretation in regression analysis

I'm considering the case of a factor which is detrimental to performance but including also, a quadratic term to allow for the existence of a non-linear relationship. In all cases, both linear and ...
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1answer
199 views

Proof that $\mathrm{Cov}(x^TAx,x^TBx) = 2 \mathrm{Tr}(A \Sigma B \Sigma) + 4 \mu^TA \Sigma B \mu$

Suppose $\vec x \sim N(\vec \mu, \Sigma)$ is multivariate normal. I want to see that $\mathrm{Cov}(\vec x^TA\vec x,\vec x^TB\vec x) = 2 \mathrm{Tr}(A \Sigma B \Sigma) + 4 \vec \mu^TA \Sigma B \vec \...
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If $\vec y \sim N(\vec 0,\Sigma)$, what is the distribution of $y^TAy$, when $A$ is indefinite?

Suppose $\vec y \sim N(\vec 0,\Sigma)$ with $\Sigma$ singular. Is the distribution of $y^TAy$ known in the case that $A$ is indefinite?
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Mix Design: Failure to predict response

I have performed a mixture design containing 3 terms and I have carried out 6 experiments needed to adjust a quadratic model. It fitted the data quite well: ...
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147 views

Quantiles of linear combination of independent $\chi^2_1$ random variables [duplicate]

I want to work out the quantiles of a linear combiation of chi square random variables. Suppose $\lambda_i \in \mathbb{R}$ for all $i \in \{1,2,\cdots,n\}$ and $Z = \sum_{i = 1}^n \lambda_iX_i$ and $...
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1answer
139 views

Calculate average slope in a quadratic

Let's say I have crop yield on y-axis and a measure of heat-stress on x-axis According to this graph, as heat increases, the yield also increases and after reaching a maximum (at heat = 0.67), the ...
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1answer
70 views

Modeling potentially unimodal data

My data: I surveyed for animal densities across an elevation gradient. Let's say I surveyed from 0 to 1000 meters elevation. My models (one for each species of interest): density ~ elevation + other ...
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93 views

How to show fitted curve for random effects?

I have a data set studying the difference pollen production among eight varities. The experiment was carried out in two locations, and pollen was collected and weighted once per week. The experiment ...
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1answer
320 views

Interaction between a categorical and a quadratic continuous variable

I have three main hypotheses: Independent variable $X_1$, which is a categorical variable of two levels (A and B), may have an effect on the response $Y$. Independent variable $X_2$, a continuous ...
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172 views

Quadratic model - how to test

For my research I have a hypothesis containing a categorical IV (positive vs negative article that the participants have to read) and a continuous moderator (ideology level of the participant). The x ...
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Non linear regression? Which model to use?

For my analysis I am predicting an effect like the following (the IV being 'ideology' on a continuous scale). I'm predicting a different effect for when it's low, middle or high. Unfortunately all I ...
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2answers
697 views

Interpreting interactions in a linear model vs quadratic model

I have a crop yield data collected across many locations for 1996-2006 period and a predictor measuring heat availability. I formulated a model like this; ...
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Interpretation: Mediation with a squared regression model (U-shape)

I did a mediation analysis with Person-Organization Fit as IV, Organizational Attraction as Mediator and Organizational Choice Intention as DV. As the regression assumption 'Linearity in the ...
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85 views

Interpretation of interaction in presence of squared terms

I am running a fixed effects model with two independent variables, their quadratic terms, and the interaction of the two variables. IV = b0 + b1*X + b2*Y + b3*X_sq + b4*Y_sq + b5*XY How do I ...
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108 views

Standardization of variables in Bayesian model

I'm running a model to understand the relationship between children education expenditures and income. I have three variables $y$ the response variable that represents the proportion of income spent ...
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83 views

Odd Behavior in Cutoff of Soft Margin SVM

In soft margin svm, we solve the following quadratic programming problem. $$ \text{maximize}\ \sum_{i=1}^N \alpha_i-\sum_{i=1}^N\sum_{j=1}^N\alpha_i\alpha_jy_iy_jX_i^tX_j,\\ \text{subject to}\ \sum_{i=...
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748 views

How can I interpret coefficients of quadratic and linear term?

It may be a basic statistic question for someone, but I'm struggling with this. I'm trying to interpret a regression analysis. Here is examples. ...