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3
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2answers
106 views

Quadratic Programming and Lasso

I'm trying to perform a lasso regression, which has following form: Minimize $w$ in $(Y - Xw)'(Y - Xw) + \lambda \;\text{norm}(w,1)$ Given a $\lambda$, I was advised to find the optimal $w$ with ...
1
vote
1answer
39 views

How should I interpret a factor that is significant both (1) in linear form in interaction with another factor and (2) in its quadratic form?

Our study aims to identify and understand how several ecological factors relate to parasite abundance in a colonial animal. However, we are uncertain on how to interpret a factor (density) that is ...
4
votes
3answers
120 views

What is the PDF of $[(X-a)^2 + (Y-b)^2]^{1/2}$ where $X$ and $Y$ are two non-standard normal random variables?

I have to conduct an experiment getting data from a system. These data are the estimated values, provided by the system, of a true value that we know beforehand. I then compare the estimated values ...
0
votes
0answers
24 views

How to standardize a variable an its power term?

In an regression analysis it is sometimes valuable to introduce power terms as predictor variables. Age is a good example in the social sciences. I know there is an general debate about introducing ...
0
votes
0answers
49 views

Interpretation when only the squared term is significant

I am using a Poisson model. My theory suggests a positive $X_1$ and a negative $X_1^2$. However, the results show an insignificant, positive estimate for $X_1$ and a significant, positive estimate for ...
4
votes
2answers
288 views

Independence of a linear and a quadratic form

How can I prove the following lemma? Let $\mathbf{X}^ \prime$ = $ \left[ X_1 , X_2 , \ldots, X_n \right]$ where $ X_1, X_2, \ldots X_n $ are observations of a random sample from a distribution which ...
0
votes
0answers
29 views

Second order terms in regressions

I have a linear regression, with 3 covariates, one of which is categorical with 4 categories. I want to determine if second order terms have a significant effect in the model. I would like to ask if ...
1
vote
1answer
63 views

Linear Kernel in Baysian Linear Regression

I came up with http://mlg.eng.cam.ac.uk/duvenaud/cookbook/index.html and it is actually very useful. At some point it says If you use just a linear kernel in a GP, you're simply doing Bayesian ...
1
vote
0answers
64 views

Comparison between normal glm and glm.nb regression with quadratic term?

Let's say I have a function to simulate data for negative binomial regression: ...
0
votes
0answers
14 views

What is “the smallest solution of an equation” mean?

Suppose you have an quadratic equation (n) of q, what's the meaning of this statement "...This implies that q is the smallest solution to equation (n)"? What is the ...
0
votes
0answers
18 views

How can a quadratic relationship be written two ways

In this answer, under: "Why include the linear term?" in the answer, it mentions the relationship can be written two ways. Later it says that the point where $x=b$ is the vertex of the parabola. ...
1
vote
2answers
100 views

Adding quadratic term changes the sign of the variable

Number of books published in a year (noBook) is my dependent variable and I have independent variables including the age of the author (age). The coefficient of age is positive and it is significant. ...
1
vote
1answer
383 views

Transforming a relationship from quadratic to linear

Assume that I'm running a path analysis and I have discovered that certain relationships, empirically, are quadratic rather than linear. In order to model the data as such, I want to transform the ...
2
votes
1answer
141 views

Comparison of two linear regression models with squared regressors

I need to compare two linear regression models that include the regressors in levels and in squares: $Y=a_1x^2+b_1x+c_1$ and $Y=a_2x^2+b_2x+c_2 $ Specifically, I need to test whether these ...
7
votes
2answers
487 views

How to include a linear and quadratic term when also including interaction with those variables?

When adding a numeric predictor with categorical predictors and their interactions, it is usually considered necessary to center the variables at 0 beforehand. The reasoning is that the main effects ...
8
votes
2answers
362 views

Sum of two normal products is Laplace?

It is apparently the case that if $X_i \sim N(0,1)$, then $X_1 X_2 + X_3 X_4 \sim \mathrm{Laplace(0,1)}$ I've seen papers on arbitrary quadratic forms, which always results in horrible non-central ...
4
votes
1answer
95 views

Quadratic form of a bivariate normal

This is a homework problem. Let $(X,Y)\sim N(\mu_1,\mu_2,\sigma^2_1,\sigma^2_2,\rho)$. Show that if $\sigma_1,\sigma_2 >0,|\rho|<1$, then $$ ...
4
votes
0answers
109 views

How to do centering if I have a quadratic term?

I have been trying to run a multilevel model with both a linear and a quadratic term for income as my main variables of interest. It looks something like: \begin{eqnarray} ...
0
votes
1answer
2k views

Quadratic models with R. The use of poly(..) and I(..) functions (R-language)

What causes the different results below? ...
3
votes
2answers
1k views

Expected Value of Quadratic Form

On page 9 of Linear Regression Analysis 2nd Edition of Seber and Lee there is a proof for the expected value of a quadratic form that I don't understand. Let $X = (X_i)$ be an $n \times 1 $ random ...
5
votes
1answer
284 views

Adding a quadratic term: should I use the squared original (and not the squared standardized?)

In a multiple logistic regression I need to standardize one of the variables because I need to add a quadratic term. Whether I add the quadratic term as the squared original or the squared ...
3
votes
1answer
130 views

Covariance of $\mathbf{x}$ and $\mathbf{x}^\prime A \mathbf{x}$ when $\mathbf{x} \sim N_{m}(\mu, \Sigma)$

This is part of problem 5.23 in A First Course in Linear Model Theory, Dey and Ravishanker. It was on a previous midterm and I didn't know how to do it, but now I am studying for the final and would ...
5
votes
1answer
149 views

Optimal sampling to estimate a quadratic function

Suppose I have a quadratic function: $$ f(x) = a^T x + x^TBx $$ with $x \in \mathbb{R}^n$. Given a point $x$ I can measure $f(x)$ up to some noise, that is I can get a measurement: $$ \hat{f}(x) = ...
2
votes
2answers
404 views

Interpretation for simple slope analysis for curvilinear regression with interaction effects

When regressing income ($Y$) on age ($X$) moderated by gender ($Z$), I not only find significant effects for age ($X$), age squared ($X^2$), gender ($Z$), the interaction of age and gender ($XZ$), and ...
3
votes
1answer
313 views

Exponentiate a quadratic form, which MVN?

I read that if you have a quadratic $f(x) = ax^2+bx+c$ and providing the leading coefficient $a < 0$, then $e^{f(x)}$ is the pdf of a normal distribution with mean $\mu = -\frac{b}{2a}$ and ...
2
votes
1answer
82 views

How to specify and estimate the parameters of a model that is quadratic in several variables

I am trying to find how the quadratic model for a multiple featured dataset will be. Suppose that my training set is $X_1,...X_n$ with each $X$ of dimension $4$. Now suppose I want to fit a quadratic ...
2
votes
0answers
137 views

What is the distribution of the 'achieved' $R^2$?

I am interested in the distribution of/performing inference on the 'achieved' $R^2$ coefficient in multiple linear regression. Suppose that $y\sim x\beta + \epsilon$ with $\epsilon \sim ...