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Questions tagged [linear-regression]

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0
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0answers
46 views

How can least squares regression be modified to penalize errors more heavily for small values?

As usual, $f(x_i)$ is some linear combination of the variables, with the error total: $$\sum_{i=1}^N (y_i - f(x_i))^2$$ I would like to penalize errors near $0$ more heavily. One conceivable ...
8
votes
3answers
198 views

Regularized linear vs. RKHS-regression

I'm studying the difference between regularization in RKHS regression and linear regression, but I have a hard time grasping the crucial difference between the two. Given input-output pairs $(x_i,y_i)...
0
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2answers
66 views

Unique parameterization of size MxN matrices of rank k

Any rank k matrix $Y\in\mathbb{R}^{m\times n}$ can be written as: $$ Y = UV'$$ Where $U\in \mathbb{R}^{m\times k}, V\in \mathbb{R}^{n\times k}$. This factorization is not unique since for any ...
4
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2answers
188 views

Given the joint probability distributions of $X$ and $Y$ for $Y = R\,X+C$, find the probability distributions of $R$ and $C$

Let $R$, $C$, and $X$ be independent random variables defined on $(0,\infty)$ and $$Y=\underbrace{R\, X}_{Z}+C.$$ We are given the joint probability distribution of $X$ and $Y$, $P_{XY}(x,y)$ and ...
3
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1answer
75 views

Why to multiply the penalty by $n$ in the penalized least squares and likelihood?

In the SCAD paper by Fan and Li (2001), there exist two forms of penalized least squares as follows: $$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$ ...
2
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0answers
35 views

increasing inter-class distances results in decreasing linear regression error

Let $\{\mathbf{x}_i, y_i \}$ be a set of binary-labeled samples ($\mathbf{x}_i \in \mathbb{R}^d, y_i \in \{a,b\}, a,b\in\mathbb{R}$). Let $\{ \mathbf{x}'_i, y_i \}$ be also such a set. Define $\mathbf{...
2
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1answer
45 views

matrix regression under side conditions

I want to solve the folowing problem B*M=V, where B is the unknown of size 3x3, M of size 3xN and V of size 3xN. The difficulty is, that B has to be unitary. N is in the range of 500. All matrices ...
1
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0answers
77 views

A different objective function in liner regression analysis

I'm an undergraduate student who is green in statistics. I have a problem in the chose of objective function when estimating the parameters. Let $Y = \beta^TX + \epsilon $ be the standard liner ...
0
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0answers
135 views

Solving overdetermined, “polynomial time dependent” linear system

Hello & happy new year, let $n \geq m$ be two natural numbers. Furthermore let $W \in \mathbb{R}^{n \times m}, Y \in \mathbb{R}^{n \times n}, \lambda \in \mathbb{R}^n, \mu \in \mathbb{R}^m$. ...
3
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1answer
5k views

Gauss-Newton vs Gradient Descent vs Levenberg-Marquadt for least squared method

I need to clarify some idea I have in my mind about linear and non-linear regressions. Whatever I now about this topic comes from the book of Taylor "Introduction to error analysis": a set of ...
1
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0answers
56 views

Posterior consistency of non linear model

This is possibly a reference request. Let $G$ : $\mathbb{R}^p \to \mathbb{R}^q$ be a continuous injective/bijective function. Let $\mu$(we may also assume this to be a non degenerate Gaussian) be ...
2
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0answers
130 views

Derivation of gradient of SSE in Geodesic Regression

On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
2
votes
3answers
1k views

When does a Vandermonde-like matrix have full rank

I have a matrix which is similar to Vandermonde matrix except that the entries are monomials of degree $d$ polynomial in 2 variables. Each row has the following form: $X_{i}= [1, x_{i}, y_{i}, x_{i}^...
2
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0answers
218 views

Is there an efficient way to compute the “complete subset regression”?

Background: Let $X \in \mathbb{R}^{N\times K}$ and $y \in \mathbb{R}^{N\times 1}$ be data for a regression problem. The aim is to find $\beta \in \mathbb{R}^{K\times 1}$ such that $X\beta \approx y$ ...
1
vote
1answer
131 views

Checking the intersection of two sets

Let $E\subset{\mathbb R}^n$ be a set of the type $I_1\times \dots \times I_n$, where $I_k$ are real intervals, and $X$ be and $n\times p$ real matrix. Suppose also that $rank(X)=p$ and $n>p$. Is ...
-1
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1answer
212 views

Fitting a quadratic using regression when the y-intercept needs to be 0 [closed]

I'm trying to fit a quadratic $a_0 + a_1x + a_2x^2$ by Polynomial Regression: $$ \begin{pmatrix} n & \Sigma x_i & \Sigma x_i\\ \Sigma x_i & \Sigma x_i^2 & \Sigma x_i^3\\ \Sigma ...
1
vote
0answers
151 views

How to find all least-square solutions [closed]

I was looking at numpy's lstsq to find a least squares solution of an equation system when the following occurred to me: Given the points (0,0), (3,4), (4,3), if I ...
3
votes
1answer
64 views

Regression with correlation structure

I have a theoretical question about regression models. Let's say I measured multiple responses from $n$ subjects and these responses are correlated with each other. For example, let's say I measured ...
1
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1answer
2k views

Minimizing sum of absolute deviations

Suppose we want to find coefficients $b$ in $\underset{b}{\operatorname{argmin}} \displaystyle\sum\limits_{i=1}^n | y_{i}-b_{1}x_{i}-b_{0}\mid$. If we rewrite this problem in terms of linear ...
10
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0answers
268 views

Testing contrasts in statistics: Is this provably a hard problem, or not?

Scheffé's method for identifying statistically significant contrasts is widely known. A contrast among the means $\mu_i$, $i=1,\ldots,r$ of $r$ populations is a linear combination $\sum_{i=1}^r c_i \...