Questions tagged [linear-regression]

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Is this regression problem solvable? [duplicate]

I have a random vector $\pmb{x}=(X_1,...,X_p)^T\in \mathbb{R}^p$, a symmetric matrix $$\Theta = \left(\begin{matrix}0 & \theta_{12} & \theta_{13} & \cdots & \theta_{1p}\\ \theta_{12} &...
Hepdrey's user avatar
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How to extract 'top k' multiple solutions from a quadratic optimization problem?

Imagine we are interested in the following problem: $$ \min_{w} \left( w^T V w + \lambda \|w\| \right) \\ \text{s.t. } w^T R \geq c $$ Where 𝑤 is an $N \times 1$ vector, $V$ is an $N \times N$ ...
Azmy Rajab's user avatar
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1 answer
61 views

Handling the $\ell^2$ norm of a matrix expression in a linear regression

I am reading a scientific article in which matrices are handled (which I do not use often). We consider a matrix $X\in\mathbb R^{n\times p}$ and a vector $y\in\mathbb R^n$. The authors show that the ...
Zach's user avatar
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0 answers
88 views

Weighted least squares regression: Iterative modeling of variance

In chemical analysis, the instrument's signal are plotted as a function of chemical concentration. In general, higher the concentration higher is the response and the relationship is linear. At ...
AChem's user avatar
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28 views

Adjust X to strengthen the linearity to Y, in regression model

Assume that we have 2 series X and Y, and obvious we can fit a linear regression model and get all the statistics. I am seeking for some transformation / adjustment which will adjust the value of X, ...
Pique's user avatar
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2 votes
1 answer
119 views

Justification of the use of residual plot

$\DeclareMathOperator\Cov{Cov}$Backround of my Question Let $Y$ be the response variable, $\mathbb{X}$ be the explanatory variables. The ultimate goal of prediction is finding a function $f^{*}$ that ...
Cheng-Yu's user avatar
1 vote
0 answers
61 views

Least squares regression with nonnegative error

I'm looking for algorithms to solve a special quadratic programming problem, but I don't know its name or related keywords. Can anyone give me some clues? The problem reads \begin{equation} {\min}_x \...
Duo Zhang's user avatar
4 votes
1 answer
166 views

Least squares problem with left and right unknowns

For $i=1,...,n$, let $b_i$ be a scalar and $A_i$ be an $k\times l$ matrix. Is there a closed form solution for the following problem assuming $n>k+l$? $$\min_{x\in \mathbb{R}^k ,y\in \mathbb{R}^l} \...
dff's user avatar
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0 answers
67 views

Under what conditions is the least-squares approximation bounded with the same Lipschitz gradient constants?

Let $f(x):\mathbb{R}^K\Longrightarrow \mathbb{R}^L$ denote a multivariate continuously differentiable function. All the partial derivatives of $f$ (all its Jacobian elements) are bounded from above ...
Yarden Levy's user avatar
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33 views

Normalizing a parameter in a regression

I am thinking about the possibility of making a parameter in my regression, let's say the $\lambda$ in a ridge regression, somehow, inside a range : $\lambda \in [0,1]$. Do you have any ideas how I ...
SUMQXDT's user avatar
1 vote
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Fitting a model

I have a function expressed as the ratio of two exponential series with certain parameters $$\frac{\sum\limits_{j=1}^{i-1} \frac{e^{-ar_jt}}{\prod_{l=1\\l \ne j}^{i-1} (b^j-b^l)}}{\sum\limits_{j=1}^{i}...
Ahmed Khan's user avatar
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semi-parametric regression

Suppose the observation $(X_1, Y_1), \ldots, (X_n, Y_n)$ satisfies the following semi-parametric model $$Y_t = m(X_t, \alpha) + \sigma(X_t, \beta) U_t,$$ where $U_t$ is independent with $X_t$ with ...
香结丁's user avatar
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1 answer
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The nonparametric estimation in generalized regression model

Let $Y_t \in \mathbb{R}$ be a response variable and $X_t$ a $d$-dimensional explanatory variable. Assume we observe the process that $(X_1, Y_1), \cdots, (X_n, Y_n)$. \begin{equation} Y_{t} = \mu(...
香结丁's user avatar
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1 answer
487 views

Hanson-Wright inequality with random matrix

I'm interested in bounding the tail probabilities of a quadratic form $x^t A x$ where $x\in \mathbb{R}^n$ is a sub-Gaussian vector with independent entries. $A\in \mathbb{R}^{n\times n}$ is a matrix. ...
Puzzler's user avatar
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1 vote
3 answers
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RKHS/non-parametric regression with missing response values

I am interested in doing RKHS regression with missing response variables. Given input-output pairs $(x_i,y_i)$, I want to estimate a function $f(\cdot)$ as follows \begin{equation}f(x)\approx u(x)=\...
MthQ's user avatar
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128 views

How to compress variables in a linear regression

I have a large linear regression where all the independent variables are logical (ie TRUE/FALSE) and sparse. The data has roughly 10,000 variables and 10 million observations, on average around 20 ...
quarague's user avatar
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1 answer
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Does the intercept converge if we fit a best fit line to points with prime coordinates?

A few months ago I asked this question on Mathematics Stack Exchange but it has received little attention. Perhaps the question is more applicable here. Let $p_k$ denote the $k$th prime such that $...
TheSimpliFire's user avatar
8 votes
3 answers
341 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)...
MthQ's user avatar
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2 answers
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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 ...
Patrick's user avatar
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4 votes
2 answers
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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 ...
stochastic's user avatar
3 votes
1 answer
121 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: $$\frac{1}{2}\left \| y-X\beta \right \|^2+\lambda \sum_{j=1}^{d}p_j (\left | \beta _j \right |),$$ ...
Tim Xu's user avatar
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2 votes
0 answers
46 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{...
le4m's user avatar
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1 answer
56 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 ...
yar's user avatar
  • 231
1 vote
0 answers
92 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 ...
R. Qian's user avatar
  • 11
9 votes
1 answer
19k 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 know about this topic comes from the book of Taylor "Introduction to error analysis": a set ...
Stefano Fedele's user avatar
1 vote
0 answers
64 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 ...
Madhuresh's user avatar
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2 votes
0 answers
147 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 ...
ABIM's user avatar
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3 votes
3 answers
2k 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}^...
TravisJ's user avatar
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2 votes
0 answers
314 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 $X\beta \approx y$ ...
svangen's user avatar
  • 326
1 vote
1 answer
167 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 ...
Linger's user avatar
  • 13
-1 votes
1 answer
648 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 ...
Ed King's user avatar
  • 101
1 vote
0 answers
273 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 ...
nh2's user avatar
  • 111
4 votes
1 answer
211 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 ...
user1701545's user avatar
1 vote
1 answer
3k 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 ...
Math_manul's user avatar
10 votes
0 answers
300 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 \...
Michael Hardy's user avatar