Questions tagged [weighted-least-squares]

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Weighted least squares with nuclear norm minimization

Nuclear norm minimization is very popularization and formulation is least squares term with nuclear norm term as following, $$\min\limits_{X} \frac{1}{2}\|X_{3\times3}-Y_{3\times3}\|_F^2+\lambda\|X_{...
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Stuck in concavity proof of least squares cost as a function of weights

This problem comes from Boyd & Vandenberghe Convex Optimization, example 3.9 in page 81. All derivations make sense for me except the last step which says: $$g(w)=b^TWb-b^TWA(A^TWA)^{-1}A^TWb ...
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Matrix notation for weighted sum of squares

While going through page 1 of Lecture 24: Weighted and Generalized Least Squares [PDF], I got the following questions. Weighted sum of squares is defined as below: $$ \sum_{i = 0}^{n}{w_i(Y_i - X_ib)...
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Sensitivity of weighted least squares estimation method

I am trying to understand the weighted least squares estimation method, and I'd really appreciate it if you could shed some light on me. Let me explain my problem briefly: Consider a linear model in ...
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From non-linear least squares to weighted linear least squares

Given an overdetermined linear system $A \in \mathbb{R}^{m \times n}$, $b \in \mathbb{R}^{m \times 1}$. And a non-linear function $f(x)$. Given a non-linear least squares: $$ e^* = \min_g \left\lVert ...
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Iterative least squares for TV-like regularizer

I am trying to implement a TV-like surface regularization in a least squares solver. The formulation is as follows: $E_{surface} = \sum |dA_\theta| $ Background $dA_{\theta} = \frac{\theta_1}{f_x ...
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31 views

Relationship between Fundamental Lemma of Calculus of Variations and the Weighted-Residual Statement

The Weighted-Residual Method states that the integral of the Residual R(x) times the weighting function w(x) is equal to zero which means that R(x) = 0 On the other hand, Fundamental Lemma of ...
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78 views

Nearest (with respect to weights) symmetric positive semidefinite matrix

I want to compute the nearest symmetric positive semidefinite matrix, similar as Higham did. But here also weights (given by an inverse co-variance matrix) should be taken into account. So the ...
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Weighted Least Squares and the unreal effect on the a posteriori MSE estimator

I expose the initial situacion and the questions are derived afterwards. Thanks a lot in advance, I hope the question is not too simple for this community. When solving normal or weighted least ...
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Restricted Weighted Linear Regression in R

I have to follwing issue. I would like to run a linear regression imposing a constraint on the weighted coefficients. Let me construct an example: Consider the following cross-sectional regression $...
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How to scale a weighted variable for linear regression?

I would like to scale a variable such that after weighting it is scaled to mean zero and standard deviation one: $\sum_{i} w_{i} x_{i} = 0$ and $\sum_{i} w_{i} x_{i}^2 = 1$ where $w_{i}$ are ...
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Restricted weighted OLS

I would like to run a restricted weighted OLS regression in R. Let me first state the problem mathematically: $\arg\min_{\beta} \sum_{i=1}^{n} w_{i} \lvert y_{i} - \sum_{k=1}^{m} x_{i,k} \beta_{k}\...
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Baseball: How are data sets with differing number of occurrences compared?

I have two data sets, each showing batting statistics for batters hitting against a specific pitcher. These tables show lifetime batting statistics of batters hitting against pitchers, therefore the ...
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30 views

Inverse of matrix sum, one symmetric PSD and one near-constant diagonal

Question How can split the calculation of a real matrix inverse $(S + D)^{-1}$ when I know that $S$ is symmetric and PSD and $D$ diagonal with only a handful of unique values (=diag$(a,a...a,b...b,c.....
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Iteratively Weighted Least Squares and Hessian

The question is about stationary point of a least squares being identical to the stationary point of an error function as stated in: Computer Vision: Algorithms and Applications by Szelinski et al. p....
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16 views

Weighted rank 1 approximation to matrix

I want to solve the following problem: $$\arg\min_{u,v} \|W\odot(u v^\mathsf{T}-M)\|_\mathrm{F}$$ where $u$ and $v$ are $N\times 1$ vectors, $W$ and $M$ are $N\times N$ matrices, $\odot$ represents ...
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Statistics: Higher confidence, higher weight

So we have an experiment going on where a group of colleagues evaluate each other's skill levels, ranging from 1 to 5. We have $n$ colleagues, so in the end we should get $n^2$ evaluations. However, ...
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When can Levenberg-Marquardt fitting algorithm be used with least absolute residuals (LAR) method and not Bisquare method for residual minimization?

I am sorry for asking a trivial question but though I have found an answer in the following link, I would like to know some more insight on the situations when one residual minimizing method is used ...
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61 views

Alternatives to least square, overestimate?

Ok so I have a very simple linear system of equations with three unknowns and eight equations. Is there anyone that can tell me if there is a method or some way to find a solution which overestimates ...
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45 views

Weighted least squares estimator for a non-zero intercept regression

If I have the following regression model with intercept $\alpha$ $$y=X\beta + \alpha + \epsilon$$ Is the Weighted Least Squares (WLS) estimator for $\beta$ the same as in the zero-intercept ...
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How to solve conditional least square?

I'm studying least square, so I can calculate optimal solution at overdetermined system. weighted least square But when some conditions are given, I can't calculate optimal solution(sub?) For ...
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Weighted Least Squares - Categorical Data vs. Numerical Data

Over on Stackoverflow, I am trying calculate the Weighted Least Squares (WLS) of a data set in a python library called Numpy as compared to using a library called Statsmodels. However, I noticed ...
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37 views

Minimizing a multivariable function with different vector length?

Assume that I want to minimize a function to a measured vector. $$V_{min} = \sum_{n = 0, k = 0}^{N, K}{|T_n - f(t_n, p_n, r_k)|}$$ Where $T_n, t_n, p_n$ has the same vector length $N$, but not $r_k$ ...
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Assign automatic weightage for set of negative numbers

I have set of negative numbers e.g. [-0.189, -3.55, -19.90, -0.0001] now I have to convert this set to percentage such that largest number will have highest ...
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81 views

Solving the two-parameter generalized Eigenvalue problem

Is there a solution for the problem: $$ (A_0 + \alpha \ A_1 + \beta \ A_2 ) x = 0 $$ where: $ A_0, A_1,$ and $A_2 \in \mathbb{R}^{n\times n}$, $\alpha$ and $\beta \in \mathbb{C}$, and $x \in \mathbb{...