# Questions tagged [weighted-least-squares]

This tag is for questions relating to weighted least squares, a generalization of ordinary least squares and linear regression in which the errors covariance matrix is allowed to be different from an identity matrix.

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### Weighted least squares formula

I was studying the weighted least squares algorithm and came across this formula for calculating the weighted result in terms of the original. Here $x_{LS}$ is the solution for $D=I$ I can't figure ...
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### Best way to remove a local maxima from a piecewise linear function

Let $x_1,...x_K \geq 0$ and $f$ be the piecewise-linear function given by $f(k)=x_k$ for every $1 \leq k \leq K$. Denote by $m$ the number of modes (i.e. local maxima) of $f$. Let's associate with ...
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### Order of convergence of operators

I was studying the paper "Discrete multi-projection methods for eigen problems of compact integral operator" by G Long, Gnaneshwar Nelakanti, Bijaya L Panigrahi and Mital M Sahani. for my ...
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### How to fit a ray into a set of 2D points, having the first point as pivot?

I am trying to improve the line simplification algorithm from Ramer-Douglas-Peucker by adding some kind of line fitting into it. Today, this algorithm does not provide the simplification with ...
20 views

### WLS/GLS weights when predicted response is zero?

I am working with a nonlinear ODE system where I am attempting to calculate the following: $$\widehat{\theta}_{WLS} = \text{arg}\min_{\theta} \sum_{j=1}^N w_j \left[y_j - f(t_j; \theta)\right]^2$$ ...
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### Proof of Weighted Least Square Solution

When calculating Weighted Least Square Solution, after taking the derivative, we will have the following equation: $X^\top WX\beta=X^\top Wy$ where $X_{n\times m}$ is the data matrix, with $n\geq m$ ...
29 views

### Obtaining all solutions of a linear equation by weighted generalized inverse

Consider a linear equation $$Ax=b,\quad b\in\operatorname{col}(A).$$ The vector $b$ lies in the column space of $A$ so the solution of this linear equation exists. It is known that any solution has ...
114 views

### Why is there more weight on smaller y values in transformed linear regression as compared to least squares regression for exponential models?

I was doing regression for an exponential model that follows the general formula $$y=Ae^{Bx}$$ And found that using linear regression for the linearized data would model larger values of y poorly ...
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### Taylor approximation is not optimal

My professor gave a lecture on an orthogonal polynomial based approximation and its advantage over the Taylor series expansion. And his statement was in weighted $L_2$ space, Taylor series expansion ...
1 vote
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### fractional curve fitting of the function $y=a+bx^{\alpha}$

Assume I have a set of data $(x_{i},y_{i})$, $i=1,...,m$. How can we find the best values of the parameters $a$ and $b$ and $\alpha$ such that the curve $y=a+bx^{\alpha}$ best fits the data. This ...
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### 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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### Understanding an identity for least squares regression line gradient

In section 2.2 of this paper, Gelman and Park present the following identity for the gradient of the least squares line through a set of 2D points: ...we recall a simple algebraic identity that ...
1 vote
195 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 ...