Questions tagged [least-squares]

The tag has no usage guidance.

28 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
6
votes
0answers
112 views

“Geometry of ill-conditioning” for least-squares problems

It is an idea that dates back to Demmel, 1987 that the condition number of a problem is often related to the distance to the closest ill-posed problems. In Section 3 of the above paper, the author ...
5
votes
0answers
142 views

Solve ill-posed linear system without transposing matrices?

I am attempting to use an iterative solver to solve $p$ in $$ Jp = -r $$ where $J$ is an $m\times m$ matrix ($m$ is in the order of $10^5$ and never explicitly stored). $J$ is a dense matrix ...
4
votes
1answer
92 views

Non-negative least squares with very small numbers

(I have asked this question on StackOverflow previously but it has been pointed to me that CSSE or MSE could be more appropriate) I have to solve a constrained optimization problem of the following ...
4
votes
0answers
43 views

Nonlinear least squares optimized Jacobian calculation

I have a nonlinear least squares problem, in which I am trying to minimize residuals which can be divided into four classes: $$ \min_x ||\epsilon(x)||^2 + ||\xi(x)||^2 + ||\delta(x)||^2 + ||s(x)||^2 $$...
4
votes
0answers
1k views

Using MINPACK for curve fitting: implementation?

I need to implement a non-linear fitting algorithm in Fortran and chose to use MINPACK's flavor of the Levenberg-Marquardt algorithm as a basis for the least-squares stuff. However, I seem to ...
3
votes
0answers
43 views

Least-squares fit of explicit parabolic sheet to data points

For a given set of data points $$\{(x_i, y_i, z_i)\}$$ there exists some $$f_{ABC}(x,y)=Ax^2+Bxy+Cy^2$$ that minimizes $$\sum_i(f_{ABC}(x_i,y_i)-z_i)^2$$ $A$, $B$, and $C$ can be found quickly ...
3
votes
0answers
185 views

Nonlinear least squares and regularization

Consider the nonlinear least-squares minimization of a vector of $n$ residuals $\mathbf{f}$ in $p$ parameters $\mathbf{x}$: $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ This can be done with ...
3
votes
0answers
39 views

Reweighted least squares factorization

This is a continuation of the question asked here. I want to solve numerous least squares systems of the form $$ D_i A x \approx D_i b $$ where $D_i$ are $m \times m$ diagonal matrices with positive ...
3
votes
0answers
82 views

Backward stable algorithm to get orthogonal projection onto the column space of a matrix

I have to find the orthogonal projection of a vector $b$ onto the matrix $A$ of size $m \times n$. In my application, I don't have the luxury of calculating the QR factorization. All I have are ...
3
votes
0answers
1k views

Polynomial Fitting with Least Squares using Numpy and Scipy

I am trying to fit data to a polynomial using Python - Numpy. The points, with lines sketched above them are as in the picture. I am trying to fit those points to a polynomial of 4. or 5. degree. ...
2
votes
0answers
31 views

Non-negative Least Squares to perform Inverse Laplace with weights

I'm trying to perform the inverse Laplace transform of a (noisy) dataset $y_i$ using Tikhonov regularization: $$\min \sum_{i=1}^{N} \left(\int_0^\infty e^{-s_i t} f(t) \, dt - y_i \right)^2 - \lambda^...
2
votes
0answers
41 views

Fitting a plane with the Prewitt gradient operator

Prewitt gradient operator Show that the Prewitt gradient operator can be obtained by fitting the least-squares plane through the 3 × 3 neighborhood of the intensity function. Hint: Fit a plane to ...
2
votes
0answers
129 views

Regularized least squares with QR factorization

Consider the regularized least squares problem $$ \min_x || b - A x ||^2 + \lambda^2 ||x||^2 $$ which is equivalent to $$ \min_x \left|\left| \pmatrix{b \\ 0} - \pmatrix{A \\ \lambda I} x \right|\...
2
votes
0answers
279 views

Convergence of a very large non-linear least squares optimization

(note: I also posted this question on stackoverflow before finding this community here, which seems a better place for it) I'm trying to solve the following problem: I have a lot (~80000) surface ...
2
votes
0answers
42 views

$(max(0, f(x)))^2$ or $(max(0, exp(f(x))))^2$ for soft constraints with Gauss-Newton

I need some kind of inequality constraint in my optimization problems (rude version of SVM for example or skeleton based mesh fitting). However hard constraints is not suitable for me because ...
2
votes
0answers
27 views

Choosing suitable polynomial degree based on information in advection stencil

I'm working on a finite volume advection scheme for unstructured meshes which uses a multidimensional polynomial weighted least squares fit for interpolating from cell centres onto faces. In 2D, the ...
2
votes
0answers
113 views

Calculating theoretical order of accuracy of least squares fit advection scheme

I'm familiar with finding the order of accuracy using von Neumann analysis for finite difference schemes formulated using Taylor series expansions. But is there a similar technique for finding the ...
2
votes
0answers
68 views

Fortran solver for the Sparse LSE problem

I was wondering if there is a Fortran library that contains a solver for the Sparse LSE(linear equality-constrained least squares) problem $$ min_{x}\|Cx-d\|^2 \text{ subject to } Ax=b $$ where $A$ ...
1
vote
0answers
39 views

Triangle on top of diagonal least squares

I need to solve many least squares problems with the following matrices: $$ \pmatrix{ R \\ D_i } $$ where $R$ is upper triangular and $D_i$ is diagonal. $R$ is the same for all the problems, while $...
1
vote
0answers
45 views

B-splines least squares with equality constraints

Can someone recommend the best way to solve a least squares fitting problem with B-splines, with additional equality constraints? I want to solve: $$ \min_x || b - A x ||^2, \textrm{subject to: } C x =...
1
vote
0answers
34 views

How to model a non-linear least-squares problem for triangles

I have a non-linear least-squares problem to solve and with my current modeling the solver is either very slow or does not converge to a correct solution. For the problem I need to minimize an energy ...
1
vote
0answers
34 views

fitting exponential versus exponential w/ power

I have two models which I would like to investigate for my data. One form is: \begin{equation} \label{one} f(r) = A e^{-B r} \end{equation} and the second is: \begin{equation} \label{two} g(r)...
1
vote
0answers
67 views

Stable Method of orthogonal projection onto a subspace with the help of Moore-Penrose inverse,

Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v $. Below is ...
1
vote
0answers
59 views

linear solution of curve fitting on multiple linear functions differing by a multiplier

I am facing the following problem. I know nonlinear least squares can provide a solution but I am wondering if a linear way to solve this data fitting problem may exists. This is my input dataset: I'...
0
votes
0answers
27 views

Estimating the dimension of a solution space in nonlinear least squares

Suppose I have a nonlinear least squares problem, $$ \min_{\mathbf{x}} || \mathbf{f}(\mathbf{x}) ||^2 $$ with $n$ residuals and $m$ parameters, so that $\mathbf{x} \in \mathbb{R}^m$, and $\mathbf{f} \...
0
votes
0answers
52 views

Least square approximation of a polynomial with a constraint on the derivative in Python

I'm trying to fit a polynomial of the third degree through a number of points. This could be a very simple problem when not constraining the derivative. I found some promising solutions using CVXPY to ...
0
votes
0answers
56 views

Parameter estimation using fmincon

This is a follow up to my previous question posted here. I am solving an optimization problem using fmincon in MATLAB. There are no equality constraints in my model....
0
votes
0answers
217 views

computation complexity of OLS in estimating a VAR model

Could someone tell me the computation cost of using Ordinary least squares in estimating a Vector autoregression model? I am thinking the cost is O(n) where n is the number of the training instance. ...