Questions about (linear or nonlinear) least-squares, an estimation method used in statistics, signal processing and elsewhere.
Questions tagged [least-squares]
1853 questions
2
votes
1 answer
Analytical solution of least square problem
could anyone explain:
a) $||{Ax-b}||^2$ (there is also a lowered 2): what does this two 2's mean?
b) why is the solution: $x =(A^TA)^{-1} A^Tb$ is?
Thank you very much:)
lars111
- 321
1
vote
1 answer
how to find measurement matrix for least square.
I know how to use least square for estimating a constant value given a bunch of measurements. It is the average assuming measurements have same weight of variance.
$$
\hat{x} = (H^{T}H)^{-1} H^{T}z
$$
where $H = 1$ in the case of estimating a…
CroCo
- 1,228
1
vote
2 answers
Least squares fitting issue
I deal with MRI image processing and while reading one of the articles in this field I faced with the next mathematical formula: $$ \widetilde{R_2}(t) = K_1*\overline{R_2}(t) + K_2 * \int_0^t \!\overline{R_2}(t') \mathrm{d}t'$$ (see formula A9 in…
Stepan Loginov
- 123
1
vote
1 answer
About sphere equation $z = a+bx+cy+dx^2 +ey^2$
I'm trying to fit a sphere from points. I tried a first way to estimate the sphere but I'm not satisfied.
I saw in an article a way to get a best fitting sphere from the equation :
$z = a+bx+cy+dx^2 +ey^2$
Now, I have $a,b,c,d$ and $e$ values but I…
usersss
- 11
1
vote
3 answers
Least Squares understanding
I am trying to understand the least squares method, I have found a simple enough example, but it doesn't fully explain the last part of it.
http://www.emathzone.com/tutorials/basic-statistics/example-method-of-least-squares.html
The part I am…
AdamM
- 113
1
vote
2 answers
Least squares problem regarding distance between two vectors in $\mathbb{R}^3$
I'm solving an exercise problem and was facing some confusion regarding how to solve it. The problem is (roughly translated to English):
Given the following:
$$\mathbf{A} = \begin{bmatrix} 2 & 0 \\ 1 & 1 \\ 0 & 1 \end{bmatrix},\ \mathbf{w} =…
Sean
- 1,487
1
vote
0 answers
Least Squares Intersection between multiple line segments
I'm wondering how I would go about computing the 'best fitting' intersection between multiple line segments (or even better lines of bearing) using the least squares method.
I understand how to use least squares to find the intersections of lines…
CsSingleton
- 11
1
vote
0 answers
Recursive Least Squares for Tidal Prediction
Currently I am developing an online algorithm for harmonic tidal prediction.
The measurements collected are $(y_i , t_i)$, with $y_i$ the height of the water (in meter) and $t_i$ the time of measurement (in hours). As the tide is (assumed to be) a…
Mathijs
- 21
1
vote
0 answers
Least squares with non-negative eigen values
I am trying to use least squares to solve a problem of the form
$$
u=-Kv
$$
where $u$ and $v$ are $3$-dimensional vectors, and $K$ is a $3\times3$ matrix. I want to estimate $K$, given $u$ and $v$. I have multiple data for $u$ and $v$, setup with…
1
vote
0 answers
Design matrix for multivariate Euler function
I want to use the least squares adjustment to get the parameter $c$ and $a$ of the following formula:
$$f(a,c) = c \cdot e^{-a^2 \cdot r^2}$$
For the design matrix, I used the Taylor series:
$$f(x) = c \cdot \sum_{k=0}^{\infty}…
Dennis
- 11
1
vote
1 answer
Least square solution to the system
I am trying to solve the following problem:
Let $u_1$ and $u_2$ be two orthogonal vectors in ${\rm I\!R}^n$ and
set $a_1 = u_1$, $a_2 = u_1 + \varepsilon u_2$ for $\varepsilon>0$.
Let also $A$ be the matrix with columns $a_1$ and $a_2$ and $b$…
Michael
- 125
1
vote
2 answers
How this trade-off has been calculated for Regularized least-squares in convex-optimization boyd book
I am reading this topic of boyd book from convex optimization, but the following division i-e trade-off least square and l2 norm are difficult to understand for me. If kindly someone can explain equation, that how it has been calculated. Your…
Sohail Khan
- 181
- 6
1
vote
1 answer
Solve $\min_{B \in M} \|A - B\|_F$ by $\sum_{j=1}^3 \alpha_j \langle B_i,B_j \rangle = \langle A,B_i \rangle$
Consider $M = \{ A \in \mathbb R^{2,2}: A = A^T\}$ and $\langle A,B \rangle := \operatorname{tr}(A^T B)$ which is a scalar product on $\mathbb R^{2,2}$ and induces $\|A\|_{F} = \sqrt{\langle A,A \rangle}$. The task is to find $\min_{B \in M} \|A -…
Pazu
- 1,077
1
vote
0 answers
The penrose inverse of augmented system
Assume y is the solution to a least squares problem and X $\in$ $R ^{m x n}$, where m $\geq$ n. In addition, X will have a full column rank, could somebody please explain how to take the inverse of the following 2x2 block matrix in the least squares…
gohan
- 11
1
vote
0 answers
Linear least squares with spans instead of points
I'm familiar with using linear least squares to find the best fit line of a set of points. I have a slightly different situation where I have spans as well as points.
Example:
I'm trying to find the linear best fit where the line crosses through…
m35
- 11