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1answer
30 views

Find cubic Bézier control points given four points

What I need is to generate an SVG file while having a series of (x,y) ready. P0(x0,y0) P1(x1,y1) P2(x2,y2) P3(x3,y3) P4(x4,y4) P5(x5,y5) ... I need to make a ...
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0answers
13 views

Wording: l2/SSE/Sum-of-Squares Objective Function

The least-square problem is a very common optimization problem, where the objective function describes the sum over squared residua $r_n$ with respect to a parameter vector $p$: $$p \mapsto ...
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0answers
10 views

python least square optimisation with two non-linear equal constraints [migrated]

I am looking for a way to solve the optimisation problem with two non-linear equal constraints. My cost function is ...
2
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1answer
24 views

Analytic solution for matrix factorization using alternating least squares

The standard form for ridge regression aims to minimize the following cost function. $$ \min\ \ \sum_i(y_i-x_i^T\beta)^2 + \lambda\sum_j\beta^2_j $$ As described here, it's possible to differentiate ...
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0answers
32 views

Linear optimization w/ linear and non-linear inequality constraints

Given dependent variables $Q_i$ and independent variables $x_i$, $y_i$, $z_i$ where $i=1,⋯, N $ which are related via the following system of N linear equations with parameters $P_1$, $P_2$ and ...
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1answer
42 views

MATLAB curve fitting - least squares method - wrong “fit” using high degrees

Anyone here that could help me with the following problem? The following code calculates the best polynomial fit to a given data-set, that is; a polynomial of a specified degree. Unfortunately, ...
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1answer
27 views

Am I doing this approximation correctly? (least squares method)

Here is the problem. Find the function $f$ of the type $f(x) = a\cos x + b\sin x$ which best approximates the function $g$ in the points : $$ \begin{array}{ c | c | c | c | c | c | c } x & ...
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1answer
26 views

Linear regression involving angles in a triangle.

In a survey experiment, three independent measurements $29.5^{\circ}$, $30.5^{\circ}$, $120.5^{\circ}$ are obtained from the three angles $\alpha,\beta,\gamma$ of a triangle. Formulate the ...
1
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1answer
24 views

Least Squares in Matlab

I'm stuck on part (d) I'm not sure how to code it so that it approximates that function in matlab. I'm also not sure if my (a) thru (c) are correct. But this is what I have so far. ...
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0answers
20 views

Estimate expected gain for least square

The following data indicate the gain in reading speed vs the number of weeks in the program of 10 students in a speed-reading program: weeks 2 3 8 11 4 5 9 7 5 7 Speedgain 21 42 102 130 52 57 105 85 ...
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0answers
36 views

Curve fitting and regression: reading speed

The following data indicate the gain in reading speed vs the number of weeks in the program of 10 students in a speed-reading program: weeks 2 3 8 11 4 5 9 7 ...
1
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2answers
37 views

Least Squares approximation for item prices

Let's say that $A$, $B$, $C$ are different items with different values. $R$ is a unit of currency, for simplicity I'll let it be $1$. Traders frequently trade these items on an open market. Price is ...
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2answers
31 views

Minimum value of an integral with least square?

I have a problem. The question is: given a parabola $$p(t) = a + bt + ct^2$$ I need to evaluate the least squares straight line $$A + Bt$$ for which this integral is minimum: $$ \int_{t_1}^{t_2} ...
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0answers
7 views

Range-Based Localization of a Point using LSE

Suppose that we a set of points $P = \{p_1, p_2, \ldots, p_n\}$ in 3D. The coordinates of these points are known. In addition, we have another point, called $p$, We have Euclidean distances ...
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0answers
10 views

How to get the minimum singular value of some points covariance matrix?

I'm having trouble understanding the context of my question. I have a set of points which correspond to some 3D coordinates. I guess i need a minimum of two for my question. So the points would be ...
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0answers
17 views

Best fitting circle to points in 3D

I have a set of n ≥ 3 points in 3D that are measurements of a possible circle. The measured points are "noisy" so best-fitting algorithms are involved. I'm programming in C# and have put together some ...
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0answers
27 views

Lattice fitting to points

I have a set of points (shown as little black circles) which ideally form a hexagonal lattice shape, each point having an equal distance to all of its neighboring points. (Sorry for my drawing, some ...
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0answers
29 views

Does Least squares solution exist for this case?

$ {\bf{Z}} = {\bf{H}} \cdot {\bf{S}} + {\bf{N}} $ Dimensions of the matrices are as follows: Z = m X m H = m X n S = n X m (matrix S is an orthogonal matrix) N = m X m. All the elements of the ...
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1answer
20 views

Fitting a polynomial model to the data

I want to fit a line to the following data. But the line I have obtained is far from the data. What is wrong in the following least square problem? ...
0
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1answer
37 views

Please help with linear algebra least squares problem

The problem is as follows: "Please set up (but do not solve) the normal equations for the following least squares approximation problem: Find $(a, b, c, d)$ such that the plane H described by $ax + ...
1
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1answer
16 views

Prove that Least squares and decorrelator are equivalent

Here is the problem: $$\mathbf{y}=\mathbf{Ax}+\mathbf{b}$$ where $\mathbf{y,x,b}$ are vectors, and$\mathbf{A}$ is matrix(generally rectangular, but with full column rank). The least squre solution ...
1
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1answer
34 views

Toeplitz equality constrained least-square optimization

What is the fastest known algorithm for least-square optimization problem with a linear equality constrain \begin{align*} &\min \|K x - y\|^2 + \mu \|x\|^2\\ \text{s. t. }& Q x = v ...
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1answer
41 views

Is k-means clustering guaranteed to converge if using Manhattan distance?

The k-means algorithm is an iterative clustering algorithm that partitions the data points into K clusters (with centroids {$\mu_1, ... , \mu_k$}, minimizing the Sum-of-Squared-Error: $$ SSE = ...
1
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1answer
37 views

Solving Linear Least Squares with Constraint

I would like to use linear least squares to solve for $x \in \mathbb{R}^5$ where $$ Ax = b \rightarrow x = (A^TA)^{-1}A^Tb $$ but would like to include the constraint $$ x_1^2 + x_2^2 = 1 $$ I ...
0
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1answer
15 views

How does minimum squared error relate to a linear system?

Given some system $U*x = b$, I've solved for $x^*$, the least squares solution. I then compute the minimum squared error by $||U*x^* - b||^2$. I know that the least squares solution minimizes the ...
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0answers
19 views

Different SVD results in Matlab

my question relates to calculating SVD in Matlab. I have been reading a lot and somehow I have jumbled up all the facts. It would be great if you experts could get me to the right track. My task is ...
3
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1answer
33 views

When is Block-Partitioned Matrix Invertible?

Suppose I have a block partitioned matrix \begin{equation} \begin{bmatrix} \mathbf{X}_1^{\top}\mathbf{X}_1 & \mathbf{X}_1^{\top}\mathbf{X}_2 \\ \mathbf{X}_2^{\top}\mathbf{X}_1 & ...
0
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1answer
28 views

Bivariate rational (quadratic over linear) model fitting by least squares

I am trying to fit a simple model over 2D data points, in the frame of an image formation model with perspective and optical distortion. My model is the ratio of a second degree polynomial over a ...
1
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2answers
42 views

Show $E\left(\mathbf{X}_i \otimes \mathbf{u}_i\right)=\mathbf{0}$ implies $E\left(\mathbf{X}_i^{\top}\mathbf{G}\mathbf{u}_i\right)=\mathbf{0}$

Let $\mathbf{X}_i$ be a $G \times K$ random matrix, and let $\mathbf{u}_i$ be a $G \times 1$ random vector, and suppose we have a sample of $i=1,\ldots,N$ of each. Suppose the following condition ...
1
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1answer
33 views

How to Change Summation Expression $\sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{\Omega}^{-1}\mathbf{X}_i$ into Matrix Expression

Let $\mathbf{X}_i$ be a $G \times K$ matrix, and suppose are $i=1,...,N$ of these matrices. Note that \begin{align} \sum_{i=1}^N \mathbf{X}_i^{\top}\mathbf{X}_i &= \begin{bmatrix} ...
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0answers
38 views

Which projection, in $L_\infty$ norm or $L_2$ norm, is non-expansion?

I am just wondering which projection is non-expansion? Basically, I am wondering if $F$ is a projection operator then which norm would satisfy the following non-expansion property, where for a given ...
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0answers
49 views

Matrix Decompositions: Difference between Cholesky Decomposition, Eigendecomposition and Jordan Normal Form Decomposition

I recently created a related topic about the square root matrix, in case you'd like to refer to that one. Here's what we want: Consider the matrix $\Omega=E(\mathbf{u}^{\top}\mathbf{u})$, where ...
0
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1answer
12 views

Least Squares with equality Constraint

So I have a problem in the form of Y = Ax, where A is a matrix and x and Y are vectors. A is a skinny matrix and I would like to do a least squares solution to solve for x. Lets say though that x ...
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2answers
92 views

Is the Square Root of an Inverse Matrix Equal to the Inverse of the Square Root Matrix?

I know in general that if a matrix $A$ is positive definite, then there exists a (unique?) square root matrix $B$, which is also positive definite, such that $BB=A$. Therefore, suppose $A$ is ...
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0answers
32 views

Least squares with three quadratic constraints (Ellipse fitting based on algebraic distance)

I would like to fit an ellipse to a given set of scattered data in $\mathcal{R}^2$. The fitting problem is in form least squares, minimizing the sum of squared algebraic distances \begin{equation} ...
2
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0answers
45 views

Least square regression of a vector onto a space [closed]

Suppose the basis vectors for a space are [1 0 0] and [0 1 2]. Now, I would like to find the least square projection of the vector [a b b] onto the mentioned space. How do I approach this?
0
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1answer
31 views

Well-defined $\xi$-weighted (Euclidean) norm

Suppose $\xi$ is a vector, that is used for $\parallel z\parallel_\xi$ calculation. Should every element of $\xi$ be positive, $\xi(i)>0$?
2
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1answer
78 views

Step by step LMS for learning a linear function

Disclaimer Since this is an exercise assignment I'm not looking for a complete solution but for help that enables me to solve it on my own The task Given the error function ...
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0answers
23 views

Finding a relative error measure on a data set proportional to another

I have a set of exact data points $\mathcal{X}=\{X_i\}$ and another approximate one $\mathcal{Y}=\{Y_i\}$ where there is a correspondence between $X_i$ and $Y_i$ for all $i$. If $\mathcal{Y}$ was ...
0
votes
1answer
59 views

Why does SVD provide the least squares solution to $Ax=b$?

I am studying the Singular Value Decomposition and its properties. It is widely used in order to solve equations of the form $Ax=b$. I have seen the following: When we have the equation system $Ax=b$, ...
0
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1answer
16 views

How to find variance of a vector?

I am given a set of measurements: $\tilde{y_1}=x+v_1$ $\tilde{y_2}=x+v_2$ Where $v$'s are random variables with $E\{v_1\}=E\{v_2\}=E\{v_1v_2\}=0, E\{v_1^2\}=a, E\{v_2^2\}=b$. A least squares ...
0
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1answer
37 views

Understanding how to solve a Cost Function?

I'm having trouble seeing the relationship in the following equation. Let's assume $J(0,1)$ and $m=4$. First I figure out my hypothesis function ...
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0answers
113 views

Normal equations for minimization of Frobenius norm least squares error

I'm having a hard time understanding the most efficient sequence of steps for deriving the normal equations for Frobenius norm least squares minimization. Here I want to minimize the norm of a matrix ...
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0answers
54 views

Non-linear least squares solver to solve a system of non-linear equations?

Can I use a non-linear least squares solver to find the solutions of a system of non-linear equations? From Wikipedia: "The method of least squares is a standard approach to the approximate ...
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2answers
53 views

Taking derivative with respect to a vector

From time to time, I come across with derivation operations which are executed with regard to a vector. For example, the least squares estimation method with more than one explanatory variables is ...
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0answers
27 views

Perpendicular distances versus vertical distances

Why is it better to use perpendicular distance rather than vertical distance along a particular coordinate axis when finding the best fit subspace? This is an exercise question in a chapter related to ...
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0answers
42 views

When can I solve in closed form this curve fitting problem?

I have $n$ real values $x_1,x_2,\ldots,x_n$ and $n$ real values $y_1,y_2,\ldots,y_n$; then I have a function $f(x,\boldsymbol\theta)$ from $\mathbb{R}$ to $\mathbb{R}$ and depending on $m$ parameters ...
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1answer
52 views

Prove that $E(\mathbf{u}|\mathbf{X})=\mathbf{0}$ implies $Cov(\mathbf{x},\mathbf{u})=0$

Let \begin{equation} \mathbf{y}=\mathbf{X}\mathbf{\beta}+\mathbf{u} \end{equation} where $\mathbf{y}=\begin{bmatrix}y_1 \\ \vdots \\ y_n\end{bmatrix}$, $\mathbf{X}=\begin{bmatrix}X_{11} & ...
1
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1answer
79 views

how to apply non-linear least square

I'm trying to implement the example of estimating an angle between a target $\textbf{x}$ and a sensor $x_{p}$. I'm using the example in this book. There are three available measurements of the angle ...
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0answers
14 views

GMM estimation of linear regression with intercept restriction

Say I have a time series regression as follows: $$y_t = a_i + \beta_i x_t + \varepsilon_t^i \ \ ; \ \ t = 1, 2, \cdots, T \ \ \text{for each } i$$ Now say I impose the following restriction on the ...