Least squares Definition and 158 Threads

With MATLAB or something. Basically, I just have a bunch of data points where I should do a linear least squares fit, but each of the points have error bars around them.

Homework Statement Let A= |2 -1 -1| |-1 2 -1| |-1 -1 2| and B= |1| |2| |3| Homework Equations Find the x in which minimizes ||Ax-b||2 The Attempt at a Solution I tried to solve it by using this formula (A**A)-1A**b=x but i get the inverse of A*A equal 0

How should I go about solving this problem? This is only to get a better understanding of how NLLS works. F(x;a) = (1+a1*x)/(a2+a3*x) (so n = 3) I am choosing a1,a2,a3 to be 2,3,5 respectively. I am also picking 6 data points (so m = 6): (0, 0), (-1/4, 1/4), (-1/2, 1/10), (1/4, 1/4)...

I'm going through some methods to solve the LLS method of minimization and have come upon 3 general methods to solve the problem. The 3 methods I am looking at are normal equations, QR factorization, and SVD. I've come upon a fact that I can't find an explanation for: Can anyone explain why...

Homework Statement For the system Ax = b, find least squares solution. A = (1 2) ; b = (3, 2, 1)T ----(2 4) ----(1 -2) Homework Equations I know if A is an m x n matrix of rank n, the normal equations ATAx = ATb have a unique solution x =...

Homework Statement Suppose the columns of A are not independent. How could you find a matrix B so that P=B(BTB)^-1BT does give the projection onto the column space of A? (The usual formula will fail when AT A is not invertible). T is transpose. Homework Equations The Attempt at a Solution I...

Homework Statement Suppose a set of N data points {(xk,yk)}Nk=1 appears to satisfy the relationship for some constants a and b. Find the least squares approximations for a and b. Homework Equations The Attempt at a Solution I really have no idea about this problem.

Problem: A = n by m matrix x = m by 1 vector y = n by 1 vector C = c by m matrix E = e by m matrix Alpha, gamma and theta are constants. norm(Ax-y) = min subject to: norm(Cx) = alpha norm(Ex) = gamma transpose(Cx)*Ex = (alpha^2)*(gamma^2)*cos(theta) I read a paper on how to do this with 1...

I have a simple problem. I have a set of 3D data points and I want to fit a line through them using linear least squares. I understand the basic approach required: set up two matrices such that Ax = b, then make it a square matrix A^t*Ax = A^t*b, then solve for x using a Cholesky decomposition...

Homework Statement The distance traveled by a comet is described with the following equation: r = B + re cos (θ) B (Beta) and e are constants θ 0.88 1.10 1.42 1.77 2.14 2.91 3.85 r 8.27 5.49 3.53 2.50 1.95 1.52 5.21 These were some of the measurements, which can be written in...

I'm studying for my numerical analysis final on tuesday, and I know this is going to be one of the problems, so any help is greatly appreciated. Homework Statement State and prove existence and uniqueness for the solution of the linear least squares problem. Homework Equations y \approx...

I'm doing a replication paper and having a bit of issues. The author uses 2 instruments and has 4 endogenous variables. Since he is under-identified, he runs 4 separate regressions, so that each equation has 1 overidentifying restrictions. I want to extend this and run the four equations as a...

I have the following problem: I have a set of m measurements $\mathbf{\phi}$ and I estimate a set of 3 variables $\mathbf{x}$ The estimated value for $\mathbf{\phi}$ depends linearly on $\mathbf{x}$ : Hx=\Tilde{\phi} The solution through weighted linear least squares is: $\mathbf{x}$ =...

Homework Statement Suppose you have a set S of three points in R^2, S1 = {(1, 12), (2, 15), (3, 16)} S2 = {(1, 12), (1, 15), (3, 16)} S3 = {(1, 12), (2, 15), (2, 15)} which you seek to interpolate with the quadratic polynomial p(t) = a_0 + a_1t + a_2t^2. Problem: Least-Squares...

How would one know when to find the least squares approximation?

Homework Statement http://img683.imageshack.us/img683/4744/leastsquares.jpg [PLAIN][PLAIN]http://img149.imageshack.us/img149/4793/graphwd.jpg Homework Equations The Attempt at a Solution So would these be the points? (-41,51),(-22,62),(23,63),(44,24) I'm not too sure how...

Homework Statement for vector space C[-1,1] with L^2 inner product <f,g> = \intf(x)g(x)dx find the best least squares approximation for function x^(1/3) on [-1,1] by a quadratic function q(x) = c0 + c1x + c2x^2 Homework Equations s+r = n <t^s, t^r> = \intt^ndt = { 2/(n+1)...

Can you give me a "least squares" example? Assume that, I have a function to estimate like below: f(x) = a3x3 + a2x2 + a1x1 + a0x0 After several experiments I have obtained these (x, f(x)) pairs: (x1, y1) (x2, y2) (x3, y3) (x4, y4) (x5, y5) (x6, y6) How can I estimate a0, a1, a2...

Is this possible? I've computed a multiobjective least squares solution and want to make it able to be updated recursively but I get stuck at applying the woodbury matrix identity since it's no longer a rank 1 udpate. Are there any derivations of this anywhere or is this not possible? Thanks

Hello, I am trying to write an algorithm to calculate the Least Squares Fit Line of a 3D data set. After doing some research and using Google, I came across this document, http://www.udel.edu/HNES/HESC427/Sphere%20Fitting/LeastSquares.pdf (section 2 in page 8) that explains the algorithm for...

1. Homework Statement Suppose two variables x and y are known to satisfy a relation y=Bx. That is a graph of x vs. y is a line through the origin. Suppose further that you have N measurements (xi,yi)and that the uncertainties in x are negligible and those in y are equal. Prove the best...

I did a least squares fit project for physics and now i have to say the value of G and the slope. I know that slope is m from the equation y = mx+b but how do i determine G?

I was wondering if someone could explain how to compute the Least squares fit to a straight line

I was wondering if someone could explain how to compute the Least squares fit to a straight line

I think that this is best suited here as it is linear algebra specific... sorry if I'm wrong. Please look at: I can do parts a,b and c. But I can't do part d. I've been trying to turn it into n independent least squares equations. Let me know if this is not the way to go or you have...

This test question is really boggling me and my math group. Any help would be appreciated. We know that AT*x=AT*b is the setup, but we're not so sure how to approach the problem most effectively. Here's the question: Use the data below to find an approximate formula for the life expectancy in...

Homework Statement For the following data, find the least squares fit of the given form x=1,2,3,4,6 y=14,10,8,6,5 h(x)=ae^x+be^(-x) Homework Equations The Attempt at a Solution So I tried to linearize the equation by taking the natural log of everything...

Hello, I am trying to figure out how to fit a plane passing through the origin in \mathbf{R}^3, given a cloud of N points. The points are vectors of the form (x_1^{(k)}, x_2^{(k)}, x_3^{(k)}) , where k stands for the k-th point. I want to minimize the sum of squared distances point-plane. What...

I am trying to fit a transformation from one set of coordiantes to another. x' = R + Px + Qy y' = S - Qx + Py Where P,Q,R,S are constants, P = scale*cos(rotation). Q=scale*sin(rotation) There is a well known 'by hand' formula for fitting P,Q,R,S to a set of corresponding points. But I...

I am doing a calculation involving taking three or more temperature measurements and then plotting them against another quantity (dependent). I get a relationship that is pretty linear, so I take the line of best fit to obtain an equation with a slope and an intercept. Now, my question is...

Hello everybody, I have two questions on conditional expectation w.r.t (Polynomial) OLS: Let X_t be a random variable and F_t the associated filtration, Vect_n{X_t} the vector space spanned by the polynomials of order {i, i<=n }, f(.) one function with enough regularity. I am wondering how...

All - Given a set of data {(xi, yi)| i = 1,2,...,m} and the regression equation f(x) = ax + b, I want to use the simplex method to minimize the equation Sigma [(yi - f(xi))/f(xi)]^2. However, I am stuck on how to initially organize the problem. I am not sure whether the equation, Sigma [(yi -...

Least squares regression of Y on A-D based on sample size of 506 [FONT="Courier New"] Y = 11.08 - 0.954*A - 0.134*B + 0.255*C - 0.052*D s.errs (0.32) (0.117) (0.043) (0.019) (0.006) R^2 = 0.581 problem A. Test null that coefficient on D is equal to 0 d = coefficient on...

Hey guys, long time lurker, first time poster! Just having some trouble with something..Im probably just looking at it the wrong way, but I was wondering if anyone could help me with this.. Im trying to prove that by choosing b0 and b1 to minimize...

Hello, I am a first year undergraduate university student majoring in Engineering and Computing Sc. One of my courses is Linear Algebra. We have been given an assignment in which question no. 2 is out of syllabus. It is on Least Squares Regression Analysis. This has not been taught to us. We...

Homework Statement We have the following x, y values x ||| y 1.0 -0.15 1.5 0.24 2.0 0.68 2.5 1.04 3.0 1.21 3.5 1.15 4.0 0.86 4.5 0.41 5.0 -0.08 How can you find the equation y(x) = ax^2 + bx + c by least squares? The Attempt at a Solution I know how to...

Homework Statement In the least squares method the vector x* that is the best approximation to b statisfies the Least squares equation: A^T A x^*= A^T b Prove that there's always a solution to this equation. Homework Equations - The Attempt at a Solution I distinct 2...

Could someone show me exactly how to derive the quadratic equation from the least squares method? I have no idea where to start. I will appreciate it very much. Thankyou.

Hi, I am trying to learn some numerical algebra. Now I don't understand the following. I'm finding the solution to the Linear Least Squares problem min||A\lambda-y||_{2}, which turns out to be (1,1). I did this by doing a QR factorization using Givens rotations. with: A= \[...

Does a http://en.wikipedia.org/wiki/Tikhonov_regularization" solution for least squares have to be iteratively solved? Or is there a way to perform regularization via linear algebra, the way linear regression can be done by solving the (XTX)B=XTy normal equations?

Does anyone know where to find source code for a simple and fast least squares solver written purely in C++?

Dear all, Apologies if this is in the wrong forum. I have a bit Nonlinear least squares fit problem. I have a pair of parametric equations (see attached, fairly nasty :frown: ). in it, a b c x0 y0 z0 are all constant, and they are the values I want to determine from a nonlinear least...

You must have used it couple of times while solving an engineering problem. For example in line fitting, why do we have to square? Can't we just pass the line thru the max number of points. Can someone explain. Thanks in advance.

Homework Statement I have an equation as a function of time. (eq1) C(t) = Css + a(e^.5t) + b(e^.9t) t>0 Where, Css is a constant. then I have 6 data points of time and C (Concentration of a liquid) 1. I have to find an equation to find the maximum time and contains a, b and Css...

Hi, Forgive me if the subject of this post is not accurate, I'm not quite sure what the correct terminology would be for what I'm trying to figure out. Currently I am using linear least squares via SVD to find the coefficients of a ten term polynomial, say f. This model allows me to...

I am currently working on a lab report for my physics class. During the lab, we used air tracks, gliders, and a photogate to measure the value of 'g'. Basically, we would raise one end of the air track to a certain height and let the glider slide down the frictionless track and the timer would...

My google search just turns up results telling me that one of the assumptions I have to make is that each Y is normal. My question is why do I have to assume its normal. Why does it follow that it has to be normal as opposed to some other distribution? Hope that makes sense. Edit: I thought...

Homework Statement why did you use the least squares method for finding m, rather than the standard slope formula? Homework Equations The Attempt at a Solution I am totally confused about why you have to use the least squares method

Homework Statement An experiment was conducted on a liquid at varying temperatures and the volume obtained at the differing temperatures are as follows: V/cm3 θ/oC 1.032 10 1.063 20 1.094 29.5 1.125 39.5 1.156 50 1.186 60.5 1.215 69.5 1.244 79.5 1.273 90 1.3 99 Assume that V...

Anybody know the math/theory behind linear least squares where the curve is forced to go through the first and last data points?I'm specifically dealing with cubic polynomials. In standard linear least squares formulation (i.e. ATAc = ATy) the curve doesn't, in general, go through any of...