Least squares Definition and 168 Threads

  1. N

    Comp Sci Fortran77 woes- single parameter least squares minimisation stuck

    Hey guys! Having issues with a code I'm writing. It doesn't fit the typical format and I may have already accidentally posted it in the wrong forum, so here's a copy of what was there with the newest version of the code. It'll (eventually) be a four-parameter least squares minimisation, but for...
  2. N

    Fortran Fortran77 woes- single parameter least squares minimisation stuck

    Hey guys! Having issues with a code I'm writing. It'll (eventually) be a four-parameter least squares minimisation, but for the moment I'm just trying to get the single parameter version to work. I've got the first module done but, when I run it it gets caught up in a loop and doesn't actually...
  3. N

    Nonlinear Least Squares Fitting

    Hello, So I was hoping to get some help implementing a nonlinear least squares fitting algorithm. Technically this is an extension of my previous thread, however the problem I am having now is correctly computing the algorithm So the problem definition is this: Given two sets of n 3D points...
  4. N

    Modified least squares fitting?

    Hi, I wonder if someone could give me some guidance on this problem please. I'm not a mathematician and I'm not even sure what the title of this problem should be - curve fitting, regression, function minimization? It started with something fairly simple, least squares fitting with 3...
  5. J

    Least squares estimator and distribution (simple linear regression)

    Homework Statement Under the simple linear regression model Y= A + Bx + e, where A is the intercept (a known concept), B is the slope parameter (unknown) and e is a random error term satisfying the normality assumption. If (X1,Y1)...(Xn,Yn) are the n data points observed, find the least squares...
  6. Z

    Multiple least squares regression

    Homework Statement design a regression model that will use the dataset y trial x1 x2 x3 0.08536, 1, -1, -1, -1.00000 0.09026, 2, -1, -1, -1.00000 0.10188, 1, -1, -1, -0.33333 0.09301, 2, -1, -1, -0.33333 0.10362, 1, -1, -1, 0.33333 0.09920, 2, -1, -1...
  7. Z

    Does the through-origin least squares line pass through the point (ybar, xbar)?

    Homework Statement Does the "through-origin" least squares line yhat = b*X pass through the point (ybar, xbar)? The "through-origin" model is the least squares model without the intercept. Homework Equations b = sum[YX]/sum[X^2] yhat = b*X The Attempt at a Solution when I calculate a...
  8. twoski

    Finding the Curve with Least Squares Approximation: 15 hrs

    Homework Statement Given this data: hours / value ----------- 2 | 1.6 4 | 1.5 6 | 1.45 8 | 1.42 10 | 1.38 12 | 1.36 fit a curve of the form Y ≈ ae^{-bx} What value can you predict after 15 hours? The Attempt at a Solution So i can rewrite the equation as Y ≈ log(a)-bx...
  9. K

    Does least squares solution to Ax=b depend on choice of norm

    To find the closest point to b in the space spanned by the columns of A we have: \mathbb{\hat{x}}=(A^TA)^{-1}A^T\mathbb{b} My question is, shouldn't this solution ##\hat{x}## depend on the choice of distance function over the vector space? Choosing two different distance functions might give...
  10. V

    Parameter Estimation, Non Linear Least Squares, Newton-Raphson

    Hello, Homework Statement I want to estimate a set of parameters by trying to minimize the sum of squares wrt to the parameter set given a simulation of a markov process and real data in Matlab. I have already implemented the Levenberg-Marquart (LM) Algorithm and it is converging to certain...
  11. N

    Least Squares Method- What is the Measured mean value of y ?

    Homework Statement So I'm doing a Least Squares Analysis and I'm wondering about what the 'measured mean value of y for replicate measurements of the unknown' value is supposed to be. I have no idea in the world what it's asking for. The value it is speaking of is not the same as the average...
  12. binbagsss

    The weighted least squares formula - Quick question

    The weighted least squares formula - Quick question ! m=Ʃiwi Ʃiwixiyi - Ʃiwixi Ʃiw[SIZE="2"]iy[SIZE="2"]i - In this formula, does w[SIZE="2"]i mean to include the weighting of all points - i.e. both x and y points throughout. Thanks Alot !
  13. G

    Least squares adjustment/regression - two points known distance apart

    Hi All, I'm struggling with finding a solution to an adjustment I'm working on. Thought someone else may have some thoughts? I have a kinematic time series of X,Y positions for two points (X1,Y1,X2,Y2). I know that the two points were a distance D (e.g., 100 m) apart from each other (the...
  14. B

    Substitution to turn a non-linear least squares problem into a linear one

    Hi, I want to solve an overdetermined non-linear equation with the method of least squares. Assume it's f(x) = 1 + ax + a^2 + b, and I want to estimate a and b. This is non-linear, as I said, so the derivatives of the squared residuals involve a^3 terms and are difficult to solve. Now I thought...
  15. S

    Least Squares Solutions for Inconsistent Linear Systems

    Homework Statement Hi there! First time user, so I hope I do this right. The question is: Let A be an 8x5 matrix of rank 3, and let b be a nonzero vector in N(AT). First, prove that the system Ax=b must be inconsistent. Then, how many least squares solutions will the system have...
  16. P

    Comparing Least Squares and Maximum Likelihood?

    Hi, Below is my attempt at a comparison between the two above-mentioned methods of estimation. Does anything in the table lack in validity and/or accuracy? Should any properties, advantages/disadvantages be eked out? Any suggestions/comments would be most appreciated! MLE...
  17. L

    Uncertainty for least squares fit

    I am fitting data to a parabolic equation using the least squares fit method. Each data point that goes into the fit is the average of 5 data points at that x value, so that each point has error bars that come from the standard deviation of those 5 y values. I have a fitted equation, and I...
  18. T

    Understanding Least Squares Statistics in Homework Problems

    Homework Statement I am not sure whether the answer should be b or d. It says the average of the function and the vertical scale need to be determined. In other words I won't have to determine a_0 (the vertical shift) right? Because it did not mention that? Also what is the average...
  19. D

    Linear Least Squares Excel Template

    Homework Statement I had to make a Linear Least Squares Excel template for my first week's Physics lab to analyze linear data for the rest of the semester. All of my values come out correctly, except the standard deviation values. I was wondering if someone could take a look at my template and...
  20. O

    Confidence interval from least squares fitting?

    Hello, Let me get right to my problem. I have an experimental distribution and a single-parameter theoretical distribution. I want to find the value for the best fit theoretical distribution that agrees with my experimental data for the bulk of the distributions (the tails of my...
  21. Y

    Help with applying the least squares method for solving simultaneous equations

    Hi everyone given the system of equations A1Cx + B1Cy + C1Cz = D1 A2Cx + B2Cy + C2Cz = D2 A3Cx + B3Cy + C3Cz = D3 I need to solve for Cx, Cy, Cz All other variables are known and constants. However all other variables (A,b,c,d) come from experimentally measured data and thus I cannot...
  22. N

    Mathemaica: Nonlinear Least Squares

    Hi I like to fit in Mathematica using NonlinearModelFit. When I look at the fitted parameters, there is an entry called "P-value". Here is what it means: "The p-value is the probability of observing a t-statistic at least as far from 0 as the one obtained.". I'm not quite sure what this...
  23. P

    Linear Algebra problem (Least Squares?)

    Linear Algebra problem (Least Squares? - Distance between lines) Homework Statement We have two points R = (x,x,x) and S = (y,3y,-1). All we know is that they are on lines somewhere in 3-space and that they don't cross. Need to find an x and y that minimize || R - S ||2 Homework...
  24. D

    Least squares approximation: Is smaller normal distance always better?

    I took a LA course in the spring, and was interested by the least squares method for building models. I decided to practice this concept by attempting to build a model that would predict ticket sales for the Mega Millions lottery given the jackpot amount. I have 249 data pairs of jackpot and...
  25. A

    Homogeneous least squares

    Given a homogeneous linear least squares problem: A^{T}y=0 What is the difference between minimizing y^{T}AA^{T}y (the least square error) and: y^{T}AA^{+}y=y^{T}A(A^{T}A)^{-1}A^{T}y ? Thanks.
  26. P

    Determine optimal vectors for least squares

    Hello all, I have a set of measurements that I want to fit to a linear model with a bunch of parameters. I do this as \theta=(H^TH)^{-1}H^Tx, where θ are the parameters and x is the set of measurements. The problem is that I'd like to reduce the number of parameters in the fit. I'd like to...
  27. GreenGoblin

    MHB Regression least squares (boring question? or more to it)

    yi = a + bxi + ei is the simple liner regression model as per is usual "state the assumptions on the errors ei to justify a least squares fit" ? So is this just that E(ei)=0, i can't see what else is a 'must' for this? what about that they are normally distrubited? i know the properties of the...
  28. ElijahRockers

    Why Do We Square the Deviation in Least Squares Regression?

    Homework Statement http://www.math.tamu.edu/~vargo/courses/251/HW7.pdf Given a set of points (xi,yi) and assuming f(xi) is linear, the deviation measured is F(m,b)=\sum_{i}^{n}(y_i - f(x_i))^2. There are a few different questions about this from the link above. The Attempt at a...
  29. J

    Matlab - Normal equation for linear least squares model fitting

    Homework Statement Im trying to construct a function in MATLAB that models linear least squares model fitting through the use of normal equations. Homework Equations Normal equation (A'*A*c)=(A'*y) A= [column vector of all x ; column vector of all 1] y= [column vector of all y] c = [b;a]...
  30. B

    Comp Sci Fortran Program for Least Squares Fit of Data

    (The usual format doesn't really fit this question, so I hope you won't mind if I set it up slightly differently) The task is to make a Fortran program that will read in a given set of data from a file, stopping when it reaches the end, and calculate the gradient and intercept of a best fit...
  31. C

    Least squares fitting by a constant

    say we have data set {y(t sub i), t sub i} Where i=1 2 3...m. I know how to fit these into a line of the form ax+b, but how about fitting into a constant??
  32. H

    Finding the Best Fit Line Using the Method of Least Squares

    Homework Statement number of years(x): [1,2,3,4,5,6,] number of accidents(y): [12,6,5,4,2,2] or (1,12),(2,6),(3,5),(4,4),(5,2),(6,2) Homework Equations the relation between x and y is on the form: y = a + b(1/x) by changing (1/x) to z, the model become linear instead of...
  33. X

    Calculating Least Squares: What is r and how do I use it?

    How do I calculate least squares? The way it's described is confusing to me. \hat{β}= r* sy/sx . sx and sy is the standard deviation of x and y, correct ? So let's say I have sets of numbers such as (16,19),(18,17) etc... 16 and 18 would be x and 19 and 17 would be y. So I have to calculate the...
  34. N

    Derivation of least squares containing vectors

    Hi, I'm trying to derive a result from this paper: http://www.research.yahoo.net/files/HuKorenVolinsky-ICDM08.pdf given cost function \min_{x_* , y_* } \sum_{u, i} c_{ui} (p_{ui} - x^T_u y_i)^2 + \lambda \left( \sum_u ||x_u||^2 + \sum_i ||y_||^2 \right) Where both xu and yi...
  35. M

    Least squares estimation with quadratic constraints (M*M = 0)

    Hello there, currently I am trying to solve a least squares problem of the following form: min_{M} ||Y - M*X||^2 where M is a 3x3 matrix and Y and X are 3xN matrices. However, the matrix M is of a special form. It is a rank 1 matrix which satisfies M*M = 0_{3x3} and the trace of M is zero...
  36. R

    Understanding Error Bars on Least Squares Fit Line

    Hello, What are the error bars on the least squares fit line? recall the variables are: x ; y ; x^2 ; xy ; y^2 ; yi=mx+c ; d =(y-yi) ; d^2 --------- is it d? Thanks
  37. O

    Uncertainty of a non linear least squares fit

    Hi, I have some experimental data as a function of time t and temperature T. I have done a least squares fit of the data with a function f=f(a1,a2,t,T) (the function is non linear in a1 and a2!). Optimization of e^2 = sum((yi-f(a1,a2,ti,Ti)^2) with Matlab's fminsearch gave me a1, a2 and the...
  38. T

    Quadratic least squares equation

    f(x) = –3√x, 1 ≤ x ≤ 4 (a) Find the quadratic least squares approximating function g for the function f. g(x)=?
  39. W

    Least squares - why the square?

    I have a presentation to give on the least squares method this week, and am finding the least squares method very intuitive with the exception of one thing: why is the square of the offset used, as opposed to the absolute value? I did a bit of searching, but could not find a detailed answer...
  40. T

    Finding orthonormal set using Gram-Schmidt and least squares

    Homework Statement A) Find an orthonormal set q1, q2, q3 for which q1,q2 span the column space A [1 1; 2, -1; -2,4] (this is a 3x2 matrix). B) Which fundamental subspace contains q3? C) What is the least-squares solution of Ax=b if b=[1 2 7]T? Homework Equations Gram-Schmidt...
  41. T

    Fit with least squares, Levenberg–Marquardt algorithm

    Hello guys, I need your help with understanding a fitting Algorithm, so that I could make it in a C++ program. I have the following function: g(f; f0, phi0, a, b) = phi0 + a ArcTan((f-f0)/b) Which is a function of "f", frequency. I would like to fit this function with the...
  42. R

    Calculating uncertainty using the least squares method.

    Can someone point me in the right direction or give me a rundown on how to use the least squares method in calculating the uncertainty in a gradient of a graph? We have been given a theoretical experiment with data and so on supplied and we need to find the uncertainty of the graph we have...
  43. X

    Least Squares Derivation question

    So I am learning how to use the method of least squares to find the best fit straight line and there is this one part in the derivation I do not fully understand and I was wondering if anyone could help me out. So basically we start out with the residuals of the equation of a straight line...
  44. K

    Forcing a Least squares Polynomial through a fixed point

    Hi guys, Thanks for taking the time to read the post. I have a question related to curve fitting and polynomials that i was hoping someone might be able to help me with. I have a set of x and y data points, all on a graph. I have then calculated the 4th order least squares polynomial...
  45. S

    Optimal Filter Coefficients: Correlation versus Least Squares

    I found a claim in a paper (BSSA, Vol 81, No. 6: "A Waveform Correlation Method for Identifying Quarry Explosions", By D.B. Harris) concerning finding filter coefficients. The statement is given without proof. I cannot locate a reference or theorem for the following, and have not been able thus...
  46. B

    Question about Least Squares Fitting

    Hey, I have a graph for which I am supposed to fit two linear least squares line and minimize the combined residuals (the lines intersect)... I would really appreciate some info about how to do this or what this type of data analysis is called so i can google the step-by-step method. Thanks!
  47. B

    Question about Least Squares Fitting

    Hey, Not sure if this is the right section to post this but ... I have a graph for which I am supposed to fit two linear least squares line and minimize the combined residuals (the lines intersect)... I would really appreciate some info about how to do this or what this type of data analysis...
  48. K

    Can Least Squares be Applied to Matrices for Solving a Complex Equation?

    Hi, as no unique unique analytical solution exists to my problem (as another poster pointed out), I have taken to solving it through a least squares method. My equation is as follows: (s1x1 + s2x2 - I).^2 = min. Here s1 and s2 are shift matrices (I know them), and I is a matrix of size...
  49. M

    Least Squares solution and residuals

    Homework Statement Solve the following linear equations simultaneously by the Least Squares solution and calculate the residuals. Homework Equations 3x + 2y + z = 5 x + 6y - z = -7 x - y + 2z = 3 5x - 2y = 1 The Attempt at a Solution This is a question from this guy at...
  50. L

    Weighted least squares fitting

    Hello y'all, If I have n data points (xi, yi) each with error bars in both x and y (xi_err, yi_err), should I use 1/(xi_err^2+yi_err^2) as the weight in a weighted least squares linear fit, or should the weight be a different value that has nothing to do with error bars? I've never used WLS...
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