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Topic: Linear least squares

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	Linear least squares - Wikipedia, the free encyclopedia
		Linear least squares is a mathematical optimization technique to find an approximate solution for a system of linear equations that has no exact solution.
		It is widely and erroneously thought that the word linear in the term linear regression refers to the linear or affine nature of the fitted function.
		is still a linear regression model, for the right-hand side is a linear combination of the parameters α, β, and γ; moreover, the least-squares estimates of those parameters are linear in the vector of observed y-values.
	en.wikipedia.org /wiki/Linear_least_squares (694 words)

	Least squares - Wikipedia, the free encyclopedia
		Least squares is a mathematical optimization technique which, when given a series of measured data, attempts to find a function which closely approximates the data (a "best fit").
		+ bx + c, estimating a, b, and c by least squares, is an instance of linear regression because the vector of least-square estimates of a, b, and c is a linear transformation of the vector whose components are f(x
		Least squares estimation for linear models is notoriously non-robust to outliers.
	en.wikipedia.org /wiki/Least_squares (855 words)

	PlanetMath: least squares
		The general problem to be solved by the least squares method is this: given some direct measurements y of random variables, and knowing a set of equations f which have to be satisfied by these measurements, possibly involving unknown parameters x, find the set of x which comes closest to satisfying
		M.L. Ralston and R.I. Jennrich, Dud, a Derivative-free Algorithm for Non-linear Least Squares, Technometrics 20-1 (1978) 7.
		This is version 3 of least squares, born on 2002-01-03, modified 2003-07-12.
	www.planetmath.org /encyclopedia/LeastSquaresProblem2.html (324 words)

	Least Squares Parameter Estimation (Regression Analysis)
		The method of linear least squares is used for all regression analysis performed by Weibull++, except for the cases of the three-parameter Weibull, mixed Weibull, gamma and generalized gamma distributions where a non-linear regression technique is employed.
		The method of least squares requires that a straight line be fitted to a set of data points, such that the sum of the squares of the distance of the points to the fitted line is minimized.
		Least squares is generally best used with data sets containing complete data, that is, data consisting only of single times-to-failure with no censored or interval data.
	www.weibull.com /LifeDataWeb/least_squares.htm (566 words)

	4.1.4.1. Linear Least Squares Regression
		The "method of least squares" that is used to obtain parameter estimates was independently developed in the late 1700's and the early 1800's by the mathematicians Karl Friedrich Gauss, Adrien Marie Legendre and (possibly) Robert Adrain [Stigler (1978)] [Harter (1983)] [Stigler (1986)] working in Germany, France and America, respectively.
		In the least squares method the unknown parameters are estimated by minimizing the sum of the squared deviations between the data and the model.
		The estimates of the unknown parameters obtained from linear least squares regression are the optimal estimates from a broad class of possible parameter estimates under the usual assumptions used for process modeling.
	www.itl.nist.gov /div898/handbook/pmd/section1/pmd141.htm (883 words)

	Logistic Regression
		Among other situations, linear least squares regression is the thing to do when one asks for the best way to estimate the response from the predictor variables when they all have a joint multivariate normal distribution.
		With linear least squares regression, the response variable is a quantitative variable.
		A multiple linear least squares regression was fitted with a 0/1 variable as a response.
	www.tufts.edu /~gdallal/logistic.htm (992 words)

	PlanetMath: linear least squares
		This is clearly an inconsistent system of linear equations, with more equations than unknowns, a frequently occurring problem in experimental data analysis.
		Cross-references: equations, system of linear equations, inconsistent, trajectories, lines, straight, vertex, point, singular value decomposition, QR decomposition, ill-conditioned, normal equations, column, vector, residual, Euclidean norm, least squares, solution, matrix
		This is version 1 of linear least squares, born on 2002-01-03.
	www.planetmath.org /encyclopedia/LinearLeastSquares.html (254 words)

	Least squares (Site not responding. Last check: 2007-11-01)
		Least squares is a mathematical optimization technique that attempts to find a fit" to a set of data by to minimize the sum of the squares the differences (called residuals) between the fitted function and the
		Many other optimization problems can also expressed in a least squares form either energy or maximizing entropy.
		There's a revolution of sorts in statistical modeling in recent years, first from recognizing the effects of model selection bias on inference and prediction, then from the impetus of boosting and then bagging from cs, statistical modeling is probably not...
	www.freeglossary.com /Method_of_least_squares (392 words)

	Weighted linear least squares (Site not responding. Last check: 2007-11-01)
		Linear least squares problems are very common in applications.
		The classical example of a linear least squares problem is the problem of finding the best fit of a line to some measured data.
		The minimization problem attained is to minimize the sum of the square of the distances between the y-coordinate for the measured data and the line.
	www.cs.umu.se /~marten/forskning/wlsq_summary.html (209 words)

	Linear Regression (Least Squares) for the Gompertz Models (Site not responding. Last check: 2007-11-01)
		The method of least squares requires that a straight line be fitted to a set of data points.
		, then the sum of the squares of the vertical deviations from the points to the line is minimized.
		, the line is fitted to a set of data points such that the sum of the squares of the horizontal deviations from the points to the line is minimized.
	www.weibull.com /RelGrowthWeb/Linear_Regression_(Least_Squares)_for_the_Gompertz_Models.htm (152 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		Linear least square approaches are appropriate when observational noise predominates and are inappropriate when significant modeling errors are present.
		In this thesis, total least squares approaches are presented and evaluated on experimental data.
		Although total least squares based methods are computationally more demanding, simulation results show that they provide better concentration estimates in terms of Euclidean norm.
	www.cdsp.neu.edu /theses/thesis01/Sirkeci_ms_Thesis.htm (251 words)

	Least Squares Polynomials
		least squares polynomial of degree m of the form
		There are various linear system solvers that could be used for this task.
		The linear least-squares problem is stated as follows.
	math.fullerton.edu /mathews/n2003/LeastSqPolyMod.html (326 words)

	General Least-Squares - Direct Solutions and Bundle Adjustments
		The linear case is an adjustment using zero as the initial guess of all parameters.
		The linear case is mathematically the same as the general case where an adjustment is performed using zero as the initial guess of all parameters.
		Note that by using the linear form of the circle equation, the square of the distance from each point to the circle in not the value that is being minimized.
	www.orbitals.com /self/least/least.htm (2327 words)

	Linear Least Squares
		Although the proof is beyond the scope of this text, it can be shown that a least squares solution of the overdetermined linear system (3.6) must satisfy the normal equations (3.8).
		Conversely, any solution of the normal equations is a least squares solution of the overdetermined linear system (3.6).
		Notice that the problem we have stated is linear, even though we are fitting data based on a nonlinear model.
	ceee.rice.edu /Books/CS/chapter3/data34.html (847 words)

	Linear least squares - InformationBlast
		The database is read-only and using an older copy while some serious problems are fixed, sorry for the inconvenience this may cause.
		This usually happens if the number of equations is bigger than the number of variables.
		The linear least squares method can be used to find a linear function R
	www.informationblast.com /Linear_least_squares.html (306 words)

	How to perform a linear least-squares analysis
		In general it is not a // good idea to solve a set of linear equations with a matrix inversion.
		Least-squares analysis with Minuit // An objective function L is minimized by Minuit, where // L = sum_i { (y - c_0 -c_1 * x / e)^2 } // Minuit will calculate numerically the derivative of L wrt c_0 and c_1.
		// For ill-conditioned linear problems it is better to use the fact it is // a linear fit as in 2.
	root.cern.ch /root/html/examples/solveLinear.C.html (627 words)

	Linear Least Squares Regression
		The relation(s) between the predictor(s) and the criterion may be nonlinear.
		Note that in some cases, relations that are theoretically nonlinear may be transformed into linear relations.
		However, some relations cannot be transformed to linear (intractable nonlinearity).
	darkwing.uoregon.edu /~mauro/psy612/LINEAR.htm (774 words)

	Linear least squares estimation of the first order moving average parameter
		We propose an iterative procedure to minimize the sum of squares function which avoids the nonlinear nature of estimating the first order moving average parameter and provides a closed form of the estimator.
		The asymptotic properties of the method are discussed and the consistency of the linear least squares estimator is proved for the invertible case.
		We perform various Monte Carlo experiments in order to compare the sample properties of the linear least squares estimator with its nonlinear counterpart for the conditional and unconditional cases.
	ideas.repec.org /p/bar/bedcje/200280.html (276 words)

	log linear least squares method
		Linear programming Technique for finding the maximum value of some equation, subject to stated linear constraints.
		Linear regression A statistical technique for fitting a...
		Quick Links v.r.d.b growth phase log linear least squares method investment strategy committee special purpose entity forbes 500...
	www.mcclproject.org /money/log+linear+least+squares+method (348 words)

	Linear Least Squares Applet (Site not responding. Last check: 2007-11-01)
		In the linear least squares applet, enter x and y data in the text area.
		The straight line can be adjusted manually to fit the data by dragging it around: positioning the cursor toward either end of the line will allow that end to move while positioning the cursor toward the middle moves the whole line up and down.
		The least-squares fit minimizes the residual standard deviation (based on the sum of the squares of the deviations between the fit and the data).
	www.dartmouth.edu /~chemlab/info/resources/linear/linear.html (191 words)

	NTU
		In addition, the student should acquire a firm grasp of: the Kalman filter, spectral and covariance factorization, performance analysis of adaptive fillters under non-stationary conditions, and (time permitting) a factual knowledge of some basic concepts concerning fundamentals of spectrum estimation, nonstationary spectrum analysis, IIR (Laguerre-based) lattice configuration, and nonlinear adaptive filtering.
		Linear least squares filtering, spectral factorization and signal modeling, adaptive filters and their figures of merit, FIR vs. IIR adaptive filters, approaches to adaptive filtering: probabilistic (Wiener-Kalman) vs. deterministic (least-squares), applications.
		Linear prediction and the innovations process, the optimum filtering problem, error performance surface, normal equations and the principle of orthogonality, minimum mean-squared error, overview of Kalman filtering.
	www.ntu.edu /ac/accourse.asp?term_id=2004%2D5&CID=CC+763%2DF (493 words)

	Linear Least Squares
		For a general linear equation, y=mx+b, it is assumed that the errors in the y-values are substantially greater than the errors in the x-values.
		The regression form which is available submits the entered data to a perl script, which calculates the above matrices and graphs the data with the regression line.
		Notice that this theory assumes the data are in a linear form.
	www.shodor.org /UNChem/math/lls/index.html (357 words)

	MUG: non-linear least squares regression (17.7.96)
		Well, technically, yes, but fitting such a curve is really an extension of the linear theory since the normal equations for such a fit are still linear.
		Bob was absolutely correct - there is no general routine for a non-linear least squares fit since each non-linear function is unique and the fit is totally dependent on the function, itself.
		To give you some idea of the nonlinearity, for each observation one must solve Kepler's equation M = E - e*sin(E) for the quantity E (called the eccentric anomaly) which is then used to compute both the function value (ie., the radial velocity) and its derivatives (6 or 7 of them usually).
	www.math.rwth-aachen.de /mapleAnswers/html/172.html (595 words)

	4.1.4.4. LOESS (aka LOWESS)
		The polynomial is fit using weighted least squares, giving more weight to points near the point whose response is being estimated and less weight to points further away.
		The subsets of data used for each weighted least squares fit in LOESS are determined by a nearest neighbors algorithm.
		The subset of data used in each weighted least squares fit is comprised of the nq (rounded to the next largest integer) points whose explanatory variables values are closest to the point at which the response is being estimated.
	www.itl.nist.gov /div898/handbook/pmd/section1/pmd144.htm (1416 words)

	Intro. to Signal Processing:Curve fitting
		The objective of curve fitting is to find a mathematical equation that describes the signal and that is minimally influenced by the presence of noise.
		The most common approach is the "linear least squares" method, which is capable of finding the coefficients of polynomial equations that are a "best fit" to the data.
		This is a linear equation that can be fit by the least squares method in order to estimate a and b (but only approximately, because the log transformation effects the weighting of the errors due to random noise).
	www.wam.umd.edu /~toh/spectrum/CurveFitting.html (694 words)

	Linear Least-Squares Data-Fitting Utility
		These errors are mine; the original FORTRAN routines have been thoroughly tested and work properly.
		For background on linear, least-squares data fitting, please visit the Background and Worked Example page (opens in a new window).
		Enter the data pairs below, and specify the degree of the polynomial to be fit to the data.
	www.akiti.ca /LinLeastSqPoly4.html (353 words)

	Difference of two squares
		Solves unsymmetric equations, linear least squares, and damped least squares for a left-hand-side matrix that is large and sparse.
		Squares that remain magic after entries are raised to various powers.
		Generates every possible combination of fl-and-white squares in a grid of 32 x 32 beginning with all white squares and ending with all fl.
	www.omniknow.com /common/wiki.php?in=en&term=Difference_of_two_squares (1529 words)

	Linear Least Squares Parameter Estimation - a PDH Online Course for Engineers and Surveyors
		When such professionals find the linear system model too restrictive for their applications, the course will enable understanding the characterization of measurement errors that is applicable to the more general nonlinear and linear-in-the-parameters system models.
		This course presents an overview of linear least squares parameter estimation theory with a focus on six basic assumptions that can be made about the measurement errors.
		The measurement error assumption sets for which least squares is the appropriate estimation technique are clearly delineated.
	www.pdhonline.org /courses/g155/g155.htm (828 words)

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