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Topic: Least squares

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In the News (Fri 19 Jul 19)

  Least Squares Estimation Curve Fitting Program to download. Nonlinear Weighted Least Squares Regression Analysis. ...
Least Squares Estimation Curve Fitting Program to download.
This form enables applying complicated curves that are not a graph of any function, applying complicated curve equations from which none of variables can be derived as well as transforming equations to be always computable.
On the basis of input errors, the chi-sqr parameter and its standard deviation is calculated (chi-sqr expected value equals the number of degrees of freedom).
www.prz.rzeszow.pl /~janand   (1450 words) 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)

  Least Squares
The least squares criterion is a statistical approach used to provide the most accurate estimate of relationships between sets of variables in sample data.
Least squares analysis is the most popular approach to the computation of regression lines because it is relatively simple and highly accurate.
The least squares criterion produces a line in which the sum of the squared deviations of every value Y from the line are lowest.
www.referenceforbusiness.com /encyclopedia/Kor-Man/Least-Squares.html   (583 words)

 Least squares Summary
The process is called "least squares" because the procedure minimizes the sum of the squares of the differences between the observed and estimated values.
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").
Least squares estimation for linear models is notoriously non-robust to outliers.
www.bookrags.com /Least_squares   (1107 words)

 Least squares - Definition, explanation
Least squares is a mathematical optimization technique that attempts to find a "best fit" to a set of data by attempting to minimize the sum of the squares of the differences (called residuals) between the fitted function and the data.
The least squares technique is commonly used in curve fitting.
+ 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
www.calsky.com /lexikon/en/txt/l/le/least_squares.php   (766 words)

 PlanetMath: least squares
Least squares methods applied to few parameters can lend themselves to very efficient algorithms (e.g.
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 4 of least squares, born on 2002-01-03, modified 2006-10-28.
planetmath.org /encyclopedia/LeastSquares.html   (319 words)

 Method of Least Squares   (Site not responding. Last check: )
The method of least squares is an alternative to interpolation for fitting a function to a set of points.
The method of least squares is probably best known for its use in statistical regression, but it is used in many contexts unrelated to statistics.
In this regard, ordinary least squares is a generalization of ordinary interpolation.
www.riskglossary.com /articles/least_squares.htm   (475 words)

 Least Squares Parameter Estimation (Regression Analysis)
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.
The same least squares principle is applied, this time minimizing the horizontal distance between the data points and the straight line fitted to the data.
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   (574 words)

 Least Squares Approximations
This method is called the least squares fit because it finds the line that minimizes the sum of the squares of the errors.
Therefore, the sum of the squares of the errors is 27.
Using our definition of least squares "best fit," you will not be able to find a parabola that fits the data better than this one.
ceee.rice.edu /Books/LA/leastsq/index.html   (1784 words)

 * Least Squares - (Stock Market): Definition
Least squares is a common way to measure errors in statistical analysis.
Least Squares Method A technique of fitting a curve close to some given points that minimizes the sum of the squares of the deviations of the given points from the curve.
A technique of fitting a curve close to some given points to minimize the sum of the squares of the deviations of the given points from the curve.
en.mimi.hu /stockmarket/least_squares.html   (331 words)

 NMath Matrix User's Guide - 6.1 Least Squares Methods
The Cholesky least squares classes solve least square problems by using the Cholesky factorization to solve the normal equations.
Finding least squares solutions using the normal equations is often the best method when speed is the only consideration.
Finding least squares solutions via QR decomposition is the "standard" method for least squares problems, and is recommended for general use.
www.centerspace.net /doc/NMath/Matrix/user/leastsquares2.html   (240 words)

 Partial Least Squares
Ordinary least squares regression, as implemented in SAS/STAT procedures such as PROC GLM and PROC REG, has the single goal of minimizing sample response prediction error, seeking linear functions of the predictors that explain as much variation in each response as possible.
Two different formulations for partial least squares are available: the original method of Wold (1966) and the SIMPLS method of de Jong (1993).
The partial least squares method was originally developed in the 1960s by the econometrician Herman Wold (1966) for modeling "paths" of causal relation between any number of "blocks" of variables.
support.sas.com /rnd/app/da/new/dapls.html   (595 words)

 The Least Squares Approach to Regression
The method of least squares estimates the length of the spring under no load to be 439.01 cm.
The method of least squares and the regression method involve the same mathematics; but the contexts may be different.
A technical point: The least squares estimate for the length of the spring under no load was 439.01 cm.
www.analytictech.com /mb313/regress3.htm   (1376 words)

 [No title]
The equations for the L2 regression coefficients are obtained by using derivatives to minimize the sum of the squares of the error terms, and this gives closed form equations for the coefficients.
Professor Rubin is comparing "least sum of fourth powers of residuals" to _linear_ least squares, which is a direct process (not iterative) using linear algebra.
Minimizing the sum of squares, then the sum of fourth powers, then the sum of eighth powers, gives an approximate minimax fit of a model, either linear or nonlinear, to data.
www.math.ucl.ac.be /~magnus/num1a/whyleastsqnews.txt   (1215 words)

 Least Squares
The Least Squares regression line for a set of data points is the line that "best fits" the data.
To be more precise, it is the line that minimizes the sum of the squares of the vertical distances from the points to the line.
The large square has the same area as the sum of all the smaller squares, and it represents how well the line fits the data.
online.redwoods.cc.ca.us /instruct/kiyokoya/GSP/squares.html   (245 words)

 [No title]
Partial Least Squares is a linear regression method that forms components (factors, or latent variables) as new independent variables (explanatory variables, or predictors) in a regression model.
The components in partial least squares are determined by both the response variable(s) and the predictor variables.
This line is called the regression line or least squares line, because it is determined such that the sum of the squared distances of all the data points from the line is the lowest possible.
www.statsoft.com /textbook/glosp.html   (4809 words)

 More Discussion On Least Squares
Most Least Squares cave survey programs are based on an old article that appeared in the NSS Bulletin in 1970 written by V. Schmidt and J.H. Schelleng.
Weights are used by Least Squares to compensate for parts of a survey where the data is less reliable.
This is important because Least Squares is based on the assumption that the errors in the data are random.
www.fountainware.com /compass/compart2.htm   (1805 words)

 Least Squares Lines
The formulas for linear least squares fitting were independently derived by German mathematician
The least squares line is often times called the line of regression.
When you realize that two different "least squares lines" can be produced we are amazed.
math.fullerton.edu /mathews/n2003/LeastSqLineMod.html   (254 words)

 Definition of "Least Squares"
Least squares keeps track of all errors, even if some are in one direction and some are in the other direction.
According to least squares, method B is better than method A because the sum of the squares of the errors is lower for method B. Method B and method C are equally good, according to least squares.
The alerts server uses least squares and related algorithms are used to optimize a variety of models.
www.trade-ideas.com /Glossary/Least_Squares.html   (529 words)

 John Halleck's
The output of a least squares adjustment is the nearest consistent survey to the one given, where a survey is called consistent if any path you now take to a point in the survey gives the same result.
The least squares solution is an appropriately weighed average of the shots on the routes that get to each point.
If you take the view that the least squares adjustment is just a weighted average of all of the (unique, non-trivial, non-redundant) paths between two points, then it is easy to justify this.
home.utah.edu /~nahaj/cave/survey/intro/leastsquares-observations.html   (3583 words)

 * Least squares - (GIS): Definition
Least squares minimizes the sum of the squares of the error term at each point.
The algorithm is based on a novel application of the Laplacian PDE (partial differential equation) whereby a system of over determined linear equations is formulated and solved by performing a linear least squares fit to the known data and unknown...
The weighted fit leads to correlation coefficients that a least squares routine numerically fixes.
en.mimi.hu /gis/least_squares.html   (197 words)

 Least Squares Fits
Least Squares computes a set of coefficients to the specified function, that minimize the square of the difference between the original data and the predicting function.
In other words, it minimizes the square of the error between the original data and the values predicted by the equation.
During the calculation of the Power curve fit, the error is minimized on a log scale, not a linear scale.
www.synergy.com /Webhelp/Least_Squares.htm   (299 words)

 BioMed Central | Full text | Least-squares methods for identifying biochemical regulatory networks from noisy ...
The three different least squares algorithms are tested for different numbers of data points per experiment, i.e., 3, 6, 9, 12, 21, 30, 60, and the quality of the Jacobian estimations was evaluated according to a number of different definitions of estimation error, which are discussed in the Methods section.
The TLS and CTLS algorithms are extensions of the widely used least squares approach which optimally deal with the presence of uncorrelated and correlated noise in the measurements, respectively.
The TLS solution is always guaranteed to be as good or better than the least squares solution in the root mean square sense, if the number of measurements is sufficient to allow the algorithm to compute a reasonable approximation of the true correction term.
www.biomedcentral.com /1471-2105/8/8   (6796 words)

 3. Least Squares
Least squares solves the problem by finding the line for which the sum of the square deviations (or residuals) in the d direction (the noisy variable direction) are minimized.
The least squares computation for a large data set is time consuming, even with a computer.
Estimation theory says that the least square estimator is the “best linear unbiased estimator” (BLUE), since it has no bias and minimizes the variance.
www.cnel.ufl.edu /nsbook/chapter13_Least_Squares.html   (1104 words)

 Least Squares Fitting (Regression)
The counting of annual growth rings should be precise; of course the trees are probably a bit older than the count as it took a few years for the tree to grow to the height at which the core was taken.
The line is adjusted until the sum of the squares of the y deviations from the line (shown above in blue) are as small as possible.
You may be guided by the suggestions of known theory, by the requirements of a particular instructor, by standard practice (usually a least squares line), by knowledge of which points are most likely to be anomalous, or (unfortunately) by a desire to produce a particular answer.
www.physics.csbsju.edu /stats/least_squares.html   (1507 words)

 Fitting a Line to (x,y) Data [beyond OLS]
In his 1809 book on planetary orbits Carl Friedrich Gauss published the method of least squares and in addition provided a hint as to why least-squares is preferred if errors are distributed "normally" (i.e., with a Gaussian distribution).
While the method of least squares is simple and in many cases the proper method to apply, it is by no means the only proper method to fit data to lines.
The method of least square puts x and y on decidedly unequal footings: "error" means vertical (y) deviation rather than horizontal deviation (x).
www.physics.csbsju.edu /stats/fitting_lines.html   (1427 words)

 least squares definition - Dictionary - MSN Encarta
least squares definition - Dictionary - MSN Encarta
Search for "least squares" in all of MSN Encarta
It involves squaring the distance that each point is from a given curve, summing the squares, and choosing the curve for which the sum has the minimum value.
encarta.msn.com /dictionary_1861693169/least_squares.html   (113 words)

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