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The primary application of the Levenberg–Marquardt algorithm is in the least-squares curve fitting problem: given a set of empirical pairs (,) of independent and dependent variables, find the parameters of the model curve (,) so that the sum of the squares of the deviations () is minimized:
Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints. [ 4 ] [ 5 ] Curve fitting can involve either interpolation , [ 6 ] [ 7 ] where an exact fit to the data is required, or smoothing , [ 8 ] [ 9 ] in which a "smooth ...
In ordinary least squares, the definition simplifies to: =, =, where the numerator is the residual sum of squares (RSS). When the fit is just an ordinary mean, then χ ν 2 {\displaystyle \chi _{\nu }^{2}} equals the sample variance , the squared sample standard deviation .
In statistics, Deming regression, named after W. Edwards Deming, is an errors-in-variables model that tries to find the line of best fit for a two-dimensional data set. It differs from the simple linear regression in that it accounts for errors in observations on both the x- and the y- axis.
These families of basis functions offer a more parsimonious fit for many types of data. The goal of polynomial regression is to model a non-linear relationship between the independent and dependent variables (technically, between the independent variable and the conditional mean of the dependent variable).
Fitting of a noisy curve by an asymmetrical peak model () with parameters by mimimizing the sum of squared residuals () = at grid points , using the Gauss–Newton algorithm. Top: Raw data and model. Bottom: Evolution of the normalised sum of the squares of the errors.
The least squares linear fit to this plot has an intercept of 0 and a slope , where corresponds to the regression coefficient for X i of a regression of Y on all of the covariates. The residuals from the least squares linear fit to this plot are identical to the residuals from the least squares fit of the original model (Y against all the ...
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.