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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク nonlinear least squares ： ウィキペディア英語版 nonlinear least squares Non-linear least squares is the form of least squares analysis used to fit a set of ''m'' observations with a model that is non-linear in ''n'' unknown parameters (''m'' > ''n''). It is used in some forms of non-linear regression. The basis of the method is to approximate the model by a linear one and to refine the parameters by successive iterations. There are many similarities to linear least squares, but also some significant differences. == Theory == Consider a set of $m$ data points, $(x\_1,\; y\_1),\; (x\_2,\; y\_2),\backslash dots,(x\_m,\; y\_m),$ and a curve (model function) $y=f(x,\; \backslash boldsymbol\; \backslash beta),$ that in addition to the variable $x$ also depends on $n$ parameters, $\backslash boldsymbol\; \backslash beta\; =\; (\backslash beta\_1,\; \backslash beta\_2,\; \backslash dots,\; \backslash beta\_n),$ with $m\backslash ge\; n.$ It is desired to find the vector $\backslash boldsymbol\; \backslash beta$ of parameters such that the curve fits best the given data in the least squares sense, that is, the sum of squares :$S=\backslash sum\_^r\_i^2$ is minimized, where the residuals (errors) ''ri'' are given by :$r\_i=\; y\_i\; -\; f(x\_i,\; \backslash boldsymbol\; \backslash beta)$ for $i=1,\; 2,\backslash dots,\; m.$ The minimum value of ''S'' occurs when the gradient is zero. Since the model contains ''n'' parameters there are ''n'' gradient equations: :$\backslash frac=2\backslash sum\_i\; r\_i\backslash frac=0\; \backslash quad\; (j=1,\backslash ldots,n).$ In a non-linear system, the derivatives $\backslash frac$ are functions of both the independent variable and the parameters, so these gradient equations do not have a closed solution. Instead, initial values must be chosen for the parameters. Then, the parameters are refined iteratively, that is, the values are obtained by successive approximation, :$\backslash beta\_j\; \backslash approx\; \backslash beta\_j^\; =\backslash beta^k\_j+\backslash Delta\; \backslash beta\_j.\; \backslash ,$ Here, ''k'' is an iteration number and the vector of increments, $\backslash Delta\; \backslash boldsymbol\; \backslash beta\backslash ,$ is known as the shift vector. At each iteration the model is linearized by approximation to a first-order Taylor series expansion about $\backslash boldsymbol\; \backslash beta^k\backslash !$ :$f(x\_i,\backslash boldsymbol\; \backslash beta)\backslash approx\; f(x\_i,\backslash boldsymbol\; \backslash beta^k)\; +\backslash sum\_j\; \backslash frac\; \backslash left(\backslash beta\_j\; -\backslash beta^\_j\; \backslash right)\; \backslash approx\; f(x\_i,\backslash boldsymbol\; \backslash beta^k)\; +\backslash sum\_j\; J\_\; \backslash ,\backslash Delta\backslash beta\_j.$ The Jacobian, J, is a function of constants, the independent variable ''and'' the parameters, so it changes from one iteration to the next. Thus, in terms of the linearized model, $\backslash frac=-J\_$ and the residuals are given by :$r\_i=\backslash Delta\; y\_i-\; \backslash sum\_^\; J\_\backslash \; \backslash Delta\backslash beta\_s;\; \backslash \; \backslash Delta\; y\_i=y\_i-\; f(x\_i,\backslash boldsymbol\; \backslash beta^k).$ Substituting these expressions into the gradient equations, they become :$-2\backslash sum\_^J\_\; \backslash left(\; \backslash Delta\; y\_i-\backslash sum\_^\; J\_\backslash \; \backslash Delta\; \backslash beta\_s\; \backslash right)=0$ which, on rearrangement, become ''n'' simultaneous linear equations, the normal equations :$\backslash sum\_^\backslash sum\_^\; J\_J\_\backslash \; \backslash Delta\; \backslash beta\_s=\backslash sum\_^\; J\_\backslash \; \backslash Delta\; y\_i\; \backslash qquad\; (j=1,\backslash dots,n).\backslash ,$ The normal equations are written in matrix notation as :$\backslash mathbf.$ When the observations are not equally reliable, a weighted sum of squares may be minimized, :$S=\backslash sum\_^m\; W\_r\_i^2.$ Each element of the diagonal weight matrix W should, ideally, be equal to the reciprocal of the error variance of the measurement.〔This implies that the observations are uncorrelated. If the observations are correlated, the expression :$S=\backslash sum\_k\; \backslash sum\_j\; r\_k\; W\_\; r\_j\backslash ,$ applies. In this case the weight matrix should ideally be equal to the inverse of the error variance-covariance matrix of the observations.〕 The normal equations are then :$\backslash mathbf.$ These equations form the basis for the Gauss–Newton algorithm for a non-linear least squares problem. 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.