Non-linear least squares - Revision history

Vipul at 22:02, 9 May 2014

2014-05-09T22:02:33Z

← Older revision		Revision as of 22:02, 9 May 2014
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	'''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of [[linear least squares]]. It occurs frequently in the context of optimization problems.		'''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of [[linear least squares]]. It occurs frequently in the context of optimization problems.

	Consider the following setup: we have a model function <math>y = f(x,\\vec{\beta})</math> (here, <math>x</math> may be a scalar or vector variable, but <math>y</math> must be scalar; for simplicity, we will notationally treat <math>x</math> as a scalar). The vector <math>\vec{\beta}</math> is an unknown parameter vector with <math>n</math> coordinates <math>\beta_1, \beta_2, \dots, \beta_n</math>. We are given a set of <math>m</math> data points <math>(x_1,y_1), (x_2,y_2), \dots, (x_m,y_m)</math> with <math>m \ge n</math>.		Consider the following setup: we have a model function <math>y = f(x,\vec{\beta})</math> (here, <math>x</math> may be a scalar or vector variable, but <math>y</math> must be scalar; for simplicity, we will notationally treat <math>x</math> as a scalar). The vector <math>\vec{\beta}</math> is an unknown parameter vector with <math>n</math> coordinates <math>\beta_1, \beta_2, \dots, \beta_n</math>. We are given a set of <math>m</math> data points <math>(x_1,y_1), (x_2,y_2), \dots, (x_m,y_m)</math> with <math>m \ge n</math>.

	For <math>1 \le i \le m</math>, we define the residual <math>r_i</math> as follows:		For <math>1 \le i \le m</math>, we define the residual <math>r_i</math> as follows:

Vipul at 22:02, 9 May 2014

2014-05-09T22:02:13Z

← Older revision		Revision as of 22:02, 9 May 2014
Line 2:		Line 2:

	'''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of [[linear least squares]]. It occurs frequently in the context of optimization problems.		'''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of [[linear least squares]]. It occurs frequently in the context of optimization problems.

			Consider the following setup: we have a model function <math>y = f(x,\\vec{\beta})</math> (here, <math>x</math> may be a scalar or vector variable, but <math>y</math> must be scalar; for simplicity, we will notationally treat <math>x</math> as a scalar). The vector <math>\vec{\beta}</math> is an unknown parameter vector with <math>n</math> coordinates <math>\beta_1, \beta_2, \dots, \beta_n</math>. We are given a set of <math>m</math> data points <math>(x_1,y_1), (x_2,y_2), \dots, (x_m,y_m)</math> with <math>m \ge n</math>.

			For <math>1 \le i \le m</math>, we define the residual <math>r_i</math> as follows:

			<math>r_i = y_i - f(x_i,\vec{\beta})</math>

			Our goal is to find a choice of the parameter vector <math>\vec{\beta}</math> for which the sum is minimized:

			<math>\sum_{i=1}^m r_i^2</math>

			In other words, we want to minimize the sum:

			<math>\sum_{i=1}^m (y_i - f(x_i,\vec{\beta}))^2</math>

			===How linear least squares is a special case===

			The case of [[linear least squares]] is the case where the function <math>f(x,\vec{\beta})</math> is linear as a function of the vector <math>\vec{\beta}</math> for each value of <math>x</math>. It need not be linear in <math>x</math>.

Vipul: Created page with "==Definition== '''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of linear least squares. It occurs frequently in the conte..."

2014-05-09T21:43:40Z

Created page with "==Definition== '''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of linear least squares. It occurs frequently in the conte..."

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==Definition==

'''Non-linear least squares''' ('''NLLS''') is a generalized problem type related to the problem of [[linear least squares]]. It occurs frequently in the context of optimization problems.