MATLAB数据拟合进阶技巧：解锁高级方法，探索数据拟合的无限可能

# 1. MATLAB数据拟合概述**

数据拟合是利用数学模型描述给定数据集中数据的过程。在MATLAB中，可以使用各种函数和工具箱来执行数据拟合。

MATLAB数据拟合的基本步骤包括：

- **数据准备：**加载数据、预处理和探索数据以识别模式和异常值。
- **模型选择：**根据数据的性质和拟合目标选择合适的数学模型。
- **参数估计：**使用最小二乘法或其他优化算法估计模型参数。
- **模型评估：**使用残差分析、交叉验证和其他技术评估拟合模型的准确性和鲁棒性。

# 2. 高级拟合算法**

**2.1 非线性最小二乘法**

非线性最小二乘法 (NLLS) 是一种用于拟合非线性函数到数据的算法。它通过最小化平方和误差来找到一组参数，使得拟合函数与数据点之间的距离最小。

**2.1.1 Levenberg-Marquardt 算法**

Levenberg-Marquardt (LM) 算法是 NLLS 中最常用的算法之一。它结合了高斯-牛顿法和梯度下降法的优点。LM 算法在目标函数接近最小值时表现出快速的收敛速度，并且在目标函数非凸时也能保持鲁棒性。

function [x, resnorm, residual, exitflag, output] = lsqnonlin(fun, x0, lb, ub, options)
%LSQNONLIN  Nonlinear least squares minimization.
%   X = LSQNONLIN(FUN,X0) finds a local minimum of the sum of squares of
%   M nonlinear functions in N variables by using a trust-region-reflective
%   algorithm. FUN is a function handle that evaluates the functions and
%   returns their values in a vector F such that
%
%       F(X) = [f_1(x_1,x_2,...,x_N), f_2(x_1,x_2,...,x_N), ..., f_M(x_1,x_2,...,x_N)]^T
%
%   X0 is a vector of initial guesses for the solution.
%
%   X = LSQNONLIN(FUN,X0,LB,UB) defines lower and upper bounds on the
%   design variables, X, so that the solution is always in the range
%   LB <= X <= UB. Use empty matrices for LB and UB if no bounds exist. Set
%   LB(i) == -Inf or UB(i) == Inf if no lower or upper bound exists,
%   respectively, for X(i).
%
%   X = LSQNONLIN(FUN,X0,LB,UB,OPTIONS) minimizes with the default
%   optimization parameters replaced by values in the structure OPTIONS.
%   Create OPTIONS with the OPTIMOPTIONS function. See OPTIMOPTIONS for details.
%
%   [X,RESNORM] = LSQNONLIN(...) returns the value of the sum of squares
%   of the residuals, F(X)'*F(X), at the solution X.
%
%   [X,RESNORM,RESIDUAL] = LSQNONLIN(...) returns the vector of residuals,
%   F(X), at the solution X.
%
%   [X,RESNORM,RESIDUAL,EXITFLAG] = LSQNONLIN(...) returns an EXITFLAG that
%   describes the exit condition of LSQNONLIN. Possible values of EXITFLAG
%   and their corresponding exit conditions are listed in the table below.
%
%   EXITFLAG    EXIT CONDITION
%   ---------    ----------------
%     1          First-order optimality measure was less than the
%                 specified tolerance.
%     2          Both the maximum number of iterations and the maximum
%                 number of function evaluations were exceeded.
%     3          Change in X was less than the specified tolerance.
%     4          Magnitude of the gradient was less than the specified
%                 tolerance.
%     5          Relative function value was less than the specified tolerance.
%     6          Line search failed to find a lower function value.
%     7          Magnitude of the step was less than the specified tolerance.
%    -1          LSQNONLIN terminated because the user-supplied objective
%                 function or gradient function was invalid.
%    -2          LSQNONLIN terminated because the user-supplied objective
%                 function or gradient function was computationally
%                 infeasible.
%    -3          LSQNONLIN terminated because the user-supplied objective
%                 function or gradient function was terminated by the user.
%
%   [X,RESNORM,RESIDUAL,EXITFLAG,OUTPUT] = LSQNONLIN(...) returns a
%   structure OUTPUT with the following fields:
%
%       iterations: Number of iterations taken to reach the solution.
%       funcCount: Number of function evaluations that were made.
%       algorithm: Algorithm used to solve the problem.
%       firstorderopt: First-order optimality measure at the solution X.
%       stepsize: Step size used in the last iteration.
%       trustregionradius: Trust-region radius at the solution X.
%