NLS (nonlinear least squares)
Nonlinear least squares (NLS) is a mathematical optimization technique used to estimate the parameters of a nonlinear model by minimizing the sum of squared differences between the observed data and the model's predictions. Unlike linear least squares, which is used for linear models, NLS is employed when the relationship between the independent and dependent variables is nonlinear.
In NLS, the objective is to find the parameter values that minimize the sum of squared residuals, which are the differences between the observed data points and the corresponding model predictions. The residuals represent the errors in the model, and by minimizing their sum of squares, we aim to find the best-fitting parameters for the nonlinear model.
The general form of a nonlinear least squares problem can be described as follows:
minimize Σ(rᵢ(θ)²),
where θ represents the vector of unknown parameters, rᵢ(θ) denotes the residual function for the i-th observation, and the summation is taken over all observations. The goal is to find the optimal parameter vector θ* that minimizes the sum of squared residuals.
The solution to an NLS problem is typically obtained iteratively. Starting with an initial guess for the parameter vector θ, the algorithm iteratively updates the parameter values until convergence is achieved. The most commonly used algorithm for NLS is the Gauss-Newton method, which approximates the nonlinear model using a linearization around the current parameter estimates.
The Gauss-Newton algorithm involves solving a linear system of equations in each iteration. At each step, the Jacobian matrix J(θ) is computed, which contains the partial derivatives of the residual functions with respect to the parameters. The linear system can be written as:
J(θ)Δθ = -r(θ),
where Δθ represents the parameter update, and r(θ) is the vector of residuals. The Gauss-Newton algorithm aims to find the Δθ that solves this system of equations.
To improve the stability and convergence of the algorithm, the Levenberg-Marquardt method is often employed. It introduces a damping factor, which combines the Gauss-Newton and gradient descent methods. The damping factor allows for better control over the step size in each iteration, preventing overshooting and improving stability.
When implementing NLS, it is essential to consider the initialization of the parameter vector θ. Poor initial guesses can lead to convergence issues or solutions that are far from the global minimum. Often, domain knowledge or previous estimates are used to provide reasonable initial values for the parameters.
Furthermore, NLS requires careful consideration of the model's structure and the selection of appropriate functional forms. Nonlinear models can exhibit complex behaviors, such as multiple local minima or parameter identifiability issues. In some cases, reparameterization or model simplification may be necessary to improve the convergence properties of the NLS algorithm.
Another challenge in NLS is the issue of outliers and influential data points. Outliers can significantly impact the estimation process, leading to biased parameter estimates. Various techniques, such as robust regression or data weighting, can be employed to mitigate the influence of outliers and improve the robustness of the NLS algorithm.
NLS finds applications in various fields, including statistics, econometrics, engineering, and biology. It is commonly used for curve fitting, model calibration, parameter estimation, and nonlinear regression. NLS can be applied to a wide range of models, such as exponential growth models, logistic models, power law models, and many others.
In conclusion, nonlinear least squares is a powerful optimization technique for estimating the parameters of nonlinear models by minimizing the sum of squared residuals. It involves iteratively updating the parameter estimates using the Gauss-Newton or Levenberg-Marquardt algorithm. Careful consideration of model structure, initialization, and handling of outliers is essential for successful implementation of NLS. The technique finds applications in various disciplines and is widely used for curve fitting and parameter estimation.