Description: Nonlinear Least Squares (NLS) is a statistical technique used to fit models to datasets that do not follow a linear relationship. Unlike linear least squares methods, which assume that the relationship between variables is linear, NLS allows for modeling more complex, nonlinear relationships. This technique is based on minimizing the sum of the squares of the differences between observed values and values predicted by the model. NLS is particularly useful in situations where data exhibit curvilinear patterns or where the relationship between variables is inherently nonlinear. The algorithms used to solve NLS problems often involve iterative methods, such as the Gauss-Newton algorithm or the Levenberg-Marquardt method, which adjust model parameters until a satisfactory minimum is reached. The flexibility of NLS makes it a valuable tool across various disciplines, from economics to biology, where complex phenomena require a more sophisticated approach to data analysis.
History: The Nonlinear Least Squares technique has its roots in the development of regression and statistical analysis. Although the least squares method was introduced by French mathematician Adrien-Marie Legendre in 1805, the extension to nonlinear models began to take shape in the 20th century. As computing became more accessible, more sophisticated algorithms were developed to solve nonlinear fitting problems, allowing for their application in various scientific and engineering disciplines. In the 1960s, the use of computers facilitated the implementation of these methods, and since then, Nonlinear Least Squares has evolved and been integrated into modern statistical software.
Uses: Nonlinear Least Squares are used in a wide variety of fields, including economics, biology, engineering, and physics. In economics, they are applied to model complex relationships between economic variables, such as supply and demand. In biology, they are useful for fitting population growth models or analyzing experimental data that follow nonlinear patterns. In engineering, they are used to calibrate models of physical systems, such as the response of materials to different forces. Additionally, in the medical field, NLS is applied in pharmacokinetics to model drug concentration in the body over time.
Examples: A practical example of Nonlinear Least Squares is fitting a logistic model to describe the growth of a bacterial population, where the growth rate is not constant and is affected by factors such as nutrient availability. Another example is found in calibrating chemical reaction models, where the reaction rate may follow a nonlinear relationship with respect to the concentration of reactants. In the financial realm, NLS is used to fit yield curves in investment analysis, where the relationships between interest rates and time can be complex and nonlinear.
