Nonlinear Least Squares Minimization

In summary, the conversation discusses the use of the Levenberg-Marquardt algorithm for solving a problem involving F(x;a) = (1+a1*x)/(a2+a3*x) with a1=2, a2=3, a3=5 and 6 data points. Other suggested methods include treating it as a general nonlinear optimization problem and using algorithms such as BFGS or simplex search. These methods can be found in "Numerical Recipes" or through online resources.
  • #1
swartzism
103
0
How should I go about solving this problem? This is only to get a better understanding of how NLLS works.

F(x;a) = (1+a1*x)/(a2+a3*x) (so n = 3)

I am choosing a1,a2,a3 to be 2,3,5 respectively. I am also picking 6 data points (so m = 6):

(0, 0), (-1/4, 1/4), (-1/2, 1/10), (1/4, 1/4), (1, 1), (1/2, 1/2).

I'm not even sure where to begin to attack this problem.

Any suggestions?
 
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  • #2
The Levenberg-Marquardt algorithm is a widely used method.

Alternatively, you can treat NLLSQ as a general nonlinear optimization problem, and use standard optimization methods like the BFGS algorithm if the problem is fairly well behaved, or the simplex search method (which can be very slow, but will find a local minimum of pretty much anything!) if it isn't.

IIRC all these are in "Numerical Recipes", or Google for other explanations of the algorithms and computer code.
 

Related to Nonlinear Least Squares Minimization

1. What is Nonlinear Least Squares Minimization?

Nonlinear Least Squares Minimization is a mathematical method used to find the optimal values of parameters in a nonlinear mathematical model. It aims to minimize the sum of the squared differences between the predicted values from the model and the actual data points.

2. How does Nonlinear Least Squares Minimization differ from Linear Least Squares Minimization?

Nonlinear Least Squares Minimization is used when the relationship between the parameters and the predicted values is nonlinear, while Linear Least Squares Minimization is used when the relationship is linear. In Nonlinear Least Squares Minimization, the parameters are estimated iteratively using techniques such as gradient descent, while in Linear Least Squares Minimization, the parameters can be solved analytically using matrix operations.

3. What types of problems can be solved using Nonlinear Least Squares Minimization?

Nonlinear Least Squares Minimization is commonly used in various fields such as statistics, engineering, and physics, to solve problems involving nonlinear models. It can be used for curve fitting, data smoothing, parameter estimation, and optimization, among others.

4. What are the advantages of using Nonlinear Least Squares Minimization?

One of the main advantages of Nonlinear Least Squares Minimization is its ability to handle nonlinear relationships between the parameters and the predicted values. It is also a flexible method that can be applied to a wide range of problems, and it typically gives more accurate results compared to linear methods.

5. Are there any limitations to Nonlinear Least Squares Minimization?

While Nonlinear Least Squares Minimization is a powerful tool, it does have some limitations. It can get stuck in local minima, leading to suboptimal solutions, and it can also be computationally intensive for complex models with a large number of parameters. Additionally, it relies on starting values for the parameters, which may not always be readily available or accurate.

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