# Nonlinear least squares problem

• ciaran_hughes
In summary, the speaker is asking for advice on how to perform a nonlinear least squares fit for a pair of parametric equations. They are struggling to obtain a non-parametric function and are unsure if it can be obtained from the given equations. The ultimate goal is to perform a nonlinear least squares fit, but the speaker is unsure if it is possible or why it may be difficult.

#### ciaran_hughes

Dear all,

Apologies if this is in the wrong forum.

I have a bit Nonlinear least squares fit problem. I have a pair of parametric equations (see attached, fairly nasty ).

in it, a b c x0 y0 z0 are all constant, and they are the values I want to determine from a nonlinear least squares fit to a given set of (u', v') data. (u', v') are a set of pixel locations of a curve in an image, and the equations attached describe that curve.

Now, to do a nonlinear least squares, I believe I have to get a single nonparametric function (e.g. of v' in terms of u'). To do this, I follow the standard steps of solving one of the equations in terms of t, and substitute into the other. This is where I fail.

I have even used the 'solve' function in the Symbolic Toolbox in MATLAB to solve one of the equations for t, but simply get an error.

So, my questions are:
1) Am I right in saying I must get the non-parametric equation for the curve to get a nonlinear least squares fit to the data points? I assume that I do, because t is an extra variable that I a) don't need and b) have no information for.
2) Can a nonparametric equation for this curve be obtained (from the attached equations)? Why?

Ultimately, I would like to know if I can do a nonlinear least squares fit to these parametric equations. If I can't, or if it would prove very difficult, why?

Ciaran

#### Attachments

• equations.GIF
4.8 KB · Views: 556
A least squares fit produces a function based on a collection of data.

If you have the function and want to produce data, it is called evaluation.

So you're thinking about least squares backward.

## 1. What is a nonlinear least squares problem?

A nonlinear least squares problem is a mathematical optimization problem that aims to find the optimal values of one or more independent variables to minimize the sum of squared differences between the observed data and the predictions made by a nonlinear function.

## 2. How is a nonlinear least squares problem different from a linear least squares problem?

While both types of problems involve minimizing the sum of squared differences, a linear least squares problem assumes a linear relationship between the independent and dependent variables, whereas a nonlinear least squares problem allows for a nonlinear relationship.

## 3. What types of functions can be used to model a nonlinear least squares problem?

A wide range of mathematical functions can be used to model a nonlinear least squares problem, including polynomial, exponential, logarithmic, and trigonometric functions.

## 4. What methods are commonly used to solve a nonlinear least squares problem?

The most commonly used methods for solving a nonlinear least squares problem include the Gauss-Newton method, the Levenberg-Marquardt method, and the trust region method. These methods use iterative algorithms to find the optimal values of the independent variables.

## 5. What are some applications of nonlinear least squares problems in science?

Nonlinear least squares problems have many applications in science, including in physics, chemistry, biology, and engineering. They can be used to model and analyze data from various experiments and simulations, as well as to make predictions and optimize processes.