Nonlinear Least Squares Estimation

## Understanding Nonlinear Least Squares Estimation

1. **Introduction to Nonlinear Least Squares**

- Least squares estimator in linear models
- Introduction to nonlinear models with function f(theta) and error term

2. **Approach to Solving Nonlinear Least Squares**

- Approximating function f using Taylor series around initial guess Theta naught
- Using linear least squares to estimate the next iteration Theta
- Iterative process known as Gauss-Newton iteration

3. **Mathematical Representation**

- Approximation: y tilde = J(theta) + noise
- Solving for Theta tilde 1: (J^T J)^-1 J^T y tilde

4. **Iterative Process**

- Iterating to find more accurate Theta estimates (Theta 2, Theta 3, etc.)
- Convergence to defined tolerance of error

5. **Considerations**

- Faster convergence compared to gradient descent in machine learning
- Potential instability, requiring regularization
- Tikhonov regularization or Levenberg-Marquardt algorithm for stabilization

6. **Conclusion**

- Offer for a deep dive into examples from industry
- Appreciation for time and interest in the topic