5.3 Non-linear Curve Fitting

Non-linear curve fitting is crucial for modeling complex relationships in scientific data. It overcomes linear regression limitations, providing more accurate representations of phenomena like exponential growth, power laws, and sinusoidal patterns.

Advanced techniques like Gauss-Newton and Levenberg-Marquardt algorithms iteratively minimize residuals. Convergence criteria, fit quality measures, and residual analysis help ensure reliable results. These methods enable better predictive power and capture underlying processes in complex systems.

Understanding Non-linear Curve Fitting

Need for non-linear curve fitting

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• Linear regression limitations cannot model complex relationships and oversimplifies real-world phenomena
• Non-linear relationships in scientific data include exponential growth or decay (population dynamics), power laws (scaling relationships), logarithmic relationships (pH scale), and sinusoidal patterns (seasonal variations)
• Non-linear curve fitting advantages provide more accurate representation of complex systems, better predictive power for extrapolation, and capture underlying physical or biological processes (enzyme kinetics)

Linearization of non-linear models

• Logarithmic transformation used for exponential and power law relationships transforms $y = ae^{bx}$ to $\ln(y) = \ln(a) + bx$
• Reciprocal transformation useful for hyperbolic relationships changes $y = \frac{1}{ax + b}$ to $\frac{1}{y} = ax + b$
• Polynomial transformation applicable to polynomial relationships converts $y = ax^2 + bx + c$ to linear in $x$ and $x^2$
• Linearization limitations potentially distort error structure and may not always be possible for complex models (logistic growth)

Advanced Non-linear Fitting Techniques

Iterative methods for least squares

• Gauss-Newton algorithm minimizes sum of squared residuals by approximating non-linear function with linear terms and uses partial derivatives to update parameter estimates
• Levenberg-Marquardt algorithm combines Gauss-Newton and gradient descent methods, more robust for difficult problems with adaptive damping parameter for improved convergence
• Newton-Raphson method uses both first and second derivatives for faster convergence on well-behaved functions
• Implementation steps:
  1. Estimate initial parameters
  2. Calculate residuals and Jacobian matrix
  3. Update parameters iteratively
  4. Check for convergence

Convergence of fitting algorithms

• Convergence criteria assess relative change in parameter values, reduction in sum of squared residuals, and maximum number of iterations
• Fit quality measures include R-squared value, root mean square error (RMSE), and Akaike Information Criterion (AIC)
• Residual analysis involves plotting residuals vs fitted values and creating normal probability plot of residuals
• Confidence intervals for parameter estimates based on covariance matrix of parameters indicate uncertainty in fitted values
• Cross-validation techniques like K-fold and leave-one-out cross-validation assess model performance
• Sensitivity to initial parameter guesses addressed by using multiple starting points to avoid local minima and grid search for initial parameter estimation