🥖Linear Modeling Theory Unit 17 – Non-Linear Regression in Linear Modeling

What's Non-Linear Regression?

• Non-linear regression models the relationship between a dependent variable and one or more independent variables when the relationship is not linear
• Useful when the data exhibits curvature or other non-linear patterns that cannot be adequately captured by a straight line
• Involves fitting a non-linear function to the data points to minimize the difference between the predicted and observed values
• Requires iterative optimization techniques (Gauss-Newton, Levenberg-Marquardt) to estimate the parameters of the non-linear function
• Can handle a wide range of functional forms (exponential, logarithmic, polynomial, sigmoidal)
  • Exponential: $y = ae^{bx}$
  • Logarithmic: $y = a + b\ln(x)$
  • Polynomial: $y = a + bx + cx^2 + \ldots$
  • Sigmoidal: $y = \frac{a}{1 + e^{-b(x-c)}}$
• Provides more flexibility in modeling complex relationships compared to linear regression
• Requires careful selection of the appropriate non-linear function based on domain knowledge and data characteristics

Key Concepts and Terms

• Dependent variable: The variable being predicted or explained by the independent variable(s)
• Independent variable(s): The variable(s) used to predict or explain the dependent variable
• Non-linear function: A mathematical function that describes a curved relationship between variables
  • Examples: exponential, logarithmic, polynomial, sigmoidal
• Parameters: The coefficients or constants in the non-linear function that need to be estimated from the data
• Iterative optimization: The process of repeatedly adjusting the parameter estimates to minimize the difference between predicted and observed values
  • Gauss-Newton method: An iterative method that approximates the non-linear function with a linear one and solves for the parameters
  • Levenberg-Marquardt method: An extension of Gauss-Newton that introduces a damping factor to improve convergence
• Residuals: The differences between the observed values and the predicted values from the non-linear model
• Goodness-of-fit: Measures how well the non-linear model fits the data
  • Examples: R-squared, root mean squared error (RMSE), mean absolute error (MAE)
• Overfitting: When a model is too complex and fits the noise in the data rather than the underlying pattern

Types of Non-Linear Models

• Exponential model: Models exponential growth or decay
  • Equation: $y = ae^{bx}$
  • Applications: population growth, radioactive decay, compound interest
• Logarithmic model: Models a relationship where the rate of change decreases over time
  • Equation: $y = a + b\ln(x)$
  • Applications: learning curves, diminishing returns, sound intensity
• Power model: Models a relationship where one variable is proportional to a power of the other
  • Equation: $y = ax^b$
  • Applications: allometric scaling, gravity, fluid dynamics
• Sigmoidal model: Models an S-shaped curve with an initial slow change, followed by rapid change, and then a slow change again
  • Equation: $y = \frac{a}{1 + e^{-b(x-c)}}$
  • Applications: population growth with carrying capacity, dose-response curves, technology adoption
• Polynomial model: Models a relationship with one or more bends or turns
  • Equation: $y = a + bx + cx^2 + \ldots$
  • Applications: trajectory of objects, economic trends, chemical reactions
• Rational model: Models a relationship as a ratio of two polynomials
  • Equation: $y = \frac{a + bx + cx^2 + \ldots}{1 + dx + ex^2 + \ldots}$
  • Applications: enzyme kinetics, pharmacokinetics, hydraulic systems

Fitting Non-Linear Models

• Specify the non-linear function: Choose an appropriate non-linear function based on domain knowledge and data characteristics
• Initialize parameter estimates: Provide initial guesses for the parameters of the non-linear function
• Define the objective function: Typically the sum of squared residuals (SSR) between the observed and predicted values
  • SSR = $\sum_{i=1}^{n} (y_i - \hat{y}_i)^2$, where $y_i$ is the observed value and $\hat{y}_i$ is the predicted value
• Use an iterative optimization algorithm: Adjust the parameter estimates to minimize the objective function
  • Gauss-Newton method: Linearizes the non-linear function and solves for the parameters iteratively
  • Levenberg-Marquardt method: Introduces a damping factor to the Gauss-Newton method for better convergence
• Check convergence: Repeat the optimization until the change in parameter estimates or the objective function is below a specified threshold
• Assess model fit: Evaluate the goodness-of-fit using measures like R-squared, RMSE, or MAE
  • R-squared: Proportion of variance in the dependent variable explained by the model
  • RMSE: Square root of the average squared residuals
  • MAE: Average absolute difference between observed and predicted values
• Interpret the fitted model: Examine the estimated parameters and their statistical significance to understand the relationship between variables

Model Evaluation and Selection

• Residual analysis: Plot the residuals against the predicted values or independent variables to check for patterns or heteroscedasticity
  • Residuals should be randomly scattered around zero with no clear patterns
• Goodness-of-fit measures: Calculate R-squared, RMSE, or MAE to assess how well the model fits the data
  • Higher R-squared and lower RMSE or MAE indicate better fit
• Cross-validation: Divide the data into training and validation sets to assess the model's performance on unseen data
  • k-fold cross-validation: Divide the data into k subsets, use k-1 subsets for training and the remaining subset for validation, repeat k times
• Information criteria: Use Akaike Information Criterion (AIC) or Bayesian Information Criterion (BIC) to compare models with different numbers of parameters
  • Lower AIC or BIC values indicate a better balance between model fit and complexity
• Parsimony principle: Prefer simpler models with fewer parameters when they provide similar goodness-of-fit
  • Occam's razor: Among competing hypotheses, the one with the fewest assumptions should be selected
• Domain knowledge: Consider the interpretability and plausibility of the model in the context of the problem domain
• Visualization: Plot the fitted non-linear function against the data points to visually assess the model's fit and appropriateness

Challenges and Limitations

• Choosing the appropriate non-linear function: Requires domain knowledge and understanding of the underlying process
  • Misspecification of the function can lead to poor model fit and incorrect conclusions
• Initialization of parameter estimates: The choice of initial values can affect the convergence and final parameter estimates
  • Multiple local optima may exist, requiring careful initialization or the use of global optimization techniques
• Overfitting: Non-linear models with many parameters can overfit the data, leading to poor generalization performance
  • Regularization techniques (L1, L2) can be used to constrain the parameter estimates and reduce overfitting
• Interpretability: Non-linear models can be more difficult to interpret than linear models
  • The relationship between variables may not be easily summarized by a single coefficient
• Extrapolation: Non-linear models may not be reliable for predicting values outside the range of the observed data
  • The functional form may not hold beyond the observed range, leading to unrealistic predictions
• Computational complexity: Fitting non-linear models can be computationally intensive, especially with large datasets or complex functions
  • Efficient optimization algorithms and computational resources may be required
• Assumptions: Non-linear regression still relies on certain assumptions, such as independence of errors and homoscedasticity
  • Violations of these assumptions can affect the validity of the model and the accuracy of the parameter estimates

Real-World Applications

• Population growth: Modeling the growth of populations over time using exponential or logistic functions
  • Logistic function accounts for carrying capacity and resource limitations
• Pharmacokinetics: Describing the absorption, distribution, metabolism, and elimination of drugs in the body
  • Compartmental models use exponential functions to model drug concentrations over time
• Learning curves: Modeling the relationship between performance and experience or practice
  • Power law or exponential functions can capture the diminishing returns of learning
• Dose-response curves: Modeling the relationship between the dose of a drug or stimulus and the observed response
  • Sigmoidal functions (Hill equation) are commonly used to model the S-shaped response
• Enzyme kinetics: Describing the rate of enzyme-catalyzed reactions as a function of substrate concentration
  • Michaelis-Menten equation is a rational function that models enzyme saturation
• Economic trends: Modeling the relationship between economic variables, such as supply and demand, over time
  • Polynomial or exponential functions can capture non-linear trends in economic data
• Environmental modeling: Describing the relationship between environmental variables and ecological responses
  • Non-linear functions can model the complex interactions and feedback loops in ecosystems
• Material science: Modeling the stress-strain relationship of materials under different loading conditions
  • Non-linear functions (Ramberg-Osgood equation) can capture the elastic-plastic behavior of materials

Advanced Topics and Extensions

• Generalized additive models (GAMs): Extend non-linear regression by allowing the relationship between variables to be modeled as a sum of smooth functions
  • Provides more flexibility in capturing complex non-linear relationships
• Nonparametric regression: Relaxes the assumption of a specific functional form and estimates the relationship between variables directly from the data
  • Examples: kernel regression, local polynomial regression, splines
• Bayesian non-linear regression: Incorporates prior knowledge about the parameters and updates the estimates based on the observed data
  • Allows for the quantification of uncertainty in the parameter estimates and predictions
• Robust non-linear regression: Reduces the influence of outliers or heavy-tailed errors on the parameter estimates
  • Examples: M-estimators, least trimmed squares, Huber regression
• Regularization: Adds a penalty term to the objective function to constrain the parameter estimates and reduce overfitting
  • L1 regularization (Lasso): Encourages sparse solutions with some parameters set to zero
  • L2 regularization (Ridge): Shrinks the parameter estimates towards zero without setting them exactly to zero
• Multivariate non-linear regression: Models the relationship between multiple dependent variables and one or more independent variables
  • Allows for the simultaneous modeling of correlated responses
• Non-linear mixed effects models: Incorporate both fixed and random effects to account for individual variability in the parameters
  • Useful for modeling longitudinal or clustered data with repeated measurements
• Gaussian process regression: Models the relationship between variables as a Gaussian process, allowing for non-linear relationships and uncertainty quantification
  • Provides a probabilistic framework for non-linear regression with built-in model selection and uncertainty estimates