# What is Nonlinear regression? And when to use it?

## What is Nonlinear regression

Nonlinear regression is a statistical technique used to model relationships between data points that aren’t straight lines. Unlike linear regression, which assumes a constant slope between variables, nonlinear regression allows for curves, S-shapes, and more complex relationships.

### What it does:

• Fits a nonlinear function to your data points. This function can involve exponents, logarithms, trigonometric functions, or any combination that captures the true relationship.
• Helps you understand curved or complex patterns in your data that linear regression might miss.
• Enables you to make more accurate predictions when the relationship between variables isn’t linear.

### When to use it:

• When you observe visible curves or non-linear trends in your data visualizations.
• When your predictions from linear regression are consistently inaccurate.
• When you’re studying phenomena known to have non-linear behavior, like population growth or chemical reactions.

### Challenges:

• More complex than linear regression. Finding the right nonlinear function and optimizing its parameters can be computationally expensive and require specialized algorithms.
• Potential for overfitting. Choosing a too-complex function can lead to a model that memorizes the data without generalizing well to new examples.

### How to choose a nonlinear model?

#### Visualize and Understand Your Data:

• Plot your data to visualize the relationship between variables. Look for curves, plateaus, or other non-linear patterns.
• Consider the nature of the variables and the domain knowledge you have. This can suggest potential model shapes.

#### Explore Common Nonlinear Models:

• Polynomial Regression: Adds higher-order terms (e.g., x^2, x^3) to capture curves.
• Exponential Models: Often used for growth or decay processes (e.g., population, radioactive decay).
• Logistic Models: Model S-shaped curves, common in probabilities and bounded growth.
• Sigmoid Models: Similar to logistic models, used in neural networks and classification.
• Power Models: Model relationships between variables with power functions (e.g., y = ax^b).
• Gompertz Models: Used for modeling sigmoidal growth with an initial exponential phase.
• Fourier Series: Represent periodic functions using sine and cosine waves.

#### Consider Theoretical Considerations:

• If you have theoretical knowledge about the underlying process, it can suggest specific model forms.
• For example, chemical reactions often follow exponential or power laws.
• Biological processes might exhibit logistic growth.

#### Experiment and Evaluate:

• Try different models and compare their performance using metrics like:
• Mean squared error (MSE)
• R-squared
• Akaike Information Criterion (AIC)
• Bayesian Information Criterion (BIC)
• Visually inspect model fits: Plot the fitted model against data points to assess how well it captures the relationship.
• Consider model complexity: Simpler models are often preferred if they provide similar performance to more complex ones.

#### Refine and Validate:

• Adjust model parameters to improve fit.
• Validate models on independent datasets to assess generalizability.
• Consider model interpretability: If understanding the underlying relationship is important, choose a model with more intuitive parameters.

### How to evaluate a nonlinear model?

#### Visual Inspection:

• Plot the fitted model curve against the actual data points.
• Look for:
• Close alignment between the curve and data points.
• Random scatter of residuals (differences between predicted and actual values) without clear patterns.
• S-shaped curves might require transformation (e.g., log transformation) to assess linearity of residuals.

#### Goodness-of-Fit Statistics:

• R-squared: Measures the proportion of variance in the dependent variable explained by the model. Higher values (closer to 1) indicate better fit.
• Adjusted R-squared: Penalizes for model complexity, useful for comparing models with different numbers of parameters.
• Mean Squared Error (MSE): Average squared difference between predicted and actual values. Lower values indicate better fit.
• Root Mean Squared Error (RMSE): Square root of MSE, in the same units as the dependent variable, for easier interpretation.

#### Residual Analysis:

• Plot residuals against predicted values and independent variables.
• Look for:
• Random scatter around zero, indicating no systematic patterns.
• Constant variance of residuals across the range of predicted values (homoscedasticity).
• If patterns or non-constant variance are present, consider model adjustments or transformations.

#### Information Criteria:

• Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC): Balance model fit and complexity. Lower values indicate better fit while penalizing for more parameters. Useful for comparing non-nested models.

#### Validation of Model:

• Assess model performance on independent datasets (not used for fitting) to evaluate generalizability and avoid overfitting.
• If performance drops significantly on new data, the model might be overfitting the training data.

#### Domain Knowledge:

• Consider whether model predictions align with theoretical understanding and expectations for the specific domain.
• Incorporate expert knowledge to assess the plausibility of parameter estimates and model behavior.

#### Important point to Remember:

• Evaluation is an iterative process.
• Refine the model, try different estimation methods, or consider alternative model forms if initial evaluation suggests issues.
• Balance statistical measures with visual inspection and domain knowledge for comprehensive assessment.