🤖Statistical Prediction Unit 6 – Non-Linear Models: Splines, GAMs & Local Reg.

Key Concepts

• Non-linear models capture complex relationships between predictors and response variables that cannot be adequately represented by linear models
• Flexibility allows non-linear models to adapt to the underlying patterns in the data, resulting in improved predictive performance
• Overfitting is a risk with non-linear models due to their increased complexity compared to linear models
  • Regularization techniques help mitigate overfitting by constraining the model's flexibility
• Interpretability can be more challenging with non-linear models as the relationships between predictors and the response variable are not always straightforward
• Non-linear models are particularly useful when dealing with datasets that exhibit non-linear patterns, such as curves, thresholds, or interactions between predictors

Types of Non-Linear Models

• Splines represent a class of non-linear models that use piecewise polynomial functions to fit the data
  • Polynomial regression is a special case of splines where a single polynomial function is used to model the entire range of the predictor variable
• Generalized Additive Models (GAMs) extend generalized linear models by allowing non-linear relationships between predictors and the response variable through smooth functions
• Local regression techniques, such as LOESS (locally estimated scatterplot smoothing) and LOWESS (locally weighted scatterplot smoothing), fit a separate model for each data point based on its neighboring observations
• Decision trees and random forests are non-linear models that recursively partition the feature space based on the most informative predictors
• Neural networks are highly flexible non-linear models inspired by the structure and function of biological neural networks, capable of learning complex patterns and relationships in the data

Splines: Basics and Applications

• Splines divide the range of a predictor variable into smaller intervals and fit separate polynomial functions within each interval
• Knots are the points where the intervals meet, and the polynomial functions are connected to ensure a smooth transition between intervals
• Basis functions, such as B-splines or natural cubic splines, are used to represent the piecewise polynomial functions and ensure continuity and smoothness at the knots
• The number and placement of knots can significantly impact the flexibility and fit of the spline model
  • Too few knots may result in underfitting, while too many knots may lead to overfitting
• Penalized splines introduce a penalty term to the model's objective function to control the smoothness of the fitted curve and prevent overfitting
• Splines are commonly used in various applications, such as modeling non-linear trends in time series data (stock prices) or capturing the relationship between age and a health outcome (blood pressure)

Generalized Additive Models (GAMs)

• GAMs extend generalized linear models by replacing the linear predictors with a sum of smooth functions of the predictor variables
• The smooth functions in GAMs can be represented using various basis functions, such as splines, to capture non-linear relationships
• The flexibility of GAMs allows for modeling complex patterns and interactions between predictors without specifying the functional form explicitly
• The additive structure of GAMs enables easier interpretation of the effects of individual predictors on the response variable compared to other non-linear models
• GAMs can handle various types of response variables, including continuous (regression), binary (classification), and count data (Poisson regression)
• The smoothness of the component functions in GAMs is controlled by smoothing parameters, which can be estimated using methods like generalized cross-validation (GCV) or restricted maximum likelihood (REML)

Local Regression Techniques

• Local regression techniques fit a separate model for each data point, considering only a subset of neighboring observations
• The local nature of these techniques allows for capturing complex non-linear patterns that may vary across the range of the predictor variables
• LOESS and LOWESS are popular local regression methods that use weighted least squares to fit a polynomial function to each data point's neighborhood
  • The weights assigned to the neighboring observations decrease with their distance from the target data point, giving more influence to closer observations
• The size of the neighborhood, often referred to as the bandwidth or span, determines the smoothness of the fitted curve
  • A larger bandwidth results in a smoother curve but may miss local patterns, while a smaller bandwidth captures more local details but may be sensitive to noise
• Local regression techniques are particularly useful for exploratory data analysis and visualizing non-linear relationships in the data
• However, local regression models may not provide a global parametric form and can be computationally intensive for large datasets

Model Fitting and Evaluation

• Fitting non-linear models involves estimating the model parameters that minimize a chosen loss function, such as least squares or maximum likelihood
• Regularization techniques, such as L1 (lasso) or L2 (ridge) penalties, can be incorporated into the fitting process to control the model's complexity and prevent overfitting
• Cross-validation is commonly used to assess the predictive performance of non-linear models and select the optimal hyperparameters (number of knots, smoothing parameters)
  • K-fold cross-validation divides the data into K subsets, trains the model on K-1 subsets, and evaluates it on the remaining subset, repeating the process K times
• Residual analysis helps assess the adequacy of the fitted model by examining the distribution and patterns of the residuals (differences between observed and predicted values)
• Visual inspection of the fitted curves or surfaces can provide insights into the captured non-linear relationships and identify potential issues (wiggles, overfitting)
• Model comparison techniques, such as the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC), can be used to select among different non-linear models based on their fit to the data and complexity

Practical Examples and Use Cases

• Non-linear models are widely used in various domains, including finance, healthcare, marketing, and environmental sciences
• In finance, splines can be used to model the non-linear relationship between interest rates and bond prices, capturing the term structure of interest rates
• GAMs are commonly employed in ecological studies to model the distribution of species abundance as a function of environmental variables (temperature, precipitation)
• Local regression techniques are useful for identifying trends and patterns in real estate prices across different neighborhoods or regions
• In marketing, non-linear models can capture the diminishing returns of advertising expenditure on sales, helping optimize marketing strategies
• Non-linear models are also applied in machine learning tasks, such as image recognition and natural language processing, where complex patterns and relationships need to be learned from high-dimensional data

Limitations and Considerations

• Non-linear models are more complex and computationally intensive compared to linear models, which can be a challenge when dealing with large datasets or real-time applications
• The increased flexibility of non-linear models comes with a higher risk of overfitting, requiring careful model selection and regularization techniques to mitigate this issue
• Interpreting non-linear models can be more challenging than linear models, as the relationships between predictors and the response variable may not be easily summarized or visualized
• The choice of basis functions, knots, or smoothing parameters in non-linear models can have a significant impact on the model's performance and requires domain knowledge and experimentation
• Non-linear models may not always provide a clear understanding of the underlying mechanisms or causal relationships, as they focus on capturing patterns and making predictions
• The assumptions and limitations of specific non-linear models should be carefully considered when applying them to real-world problems, and the results should be interpreted in the context of the domain and the available data