What is local regression and how does it compare to global regression?

Local regression (LOWESS/LOESS) fits models to subsets of data, adapting to local changes without assuming a global function, thus it's non-parametric and flexible. Global regression, like ordinary least squares (OLS), fits a single model to all data, assuming a specific form (e.g., linear), making it parametric and interpretable but less flexible. Local regression is computationally intensive and not suitable for extrapolation, while global regression is simpler computationally, allows extrapolation, and requires fewer data points.

The selection of the smoothing parameter for instance, λ (lambda) influences the bias-variance trade-off. Lambda determines the width of the local neighborhood : a large λ reduces variance but increases bias, while a smaller lambda decreases bias but increases variance. Moreover, variance in local regression is inversely related to the local density. In other words, as data becomes sparse in a given neighborhood , variance increases, and vice versa. This is particularly challenging for local regression at dataset's periphery, a known phenomenon called the boundary effect. On the other hand, local regression is much easier to interpret and intuitive as oppose to other nonlinear methods that produce complex models.

Advantages of local regression include its flexibility to model complex, nonlinear relationships and robustness to functional form mis-specification. Disadvantages are its computational intensity, difficulty in extrapolating beyond the data range, and greater data requirement for reliable estimates. Local regression is less influenced by outliers as it focuses on subsets of the data for fitting.

To apply local regression, select a bandwidth, which determines how much of the data is used to fit at each point. Use a kernel function to weigh data points, giving more weight to points closer to the target x-value. Fit a simple model (like linear regression) to the weighted points to estimate the curve at the target x-value. Repeat for each x-value of interest. The bandwidth and kernel can be chosen through cross-validation to minimize prediction error.

Local regression is used in finance for predicting stock prices by capturing short-term trends, in epidemiology for modeling disease rates over time or space, and in quality control for adapting to changes in manufacturing processes. It's also applied in environmental science to assess the impact of pollutants over regions, and in econometrics for non-linear trend analysis. This technique suits any scenario requiring flexible fit to localized patterns in data without assuming a global model.

1. Data Size: Large datasets can slow down the process significantly. 2. Dimensionality: Local regression works best with fewer predictors due to the curse of dimensionality. 3. Noise Level: It can overfit noisy data; hence, smoothing parameters must be carefully chosen. 4. Computational Resources: Ensure adequate computational power and time are available. 5. Interpretability: The resulting model may be less interpretable than global models. 6. Bandwidth Selection: Picking the right bandwidth is crucial for balancing bias and variance. 7. Application Domain: Understand domain-specific considerations that might affect model performance.