Local patterns, such as piecewise constant structures, frequently appear in real-world data. Many applications demand models capable of capturing these localized variations, where data may change abruptly or remain constant across different local regions. Statistical models such as fused lasso and nearest-neighbor algorithms are particularly effective in addressing this localized complexity. The fused lasso encourages sparsity in differences between adjacent observations over a graph structure, making it well-suited for identifying and estimating local patterns. Nearest-neighbor algorithms focus on nearby data points, offering a flexible and locally adaptive framework to capture varying structures within the data.

This dissertation develops innovative statistical models that integrate fused lasso and nearest-neighbor algorithms to address research problems in quantile regression and causal inference, as demonstrated in the following projects. The first project introduces a non-parametric quantile regression framework using the $K$-nearest-neighbor fused lasso to provide robust and locally adaptive estimation of multivariate functions at different quantile levels. The second project presents a score-based approach for estimating heterogeneous treatment effects, combining propensity and prognostic scores with nearest-neighbor matching to stratify and estimate treatment effects over a two-dimensional grid. The third project proposes graph-based fused lasso models to estimate heterogeneous treatment effects over graph structures. These projects highlight the versatility and effectiveness of locally adaptive models that build on fused lasso and nearest-neighbor algorithms. Validated through simulation studies and real-world applications, these models offer valuable insight for fields such as economics, medicine, and social science.