In brtang63/StatComp21060: Final project of statistical computing

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Introduction to regression models for network-linked data

This package carries out regression model fitting for network-linked data. We briefly reviews these models at first.

In our models, network structure among observations is always assumed existing and being observed. Specifically, each sample is a node of a undirected unweighted graph and the graph is given. We aim to make use of the network structure to construct better regression models, which encourage similarity between linked nodes. In this package, linear regression and quantile regression are considered.

Formally, Denote by $L$ the Laplacian matrix of the graph. We solve the following optimizing problems, corresponding to linear regrssion and quantile regression, respectively.

$$ \min_{\alpha, \beta} L(\boldsymbol{\alpha}, \boldsymbol{\beta})=\|\boldsymbol{Y}-X \boldsymbol{\beta}-\boldsymbol{\alpha}\|^{2}+\lambda \boldsymbol{\alpha}^{T} L \boldsymbol{\alpha}. $$

$$ \min_{\alpha(\tau), \beta(\tau)} (\alpha(\tau), \beta(\tau))=\sum_{i=1}^{n} \rho_{\tau}\left(y_{i}-\alpha_{i}(\tau)-x_{i}^{T} \beta(\tau)\right)+\alpha(\tau)^{T} L \alpha(\tau). $$

Here $\alpha$ or $\alpha(\tau)$ are vectors of network cohesion, which is the key concept in our model. We regularize on differences between network cohesion to encourage similarity of

Quick Start

Now we provide a basic example on usage of this package. Note that we provide in this package a \code{generate.data} function to generate simulated dataset with network structure from the Stochastic block model. We generate a dataset with 100 nodes and 3 communities.

library(StatComp21060)
data <- generate.data(100, 3, 4, 0.5, 0.02)
X <- data$X
Y <- data$Y
Adj <- data$Adj

Before fitting regression models, we visualize the network structure to provide more intuition.

library(igraph)
g <- graph_from_adjacency_matrix(Adj)
e <- as_edgelist(g)

## Visualization
plot(g, vertex.size = 6)

Now we fit our regression models on this data.

coef_lm <- lm.net(X, Y, Adj, 0.1, gamma = 0.01)
coef_rq <- rq.net(X, Y, Adj, 0.1)

cohesion_lm <- coef_lm$alpha
beta_lm <- coef_lm$beta
cohesion_rq <- as.vector(coef_rq$alpha)
beta_rq <- as.vector(coef_rq$beta)

The fitted network cohesion is as follows.

knitr::kable(head(cbind(cohesion_lm, cohesion_rq)))

The fitted regression coefficients are as follows.

knitr::kable(cbind(beta_lm, beta_rq))

brtang63/StatComp21060 documentation built on Dec. 24, 2021, 1:25 a.m.