Construct Design Matrix for Alpha Updates in ALS

Builds the design matrix for updating sender effects (alpha parameters) in the Alternating Least Squares algorithm, holding receiver effects (beta) fixed.

Usage

cpp_construct_Wbeta_design(W, X, beta)

Arguments

W

Three-dimensional array (m x m x p) of influence covariates. These parameterize how influence flows through the network.

X

Three-dimensional array (m x m x T) of network states over time. Typically contains lagged outcomes that carry influence forward.

beta

Vector (p x 1) of current receiver effect parameters. These are held fixed while updating alpha in this ALS step.

Value

Matrix (m*m*T x p) that serves as the design matrix for GLM estimation. Each column corresponds to one influence covariate, rows match vectorized Y.

Details

In the ALS algorithm, when updating alpha with beta fixed, the model becomes linear in alpha. The design matrix for this GLM sub-problem has columns corresponding to each influence covariate.

For covariate k and observation (i,j,t), the design matrix element is: [W[,,k] * X[,,t] * B'][i,j]

Where B = sum_l beta[l] * W[,,l] is the current receiver effects matrix.

The algorithm: 1. Compute B from current beta and W 2. For each covariate k: - Calculate W[,,k] * X * B' for all time points - Flatten to match the vectorized Y 3. Combine into design matrix

This C++ implementation is 10-100x faster than the equivalent R code using loops or apply functions, making ALS feasible for larger networks.

Note

This function is called once per ALS iteration. The resulting matrix can be large (m^2 * T x p), so memory usage should be considered for big networks.

Examples