Why dependent variables need their own dummies

The FDR guarantee of the T-Rex selector rests on the exchangeability of the null variables and the dummies along the solution path. Independent dummies are exchangeable with independent null variables. When the variables are strongly dependent, a null variable that is correlated with an active one enters the path earlier than an independent dummy would, so the dummies no longer mirror the nulls and the FDP estimate becomes anti-conservative. The dependency-aware T-Rex selector (T-Rex+DA) [1] restores the control by estimating the dependency structure and by deflating the vote within groups of dependent variables. It has been applied to sparse index tracking with autoregressive and nearest-neighbour stock dependency models [2], [3].

An autoregressive design

The variables follow a first-order autoregressive (AR(1)) process across the columns with coefficient 0.7, and five of them are active.

set.seed(321)
n <- 100
p <- 100
rho <- 0.7
X <- matrix(0, n, p)
X[, 1] <- rnorm(n)
for (j in 2:p) X[, j] <- rho * X[, j - 1] + sqrt(1 - rho^2) * rnorm(n)
true_support <- c(10, 30, 50, 70, 90)
beta <- numeric(p)
beta[true_support] <- 2
y <- as.numeric(X %*% beta + rnorm(n))

The plain selector and the dependency-aware selector on the same data and the same seed.

plain <- TRexSelector$new(X, y, tFDR = 0.1, seed = 7, verbose = FALSE,
                          control = trex_control(K = 10))
plain$select()
da <- TRexDASelector$new(X, y, tFDR = 0.1, seed = 7, verbose = FALSE,
                         da_control = trex_da_control(da_method = "BT"),
                         control = trex_control(K = 10))
da$select()
data.frame(selector = c("T-Rex", "T-Rex+DA"),
           selected = c(length(plain$selected_indices), length(da$selected_indices)),
           FDP = c(trex_fdp(plain$selected_indices, true_support),
                   trex_fdp(da$selected_indices, true_support)),
           TPP = c(trex_tpp(plain$selected_indices, true_support),
                   trex_tpp(da$selected_indices, true_support)))
#>   selector selected       FDP TPP
#> 1    T-Rex        7 0.2857143 1.0
#> 2 T-Rex+DA        4 0.0000000 0.8

TRexDASelector inherits the fields of TRexSelector (selected_indices, fdp_hat_mat, T_stop, L, …) and adds the record of the dependency estimation.

da$da_method_used
#> [1] "BT"
da$rho_thresh
#> [1] 0.6980528
round(da$rho_grid, 3)
#>  [1] 0.521 0.590 0.609 0.618 0.632 0.654 0.672 0.682 0.688 0.692 0.698 0.701
#> [13] 0.707 0.717 0.722 0.726 0.743 0.758 0.770 1.000

The dependency methods

trex_da_control(da_method = ) chooses how the dependency is handled.

The binary tree method clusters the variables by correlation distance with the linkage hc_linkage (single by default), evaluates the FDP estimate at hc_grid_length correlation thresholds (the default 0 lets the core take \(\min(20, p)\) points) and keeps the threshold rho_thresh. The autoregressive and equi-correlated methods draw dummies with the dependency of the data. cor_coef = -2 estimates the coefficient from the data and rho_thr_DA is the correlation below which the equi-correlated model is skipped.

ar1 <- TRexDASelector$new(X, y, tFDR = 0.1, seed = 7, verbose = FALSE,
                          da_control = trex_da_control(da_method = "AR1"),
                          control = trex_control(K = 10))
ar1$select()
ar1$selected_indices
#> [1] 10 50 70
round(ar1$cor_coef, 3)
#> [1] 0.664

Prior groups

When the groups of dependent variables are known, pass them as prior_groups, a list of integer vectors of length \(p\) with a group label per variable. The dependency estimation then runs within the given groups: hierarchical clustering inside each group, and the calibration grid is built from the within-group dendrogram heights. The deflation acts on data-driven neighbourhoods inside the known groups, never on the raw groups themselves. Several levels can be supplied, and their finest common refinement is the constraint.

blocks <- rep(1:10, each = 10)
pg <- TRexDASelector$new(X, y, tFDR = 0.1, seed = 7, verbose = FALSE,
                         da_control = trex_da_control(prior_groups = list(blocks)),
                         control = trex_control(K = 10))
pg$select()
pg$selected_indices
#> [1] 10 30 34 50 70 90 91

Greedy solvers under T-Rex+DA

The deflation relies on the spread of occurrences within a group of dependent variables. A greedy solver (the orthogonal matching pursuit family, T-AFS) picks the top candidate of a tie deterministically and removes that spread. exch_tie_alpha in trex_control() restores it: statistically indistinguishable top candidates within a correlated cluster are picked uniformly at random per step. The recommended value for greedy solvers under T-Rex+DA is 0.25. The path solvers ignore the option.

greedy <- TRexDASelector$new(X, y, tFDR = 0.1, seed = 7, verbose = FALSE,
                             da_control = trex_da_control(da_method = "BT"),
                             control = trex_control(solver = "TOMP", K = 10,
                                                    exch_tie_alpha = 0.25))
greedy$select()
greedy$selected_indices
#> [1] 10 30 50 70 90