Date: 2025-11-04

# SelectBoost.beta algorithms

#### Frédéric Bertrand

#### 2025-11-04

## Motivation

`SelectBoost.beta` re-uses the correlated-resampling machinery introduced by the original SelectBoost package and combines it with Beta-regression selectors. This vignette summarises the main routines and presents pseudo-code for their internal logic. The goal is to make it easy to re-implement or extend the algorithms in other contexts.

## Building blocks

The following helpers expose the canonical SelectBoost stages.

- `sb_normalize()` centres and \(\ell_2\)-normalises the design matrix columns.
- `sb_compute_corr()` computes a correlation (or user-supplied association) matrix from the normalised design.
- `sb_group_variables()` converts the correlation matrix into groups of highly associated predictors for a given threshold \(c_0\).
- `sb_resample_groups()` regenerates correlated predictors for each group by drawing from a multivariate normal approximation and re-normalising. When all groups are singletons it now warns and simply returns repeated copies of the normalised design.
- `sb_apply_selector_manual()` applies a selector to each resampled design and collects the resulting coefficient vectors. Set `keep_template = TRUE` (the default) to retain the base fit as column `sim0` without recomputing it on the first resample.
- `sb_selection_frequency()` converts the matrix of coefficients into selection frequencies while respecting the selector’s coefficient convention.

## Pseudo-code: manual workflow

The manual SelectBoost workflow follows the same steps regardless of the base selector. Pseudo-code for producing selection frequencies at a single threshold is given below.

    Procedure ManualSelectBoost(X, Y, selector, c0, B):
      1. X_norm <- sb_normalize(X)
      2. Corr <- sb_compute_corr(X_norm)
      3. Groups <- sb_group_variables(Corr, c0)
      4. Resamples <- sb_resample_groups(X_norm, Groups, B)
      5. CoefMatrix <- sb_apply_selector_manual(X_norm, Resamples, Y, selector)
      6. Frequencies <- sb_selection_frequency(CoefMatrix, version = "glmnet")
      7. Return Frequencies

In practice `sb_resample_groups()` preserves singletons untouched. Only groups with two or more predictors receive correlated draws.

## Pseudo-code: correlation grid driver

`sb_beta()` extends the manual workflow by iterating over a grid of correlation thresholds. The following pseudo-code matches the behaviour of the exported function.

    Algorithm sb_beta(X, Y, selector, B, step.num, steps.seq, version, squeeze):
      1. If squeeze, transform Y into the open unit interval.
      2. X_norm <- sb_normalize(X)
      3. Corr <- sb_compute_corr(X_norm)
      4. Grid <- {1} ∪ .sb_c0_sequence(Corr, step.num, steps.seq) ∪ {0}
      5. For each c0 in Grid:
           a. Groups <- sb_group_variables(Corr, c0)
           b. If every group has size 1:
                i. CoefMatrix <- selector(X_norm, Y)
              Else:
                i. Resamples <- sb_resample_groups(X_norm, Groups, B)
               ii. For each design in Resamples:
                      - CoefMatrix[, b] <- selector(design, Y)
           c. Freq[c0, ] <- sb_selection_frequency(CoefMatrix, version)
      6. Attach attributes (B, selector, c0 sequence) and return Freq

The selector argument can be any function returning a numeric vector of coefficients with optional names. When `version = "glmnet"`, the first entry is interpreted as the intercept and excluded from the selection frequencies.

The squeezing step enforces the usual SelectBoost transformation that pushes all responses inside `(0, 1)`. Keep it enabled unless you already pre-processed the outcome; otherwise zero or one values will cause the selectors to abort.

## Extending the algorithms

The modular helpers are designed to be recomposed. For example, it is possible to plug in a custom grouping routine before calling `sb_resample_groups()` or to supply a selector that implements cross-validation or penalisation strategies. Because each helper only relies on basic R primitives, the pseudo-code above translates readily into other languages.

## Conference communications

The SelectBoost4Beta concepts described here were showcased by Frédéric Bertrand and Myriam Maumy in 2023 at:

- Joint Statistical Meetings 2023 (Toronto, Canada): “Improving variable selection in Beta regression models using correlated resampling”.
- BioC2023 (Boston, USA): “SelectBoost4Beta: Improving variable selection in Beta regression models”.

These communications detailed how correlation-aware resampling strengthens variable selection performance for Beta regression under strong predictor dependencies.