l1rotation: Identify Loading Vectors under Sparsity in Factor Models

Description

Simplify the loading matrix in factor models using the l1 criterion as proposed in Freyaldenhoven (2025) doi:10.21799/frbp.wp.2020.25. Given a data matrix, find the rotation of the loading matrix with the smallest l1-norm and/or test for the presence of local factors with main function local_factors().

Author(s)

Maintainer: Ryan Kobler kobleary@gmail.com

Authors:

• Simon Freyaldenhoven simon.freyaldenhoven@phil.frb.org [copyright holder]

See Also

Useful links:

• https://kobleary.github.io/l1rotation/

• https://github.com/SimonFreyaldenhoven/l1rotation

• Report bugs at https://github.com/SimonFreyaldenhoven/l1rotation/issues

Pipe operator

Description

See magrittr::%>% for details.

Usage

lhs %>% rhs

Arguments

Value

The result of calling rhs(lhs).

Example data with two factors from the replication files of Freyaldenhoven (2025).

Description

Example data with two factors from the replication files of Freyaldenhoven (2025).

Usage

example_data

Format

example_data

A matrix with 224 rows and 207 columns.

Source

Dataset available as a .mat file can be found under local_factors.zip at https://simonfreyaldenhoven.github.io/software/

Find the rotation of the loading matrix with the smallest l1-norm, as in local_factors(), with additional flexibility.

Description

Find the most sparse rotation of an orthonormal basis of the loading space of a t by n matrix X. Additional flexibility with the initial_loadings argument allows the user to specify any orthonormal basis rather than defaulting to PCA.

Usage

find_local_factors(X, r, initial_loadings, parallel = FALSE, n_cores = NULL)

Arguments

Value

Returns a list with the following components:

• initial_loadings Principal Component estimate of the loading matrix (if not supplied).

• rotated_loadings Matrix that is the rotation of the loading matrix that produces the smallest l1-norm.

• rotation_diagnostics A list containing 3 components:

  • R Rotation matrix that when used to rotate initial_loadings produces the smallest l1-norm.

  • l1_norm Vector of length r containing the value of the l1 norm each solution generates.

  • sol_frequency Vector of length r containing the frequency in the initial grid of each solution.

Examples

# Minimal example with 2 factors, where X is a 224 by 207 matrix
r <- 2
M <- nrow(example_data)
n <- ncol(example_data)

# Compute PCA estimates
basis <- svd(example_data / sqrt(M), nu = M, nv = n)
initial_loadings <- sqrt(n) * basis$v[, 1:r]

# Find minimum rotation using orthonormal basis initial_loadings
rotation_result <- find_local_factors(X = example_data, r = r, initial_loadings = initial_loadings)

Check whether local factors are present and find the rotation of the loading matrix with the smallest l1-norm.

Description

local_factors tests whether local factors are present and returns both the Principal Component estimate of the loadings and the rotation of the loadings with the smallest l1-norm. It also produces graphical illustrations of the results.

Usage

local_factors(X, r, parallel = FALSE, n_cores = NULL)

Arguments

Value

Returns a list with the following components:

• has_local_factors A logical equal to TRUE if local factors are present.

• initial_loadings Principal component estimate of the loading matrix.

• rotated_loadings Matrix that is the rotation of the loading matrix that produces the smallest l1-norm.

• rotation_diagnostics A list containing 3 components:

  • R Rotation matrix that when used to rotate initial_loadings produces the smallest l1-norm.

  • l1_norm Vector of length r containing the value of the l1 norm each solution generates.

  • sol_frequency Vector of length r containing the frequency in the initial grid of each solution.

• pc_plot Tile plot of the Principal Component estimate of the loading matrix.

• rotated_plot Tile plot of the l1-rotation of the loading matrix estimate.

• small_loadings_plot Plot of the number of small loadings for each column of the l1-rotation of the loading matrix estimate.

Examples

# Minimal example with 2 factors, where X is a 224 by 207 matrix
lf <- local_factors(X = example_data, r = 2)

# Visualize Principal Component estimate of the loadings
lf$pc_plot

# Visualize l1-rotation loadings
lf$pc_rotated_plot

Test for the presence of local factors, as in local_factors(), with additional flexibility.

Description

Test for the presence of local factors, as in local_factors(), with additional flexibility.

Usage

test_local_factors(X, r, loadings = NULL)

Arguments

Value

Returns a list with the following components:

• has_local_factors Logical equal to TRUE if local factors are present.

• n_small Integer denoting the number of small loadings in sparse rotation.

• gamma_n Integer denoting the critical value to compare n_small to.

• h_n Number denoting the cutoff used to determine which loadings are small.

• loadings Matrix that is the rotation of the loadings that produces the smallest l1-norm (if not supplied).

Examples

# Minimal example with 2 factors, where X is a 224 by 207 matrix
r <- 2
M <- nrow(example_data)
n <- ncol(example_data)

# Find minimum rotation
rotation_result <- find_local_factors(X = example_data, r)

# Test if sparse basis has local factors
test_result <- test_local_factors(
   X = example_data,
   r = r,
   loadings = rotation_result$rotated_loadings
)

test_result$has_local_factors