ddn3.bcd

Block coordinate descent for DDN

This module implements the original block coordinate descent in [1], as well as various accelerated versions. Most methods are also Numba accelerated, except for bcd_org_old, which is left for comparison purposes.

For data with lots of samples, consider using bcd_corr. For data with lots of features, the bcd_residual is faster.

[1] Zhang, Bai, and Yue Wang. “Learning structural changes of Gaussian graphical models in controlled experiments.” UAI (2010).

Module Contents

Functions

bcd_org_old(beta_in, y, X, lambda1, lambda2[, n1, n2, ...])

The BCD algorithm in DDN 2.0.

bcd_org(beta_in, y, X, lambda1, lambda2[, n1, n2, ...])

The BCD algorithm in DDN 2.0, with Numba acceleration and some simplification.

bcd_residual(beta_in, X1, X2, y1_resi, y2_resi, ...[, ...])

BCD algorithm for DDN using residual update strategy

bcd_corr(beta_in, cur_node, lambda1, lambda2, ...[, ...])

BCD algorithm for DDN using correlation matrix update strategy

solve2d(rho1, rho2, lambda1, lambda2)

Optimize for two variables corresponding to one node
ddn3.bcd.bcd_org_old(beta_in, y, X, lambda1, lambda2, n1=1, n2=1, threshold=1e-06, max_iter=100000)

The BCD algorithm in DDN 2.0.

See https://arxiv.org/abs/1203.3532 for details Denote P be the number features. N1 be the sample size for condition 1, and N2 for condition 2.

Parameters:

  • beta_in (array_like, length 2P) – Initial beta. If initialization is not needed, use an array of all zeros

  • y (array_like, size (N1+N2) by 1) – The samples for the node serving as response.

  • X (array_like, size (N1+N2) by (P-2)) – The block diagonal matrix with all the predictive nodes. Each condition has P-1 nodes. The top left block has size N1(P-1), while the bottom right block has size N2(P-1). Other elements are all zeros.

  • lambda1 (float) – DDN parameter lambda1.

  • lambda2 (float) – DDN parameter lambda2.

  • n1 (int) – Sample size for condition 1, N1

  • n2 (int) – Sample size for condition 2, N2

  • threshold (float) – Convergence threshold.

  • max_iter (int) – Maximum number of iterations

Returns:

  • beta (ndarray, shape 2P) – Estimated beta for two conditions on node CurrIdx

  • r (int) – Number of iterations taken

  • betaerr (float) – The error term

ddn3.bcd.bcd_org(beta_in, y, X, lambda1, lambda2, n1=1, n2=1, threshold=1e-06, max_iter=100000)

The BCD algorithm in DDN 2.0, with Numba acceleration and some simplification.

See https://arxiv.org/abs/1203.3532 for details Denote P be the number features. N1 be the sample size for condition 1, and N2 for condition 2.

Parameters:

  • beta_in (array_like, length 2P) – Initial beta. If initialization is not needed, use an array of all zeros

  • y (array_like, size (N1+N2) by 1) – The samples for the node serving as response.

  • X (array_like, size (N1+N2) by (P-2)) – The block diagonal matrix with all the predictive nodes. Each condition has P-1 nodes. The top left block has size N1(P-1), while the bottom right block has size N2(P-1). Other elements are all zeros.

  • lambda1 (float) – DDN parameter lambda1.

  • lambda2 (float) – DDN parameter lambda2.

  • n1 (int) – Sample size for condition 1, N1

  • n2 (int) – Sample size for condition 2, N2

  • threshold (float) – Convergence threshold.

  • max_iter (int) – Maximum number of iterations

Returns:

  • beta (ndarray, shape 2P) – Estimated beta for two conditions on node CurrIdx

  • r (int) – Number of iterations taken

  • betaerr (float) – The error term

ddn3.bcd.bcd_residual(beta_in, X1, X2, y1_resi, y2_resi, CurrIdx, lambda1, lambda2, threshold, max_iter=10000)

BCD algorithm for DDN using residual update strategy

The algorithm allows warm start, which requires initial beta_in, y1_resi, and y2_resi. See run_resi on how to prepare these inputs. Denote P be the number features. N1 be the sample size for condition 1, and N2 for condition 2.

Parameters:

  • beta_in (array_like, length 2P) – Initial beta. If initialization is not needed, use an array of all zeros

  • X1 (array_like, shape N1 by P) – The data from condition 1

  • X2 (array_like, shape N2 by P) – The data from condition 2

  • y1_resi (array_like, shape N1 by 1) – The initial residual signal for condition 1. If warm start is not used, it is column CurrIdx of X1.

  • y2_resi (array_like, shape N2 by 1) – The initial residual signal for condition 2. If warm start is not used, it is column CurrIdx of X2.

  • CurrIdx (int) – Index of the current node that serve as the response variable.

  • lambda1 (float) – DDN parameter lambda1.

  • lambda2 (float) – DDN parameter lambda2.

  • threshold (float) – Convergence threshold.

  • max_iter (int) – Maximum number of iterations

Returns:

  • beta (ndarray, shape 2P) – Estimated beta for two conditions on node CurrIdx

  • r (int) – Number of iterations taken

  • betaerr (float) – The error term

ddn3.bcd.bcd_corr(beta_in, cur_node, lambda1, lambda2, corr_matrix_1, corr_matrix_2, threshold=1e-06, max_iter=100000)

BCD algorithm for DDN using correlation matrix update strategy

This approach is more suitable for larger sample sizes. The algorithm allows warm start, which requires initial beta_in. Denote P be the number features. N1 be the sample size for condition 1, and N2 for condition 2.

Parameters:

  • beta_in (array_like, length 2P) – Initial beta. If initialization is not needed, use an array of all zeros

  • cur_node (int) – Index of the current node that serve as the response variable.

  • lambda1 (float) – DDN parameter lambda1.

  • lambda2 (float) – DDN parameter lambda2.

  • corr_matrix_1 (array_like, P by P) – Correlation matrix for condition 1

  • corr_matrix_2 (array_like, P by P) – Correlation matrix for condition 2

  • threshold (float) – Convergence threshold.

  • max_iter (int) – Maximum number of iterations

Returns:

  • beta (ndarray, shape 2P) – Estimated beta for two conditions on node CurrIdx

  • r (int) – Number of iterations taken

  • delta_beta (float) – The error term

ddn3.bcd.solve2d(rho1, rho2, lambda1, lambda2)

Optimize for two variables corresponding to one node

The details can be found in https://arxiv.org/abs/1203.3532

Parameters:

  • rho1 (float) – The rho from data1

  • rho2 (float) – The rho from data2

  • lambda1 (float) – DDN parameter lambda1

  • lambda2 (float) – DDN parameter lambda2

Returns:

  • beta1 (float) – Optimal coefficient for data1

  • beta2 (float) – Optimal coefficient for data2