This python package implements the CDPA method proposed in the paper [1]. See example.py for details, with Python 3.6.3 (or above) and the lapjv package (pip install lapjv).
Let 
be two datasets measured on a common set of 
objects, where 
is the number of variables in the 
-th dataset. The CDPA method conducts the following decomposition:
where 
is obtained from the D-CCA method. Specifically,
: the signal matrix,
: the common-source matrix,
: the distinctive-source matrix,
: the noise matrix,
: the matrix with zero padding and permutation matrix 
if necessary,
: the common-pattern matrix 
rescaled with the magnitude of ,
: the distinctive-pattern matrix.
Use the CDPA function:
X_1_hat, X_2_hat, C_1_hat, C_2_hat, D_1_hat, D_2_hat, r_1_hat, r_2_hat, r_12_hat, \
ccor_hat, ctheta_hat, C_mat_hat, C_mat_neg_hat, pcor_hat, ptheta_hat, P_mat_hat \
= dcca.CDPA(Y_1, Y_2, r_1=None, r_2=None, r_12=None, method=None, P_set = 1, P_mat=None, assignment=1)
or
X_1_hat, X_2_hat, C_1_hat, C_2_hat, D_1_hat, D_2_hat, r_1_hat, r_2_hat, r_12_hat, \
ccor_hat, ctheta_hat, C_mat_hat, C_mat_neg_hat, pcor_hat, ptheta_hat \
= dcca.CDPA(Y_1, Y_2, r_1=None, r_2=None, r_12=None, method=None, P_set = 0)
with the function parameters:
-
r_1, r_2: the ranks of
and 
. -
method: If r_1 and r_2 are None, then for the selection of r_1 and r_2, method='ED' and method=None use the ED method [3], and method='GR' uses the GR method [4].
-
r_12: the rank of
. If r_12=None, then r_12 is automatically selected by the MDL-IC method [5]. -
P_set, P_mat, assignment: If P_set=0, no row matching of
and 
. If P_set=1, then the row matching is implemented by permuting to be 
: in particular, if P_mat=None, then assignment=1 and assignment=0 respectively use a greedy algorithm [6] (fast but less accurate) and the Jonker-Volgenant algorithm [7] for the DSPFP method [6] for the row matching, and otherwise P_mat can be assigned to be a given permutation matrix.
with output:
-
X_1_hat, X_2_hat, C_1_hat, C_2_hat, D_1_hat, D_2_hat: the D-CCA matrix estimates [2] for
. -
r_1_hat, r_2_hat, r_12_hat: estimates of r_1, r_2, r_12.
-
ccor_hat, ctheta_hat: the estimated canonical correlations and associated angles between the respective latent-factor spaces of
and . -
C_mat_hat: the estimate for the unscaled common-pattern matrix
of and . -
C_mat_neg_hat: the estimate for the unscaled common-pattern matrix
of and . -
ptheta_hat, pcor_hat: the estimated principal angles and their cosines between the respective column spaces of
and 
that are coefficient matrices of
and 
on the common latent factors of and . -
P_mat_hat: the estimate fo the permutation matrix
for .
[1] Shu, H., & Qu, Z. (2019). CDPA: Common and Distinctive Pattern Analysis between High-dimensional Datasets. arXiv preprint arXiv:1912.09989.
[2] Shu, H., Wang, X., & Zhu, H. (2019) D-CCA: A Decomposition-based Canonical Correlation Analysis for High-dimensional Datasets. Journal of the American Statistical Association, DOI: 10.1080/01621459.2018.1543599
[3] Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 92(4), 1004-1016.
[4] Ahn, S. C., & Horenstein, A. R. (2013). Eigenvalue ratio test for the number of factors. Econometrica, 81(3), 1203-1227.
[5] Song, Y., Schreier, P. J., Ramírez, D., & Hasija, T. (2016). Canonical correlation analysis of high-dimensional data with very small sample support. Signal Processing, 128, 449-458.
[6] Lu, Y., Huang, K., & Liu, C. L. (2016). A fast projected fixed-point algorithm for large graph matching. Pattern Recognition, 60, 971-982.
[7] Jonker, R., & Volgenant, A. (1987). A shortest augmenting path algorithm for dense and sparse linear assignment problems. Computing, 38(4), 325-340.