Sparse HJ Biplot: A New Methodology via Elastic Net
Abstract
:1. Introduction
2. Materials and Methods
2.1. Biplot and HJBiplot
 $X$: is the data matrix
 $U$: is the matrix of data whose columns contain the eigenvectors of $X{X}^{T}$
 $V$: is the matrix of data whose columns contain the eigenvectors of ${X}^{T}X$
 $D$: is the diagonal matrix containing the eigenvalues of $X$
 $U$ and $V$ must be orthonormal, that is, ${U}^{T}U=I$ and ${V}^{T}V=I,$ to guarantee the uniqueness of the factorisation.
 The proximity between the points that represent the row markers is interpreted as the similarity between them. Consequently, nearby points allow the identification of clusters of individuals with similar profiles.
 The standard deviation of a variable can be estimated by the module of the vector which represents it.
 Correlations between variables can be captured from the angles between vectors. If two variables are correlated, they will have an acute angle; if the angle they form is obtuse the variables will present a negative correlation; and, if the angle is a right angle it indicates that the variables are not correlated.
 The points orthogonally projected onto a variable approximates the position of the sample values in that variable.
2.2. Disjoint HJ Biplot
2.3. Sparse HJ Biplot
Algorithm 1 Sparse HJ biplot algorithm using elastic net regularisation. 
1. Consider a $nxp$ data matrix. 
2. A tolerance value is set (1 × 10^{−5}). 
3. The data is transformed (centred or standardised). 
4. Decomposition of the original data matrix is performed via SVD. 
5. A is taken as the loadings of the first k components V[, 1:k]. 
6. ${\beta}_{j}$ is calculated by:
$${\beta}_{j}={({\alpha}_{j}{\beta}_{j})}^{T}{X}^{T}X\left({\alpha}_{j}{\beta}_{j}\right)+{\lambda}_{2}\left\right{\beta}_{j}\left\right{}^{2}+{\lambda}_{1,j}\left\right{\beta}_{j}\left\right{}_{1}$$

7. A is updated via SVD of ${X}^{T}X\beta $:
$${X}^{T}X\beta =UD{V}^{T}\to A=U{V}^{T}$$

8. The difference between A and B is updated:
$$di{f}_{AB}=\frac{1}{p}{\displaystyle \sum}_{i=1}^{p}\frac{1}{{\left{\beta}_{i}\right}^{2}{\left{\alpha}_{i}\right}^{2}}{\displaystyle \sum}_{j=1}^{m}{\beta}_{ij}{\alpha}_{ij}$$

9. Steps 4, 5 and 6 are repeated until $di{f}_{AB}<$ tolerance. 
10. The columns are normalized using ${\widehat{V}}_{J}^{EN}=\frac{{\beta}_{j}}{\left\right{\beta}_{j}\left\right},j=1,\dots ,k$ 
11. We then calculate the row markers and column markers. 
12. The elastic net HJ biplot obtained by the previous steps is plotted. 
2.4. Software
3. Illustrative Example
4. Conclusions and Discussion
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Proteins  HJ Biplot  Disjoint Biplot  Elastic Net HJ Biplot  

D1  D2  D3  D1  D2  D3  D1  D2  D3  
1433_epsilon  9.835  −0.791  0.698  1  0  0  6.330  0  0 
4EBP1  −1.127  3.408  −0.752  0  0  1  0  0  0 
4EBP1_pS65  −2.074  6.317  −2.116  0  0  1  0  1.633  0 
4EBP1_pT37  −1.862  2.997  −5.079  0  0  1  0  0  0 
4EBP1_pT70  0.486  5.227  −1.753  0  0  1  0  0.832  0 
53BP1  −6.654  −3.875  1.235  0  0  1  −3.015  0  0 
ARaf_pS299  −4.47  2.681  −1.221  0  1  0  0  0  0 
ACC1  −4.007  −3.042  −0.894  0  0  1  0  0  0 
ACC_pS79  −4.094  −2.386  −2.147  0  0  1  0  0  0 
AMPK_alpha  −1.103  −5.287  1.69  1  0  0  0  −0.917  0 
AMPK_pT172  −0.86  −6.362  1.486  1  0  0  0  −1.075  0 
ANLN  0.877  6.213  5.348  1  0  0  0  0  2.111 
AR  −0.729  −6.68  4.084  1  0  0  0  −4.209  0 
ARID1A  −3.643  0.852  1.409  1  0  0  0  0  0 
ASNS  −4.067  8.449  −1.908  0  0  1  0  4.819  0 
ATM  −5.234  −1.222  −0.396  0  1  0  −1.118  0  0 
Akt  −5.513  −4.694  −0.288  0  0  1  −1.664  0  0 
Akt_pS473  −1.049  1.326  −7.186  0  0  1  0  0  0 
Akt_pT308  −1.782  3.054  −5.25  0  0  1  0  0  0 
Annexin_I  6.102  −0.652  −4.703  1  0  0  1.919  0  0 
BRaf  −7.829  0.995  2.763  1  0  0  −4.104  0  0 
Bak  9.633  −1.78  −1.534  1  0  0  6.042  0  0 
Bax  4.12  −1.972  −2.539  1  0  0  0  0  0 
Bcl2  1.021  −6.875  4.678  1  0  0  0  −3.623  0 
BclxL  4.824  −0.207  1.384  1  0  0  0.189  0  0 
Beclin  −3.283  4.485  6.71  1  0  0  0  0  2.459 
Bid  9.885  1.076  1.13  1  0  0  6.612  0  0 
Bim  0.715  −2.899  3.656  1  0  0  0  0  0 
CRaf  −7.355  −2.215  −0.384  1  0  0  −3.880  0  0 
CRaf_pS338  5.312  6.464  3.593  1  0  0  1.686  0  2.060 
CD31  −2.116  8.398  7.088  1  0  0  0  0  4.871 
CD49b  3.632  2.32  3.552  0  0  1  0  0  0 
CDK1  0.488  9.008  −0.729  1  0  0  0  3.351  0.892 
Caspase7_cleavedD198  1.906  6.466  −1.048  1  0  0  0  2.174  0 
Caveolin1  7.827  −6.415  −1.311  1  0  0  3.643  −0.742  −0.702 
Chk1  6.792  7.605  1.289  1  0  0  3.358  0.985  0.563 
Chk1_pS345  2.418  8.966  5.551  1  0  0  0  0  4.071 
Chk2  −6  3.914  −2.176  1  0  0  −1.815  1.190  0 
Chk2_pT68  −2.247  10.009  4.991  1  0  0  0  0.914  4.810 
Claudin7  −4.187  0.804  4.021  1  0  0  0  0  0 
Collagen_VI  8.506  −2.722  −0.097  1  0  0  4.628  0  0 
Cyclin_B1  −4.571  7.465  −2.745  1  0  0  0  4.414  0 
Cyclin_D1  8.872  −2.678  2.491  1  0  0  5.170  0  0 
Cyclin_E1  −1.927  6.257  −3.637  1  0  0  0  3.837  0 
DJ1  3.216  −5.246  2.356  1  0  0  0  −1.053  0 
Dvl3  −7.369  −0.063  −0.467  1  0  0  −3.462  0  0 
ECadherin  −4.731  1.337  3.855  1  0  0  −0.142  0  0 
EGFR  2.315  4.231  −2.325  0  0  1  0  0.244  0 
EGFR_pY1068  −0.772  1.862  −2.4  0  0  1  0  0  0 
EGFR_pY1173  8.702  1.484  1.23  0  1  0  5.321  0  0 
ERalpha  −0.686  −8.918  5.329  0  1  0  0  −6.509  0 
ERalpha_pS118  −3.542  −4.816  6.177  1  0  0  0  −2.834  0 
ERK2  −4.911  −4.53  −1.404  0  1  0  −0.903  0  0 
FOXO3a  7.666  3.783  1.127  1  0  0  4.027  0  0 
Fibronectin  1.852  −2.596  −0.897  1  0  0  0  0  0 
GAB2  −3.465  1.083  −0.098  1  0  0  0  0  0 
GATA3  −1.972  −8.738  5.216  0  1  0  0  −6.058  0 
GSK3alphabeta  −9.243  0.501  −0.787  1  0  0  −5.924  0  0 
GSK3alphabeta_pS21_S9  −6.718  −0.029  −4.263  0  0  1  −2.273  0  0 
HER2  −3.51  −0.115  0.883  0  0  1  0  0  0 
HER2_pY1248  −0.973  1.28  −0.442  0  0  1  0  0  0 
HER3  3.995  −3.859  0.899  0  1  0  0  0  0 
HER3_pY1289  5.381  0.692  −1.768  0  0  1  1.365  0  0 
HSP70  8.525  3.238  0.181  0  1  0  5.127  0  0 
IGFBP2  1.102  −3.158  1.768  1  0  0  0  0  0 
INPP4B  −2.524  −6.657  6.597  1  0  0  0  −5.098  0 
IRS1  4.045  −2.6  5.377  1  0  0  0  −0.741  0 
JNK2  −0.584  −9.036  1.984  1  0  0  0  −4.625  −0.041 
JNK_pT183_pT185  2.476  −2.463  −0.011  1  0  0  0  0  0 
KRas  10.304  0.949  0.456  1  0  0  7.040  0  0 
Ku80  −8.303  0.743  0.447  1  0  0  −4.768  0  0 
LBK1  −2.52  1.968  7.864  1  0  0  0  0  1.173 
Lck  4.528  3.158  −3.052  0  1  0  0.055  0  0 
MAPK_pT202_Y204  −0.304  −2.731  −3.607  1  0  0  0  0  0 
MEK1  2.993  −2.122  −2.706  1  0  0  0  0  0 
MEK1_pS217_S221  −5.209  −1.263  −2.992  0  0  1  −0.514  0  0 
MIG6  4.206  2.429  1.513  0  1  0  0  0  0 
Mre11  2.55  7.729  7.37  1  0  0  0  0  4.444 
NCadherin  10.669  1.608  0.799  1  0  0  7.616  0  0 
NFkBp65_pS536  −4.992  0.915  −2.066  0  0  1  −0.330  0  0 
NF2  −4.468  −1.197  1.559  0  0  1  −0.284  0  0 
Notch1  4.22  5.154  0.049  0  1  0  0  0.221  0 
PCadherin  0.692  5.044  −4.532  1  0  0  0  2.793  0 
PAI1  2.668  0.836  −1.01  1  0  0  0  0  0 
PCNA  5.345  2.069  −2.278  1  0  0  0.893  0  0 
PDCD4  −7.3  4.39  2.29  1  0  0  −2.851  0  0.807 
PDK1_pS241  −4.328  −7.468  1.103  0  0  1  −0.406  −2.068  0 
PI3Kp110alpha  −2.045  −1.893  −0.058  0  1  0  0  0  0 
PKCalpha  4.107  −2.008  −2.34  1  0  0  0  0  0 
PKCalpha_pS657  2.787  −0.643  −0.587  1  0  0  0  0  0 
PKCdelta_pS664  −1.622  3.93  6.058  1  0  0  0  0  1.686 
PR  −0.264  −6.858  5.072  0  0  1  0  −4.465  0 
PRAS40_pT246  −4.816  6.243  −2.556  0  1  0  0  0.549  0 
PRDX1  1.244  2.609  −0.483  1  0  0  0  0  0 
PTEN  −3.443  −4.129  0.798  1  0  0  0  0  0 
Paxillin  −3.448  −4.577  −2.127  1  0  0  0  0  0 
Pea15  3.573  −5.791  0.144  1  0  0  0  −0.581  0 
RBM3  −3.695  −3.794  0.379  1  0  0  0  0  0 
Rad50  −0.172  −5.001  2.657  0  1  0  0  −0.962  0 
Rb_pS807_S811  −6.574  1.905  −2.987  1  0  0  −2.009  0  0 
S6  −7.718  4.048  −0.745  1  0  0  −3.914  0.543  0 
S6_pS235_S236  −2.383  2.566  −6.953  1  0  0  0  0.644  0 
S6_pS240_S244  −2.97  2.075  −7.177  1  0  0  0  0.350  0 
SCD1  −4.54  6.624  4.302  1  0  0  0  0  2.683 
STAT3_pY705  4.967  −2.48  −2.268  1  0  0  0.564  0  0 
STAT5alpha  −4.779  −3.47  −2.199  0  0  1  −0.608  0  0 
Shc_pY317  −4.083  3.86  2.936  0  1  0  0  0  0.740 
Smad1  0.205  −0.579  2.104  1  0  0  0  0  0 
Smad3  4.219  −5.98  2.297  1  0  0  0  −1.414  0 
Smad4  9.908  −0.979  −1.163  1  0  0  6.333  0  0 
Src  5.433  1.57  −0.883  1  0  0  1.023  0  0 
Src_pY416  −2.997  8.115  2  1  0  0  0  0.303  2.653 
Src_pY527  4.385  −0.67  −5.883  1  0  0  0.282  0  0 
Stathmin  7.149  7.29  4.099  1  0  0  3.795  0  1.895 
Syk  −4.569  2.55  −4.518  0  0  1  −0.140  0.699  0 
Transglutaminase  −2.552  3.844  −0.091  0  1  0  0  0  0 
Tuberin  −6.672  −6.243  −0.289  1  0  0  −3.259  −0.161  0 
VEGFR2  −4.498  −4.125  2.757  0  1  0  −0.236  −0.168  0 
XBP1  1.892  1.07  4.482  1  0  0  0  0  0 
XRCC1  1.111  −0.622  2.963  0  0  1  0  0  0 
YAP_pS127  0.955  −1.745  −1.806  0  1  0  0  0  0 
YB1  −3.686  −0.954  1.81  0  1  0  0  0  0 
YB1_pS102  −2.047  2.807  −6.06  1  0  0  0  0  0 
alphaCatenin  −4.342  4.573  7.605  1  0  0  0  0  3.159 
betaCatenin  −6.877  −0.545  0.895  1  0  0  −2.921  0  0 
cKit  2.082  3.856  −2.432  1  0  0  0  0.042  0 
cMet_pY1235  1.579  8.612  6.828  1  0  0  0  0  4.546 
cMyc  1.778  2.82  2.362  1  0  0  0  0  0 
eEF2  −5.963  3.474  −1.39  1  0  0  −1.832  0.245  0 
eEF2K  −5.193  −5.415  2.727  0  1  0  −1.232  −1.218  0 
eIF4E  1.789  −0.64  −1.43  1  0  0  0  0  0 
mTOR  −9.332  −2.094  2.38  1  0  0  −6.073  0  0 
mTOR_pS2448  −4.804  0.348  −1.92  1  0  0  0  0  0 
p27  3.678  −0.86  2.963  1  0  0  0  0  0 
p27_pT157  −1.764  7.218  4.386  1  0  0  0  0  2.536 
p27_pT198  −2.494  5.968  −1.246  1  0  0  0  1.908  0 
p38_MAPK  0.767  −5.788  −0.57  1  0  0  0  0  −0.006 
p38_pT180_Y182  0.215  −1.303  −3.117  1  0  0  0  0  0 
p53  −2.966  9.064  4.314  1  0  0  0  0.846  3.663 
p70S6K  −4.922  −1.661  1.338  1  0  0  −0.838  0  0 
p70S6K_pT389  5.575  1.371  −1.5  1  0  0  1.619  0  0 
p90RSK_pT359_S363  −6.319  1.839  −1.653  1  0  0  −1.687  0  0 
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CubillaMontilla, M.; NietoLibrero, A.B.; GalindoVillardón, M.P.; TorresCubilla, C.A. Sparse HJ Biplot: A New Methodology via Elastic Net. Mathematics 2021, 9, 1298. https://doi.org/10.3390/math9111298
CubillaMontilla M, NietoLibrero AB, GalindoVillardón MP, TorresCubilla CA. Sparse HJ Biplot: A New Methodology via Elastic Net. Mathematics. 2021; 9(11):1298. https://doi.org/10.3390/math9111298
Chicago/Turabian StyleCubillaMontilla, Mitzi, Ana Belén NietoLibrero, M. Purificación GalindoVillardón, and Carlos A. TorresCubilla. 2021. "Sparse HJ Biplot: A New Methodology via Elastic Net" Mathematics 9, no. 11: 1298. https://doi.org/10.3390/math9111298