A New Algorithm for Computing Disjoint Orthogonal Components in the ThreeWay Tucker Model
Abstract
:1. Introduction
2. The Tucker Model and the Disjoint Approach
2.1. ThreeWay Tables
2.2. The Tucker Model
Algorithm 1: TuckerALS 
2.3. Disjoint Approach for the Tucker Model
 For all $i\phantom{\rule{4pt}{0ex}}\exists !\phantom{\rule{4pt}{0ex}}j$, such that ${x}_{ij}\ne 0$.
 For all $j\phantom{\rule{4pt}{0ex}}\exists \phantom{\rule{4pt}{0ex}}i$, such that ${x}_{ij}\ne 0$.
2.4. Illustrative Example
2.5. The DisjointPCA Algorithm
3. The DisjointTuckerALS Algorithm
3.1. The Stages of the Algorithm
 $\underline{\mathit{X}}$: Threeway table of data;
 $P,Q,R$: Number of components in $\mathit{A}$mode, $\mathit{B}$mode, $\mathit{C}$mode, respectively;
 ALSMaxIter: Maximum number of iterations of the ALS algorithm; and
 Tol: Maximum distance allowed in the fit of the model for two consecutive iterations of the DisjointPCA algorithm.
Algorithm 2: Adapted TuckerALS 
3.2. Using the DisjointTuckerALS Algorithm
Algorithm 3: Procedure for using DisjointTuckerALS 

4. Numerical Results
4.1. Computational Aspects
4.2. Generator of Disjoint Structure Tensors
4.3. Applying the DisjointTuckerALS Algorithm to Simulated Data
4.4. Applying the DisjointTuckerALS Algorithm to Real Data
5. Conclusions, Discussion, Limitations, and Future Research
 (i)
 A new algorithm for computing disjoint orthogonal components in a threeway table with the Tucker model was proposed.
 (ii)
 A measure of goodness of fit to evaluate the algorithms presented was proposed.
 (iii)
 A optimization mathematical model was used.
 (iv)
 A numerical evaluation of the proposed methodology was considered by means of Monte Carlo simulations.
 (v)
 By using a case study with realworld data, we have illustrated the new algorithm.
 (i)
 There is no guarantee that the optimal solution is attained due to the heuristic nature of the DisjointTuckerALS algorithm.
 (ii)
 In the absence of additional constraints to those inherent to the original problem, the space of feasible solutions contains the global optimum. However, by incorporating the constraints of the DisjointTuckerALS algorithm, the space of feasible solutions is compressed, which aims to find a solution that is as close as possible to the global optimum within this new set of feasible solutions. For this reason, the fit corresponding to the solution provided by the DisjointTuckerALS algorithm is less than the fit achieved by the TuckerALS algorithm. Nevertheless, the incorporated constraints allow us to put zeros in the positions of the variables with low contribution into a component of the loading matrix, which permits us to interpret the components more clearly.
 (iii)
 The proposed algorithm takes longer than the TuckerALS algorithm, so that a tradeoff between interpretation and speed exists.
 (i)
 Functional magnetic resonance imaging (fMRI) has been successfully used by the neuroscientists for diagnosis of neurological and neurodevelopmental disorders. The fMRI has been analyzed by means of tensorial methods while using the Tucker model [23].
 (ii)
 Component analysis in threeway tables also has application in environmental sciences. For example, in [24], through the multivariate study of a sample of blue crabs, a hypothesis is tested that environmental stress weakens some organisms, since the normal immune response is not able to protect them from a bacterial infection. A Tucker model was used for this analysis.
 (iii)
 The data of the price indexes in search of behavior patterns using the Tucker decomposition were analyzed in [25]. The DisjointTuckerALS algorithm can be used for detecting these patterns.
 (iv)
 An application in economy on the specialization indexes of the electronic industries of 23 European countries of the Organisation for Economic Cooperation and Development (OECD) based on threeway tables is presented in [1]; see also http://threemode.leidenuniv.nl. Applications in stock markets and breakpoint analysis for the COVID19 pandemic can be also considered [26].
 (v)
 On the website "The ThreeMode Company” (see http://threemode.leidenuniv.nl), data sets corresponding to threeway tables, including engineering, management, and medicine, are related to (a) aerosol particles in Austria; (b) diseased blue crabs in the US; (c) chromatography; (d) coping Dutch primary school children; (e) Dutch hospitals as organizations; (f) girls’ growth curves between five and 15 years old; (g) happiness, siblings, and schooling; (h) multiple personalities; (i) parental behavior in Japan; (j) peer play and a new sibling; (k) Dutch children in the strange situations; and, (l) university positions and academics.
 (i)
 We believe that the disjoint approach can be used together with existing techniques.
 (ii)
 A study that allows for obtaining a disjoint structure in the core of a Tucker model to facilitate their interpretation is of interest.
 (iii)
 A bootstrap analysis for the loading matrices can be performed.
 (iv)
 (v)
 Other applications of the algorithm developed in the context of multivariate methods are: discriminant analysis, correspondence analysis, and cluster analysis, as well as the already mentioned functional data analysis and PLS.
 (vi)
 There is also a promising field of applications in the socalled statistical learning; for example, for image compression.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Emotional  Sensitive  Caring  Thorough  

Anne  0.0  0.0  4.0  4.0 
Bert  0.0  0.0  2.0  2.0 
Claus  0.0  0.0  2.0  2.0 
Dolly  0.0  0.0  4.0  4.0 
Edna  0.0  0.0  2.5  2.5 
Frances  0.0  0.0  4.0  4.0 
Emotional  Sensitive  Caring  Thorough  

Anne  0.6  0.6  2.4  2.4 
Bert  0.2  0.2  1.8  1.8 
Claus  0.2  0.2  1.8  1.8 
Dolly  0.6  0.6  2.4  2.4 
Edna  0.4  0.4  2.1  2.1 
Frances  0.6  0.6  2.4  2.4 
Emotional  Sensitive  Caring  Thorough  

Anne  4.0  4.0  0.0  0.0 
Bert  1.0  1.0  1.0  1.0 
Claus  1.0  1.0  1.0  1.0 
Dolly  4.0  4.0  0.0  0.0 
Edna  2.0  2.0  0.5  0.5 
Frances  4.0  4.0  0.0  0.0 
Emotional  Sensitive  Caring  Thorough  

Anne  4.6  4.6  0.9  0.9 
Bert  1.2  1.2  1.8  1.8 
Claus  1.2  1.2  1.8  1.8 
Dolly  4.6  4.6  0.9  0.9 
Edna  2.4  2.4  1.4  1.4 
Frances  4.6  4.6  0.9  0.9 
Femininity  Masculinity  

Anne  −0.518560994  0.2282689980 
Bert  −0.2167812130  −0.619921659 
Claus  −0.2167812130  −0.619921659 
Dolly  −0.518560994  0.2282689980 
Edna  −0.315111562  −0.2739964750 
Frances  −0.518560994  0.2282689980 
Femininity  Masculinity  

Anne  −0.545737149  0 
Bert  0  −0.707106781 
Claus  0  −0.707106781 
Dolly  −0.545737149  0 
Edna  −0.326363131  0 
Frances  −0.545737149  0 
Emotionality  Conscientiousness  

Emotional  −0.588715219  −0.3916814920 
Sensitive  −0.588715219  −0.3916814920 
Caring  −0.3916814920  0.588715219 
Thorough  −0.3916814920  0.588715219 
Emotionality  Conscientiousness  

Emotional  −0.707106781  0 
Sensitive  −0.707106781  0 
Caring  0  −0.707106781 
Thorough  0  −0.707106781 
Social Situations  Performance Situations  

Applying for an examen  −0.3331986930  0.770591276 
Giving a speech  −0.3104255140  0.435143574 
Family picnic  −0.538191124  −0.3872072680 
Meeting a new date  −0.709200215  −0.2586690690 
Social Situations  Performance Situations  

Applying for an examen  0  −0.827376571 
Giving a speech  0  −0.561647585 
Family picnic  −0.633810357  0 
Meeting a new date  −0.773488482  0 
Social Situations  Performance Situations  

Emotionality  Conscientiousness  Emotionality  Conscientiousness  
Femininity  −17.09837769  −0.489491858  0.50606476  −12.25957342 
Masculinity  −0.623702797  4.106153763  0.54320104  0.72835096 
Social Situations  Performance Situations  

Emotionality  Conscientiousness  Emotionality  Conscientiousness  
Femininity  −15.55006689  −2.25788715  −0.883942787  −12.28278827 
Masculinity  −3.12399307  −4.052179249  −0.224659034  −5.33143759 
Disjoint Orthogonal Components  Fit (in %)  Runtime (in min) 

None (TuckerALS)  99.84  0.0038 
$\mathit{A}$ (DisjointTuckerALS)  99.25  0.0831 
$\mathit{B}$ (DisjointTuckerALS)  99.84  0.0797 
$\mathit{C}$ (DisjointTuckerALS)  98.67  0.0672 
$\mathit{A}$, $\mathit{B}$ (DisjointTuckerALS)  99.25  0.0989 
$\mathit{A}$, $\mathit{C}$ (DisjointTuckerALS)  98.10  0.0866 
$\mathit{B}$, $\mathit{C}$ (DisjointTuckerALS)  98.67  0.0913 
$\mathit{A}$, $\mathit{B}$, $\mathit{C}$ (DisjointTuckerALS)  98.10  0.1012 
Algorithm  Fit (in %)  Runtime (in min) 

TuckerALS  94.73  0.0717 
DisjointTuckerALS  92.79  0.8735 
${\mathit{sy}}_{1}$  ${\mathit{sy}}_{2}$  ${\mathit{sy}}_{3}$  

${\mathit{sx}}_{\mathbf{1}}$  0.48932151  0  0 
${\mathit{sx}}_{\mathbf{2}}$  0.44138634  0  0 
${\mathit{sx}}_{\mathbf{3}}$  0.38687825  0  0 
${\mathit{sx}}_{\mathbf{4}}$  0.40775562  0  0 
${\mathit{sx}}_{\mathbf{5}}$  0.49980310  0  0 
${\mathit{sx}}_{\mathbf{6}}$  0  0.39376793  0 
${\mathit{sx}}_{\mathbf{7}}$  0  0.36369995  0 
${\mathit{sx}}_{\mathbf{8}}$  0  0.39165864  0 
${\mathit{sx}}_{\mathbf{9}}$  0  0.37564528  0 
${\mathit{sx}}_{\mathbf{10}}$  0  0.43001693  0 
${\mathit{sx}}_{\mathbf{11}}$  0  0.31146907  0 
${\mathit{sx}}_{\mathbf{12}}$  0  0.36910128  0 
${\mathit{sx}}_{\mathbf{13}}$  0  0  −0.32559414 
${\mathit{sx}}_{\mathbf{14}}$  0  0  −0.33366963 
${\mathit{sx}}_{\mathbf{15}}$  0  0  −0.31117803 
${\mathit{sx}}_{\mathbf{16}}$  0  0  −0.36099787 
${\mathit{sx}}_{\mathbf{17}}$  0  0  −0.34156470 
${\mathit{sx}}_{\mathbf{18}}$  0  0  −0.38518722 
${\mathit{sx}}_{\mathbf{19}}$  0  0  −0.37839893 
${\mathit{sx}}_{\mathbf{20}}$  0  0  −0.38377130 
${\mathit{vy}}_{1}$  ${\mathit{vy}}_{2}$  ${\mathit{vy}}_{3}$  ${\mathit{vy}}_{4}$  

${\mathit{vx}}_{\mathbf{1}}$  −0.63185677  0  0  0 
${\mathit{vx}}_{\mathbf{2}}$  −0.53495062  0  0  0 
${\mathit{vx}}_{\mathbf{3}}$  −0.56087864  0  0  0 
${\mathit{vx}}_{\mathbf{4}}$  0  −0.54058450  0  0 
${\mathit{vx}}_{\mathbf{5}}$  0  −0.51277877  0  0 
${\mathit{vx}}_{\mathbf{6}}$  0  −0.54593785  0  0 
${\mathit{vx}}_{\mathbf{7}}$  0  −0.38311642  0  0 
${\mathit{vx}}_{\mathbf{8}}$  0  0  −0.43631239  0 
${\mathit{vx}}_{\mathbf{9}}$  0  0  −0.46432965  0 
${\mathit{vx}}_{\mathbf{10}}$  0  0  −0.45681443  0 
${\mathit{vx}}_{\mathbf{11}}$  0  0  −0.45592570  0 
${\mathit{vx}}_{\mathbf{12}}$  0  0  −0.42128590  0 
${\mathit{vx}}_{\mathbf{13}}$  0  0  0  0.48405851 
${\mathit{vx}}_{\mathbf{14}}$  0  0  0  0.47246029 
${\mathit{vx}}_{\mathbf{15}}$  0  0  0  0.36140660 
${\mathit{vx}}_{\mathbf{16}}$  0  0  0  0.40851941 
${\mathit{vx}}_{\mathbf{17}}$  0  0  0  0.35273551 
${\mathit{vx}}_{\mathbf{18}}$  0  0  0  0.34719369 
${\mathit{ty}}_{1}$  ${\mathit{ty}}_{2}$  ${\mathit{ty}}_{3}$  ${\mathit{ty}}_{4}$  ${\mathit{ty}}_{5}$  

${\mathit{tx}}_{\mathbf{1}}$  0.56826144  0  0  0  0 
${\mathit{tx}}_{\mathbf{2}}$  0.82284806  0  0  0  0 
${\mathit{tx}}_{\mathbf{3}}$  0  0.50066252  0  0  0 
${\mathit{tx}}_{\mathbf{4}}$  0  0.57846089  0  0  0 
${\mathit{tx}}_{\mathbf{5}}$  0  0.64398760  0  0  0 
${\mathit{tx}}_{\mathbf{6}}$  0  0  0.62935973  0  0 
${\mathit{tx}}_{\mathbf{7}}$  0  0  0.65899658  0  0 
${\mathit{tx}}_{\mathbf{8}}$  0  0  0.41186143  0  0 
${\mathit{tx}}_{\mathbf{9}}$  0  0  0  0.40455206  0 
${\mathit{tx}}_{\mathbf{10}}$  0  0  0  0.52191470  0 
${\mathit{tx}}_{\mathbf{11}}$  0  0  0  0.47595483  0 
${\mathit{tx}}_{\mathbf{12}}$  0  0  0  0.58086976  0 
${\mathit{tx}}_{\mathbf{13}}$  0  0  0  0  0.50933537 
${\mathit{tx}}_{\mathbf{14}}$  0  0  0  0  0.44032157 
${\mathit{tx}}_{\mathbf{15}}$  0  0  0  0  0.34018158 
${\mathit{tx}}_{\mathbf{16}}$  0  0  0  0  0.36337834 
${\mathit{tx}}_{\mathbf{17}}$  0  0  0  0  0.54674224 
Disjoint Orthogonal Components  Fit (in %)  Runtime (in min) 

NONE (TuckerALS)  42.63  0.0711 
$\mathit{A}$ (DisjointTuckerALS)  38.92  0.4171 
$\mathit{B}$ (DisjointTuckerALS)  40.24  0.4808 
$\mathit{A}$, $\mathit{B}$ (DisjointTuckerALS)  36.79  0.7136 
Chopin’s Preludes  Comp1  Comp2 

(1) C major Agitato  −0.033953528  0.184842162 
(2) a minor Lento  −0.137757855  −0.362945929 
(3) G major Vivace  0.183142589  0.335407766 
(4) e minor Largo  −0.046034600  −0.290603627 
(5) D major Allegro  0.204062480  0.200322397 
(6) b minor Lento assai  −0.109263322  −0.337194080 
(7) A major Andantino  0.322239475  −0.165253184 
(8) f# minor Molto agitato  −0.131497358  0.103731974 
(9) E major Largo  −0.191131130  −0.199220960 
(10) c# minor Allegro molto  0.060345767  0.118857312 
(11) B major Vivace  0.270309648  −0.021135215 
(12) g# minor Presto  −0.174159755  0.154464730 
(13) F# major Lento  0.124820434  −0.196316896 
(14) eb minor Allegro  −0.200476583  0.134265782 
(15) Db major Sostenuto  0.317539137  −0.189612402 
(16) bb minor Presto con fuoco  −0.13411733  0.374842648 
(17) Ab major Allegretto  0.039035613  −0.099661458 
(18) f minor Allegro molto  −0.297127956  0.124019724 
(19) Eb major Vivace  0.228844926  0.119167072 
(20) c minor Largo  −0.225976898  −0.225192986 
(21) Bb major Cantabile  0.144306691  −0.085136044 
(22) g minor Molto agitato  −0.269583704  0.074670854 
(23) F major Moderato  0.313700444  0.137709411 
(24) d minor Allegro appasionato  −0.257267184  0.109970949 
Chopin’s Preludes  Fast + Minor, Slow + Major  Fast + Major, Slow + Minor 

(1) C major Agitato  0.153  0.109 
(2) a minor Lento  −0.155  −0.356 
(3) G major Vivace  0.103  0.368 
(4) e minor Largo  −0.170  −0.240 
(5) D major Allegro  0.006  0.286 
(6) b minor Lento assai  −0.157  −0.318 
(7) A major Andantino  −0.346  0.107 
(8) f# minor Molto agitato  0.167  0.018 
(9) E major Largo  −0.003  −0.276 
(10) c# minor Allegro molto  0.040  0.127 
(11) B major Vivace  −0.208  0.174 
(12) g# minor Presto  0.232  −0.011 
(13) F# major Lento  −0.227  −0.053 
(14) eb minor Allegro  0.237  −0.044 
(15) Db major Sostenuto  −0.360  0.086 
(16) bb minor Presto con fuoco  0.358  0.174 
(17) Ab major Allegretto  −0.098  −0.044 
(18) f minor Allegro molto  0.299  −0.119 
(19) Eb major Vivace  −0.080  0.245 
(20) c minor Largo  −0.004  −0.319 
(21) Bb major Cantabile  −0.163  0.040 
(22) g minor Molto agitato  0.245  −0.135 
(23) F major Moderato  −0.128  0.318 
(24) d minor Allegro appasionato  0.261  −0.101 
Chopin’s Preludes  Fast + Minor, Slow + Major  Fast + Major, Slow + Minor 

(1) C major Agitato  0.091495243  0 
(2) a minor Lento  0  −0.349499568 
(3) G major Vivace  0  0.380083836 
(4) e minor Largo  0  −0.216381495 
(5) D major Allegro  0  0.322563112 
(6) b minor Lento assai  0  −0.306829783 
(7) A major Andantino  −0.391407119  0 
(8) f# minor Molto agitato  0.170783607  0 
(9) E major Largo  0  −0.308357327 
(10) c# minor Allegro molto  0  0.130529712 
(11) B major Vivace  −0.292993592  0 
(12) g# minor Presto  0.231153448  0 
(13) F# major Lento  −0.191308850  0 
(14) eb minor Allegro  0.252789215  0 
(15) Db major Sostenuto  −0.393591840  0 
(16) bb minor Presto con fuoco  0.254710019  0 
(17) Ab major Allegretto  −0.071449806  0 
(18) f minor Allegro molto  0.352314932  0 
(19) Eb major Vivace  0  0.300266062 
(20) c minor Largo  0  −0.359417233 
(21) Bb major Cantabile  −0.178560495  0 
(22) g minor Molto agitato  0.308258927  0 
(23) F major Moderato  0  0.396120182 
(24) d minor Allegro appasionato  0.305864985  0 
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MartinBarreiro, C.; RamirezFigueroa, J.A.; NietoLibrero, A.B.; Leiva, V.; MartinCasado, A.; GalindoVillardón, M.P. A New Algorithm for Computing Disjoint Orthogonal Components in the ThreeWay Tucker Model. Mathematics 2021, 9, 203. https://doi.org/10.3390/math9030203
MartinBarreiro C, RamirezFigueroa JA, NietoLibrero AB, Leiva V, MartinCasado A, GalindoVillardón MP. A New Algorithm for Computing Disjoint Orthogonal Components in the ThreeWay Tucker Model. Mathematics. 2021; 9(3):203. https://doi.org/10.3390/math9030203
Chicago/Turabian StyleMartinBarreiro, Carlos, John A. RamirezFigueroa, Ana B. NietoLibrero, Víctor Leiva, Ana MartinCasado, and M. Purificación GalindoVillardón. 2021. "A New Algorithm for Computing Disjoint Orthogonal Components in the ThreeWay Tucker Model" Mathematics 9, no. 3: 203. https://doi.org/10.3390/math9030203