2011_06_15_Daily Paper Review (Aamir Alaud-din)
Title and Authors
Title: The equivalence of Tucker3 and Parafac models with two components
Authors: P.M. Kroonenberga,* , J.M.F. ten Bergeb
a Child and Family Studies, Institute of Education and Child Studies, Leiden University, Wassenaarseweg 52, 2333 AK Leiden, The Netherlands
b Department of Psychology, Heymans Institute, University of Groningen, Grote Kruisstraat 2/1, 9712 TS Groningen, The Netherlands

Summary of Paper:
In this paper, equivalency of Tucker3 and Parafac models is shown. It happens when the core array of 2 × 2 × 2 Tucker3 model has rank two. In such a case, both models show same results. But, if the core array of 2 × 2 × 2 Tucker3 model has rank more than two, then the results of both models differ. To show this two data sets Osgood data and Kojima girls data are used.
A Tucker3 models can be written as:
X_ijk=∑_(p=1)^P▒∑_(q=1)^Q▒∑_(r=1)^R▒?(a_ip b_jq c_kr)g_pqr+e_ijk ?
A Parafac model can be written as:
X_ijk=∑_(s=1)^S▒?(a_is b_js c_ks)g_sss+e_ijk ?
Osgood Data: The results of both models are same for this data set. Table 1 below shows the results of both models
Kojima Girls Data: The results of both models for this data sets are shown in table 2. For this data set, Parafac core array elements are shown bold.

Table 1
Applied to    2×2×2 Tucker3 Model    Two-component Parafac model    Two-component Parafac model
    Full data set    Full data set    Tucker3 core array
    1    2    1    2    1    2
Slice G1                        
1    38.4    2.2    37.5    0.0    37.5    0.0
2    1.0    8.0    0.0    0.0    0.0    0.0
                        
Slice G2                        
1    -0.9    10.8    0.0    0.0    0.0    0.0
2    31.7    -2.6    0.0    36.0    0.0    36.0

Table 2
    2×2×2 Tucker3 core array    Parafac core array 2 components    Parafac core array 3 components
    1    2    1    2    1    2    3
Slice G1    1.67    -.27    1390    -1421    1.45    -1.57    1.28
1    .27    1.56    -1421    1421    -1.58    1.70    -1.35
2                    1.26    -1.32    .25
                            
Slice G2                            
1    -.12    -.11    1421    -1421    1.83    -1.97    1.41
2    .20    .08    -1421    1390    -1.98    2.13    -1.46
                    1.39    -1.44    -.23
                            
Slice G3                            
                    -2.49    2.59    -.12
                    2.60    -2.70    -.13
                    -.12    -.14    5.32

Hence, Parafac and Tucker3 have same results when core array of 2×2×2 Tucker3 model has rank 2 and Parafac has two components.