three-way component models 880305- pages 66-76 by: maryam khoshkam 1
TRANSCRIPT
THREE-WAYCOMPONENT MODELS
880305- pages 66-76
By: Maryam Khoshkam
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Tucker component models
Ledyard Tucker was one of the pioneers in multi-way analysis.
He proposed a series of models nowadays called N-mode PCA or Tucker models [Tucker 1964- 1966]
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TUCKER3 MODELS
: nonzero off-diagonal elements in its core.
In Kronecker product notation the Tucker3 model
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PROPERTIES OF THE TUCKER3 MODEL
Tucker3 model has rotational freedom.
TA : arbitrary nonsingular matrix
Such a transformation of the loading matrix A can be defined similarly for B and C, using TB and TC, respectively
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Tucker3 model has rotational freedom, But: it is not possible to rotate Tucker3 core-array to a superdiagonal form (and to obtain a PARAFAC model.!
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The Tucker3 model : not give unique component matrices it has rotational freedom.
rotational freedom Orthogonal component matrices (at no cost in fit by defining proper matrices TA, TB and TC)
convenient : to make the component matrices orthogonal
easy interpretation of the elements of the core-array and of the loadings by the loading plots
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SS of elements of core-array
amount of variation explained by combination of factors in different modes.
variation in X: unexplained and explained by model
Using a proper rotation all the variance of explained part can be gathered in core.
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The rotational freedom of Tucker3 models can also be used to rotate the core-array to a simple structure as is also common in two-way analysis (will be explained).
Imposing the restrictions A’A = B’B = C’C = I : not sufficient for obtaining a unique solution
To obtain uniqe estimates of parameters, 1. loading matrices should be orthogonal, 2. A should also contain eigenvectors of X(CC’ ⊗ BB’)X’ corresp. to decreasing eigenvalues of that same matrix; similar restrictions should be put on B and C
[De Lathauwer 1997, Kroonenberg et al. 1989].
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Unique Tucker
2 4 6 80
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1Simulated data:
Two components,PARAFAC model
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UniqueTucker3 component model
P=Q=R=3
Only two significant elements in core
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all three modes are reduced
In tucker 3
models where only two of the three modes are reduced, :Tucker2 models.
a Tucker3 model is made for X (I × J × K) C is chosen to be the identity matrix I, of size K × K. no reduction sought in the third mode (basis is not changed.
↘Tucker2 model :
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Tucker2 has rotational freedom:G : postmultiplied by U⊗V (B⊗A) : premultiplied by (U⊗V)−1 =>(B(U’)−1 ⊗A(V’)−1) without changing the fit.
component matrices A and B can be made orthogonal without loss of fit. (using othog U and V)16
Tucker1 models : reduce only one of the modes.
+ X (and accordingly G) are matricized :
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different models [Kiers 1991, Smilde 1997].
Threeway component models for X (I × J × K), A : the (I × P) component matrix (of first (reduced) mode,
X(I×JK) : matricized X; A,B,C : component matrices; G : different matricized core-arrays ; I :superdiagonal array (ones on superdiagonal. (compon matrices, core-arrays and residual error arrays : differ for each model
=> PARAFAC model is a special case of Tucker3 model.