Gifi Theory

Theory in a Nutshell

The data are collected in an \(n \times m\) data frame. Whereas the homals package uses the homogeneity loss function (see De Leeuw and Mair 2009), the Gifi package uses and solves the following loss function (meet loss, see Gifi (1990), Sec. 4.4.), with SSQ as the sum-of-squares of a matrix:

\[\sigma(X,Z,A)=\frac{1}{mp}\sum_{j=1}^m\text{SSQ}\ (X-\sum_{i\in I_j}H_iZ_iA_i)\]

The index set \(\mathcal{N}=\{1,2,\cdots,N\}\), where \(N\) is the total number of active variables (see below) in the analysis, is partitioned into the \(m\) index sets \(I_j\), with \(I_j\cap I_l=\emptyset\) and \(\bigcup_{j=1}^m I_j=\mathcal{N}\).

\(X\) is \(n\times p\) matrix of object scores (\(p\) being the number of dimensions). \(X\) is centered \(e'X=0\) and orthonormal \(X'X=I\). For each variable \(i\), the meet loss involves:

• \(H_i\) as the basis, \(n \times k_i\), observations by basis, known and fixed;
• \(Z_i\) as the coefficients, \(k_i\times l_i\), basis by copies;
• \(A_i\) as the loadings, \(l_i\times p\), copies by dimensions.

Instead of using rank restrictions like homals, Gifi uses the idea of copies, first introduced by De Leeuw (1984), which are literally copies (or duplicates) of variables that enter the loss. Overall, this concept called multiple quantifications in the original Gifi terminology, makes the system more flexible.

In addition, in homals all data were categorical and the basis was always an indicator matrix. In Gifi the basis is either categorical, or a B-spline basis (van Rijckevorsel 1988) for which the user needs to specify the knots implying that the data must be numerical.

For each variable \(i\), the following matrices are returned by various Gifi functions like homals() and princals():

• transformations \(T_i=H_iZ_i\), orthogonalized, observations by copies, \(n\times l_i\);
• loadings \(A_i\), copies by dimensions, \(l_i\times p\);
• scores \(S_i=T_iA_i=H_iZ_iA_i\), observations by dimensions, \(n\times p\);
• quantifications \(Q_i=Z_iA_i\), degrees by dimensions, \(k_i\times p\);
• coefficients \(Z_i\), basis by copies, \(k_i\times l_i\).

The Gifi loss is solved using alternating least squares (ALS), combined with majorization. The gifiEngine() function alternates over \(X\), and \(Z_i\) and \(A_i\).

Gifi also allows for declaring variables as active vs. passive. Active variables are all variables of main interest, contributing to the loss and to the ALS step that updates \(X\). Passive (or supplementary) variables don’t contribute to these components; each of them is scaled in a separate step via \(\text{SSQ}\ (X-H_iZ_iA_i)\) using the optimal \(X\).

Gifi provides several options for handling missing values:

• single: add a single 0/1 column to the basis with 1 for missing;
• multiple: add a 0/1 column to the basis for each missing observation, i.e. append an identity matrix.
• average: use a basis row with all elements equal to the mean of the non-missing rows.

Implementation

So far, the following wrapper functions are implemented. Internally they all use the same gifiEngine() function to solve the loss from above. The main difference between these functions are the default settings in terms of the number of copies and the number of sets:

• princals(): one variable per set, all variables one copy.
• homals(): one variable per set, all variables \(p\) copies.
• morals(): two sets, one set has a single variable with one copy (\(p = 1\)).

princals() is designed to fit ordinal or mixed PCA in a user-friendly way, whereas homals() is designed for multiple correspondence analysis. However, if the default settings for princals() and homals() are changed accordingly, they both give the same result. morals() performs multiple (monotone) regression analysis within the Gifi system. For these 3 functions applied vignettes are provided.