Proximity mapping: Model

With the Model tab, you can specify the individual differences model to be applied when analyzing multiple proximity sources. It also defines how proximity transformations are handled across sources: either uniformly (unconditional) or independently (matrix-conditional).

Three-Way Model

Settings in Three-Way Model determines how the proximities from multiple sources are mapped into a common spatial representation, by using space weights. The model determines how the space weights are applied to the common space. For some models, space weights are equivalent to space weights. When only a single source is defined, all options except Identity are disabled.

Option	Description
Identity	Assumes all sources share the same configuration with no source-specific scaling. Only available when a single source is specified.
Dilation (Weighted Identity)	Each source is modeled with a single scaling factor applied uniformly across all dimensions. With this option, each source can perceive the common space at a different overall scale.
Weighted Euclidean (Diagonal)	Each source assigns different weights to the dimensions of the common space. This allows sources to emphasize different spatial features.
Generalized Euclidean	Each source is allowed a rotated and reweighted (scaled) version of the common space. This is the most flexible model and accommodates a wide range of individual differences. Full Rank: Each source uses the full dimensionality of the common space. Reduced Rank: When you apply a reduced rank constraint, each source's configuration (individual space, with rank = number of dimensions – 1) is represented in a lower-dimensional subspace of the common space. This approach involves an orthogonal rotation of the common configuration, followed by a projection on a subspace. Here, the rotation and projection together form the so-called space weights.

Conditionality / Transformation

This pane controls how proximities are transformed prior to fitting distances in the configuration.

• With the option Unconditional, you can apply a single transformation function across all sources. This option is appropriate when the sources are assumed to reflect a common structure.

  The default transformation is displayed under Transformation (ordinal with ties allowed to be untied).

  Click Define Transformation to open a dialog where you can choose between all transformation types (Linear, Power, Monotonic, Spline, Ordinal), handling of ties, and other parameters.

• With the option Matrix-Conditional you can choose different transformation functions for different sources or groups of sources. When selected, each source is listed with the default transformation function.

  Click Define Transformation to choose another function to be applied to the selected sources. This option is only enabled when more than one source is defined.

In the Define Transformation, you can select how proximities are optimally transformed before being matched to distances in the spatial configuration. The following options are available:

Option	Description
None	No transformation is applied. Proximities are used as-is.
Linear	Applies a straight-line transformation to the proximities. Intercept (checkbox) is to add a constant to the transformation, allowing for interval-scale modeling. If unchecked, the transformation is ratio-scaled (no constant added).
Power	Applies a power transformation. Intercept (checkbox) is to add a constant shift to the power function, if enabled.
Spline	Fits a smooth piecewise polynomial (spline) function to the proximities. Intercept includes a constant term in the spline basis. Degree sets the complexity of the polynomial segments (for example, 1 = linear, 2 = quadratic). Interior Knots controls the number of interior points used to define the spline shape.
Knot Sequence	Equal intervals: Knots are evenly spaced along the proximity range. Percentiles: Knots are placed at empirical quantiles of the proximity distribution.
Ordinal	Applies a rank-preserving transformation (monotonic). Allow ties to be untied: Tied proximities may be transformed to different values. Keep ties tied: Tied proximities remain tied after transformation.