betaijk

Betaijk is a symbolic notation used across disciplines to denote a coefficient or tensor component indexed by i, j, and k. There is no universal definition for betaijk; its meaning is determined by the mathematical model or application in which it appears.

In mathematics and physics, β_ijk often denotes a component of a rank-3 tensor. Such tensors can describe directional dependencies in anisotropic materials, third-order constitutive relations, or fields in theoretical physics. The indices i, j, and k enumerate the component within the three-dimensional array and indicate its position in the tensor structure.

In statistics and econometrics, β_ijk may represent a three-way interaction coefficient in models with three categorical or ordinal factors. Here the value expresses how the effect of one factor changes across the levels of the other two, beyond any pairwise interactions. This usage helps capture complex dependencies that arise when multiple factors interact simultaneously.

In machine learning and data analysis, β_ijk commonly appears as elements of a core tensor in tensor factorization (for example, Tucker or CP decompositions) or as parameters in models that include three-way interactions. The triple-indexed form encodes multi-way dependencies among latent factors associated with each mode, facilitating compact representations of high-dimensional data.

In practice, authors choose the notation β_ijk for readability; the capitalization, index conventions, and whether indices start at 0 or 1 vary by field and software.