Whereas structure - property relationships have long guided the discovery and optimization of novel materials(1-3), formal quantitative methods to identify such relationships in crystalline systems are beginning to emerge(4-8). Among them is cluster expansion(9-13), which has been successfully used to parametrize the configurational dependence of important scalar physical properties such as bandgaps(14), Curie temperatures(15), equationofstate parameters(16,17) and densities of states(18-20). However, cluster expansion is currently unable to handle anisotropic properties, a key distinguishing feature of crystalline systems central to the design of modern epitaxial structures and devices. Here, I introduce a tensorial cluster expansion enabling the prediction of fundamental tensor- valued material properties such as elasticity, piezoelectricity, dielectric constants, optoelectric coupling, anisotropic diffusion coefficients, surface energy and stress. As an application, I develop predictive ab initio models of anisotropic properties relevant to the design and optimization of III - V semiconductor epitaxial optoelectronic devices(21,22).