Speech signals are usually affected by multiple acoustic factors, such as speaker characteristics and environment differences. Usually, the combined effect of these factors is modelled by a single transform. Acoustic factorisation splits the transform into several factor transforms, each modelling only one factor. This allows, for example, estimating a speaker transform in a noise condition and ... [Show full abstract] applying the same speaker transform in a different noise condition. To achieve this factorisation, it is crucial to keep factor transforms independent of each other. Previous work on acoustic factorisation relies on using different forms of factor transforms and/or the attribute of the data to enforce this independence. In this work, the independence is formulated in mathematically, and an explicit constraint is derived to enforce the independence. Using factorised cluster adaptive training (fCAT) as an application, experimental results demonstrates that the proposed explicit independence constraint helps factorisation when imbalanced adaptation data is used.