  Single molecule data made of on and off events are ubiquitous. Famous
examples include enzyme turnover, probed via fluorescence, and opening and
closing of ion-channel, probed via the flux of ions. The data reflects the
dynamics in the underlying multi-substate on-off kinetic scheme (KS) of the
process, but the determination of the underlying KS is difficult, and sometimes
even impossible, due to the loss of information in the mapping of the
mutli-dimensional KS onto two dimensions. A way to deal with this problem
considers canonical (unique) forms. (Unique canonical form is constructed from
an infinitely long trajectory, but many KSs.) Here we introduce canonical forms
of reduced dimensions that can handle any KS (i.e. also KSs with symmetry and
irreversible transitions). We give the mapping of KSs into reduced dimensions
forms, which is based on topology of KSs, and the tools for extracting the
reduced dimensions form from finite data. The canonical forms of reduced
dimensions constitute a powerful tool in discriminating between KSs.

Quantitative Methods;Other Condensed Matter;Other Quantitative Biology