A matrix-based approach to solving the inverse Frobenius–Perron problem using sequences of density functions of stochastically perturbed dynamical systems

•The paper introduces new method of reconstructing an unknown one-dimensional transformation that is subject to constantly applied stochastic perturbations based on temporal sequences of probability density functions.•The main assumption is that the one-dimensional transformation that generated the densities is piecewise-linear, semi-Markov.•A matrix approximation of the transfer operator associated with the stochastically perturbed transformation, which forms the basis for the reconstruction algorithm, is introduced.•A practical algorithm to estimate the matrix-representation of the Frobenius-Perron operator associated with the unperturbed transformation and reconstruct the onedimensional map is proposed.•The algorithm is extended to nonlinear continuous maps.•Numerical simulation examples are provided to demonstrate the performance of the approach and to compare it with that of an existing algorithm. The paper introduces a matrix-based approach to estimate the unique one-dimensional discrete-time dynamical system that generated a given sequence of probability density functions whilst subjected to an additive stochastic perturbation with known density.