We discuss discrete-time dynamical systems depending on a parameter μ. Assuming that the system matrix A(μ) is given, but the parameter μ is unknown, we infer the most-likely parameter μ_m≈μ from an observed trajectory x of the dynamical system. We use parametric eigenpairs (v_i(μ),lambda_i(μ) of the system matrix A(μ) computed with Newton's method based on a Chebyshev expansion. We then represent x in the eigenvector basis defined by the v_i(μ) and compare the decay of the components with predictions based on the lambda_i(μ). The resulting estimates for μ are combined using a kernel density estimator to find the most likely value for μ_m and a corresponding uncertainty quantification.