A Low-rank Spectral Estimation of Markov Process

As the method for spectral estimation of Markov process makes use of nonnegativity-preserving step, the spectral estimator does not necessarily satisfy low-rank condition. Motivated by this, a low-rank spectral estimation algorithm (LRSEA) is proposed. First of all, the local Lipschitzian type error bound of the rank-constrained state transition matrix set is established, and an approximate projection matrix that satisfies the error bound inequality is given. Then, using the approximate projection matrix to modify the spectral estimation method, the LRSEA is proposed, and the statistical error bound for the proposed estimation method is provided. Numerical comparisons on the synthetic data with empirical estimator and spectral estimator show that the LRSEA has the lowest estimation error. Finally, the LRSEA together with k-means algorithm is used to analyze the dataset of Manhattan taxi trips.