We consider the problem of constructing a vector-valued linear Markov process in continuous time, such that its first coordinate is in good agreement with given samples of the scalar autocorrelation function of an otherwise unknown stationary Gaussian process. This problem has intimate connections to the computation of a passive reduced model of a deterministic time-invariant linear system from given output data in the time domain. We construct the stochastic model in two steps. First, we employ the AAA algorithm to determine a rational function which interpolates the z-transform of the discrete data on the unit circle and use this function to assign the poles of the transfer function of the reduced model. Second, we choose the associated residues as the minimizers of a linear inequality constrained least squares problem which ensures the positivity of the transfer function’s real part for large frequencies. We apply this method to compute extended Markov models for stochastic processes obtained from generalized Langevin dynamics in statistical physics. Numerical examples demonstrate that the algorithm succeeds in determining passive reduced models and that the associated Markov processes provide an excellent match of the given data.

我们考虑在连续时间内构造向量值线性马尔可夫过程的问题，使其第一个坐标与未知平稳高斯过程的标量自相关函数的给定样本非常一致。该问题与根据时域中给定输出数据计算确定性时不变线性系统的被动简化模型密切相关。我们分两步构建随机模型。首先，我们采用 AAA 算法来确定一个有理函数，该函数将离散数据的 z 变换插值到单位圆上，并使用该函数来指定简化模型的传递函数的极点。其次，我们选择相关的留数作为线性不等式约束最小二乘问题的最小化器，这确保了传递函数的实部对于大频率的正性。我们应用这种方法来计算从统计物理学中的广义朗之万动力学获得的随机过程的扩展马尔可夫模型。数值示例表明，该算法成功地确定了被动简化模型，并且相关的马尔可夫过程提供了给定数据的出色匹配。