Automatic numerical differentiation by maximum likelihood estimation of a linear Gaussian state space model

A linear Gaussian state-space smoothing algorithm is presented for off-line estimation of derivatives from a sequence of noisy measurements. The algorithm uses numerically stable square-root formulas, can handle simultaneous independent measurements and non-equally spaced abscissas, and can compute state estimates at points between the data abscissas. The state space model's parameters, including driving noise intensity, measurement variance, and initial state, are determined from the given data sequence using maximum likelihood estimation computed using an expectation maximisation iteration. In tests with synthetic biomechanics data, the algorithm is found to be more accurate compared to a widely used open source automatic numerical differentiation algorithm, especially for acceleration estimation.