Streaming and distributed nonlinear parameter estimation by least squares in expansion coefficients

Abstract

Relevance. Nonlinear least squares rereads every sample at every iteration, which suits neither streams nor distributed archives. Recursive filters need preset noise covariance’s; distributed frameworks merge the statistics of linear models only. Aim and objectives. The aim is to develop a recursive and an additive form of least squares in expansion coefficients, the equal-areas criterion and least squares on Legendre coefficients, for nonlinear estimation on streams and archives. The objectives are to establish when the fit from an additive image of the data coincides with batch least squares, to build on it a window filter compared with the extended Kalman filter, to equip the filter for gross anomalies, missing samples and parameter changes, and to apply the additive form to large archives. Methods used. The data projection on a block or Legendre basis, its Gram matrix and the sample sums form an additive image, to which the model is fitted by whitened least squares. The filter is a damped information-form update with a noise level from the window, residual clipping and an innovation test on every sample. Results. The fit from the image coincides with batch least squares when the model’s sensitivities lie in the span of the basis, and merged images reproduce it to rounding error. Under white noise the window filter is level with or ahead of a static Kalman filter on most plants and behind it on a sinusoid, keeps as many runs as a well-gated Kalman filter under gross anomalies and flags parameter jumps promptly, mostly by testing every sample. Public archives were reduced once and refitted from the images on a single-board computer, a nonlinear decay time included. Conclusions. One additive statistic serves streams, archives and small devices. The filter is comparable in accuracy to a well-tuned Kalman filter and costs more per step, but needs no noise covariance’s, resists gross errors without a gate and tests for a change on every sample.