Abstract
Special system identification algorithms are required if there are significant amounts of data missing. Some such algorithms have been developed previously and typically result in iterative procedures for the parameter estimation. Since missing data can be viewed as irregular sampling (decimation) of the signals, it is obvious that there is a risk for aliasing. In system identification aliasing manifests itself as potential multiple global optima of the identification loss function. The aim of this paper is to investigate under what circumstances this may in fact occur. The focus of the paper is on periodic missing data patterns. It is shown that it is, in fact, not the fraction of missing data that is important, but rather what time lags of the input and output correlation and cross-correlation functions that can be estimated. For ARX models with all input data observed we verify that there is indeed only one global optimum.
Highlights
Often data sets used for system identification are incomplete
If we extend the model with a time series input model, for example, AR or ARMA, it is still possible to write the model on innovation form with different matrices A, C, and F
We have studied the existence of multiple global optima of the likelihood function when identifying parameters of AR and ARX models
Summary
Often data sets used for system identification are incomplete. Some observations are missing, either according to a periodical pattern or at random. As identification experiments are expensive and time consuming, methods that can cope with missing data are attractive. They make it possible to use all data sets that are available. The missing data problem has been studied extensively in statistics, but less so in engineering literature. The specific problem studied in this paper is the existence of multiple global optima of the system identification procedure. That multiple optima can occur is obvious realizing that missing data can be viewed as sampling (or decimation), albeit often irregular, of the signals. The sampling theorem indicates that aliasing may occur It is, not entirely obvious for what combinations of missing data pattern and model order there may be more than one system that optimally predicts the observed data. This paper presents some results for linear time invariant systems with and without input and gives some examples for autoregressive (AR) and autoregressive models with an exogenous input (ARX)
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