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We describe a novel approach, called data driven local coordinates (DDLC), for parametrizing linear systems in state space form, and we analyze some of its properties which are relevant for e.g. maximum likelihood estimation. In addition we describe how this idea can be used for a concentrated likelihood function, obtained by a least squares type concentration step, which gives the so called sls (separable least squares) DDLC approach. Both approaches give favourable results in numerically optimizing the likelihood function in simulation studies.