Hybrid system identification is composed of two subproblems: estimate the discrete state or mode for each data point, and estimate the submodel governing the dynamics of the continuous state for each mode. For linear hybrid systems, the paper proposes to tackle these problems in a single step by simultaneously approximating each submodel while associating data points to each of these. The method borrows ideas from bounded-error approaches and Support Vector Regression to extend the algebraic procedure. The algorithm can easily deal with noise by fixing a predefined accuracy threshold. This bound on the error can be different for each mode when the noise level is considered to switch with the system. An extension of the algorithm to automatically tune itself in accordance with the noise level of each mode is also provided. The method can be seen as an an extension of the algebraic approach, but also as an alternative to the bounded error approach when a predefined or preferred model structure is given or when the noise level is unknown. An extension of the method to nonlinear submodels is provided in the companion paper (Lauer and Bloch, 2008).