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We first investigate the structure of the systems derived from the  gPC based stochastic Galerkin method for the nonlinear hyperbolic  systems with random inputs. This method adopts a generalized  Polynomial Chaos (gPC) approximations in the stochastic Galerkin  framework, but such approximations to the nonlinear hyperbolic  systems do not necessarily yield hyperbolic systems. Thus based on the work in [A framework on moment model reduction for kinetic equation, Cai, Z. N., Fan, Y. W. and Li, R. (2015)], we  propose a framework to carry out the model reduction for the general  nonlinear hyperbolic system to derive a final global system. Within  this framework, the nonlinear hyperbolic system in one space  dimension and the symmetric hyperbolic system in multiple space  dimensions are reduced into a symmetric hyperbolic system based on  the stochastic Galerkin method. We note that the basis functions in  the expansion are not restricted to the random-dependent polynomials  as that in gPC method and there is no restriction on the dimensions  of the random variables neither.