  We present a new framework of $h^{2}$ optimal model reduction for linear
discrete-time systems. Our main contribution is to create optimal reduced order
models in the $h^{2}$-norm sense directly from the measurement data alone,
without using the information of the original system. In particular, we focus
on the fact that the gradient of the $h^{2}$ model reduction problem is
expressed using the discrete-time Lyapunov equation and the discrete-time
Sylvester equation, and derive the data-driven gradient. In the proposed
algorithm, the initial point is chosen as the output of the existing
data-driven methods. Numerical experiments are conducted to show that the
proposed method produce better reduced order models in the $h^{2}$-norm sense
than other data-driven model order reduction approaches.
