This monograph deals with approximation and noise cancellation of dynamical systems which include linear and nonlinear input/output relations. It will be of special interest to researchers, engineers and graduate students who have specialized in altering theory and system theory. From noisy or noiseless data, reduction will be made. A new method which reduces noise or models information will be proposed. Using this method will allow model description to be treated as noise reduction or model reduction. As proof of the efficacy, this monograph provides new results and their extensions which can also be applied to nonlinear dynamical systems. To present the effectiveness of our method, many actual examples of noise and model information reduction will also be provided. Using the analysis of state space approach, the model reduction problem may have become a major theme of technology after 1966 for emphasizing efficiency in the fields of control, economy, numerical analysis, and others. Noise reduction problems in the analysis of noisy dynamical systems may have become a major theme of technology after 1974 for emphasizing efficiency in control.However, the subjects of these researches have been mainly concentrated in linear systems. In common model reduction of linear systems in use today, a singular value decomposition of a Hankel matrix is used to find a reduced order model. However, the existence of the conditions of the reduced order model are derived without evaluation of the resultant model. In the common typical noise reduction of linear systems in use today, the order and parameters of the systems are determined by minimizing information criterion. Approximate and noisy realization problems for input/output relations can be roughly stated as follows: A. The approximate realization problem. For any input/output map,?nd one mathematical model such that it is similar to the input/output map and has a lower dimension than the given minimal state space of a dynamical system which has the same behavior to the input/outputmap. B. The noisy realization problem.