The proposed approach is tested on simulations of linear stochastic processes, first in a theoretical analysis aimed at relating multiscale complexity to the dynamical properties of the process without the unavoidable errors connected with estimation procedures and then in realizations of these simulated processes generated to assess the computational reliability of the LMSE compared with the traditional RMSE estimator.
Note that MSE measure defined in (17), which we denote as linear MSE (LMSE), is relevant to a time scale given by the rational number [[tau].sub.s] = [tau]/s, obtained setting integer values for the scale factor r and the upsampling factor s.
Practical Implementation and Applicability of LMSE and RMSE.
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The usefulness of the proposed MOOI detections is demonstrated by applying the detected MOOI to the content-aware image resizing with the comparison of the LMSE method  and to the image size reduction as a preprocessing for the H.264/AVC compressions.
For comparisons, the proposed method and the LMSE method in Vo et al.
Figure 7 and Figure 8 show the results of the size reduction by the LMSE in Vo et al.
As illustrated in figure 8 (noise free and noisy cases) the output prediction LMSE
becomes negligible after approximately just twenty optimizing iterations; and as depicted in figures 10, the searched materials permittivity, are estimated with a good precision, which shows the accuracy and efficiency of the PSO-based dielectric characterization of multilayered inhomogeneous media.
This division, however, may produce noisier estimates for low-frequency classes, resulting in a set of estimates that are not really optimized in an LMSE
sense (as the net output are).