Quality noise estimation

The previous method supposes complete knowledge of the system and depends on the measurement quality of instruments (errors, availability), leading to severe effects on the accuracy of the on-line estimates. Therefore, a good noise filtration algorithm (like the Kalman filter or derivative) should be employed to improve the reliability of the estimated values before their use. 

Summarizing, the statistical characterisation of the random process (mean and covariance) can be projected through the interval tk < t < tk+1, and in this process there is an input noise that will increase the error, damaging the quality of the estimate. 

One of the more challenging unsolved problems is the representation of transient events, such as attacks in musical percussive sounds and plosives in speech, which are neither quasi-periodic nor random. The residual which results from the deterministic/stochastic model generally contains everything which is not deterministic, i.e., everything that is not sine-wave-like. Treating this residual as stochastic when it contains transient events, however, can alter the timbre of the sound, as for example in time-scale expansion. A possible approach to improve the quality of such transformed sounds is to introduce a second layer of decomposition where transient events are separated and transformed with appropriate phase coherence as developed in section 4.4. One recent method performs a wavelet analysis on the residual to estimate and remove transients in the signal [Hamdy et al., 1996] the remainder is a broadband noise-like component. 

If the experimental errors are underestimated, it can lead to tight but inaccurate distance bounds. Conversely, the overestimated errors can lead to unnecessarily imprecise distance bounds. RANDMARDI takes into account two types of experimental errors relative integration errors and absolute errors due to spectral noise. The first kind can be estimated, for example, by comparing intensities of symmetric peaks below and above the diagonal, and the second type can be estimated as 50-200% of the lowest quantifiable peak, depending on the spectrum quality. 

Measurements are often corrupted by high frequency, zero-mean, random variations (i.e., measurement noise). These variations can be caused by high-frequency disturbances from the process (e.g., vortices shedding in flow measurement) or electrical interference during signal transmission. Plant measurements should be filtered to reduce the impact of high-frequency variations on the quality of the estimated parameters and as a result, the predicted optimal plant operation. 

Next we examine the quality of these approximate confidence intervals for this problem. Figure 9.22 shows the results of a Monte Carlo simulation study. In this study we generate 5Q0 datasets by adding zero-mean measurement noise with variance 0.01 to the model solution with the correct parameters. For each of these 500 datasets, we solve the optimization problem to obtain the parameter estimates. We also produce a value for the Hessian for each dataset, and we use the mean of these for H. Finally we calculate what fraction of these... 

After a first paper in 1988, " Bretthorst paved the way for the application of Bayesian probability theory (BPT) in a series of three papers.- The first presents the connection between the theory and the case of NMR phenomena and discusses parameter estimation and detection of quadrature signals. The second " shows the ability of Bayesian methods to measure the quality of a model. The third " provides examples of applications to experimental data where decaying sinusoids are assumed. A fourth paper, produced during the following two years, discusses computer time requirements and noise, and it is shown to be important to include knowledge in the analysis," " while a fifth publication is devoted to amplitude estimations for multiplets of well-separated resonances. 

All prior knowledge about the system under study is utilized in the model specification task. This includes results from specially designed preliminary experiments, which can be used to establish, for instance, whether the system is static or dynamic, linear or nonlinear, and so on. The model estimation task, however, relies on the quality of the available data (e.g., spectral characteristics, noise conditions, data length) and may set specific data-collection requirements. 

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