Stochastic noise

Partial order ranking (POR) is based on elementary methods of discrete mathematics (e.g., Hasse diagrams) — if A < B and B < C, then A < C in the ranking procedures. POR does not assume linearity or any assumptions about distribution properties such as normality. The disadvantage is that often a preprocessing of data is needed to avoid the effects of stochastic noise. Combining POR with PCA may improve its usefulness. POR can only be applied for interpolation. 

T. R. Chay and H. S. Kang Role of singlechannel stochastic noise on bursting clusters of pancreatic / -cells. Biophys. J. 1988, 54 427-435. 

F(f) is usually named multiplicative stochastic noise, since it multiplies the function g x), which depends on the variable of interest x. Due to the multiplicative nature of the stochastic force F t), it is not immediately... 

I is an important parameter of this theory which expresses the ratio of the correlation time of the multiplicative stochastic noise to that of the velocity V. The second term on the right-hand side of Eq. (1.18) is a potential term independent of the intensity of the noise < )eq, the role of which becomes increasingly irrelevant as y - oo. We cannot, however, neglect the last two terms, which express the lowest-order contribution coming from the noise i. 

Figure 4. Steady state probability distribution

In several respects the finite inertia of the system of Eq. (1.7) produces effects similar to those of a weak additive stochastic force. In particular, as a consequence of finite inertia, the escape from the well is also possible in the absence of the additive stochastic noise. These effects are not taken into... 

One cannot expect that such an agreement covers also the large-Q region. Two main reasons have already beoi singled out inotia and weak residual stochastic forces of additive kind. In particular, as shown in Figs. 6 and 7, the additive stochastic noise forces to increase when the large-0 re-... 

Axxrid the line frequency and harmonics Modem impedance instniments provide very effective filters for stochastic noise, but these filters are generally inadequate for measiuements conducted at the line frequency. The resulting meastuements generally appear as outliers in an impedance spectrum, and such outliers have a profovmd impact on nonlinear regression used to extract parameters from the data. Measurement of impedance should be avoided at line frequency and its first harmonic, i.e., 60 5 Hz and 120 5 Hz in the United States and 50 5 Hz and 100 5 Hz in Europe. 

As discussed in Chapter 21, the variances of stochastic errors are equal for real and imaginary parts of the impedance. Thxis, another advantage of presenting real and imaginary parts of the impedance as a function of frequency is that comparison between data and levels of stochastic noise can be easily represented. 

The difficulty with using the Ohmic-resistance-corrected Bode plots presented in the previous section is that an accurate estimate is needed for the electrolyte resistance and that, at high frequencies, the difference Zr — Re,est is determined by stochastic noise. As discussed in Section 16.1.4, these difficulties can be obviated by plotting the real and imaginary components of the impedance. 

The regression procedure is strongly influenced by stochastic errors or noise in the measurement. One effect is illustrated in comparison of Figure 19.1 to Figure 19.3, in which stochastic noise with a standard deviation equal to 1 percent of the modulus was added to the synthetic data. Solid lines have been drawn on the bottom contour map to indicate the values for which the function is inmimized. The presence of stochastic errors in the data does not introduce roughness in the parabolic... 

Figure 19.3 The objective function, equation (19.3), for an RC circuit as a function of parallel resistor and RC-time constant values. The circuit was the same as presented in Figure 19.1 with the exception that stochastic noise was added to the synthetic data with a standard deviation equal to 1 percent of the modulus, a) 3-D perspective drawing of the contour surface b) 2-D representation of the contour surface.

Figure 19.5 The impedance data used for Figure 19.4 in impedance-plane format. The solid line represents the noise-free data, and the symbols represent the data with added stochastic noise with a standard deviation equal to 1 percent of the modulus. Note that the third line-shape, with parameters R3 = 5 flcm and t — 0.05 s, is not readily seen, even for the noise-free solid line.

In the more practical case where the impedance is sampled at a finite number of frequencies, r, x) represents the error between an interpolated function and the "true" impedance value at frequency x. This error is seen in Figure 22.3, where a region of Figure 22.2 was expanded to demonstrate the discrepancy between a straight-line interpolation between data points and the model that conforms to the interpolation of the data. This error is composed of contributions from the quadrature and/or interpolation errors and from the stochastic noise at the measurement frequency (v. Effectively, equation (22.75) represents a constraint on the integration procedure. In the limit that quadrature and interpolation errors are negligible, the residual errors r( c) should be of the same magnitude as the stochastic noise r(o ). 

The entire system evolves unitarily but a noise term is present in the ancilla Hamiltonian, see Fig. 1 (b). This is not, in fact, a measurement in the sense considered in this work nonetheless, it provides similar physics since a stochastic noise acting on the ancilla produces coherence loss to the combined system. 

The similarity of stochastic mechanics with quantum mechanics has been pointed out and used by several authors [7-9]. The relationship goes back to Nelson [10, 11], who published in the mid-1960s a stochastic theory that eventually led to SchrPdinger s equation. At that time. Nelson s theory did not become very popular, because it did not explain the origin of the stochastic noise, which is essential for this theory. However, nowadays this theory seems to have been rediscovered [12]. 

The notion of sines plus noise modeling was posed and implemented by Xavier Serra and Julius Smith in the Spectral Modeling Synthesis (SMS) system. They called the sinusoidal components the deterministic component of the signal, and the leftover noise part the residual or stochastic component. Figure 6.12 shows the decomposition of a sung ahh sound into deterministic (harmonic sinusoidal) and stochastic (noise residue) components. 

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