Differential nonlinearity

So we seem to have identified a key characteristic of chemometric modeling that influences the capabilities of the models that can be achieved not nonlinearity per se, because simple nonlinearity could be accommodated by a suitable transformation of the data, but differential nonlinearity, which cannot be fixed that way. In those cases where this type of differential, or non-uniform, nonlinearity is an important characteristic of the data, then selecting those wavelengths and only those wavelengths where the data are most nearly linear will provide better models than the full-spectrum methods, which are forced to include the non-linear regions as well, are capable of. 

Solutions are presented in the form of equations, tables, and graphs—most often the last. Serious numerical results generally have to be obtained with computers or powerful calculators. The introductory chapter describes the numerical procedures that are required. Inexpensive software has been used here for integration, differentiation, nonlinear equations, simultaneous equations, systems of differential equations, data regression, curve fitting, and graphing. 

It was assumed that a description of evolution of deterministic systems required a solution of the equations of motion, starting from some initial conditions. Although Poincare [1] knew that it was not always true, this opinion was common. Since the work of Lorenz [2] in 1963, unpredictability of deterministic systems described by differential nonlinear equations has been discovered in many cases. It has been established that given infinitesimally different initial conditions, the outcomes can be wildly different, even with the simplest equations of motion. This feature means the occurrence of deterministic chaos. The literature devoted to this multidisciplinary and rapidly developing discipline of science is huge. There are many excellent textbooks, monographs, and collections of main papers, and we mention only a few [3-8]. 

FIGURE 1 Velocity calibration of a M ssbauer spectrometer. The spectrum shown (a) is of metallic iron at room temperature. Line positions are given in channels and line widths in mm/s. The velocity calibration constant (b) is derived from the known energy differences between various components of the magnetic hyper fine spectrum. In the present data a differential nonlinearity of about 1 percent is observed. Such spectrometer nonlinearity may become a source for significant systematic errors in high-resohition experiments. 

Differential nonlinearity is a measure of the change in amplifier gain as a function of amplifier input signal. Referring to Fig. 10.38, the differential... 

The accuracy of the ADC is expressed in terms of its differential and integral nonlinearity. The differential nonlinearity describes the uniformity of address widths over the entire range of the ADC. To make this point better... 

Commercial ADCs have differential nonlinearity of the order of 0.5 percent to 1 percent. 

Figure 4.8 shows the differential nonlinearity (left) and the electronic instrument response function (right) of an SPC-134 TCSPC module (Becker Flickl, Berlin) for a different number of DAC bits, Ndac. The nonlinearity curves were recorded by detecting a continuous, unmodulated light signal. The instrument response functions were measured by using 1 ns pulses from a pulse generator at the photon pulse and reference inputs. 

The same delay eontrol voltage that results in a delay differenee of To in the reference structure is also applied to the measurement Vernier eireuit. Both delay structures are identieal in their strueture and layout and are implemented on one chip. Therefore the overall delay differenee in the measurement Vernier structure is very close to To, and the time seale is elose to To/n per gate. A Vernier TDC with 128 stages aehieved a resolution of 30 ps [146], yet with a differential nonlinearity of almost 100% peak-to-peak. The resolution appears to be limited mainly by the inerease of differential nonlinearity and by the jitter in a long active delay line. 

The differential nonlinearity (DNL) is the nonuniformity of the time ehannel width in a TGSPG system. Beeause the number of photons eolleeted in the ehannels is proportional to the ehannel width, any nonuniformity appears as a modulation in the reeorded photon distributions. DNL is the most important souree of systematie errors in TGSPG measurements. 

Differential Nonlinearity. The nonuniformity of the voltage increments in an analog-to-digital converter, or the nonuniformity of the time-channels of a photon counter... 

Ideally, as you increase the level of voltage appHed to a DAQ board, the digital codes from the ADC should also increase linearly. If you were to plot the voltage vs. the output code from an ideal ADC, the plot would be a straight Hne. Deviations from this ideal straight fine are specified as the nonhnearity. Three specifications indicate how linear a DAQ boards transfer function is differential nonlinearity, relative accuracy, and integral nonlinearity. 

Differential nonlinearity (DNL) A measure in LSB of the worst-case deviation of code widths from their ideal value of 1 LSB. 

DIFFERENTIAL NONLINEARITY A measure of quality of an ADC. In essence, it is a measure of the constancy of channel width. 

These modified inputs are summed on entry to the neurode. This net input is then modified by an internal transfer function. This might be a step function that passes a signal if a certain threshold is exceeded. More often, a function (such as a sigmoidal or hyperbolic tangent function) that produces a continuous, differentiable nonlinear signal is used (F. 2). The output is passed on either as the input for other neurodes, or as an output carrying a result. 

Jain has also developed an empirically derived scoring function based on 34 diverse protein-ligand complexes. The primary terms of the function arise for hydrophobic and polar complementarity, with additional factors for entropic and solvation effects. From this, Jain has constructed a sufficiently fast continuously differentiable nonlinear function, such that optimization of alignment/conformation of the ligand within the receptor, based on the predicted affinity, may be readily achieved. 

The primary characteristics of ADCs are their number of bits, resolution (step size), conversion speed, and power. The most significant performance parameters for a given ADC are noise, integral nonlinearity (INL), and differential nonlinearity (DNL). 

In the preceding sections, we considered only unconstrained optimization problems in which X may take any value. Here, we extend these methods to constrained minimization problems, where to be acceptable (or feasible), x must satisfy a number e of equality constraints gi (x) = 0 and a number n of inequality constraints hj x) > 0, where each g, (x) and hj(x) are assumed to be differentiable nonlinear functions. This constrained optimization problem... 

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