Standard deviation of noise

Standard deviation of noise, On, which be roughly estimated by means of the peak-to-peak distance according to... 

For the second situation, the point P4 is defined as the lowest point among 5 points whose intensities are smaller than the standard deviation of noise at the right end in the neighbourhood of a peak. Then the straight line between P2 (which is previously defined) and P4 is set as the new base line (case 3). 

Step 2 The peak detection limit is set at 20 times the standard deviation of noise. If the value is greater than 2000, the limit is set at 2000. These parameters have been determined empirically. 

The Standard Deviation of Noise Values - This is the noise itself divided by the square root of 2J, where J is the number of independent samples used in determining the noise (Mandel, 1964) see Equation 11.7. 

The detection limits in the table correspond generally to the concentration of an element required to give a net signal equal to three times the standard deviation of the noise (background) in accordance with lUPAC recommendations. Detection limits can be confusing when steady-state techniques such as flame atomic emission or absorption, and plasma atomic emission or fluorescence, which... 

A study was conducted to measure the concentration of D-fenfluramine HCl (desired product) and L-fenfluramine HCl (enantiomeric impurity) in the final pharmaceutical product, in the possible presence of its isomeric variants (57). Sensitivity, stabiUty, and specificity were enhanced by derivatizing the analyte with 3,5-dinitrophenylisocyanate using a Pirkle chiral recognition approach. Analysis of the caUbration curve data and quaUty assurance samples showed an overall assay precision of 1.78 and 2.52%, for D-fenfluramine HCl and L-fenfluramine, with an overall intra-assay precision of 4.75 and 3.67%, respectively. The minimum quantitation limit was 50 ng/mL, having a minimum signal-to-noise ratio of 10, with relative standard deviations of 2.39 and 3.62% for D-fenfluramine and L-fenfluramine. 

Measurement noise covariance matrix R The main problem with the instrumentation system was the randomness of the infrared absorption moisture eontent analyser. A number of measurements were taken from the analyser and eompared with samples taken simultaneously by work laboratory staff. The errors eould be approximated to a normal distribution with a standard deviation of 2.73%, or a varianee of 7.46. 

We will now add random noise to each concentration value in Cl through C5. The noise will follow a gaussian distribution with a mean of 0 and a standard deviation of. 02 concentration units. This represents an average relative noise level of approximately 5% of the mean concentration values — a level typically encountered when working with industrial samples. Figure 15 contains multivariate plots of the noise-free and the noisy concentration values for Cl through C5. We will not make any use of the noise-free concentrations since we never have these when working with actual data. 

To better understand this, let s create a set of data that only contains random noise. Let s create 100 spectra of 10 wavelengths each. The absorbance value at each wavelength will be a random number selected from a gaussian distribution with a mean of 0 and a standard deviation of 1. In other words, our spectra will consist of pure, normally distributed noise. Figure SO contains plots of some of these spectra, It is difficult to draw a plot that shows each spectrum as a point in a 100-dimensional space, but we can plot the spectra in a 3-dimensional space using the absorbances at the first 3 wavelengths. That plot is shown in Figure 51. 

The limit of detection is the smallest amount of an analyte that is required for reliable determination, identification or quantitation. More mathematically, it may be defined as that amount of analyte which produces a signal greater than the standard deviation of the background noise by a defined factor. Strictly for quantitative purposes, this should be referred to as the limit of determination . The factor used depends upon the task being carried out and for quantitative purposes a higher value is used than for identification. Typical values are 3 for identification and 5 or 10 for quantitation. 

Thus, one can be far from the ideal world often assumed by statisticians tidy models, theoretical distribution functions, and independent, essentially uncorrupted measured values with just a bit of measurement noise superimposed. Furthermore, because of the costs associated with obtaining and analyzing samples, small sample numbers are the rule. On the other hand, linear ranges upwards of 1 100 and relative standard deviations of usually 2% and less compensate for the lack of data points. 

The synthetic data have been obtained by adding random noise with standard deviation of about 0.4 )0.g 1 to the theoretical plasma concentrations. As can be seen, the agreement between the estimated and the computed values is fair. Estimates tend to deteriorate rapidly, however, with increasing experimental error. This phenomenon is intrinsic to compartmental models, the solution of which always involves exponential functions. 

A random noise with standard deviation of 0.4 pg 1 has been added to the theoretical values in order to produce a realistic example. The specifications of the model are in part the same as those used for the one-compartment models which have been discussed above. The major distinction between this model and the... 

We reconsider the data used previously in Section 39.1.2 in the discussion of the two-compartment system for extravascular administration (e.g. oral, subcutaneous, intravascular). The data are truncated at 120 minutes in order to obtain a realistic case. It is recalled that these data have been synthesized from a theoretical model and that random noise with a standard deviation of about 0.4 pg T has been superimposed. 

The DL and QL for chromatographic analytical methods can be defined in terms of the signal-to-noise ratio, with values of 2 1-3 1 defining the DL and a value of 10 1 defining the QL. Alternatively, in terms of the ratio of the standard deviation of the blank response, the residual standard deviation of the calibration line, or the standard deviation of intercept (s) and slope (5) can be used [40, 42], where ... 

Figure 42-3 Comparison of the exact (upper curve equation 42-37) and approximate (lower curve Ingle and Crouch equation 5-45) expressions for the standard deviation of A A/A as a function of %T. Noise-to-signal is set at 0.01.

The family of curves obtained, and presented in Figure 43-6, show that, not surprisingly, the controlling parameter of the family of curves is the standard deviation of the noise the maximum value of the multiplication factor occurs at a given fraction of the standard deviation of the energy readings. Successive approximations show that the maximum multiplier of approximately 1.28 occurs when If is approximately 2.11 times sigma, the standard deviation of AEr. 

Figure 44-10a Transmittance noise as a function of transmittance, for different values of reference energy S/N ratio (recall that, since the standard deviation of the noise equal unity, the set value of the reference energy equals the S/N ratio), (see Color Plate 11)...

The results are shown in Figure 45-11. It is obvious that for values of Ex greater than five (standard deviations of the noise), the optimum transmittance remains at the level we noted previously, 33 %T. When the reference energy level falls below five standard deviations, however, the optimum transmittance starts to decrease. The erratic nature of the variance at these low values of Ex, however, makes it difficult to ascertain the exact amount of falloff with any degree of precision nevertheless it is clear that as much as we can talk about an optimum transmittance level under these conditions, where variance can become infinite and the actual transmittance value itself is affected, it decreases at such low values of Ex. Nevertheless, a close look reveals that when... 

In our previous development, we presented a family of curves, corresponding to different values of SD(A ). In the case of uniformly distributed noise, which is of necessity contained within a limited range of values, the well-known fact that the standard deviation of the noise equals the range/ f 2 helps us, in that it requires only one curve to display, rather than a family of curves. ([7], p. 146). For this case, then, equation 44-71 becomes equation (46-84) ... 

Poisson-distributed noise, however, has an interesting characteristic for Poisson-distributed noise, the expected standard deviation of the data is equal to the square root of the expected mean of the data ([11], p. 714), and therefore the variance of the data is equal (and note, that is equal, not merely proportional) to the mean of the data. Therefore we can replace Var(A s) with Es in equation 47-17 and Var(A r) with Et ... 

Our first chapter in this set [4] was an overview the next six examined the effects of noise when the noise was due to constant detector noise, and the last one on the list is the first of the chapters dealing with the effects of noise when the noise is due to detectors, such as photomultipliers, that are shot-noise-limited, so that the detector noise is Poisson-distributed and therefore the standard deviation of the noise equals the square root of the signal level. We continue along this line in the same manner we did previously by finding the proper expression to describe the relative error of the absorbance, which by virtue of Beer s law also describes the relative error of the concentration as determined by the spectrometric readings, and from that determine the... 

Equation 52-149 presents a minor difficulty one that is easily resolved, however, so let us do so the difficulty actually arises in the step between equation 52-148 and 52-149, the taking of the square root of the variance to obtain the standard deviation conventionally we ordinarily take the positive square root. However, T takes values from zero to unity that is, it is always less than unity, the logarithm of a number less than unity is negative, hence under these circumstances the denominator of equation 52-149 would be negative, which would lead to a negative value of the standard deviation. But a standard deviation must always be positive clearly then, in this case we must use the negative square root of the variance to compute the standard deviation of the relative absorbance noise. 

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