Gaussian distribution errors

The two sources of stochasticity are conceptually and computationally quite distinct. In (A) we do not know the exact equations of motion and we solve instead phenomenological equations. There is no systematic way in which we can approach the exact equations of motion. For example, rarely in the Langevin approach the friction and the random force are extracted from a microscopic model. This makes it necessary to use a rather arbitrary selection of parameters, such as the amplitude of the random force or the friction coefficient. On the other hand, the equations in (B) are based on atomic information and it is the solution that is approximate. For ejcample, to compute a trajectory we make the ad-hoc assumption of a Gaussian distribution of numerical errors. In the present article we also argue that because of practical reasons it is not possible to ignore the numerical errors, even in approach (A). 

It is important to verify that the simulation describes the chemical system correctly. Any given property of the system should show a normal (Gaussian) distribution around the average value. If a normal distribution is not obtained, then a systematic error in the calculation is indicated. Comparing computed values to the experimental results will indicate the reasonableness of the force field, number of solvent molecules, and other aspects of the model system. 

The normal distribution of measurements (or the normal law of error) is the fundamental starting point for analysis of data. When a large number of measurements are made, the individual measurements are not all identical and equal to the accepted value /x, which is the mean of an infinite population or universe of data, but are scattered about /x, owing to random error. If the magnitude of any single measurement is the abscissa and the relative frequencies (i.e., the probability) of occurrence of different-sized measurements are the ordinate, the smooth curve drawn through the points (Fig. 2.10) is the normal or Gaussian distribution curve (also the error curve or probability curve). The term error curve arises when one considers the distribution of errors (x — /x) about the true value. 

The root-mean-square error (RMS error) is a statistic closely related to MAD for gaussian distributions. It provides a measure of the abso differences between calculated values and experiment as well as distribution of the values with respect to the mean. 

If a large number of replicate readings, at least 50, are taken of a continuous variable, e.g. a titrimetric end-point, the results attained will usually be distributed about the mean in a roughly symmetrical manner. The mathematical model that best satisfies such a distribution of random errors is called the Normal (or Gaussian) distribution. This is a bell-shaped curve that is symmetrical about the mean as shown in Fig. 4.1. 

Comparison of Equation 10-7 with Equation 10-1 results in two important conclusions, of which the second is the less obvious. First, if one equation represents a Gaussian distribution, so does the other. Second, the standard deviation in Equation 10-7, which we have chosen to call the standard counting error, must be... 

Figure 4.51. Distribution of experimental data. Six experimental formulations (strengths 1, 2, resp. 3 for formulations A, respectively B) were tested for cumulative release at five sampling times (10, 20, 30, 45, respectively 60 min.). Twelve tablets of each formulation were tested, for a total of 347 measurements (13 data points were lost to equipment malfunction and handling errors). The group means were normalized to 100% and the distribution of all points was calculated (bin width 0.5%, her depicted as a trace). The central portion is well represented by a combination of two Gaussian distributions centered on = 100, one that represents the majority of points, see Fig. 4.52, and another that is essentially due to the 10-minute data for formulation B. The data point marked with an arrow and the asymmetry must be ignored if a reasonable model is to be fit. There is room for some variation of the coefficients, as is demonstrated by the two representative curves (gray coefficients in parentheses, h = peak height, s = SD), that all yield very similar GOF-figures. (See Table 3.4.)...

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

For a normal (Gaussian) error distribution, the RMSE is by a factor of Jl larger than the mean absolute error, also denoted as mean unsigned error. The error distribution of log Sw prediction methods appears to be somewhat less inhomogeneous than a Gaussian distribution and typically leads to a ratio of RMSE/mean absolute error... 

It can be shown [4] that the innovations of a correct filter model applied on data with Gaussian noise follows a Gaussian distribution with a mean value equal to zero and a standard deviation equal to the experimental error. A model error means that the design vector h in the measurement equation is not adequate. If, for instance, in the calibration example the model was quadratic, should be [1 c(j) c(j) ] instead of [1 c(j)]. In the MCA example h (/) is wrong if the absorptivities of some absorbing species are not included. Any error in the design vector appears by a non-zero mean for the innovation [4]. One also expects the sequence of the innovation to be random and uncorrelated. This can be checked by an investigation of the autocorrelation function (see Section 20.3) of the innovation. 

If basic assumptions concerning the error structure are incorrect (e.g., non-Gaussian distribution) or cannot be specified, more robust estimation techniques may be necessary. In addition to the above considerations, it is often important to introduce constraints on the estimated parameters (e.g., the parameters can only be positive). Such constraints are included in the simulation and parameter estimation package SIMUSOLV. Beeause of numerical inaccuracy, scaling of parameters and data may be necessary if the numerical values are of greatly differing order. Plots of the residuals, difference between model and measurement value, are very useful in identifying systematic or model errors. 

Figure 2. Histograms of Monte Carlo simulations for two synthetic analyses (Table 1) of a 330 ka sample. The lower precision analysis (A) has a distinctly asymmetric, non-Gaussian distribution of age errors and a misleading first-order error calculation. The higher precision analysis (B) yields a nearly symmetric, Gaussian age distribution with confidence limits almost identical those of the first-order error expansion.

Methods. Perhaps the best way of dealing with this thorny problem (common to not only °Th/U geochronology, but also the more classical methods of isotope geochronology as well) is to abandon the reliance on a strictly Gaussian distribution of residuals, whether arising from analytical error or geologic complexities. Robusf in the statistical sense implies insensitivity to departure of the data from the initial... 

Here xik is an estimated value of a variable at a given point in time. Given that the estimate is calculated based on a model of variability, i.e., PCA, then Qi can reflect error relative to principal components for known data. A given pattern of data, x, can be classified based on a threshold value of Qi determined from analyzing the variability of the known data patterns. In this way, the -statistic will detect changes that violate the model used to estimate x. The 0-statistic threshold for methods based on linear projection such as PCA and PLS for Gaussian distributed data can be determined from the eigenvalues of the components not included in the model (Jack-son, 1992). 

Figure 5.25. (A) Quantitative Cu map of an Al-4wt% Cu film at 230 kX, 128 x 128 pixels, probe size 2.7nm, probe current 1.9 nA, dwell time 120 msec per pixel, frame time 0.75 hr. Composition range is shown on the intensity scale (Reproduced with permission by Carpenter et al. 1999). (B) Line profile extracted from the edge-on boundary marked in Figure 5.25a, averaged over 20 pixels ( 55 nm) parallel to the boundary, showing an analytical resolution of 8nm FWTM. Error bars represent 95% confidence, and solid curve is a Gaussian distribution fitted to the data (Reproduced with permission by Carpenter...

Fig. 2.2. Average electrostatic potential mc at the position of the methane-like Lennard-Jones particle Me as a function of its charge q. mc contains corrections for the finite system size. Results are shown from Monte Carlo simulations using Ewald summation with N = 256 (plus) and N = 128 (cross) as well as GRF calculations with N = 256 water molecules (square). Statistical errors are smaller than the size of the symbols. Also included are linear tits to the data with q < 0 and q > 0 (solid lines). The fit to the tanh-weighted model of two Gaussian distributions is shown with a dashed line. Reproduced with permission of the American Chemical Society...

The standard requirements for the behavior of the errors are met, that is, the errors associated with the various measurements are random, independent, normally (i. e Gaussian) distributed, and are a random sample from a (hypothetical, perhaps) population of similar errors that have a mean of zero and a variance equal to some finite value of sigma-squared. 

In a well-behaved calibration model, residuals will have a Normal (i.e., Gaussian) distribution. In fact, as we have previously discussed, least-squares regression analysis is also a Maximum Likelihood method, but only when the errors are Normally distributed. If the data does not follow the straight line model, then there will be an excessive number of residuals with too-large values, and the residuals will then not follow the Normal distribution. It follows, then, that a test for Normality of residuals will also detect nonlinearity. 

Figure 16.6 Calibration of the radiocarbon ages of the Cortona and Santa Croce frocks the software used[83] is OxCal v.3.10. Radiocarbon age is represented on the y axis as a random variable normally distributed experimental error of radiocarbon age is taken as the sigma of the Gaussian distribution. Calibration of the radiocarbon agegivesa distribution of probability that can no longer be described by a well defined mathematical form it is displayed in the graph as a dark area on the x axis...

Indeterminate errors arise from the unpredictable minor inaccuracies of the individual manipulations in a procedure. A degree of uncertainty is introduced into the result which can be assessed only by statistical tests. The deviations of a number of measurements from the mean of the measurements should show a symmetrical or Gaussian distribution about that mean. Figure 2.2 represents this graphically and is known as a normal error curve. The general equation for such a curve is... 

The vector nk describes the unknown additive measurement noise, which is assumed in accordance with Kalman filter theory to be a Gaussian random variable with zero mean and covariance matrix R. Instead of the additive noise term nj( in equation (20), the errors of the different measurement values are assumed to be statistically independent and identically Gaussian distributed, so... 

As was shown, the conventional method for data reconciliation is that of weighted least squares, in which the adjustments to the data are weighted by the inverse of the measurement noise covariance matrix so that the model constraints are satisfied. The main assumption of the conventional approach is that the errors follow a normal Gaussian distribution. When this assumption is satisfied, conventional approaches provide unbiased estimates of the plant states. The presence of gross errors violates the assumptions in the conventional approach and makes the results invalid. 

The excited state of a molecule can last for some time or there can be an immediate return to the ground state. One useful way to think of this phenomenon is as a time-dependent statistical one. Most people are familiar with the Gaussian distribution used in describing errors in measurement. There is no time dependence implied in that distribution. A time-dependent statistical argument is more related to If I wait long enough it will happen view of a process. Fluorescence decay is not the only chemically important, time-dependent process, of course. Other examples are chemical reactions and radioactive decay. 

In a situation whereby a large number of replicate readings, not less than 5 0, are observed of a titrimetric equivalence point (continuous variable), the results thus generated shall normally be distributed around the mean in a more or less symmetrical fashion. Thus, the mathematical model which not only fits into but also satisfies such a distribution of random errors is termed as the Normal or Gaussian distribution curve. It is a bell-shaped curve which is noted to be symmetrical about the mean as depicted in Figure 3.2. 

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