Gaussian distributions random errors

Since we are not able to find the true value for any parameter, we often make do with the average of all of the experimental data measured for that parameter, and consider this as the most probable value." Measured values, which necessarily contain experimental errors, should lie in a random manner on either side of this most probable value as expressed by the normal or Gaussian distribution. This distribution i. a bell-shaped curve that represents the number of measurements N that have a specific value x (which deviates from the mean or most probable value Xq by an amount x - Xo, representative of the error). Obviously the smaller the value of x - Xo, the higher the probability that the quantity being measured lies near the most likely value xq, which is at the top of the peak. A plot of N against x, shown in Figure 10.1, is called a Gaussian distribution or error curve, expressed mathematically as ... 

The NLME function in S-Plus offers three different estimation algorithms a FOCE algorithm similar to NONMEM, adaptive Gaussian quadrature, and Laplacian approximation. The FOCE algorithm in S-Plus, similar to the one in NONMEM, was developed by Lindstrom and Bates (1990). The algorithm is predicated on normally distributed random effects and normally distributed random errors and makes a first-order Taylor series approximation of the nonlinear mixed effects model around both the current parameter estimates 0 and the random effects t). The adaptive Gaussian quadrature and Laplacian options are similar to the options offered by SAS. 

One critical factor to keep in mind when building and evaluating predictive models is that every experimental data point has an error associated with it. For example, if we measure the Log 5 of a compound as -6 and that data point has an error of 0.3 log units, the acmal value could be anywhere between -6.3 and -5.7. In a 2009 paper, Brown and coworkers [41] examined the relationship between experimental error and model performance. They carried out a series of theoretical experiments where Gaussian distributed random values were added to data to simulate experimental errors. The authors then calculated the correlation between the measured values and the same values with this simulated error. This correlation can be thought of as the maximum correlation possible given the error in the measurement. As we saw earlier. 

K 2> and constants. If specially required, selected values of p, q, r, and s can also be searched for by simultaneous regression estimation. The highest number of parameters should not exceed 16. Apart from this, a simulation can be performed while theoretical data calculated from assumed values of parameters (p, q, r, s), and K i are loaded with a random noise having the normal (Gaussian) distribution of errors. An arbitrary level of noise can be chosen by the user. 

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). 

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. 

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. 

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. 

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. 

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...

Measurements can contain any of several types of errors (1) small random errors, (2) systematic biases and drift, or (3) gross errors. Small random errors are zero-mean and are often assumed to be normally distributed (Gaussian). Systematic biases occur when measurement devices provide consistently erroneous values, either high or low. In this case, the expected value of e is not zero. Bias may arise from sources such as incorrect calibration of the measurement device, sensor degradation, damage to the electronics, and so on. The third type of measurement... 

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... 

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. 

The features of an ideal chromatogram are the same as those obtained from a normal distribution of random errors (Gaussian distribution equation (1.2), cf. 21.3). In keeping with the classical notations, fi would correspond to the retention time of the eluting peak, a to the standard deviation of the peak (a2 represents the variance) and y represents the signal, as a function of time, from the detector located at the end of the column (see Fig. 1.3). 

If an experiment is repeated a great many times and if the errors are purely random, then the results tend to cluster symmetrically about the average value (Figure 4-1). The more times the experiment is repeated, the more closely the results approach an ideal smooth curve called the Gaussian distribution. In general, we cannot make so many measurements in a lab experiment. We are more likely to repeat an experiment 3 to 5 times than 2 000 times. However, from the small set of results, we can estimate the statistical parameters that describe the large set. We can then make estimates of statistical behavior from the small number of measurements. 

Gaussian distribution Theoretical bell-shaped distribution of measurements when all error is random. The center of the curve is the mean, p, and the width is characterized by the standard deviation, a. A nortnalized Gaussian distribution, also called the normal error curve, has an area of unity and is given by... 

The distribution of errors of measurement is usually analyzed according to the Gaussian or normal distribution. This applies to sampling a population that is subject to a random distribution. The normal distribution follows the equation... 

Precision determines the reproducibility or repeatability of the analytical data. It measures how closely multiple analysis of a given sample agree with each other. If a sample is repeatedly analyzed under identical conditions, the results of each measurement, x, may vary from each other due to experimental error or causes beyond control. These results will be distributed randomly about a mean value which is the arithmetic average of all measurements. If the frequency is plotted against the results of each measurement, a bell-shaped curve known as normal distribution curve or gaussian curve, as shown below, will be obtained (Figure 1.2.1). (In many highly dirty environmental samples, the results of multiple analysis may show skewed distribution and not normal distribution.)... 

Sometimes a measurement involves a single piece of calibrated equipment with a known measurement uncertainty value o, and then confidence limits can be calculated just as with the coin tosses. Usually, however, we do not know o in advance it needs to be determined from the spread in the measurements themselves. For example, suppose we made 1000 measurements of some observable, such as the salt concentration C in a series of bottles labeled 100 mM NaCl. Further, let us assume that the deviations are all due to random errors in the preparation process. The distribution of all of the measurements (a histogram) would then look much like a Gaussian, centered around the ideal value. Figure 4.2 shows a realistic simulated data set. Note that with this many data points, the near-Gaussian nature of the distribution is apparent to the eye. 

All measurements are accompanied by a certain amount of error, and an estimate of its magnitude is necessary to validate results. The error cannot be eliminated completely, although its magnitude and nature can be characterized. It can also be reduced with improved techniques. In general, errors can be classified as random and systematic. If the same experiment is repeated several times, the individual measurements cluster around the mean value. The differences are due to unknown factors that are stochastic in nature and are termed random errors. They have a Gaussian distribution and equal probability of being above or below the mean. On the other hand, systematic errors tend to bias the measurements in one direction. Systematic error is measured as the deviation from the true value. 

Random error — The difference between an observed value and the mean that would result from an infinite number of measurements of the same sample carried out under repeatability conditions. It is also named indeterminate error and reflects the - precision of the measurement [i]. It causes data to be scattered according to a certain probability distribution that can be symmetric or skewed around the mean value or the median of a measurement. Some of the several probability distributions are the normal (or Gaussian) distribution, logarithmic normal distribution, Cauchy (or Lorentz) distribution, and Voigt distribution. Voigt distribution is... 

So far the discussion has dealt with the errors themselves, as if we knew their magnitudes. In actual circumstances we cannot know the errors by which the measurements Xj deviate from the true value Xq, but only the deviations (x,- — x) from the mean T of a given set of measurements. If the random errors follow a Gaussian distribution and the systematic errors are negligible, the best estimate of the true value Aq of an experimentally measured quantity is the arithmetic mean x. If you as an experimenter were able to make a very large (theoretically infinite) number of measurements, you could determine the true mean /jl exactly, and the spread of the data points about this mean would indicate the precision of the observation. Indeed, the probability function for the deviations would be... 

In most analytical experiments where replicate measurements are made on the same matrix, it is assumed that the frequency distribution of the random error in the population follows the normal or Gaussian form (these terms are also used interchangeably, though neither is entirely appropriate). In such cases it may be shown readily that if samples of size n are taken from the population, and their means calculated, these means also follow the normal error distribution ( the sampling distribution of the mean ), but with standard deviation sj /n this is referred to as the standard deviation of the mean (sdm), or sometimes standard error of the mean (sem). It is obviously important to ensure that the sdm and the standard deviation s are carefully distinguished when expressing the results of an analysis. 

If the assumption of a Gaussian error distribution is considered valid, then an additional method of expressing random errors is available, based on confidence levels. The equation for this distribution can be manipulated to show that approximately 95% of all the data will lie within 2 5 of the mean, and 99.7% of the data will lie within 3i of the mean. Similarly, when the sampling distribution of the mean is considered, 95% of the sample means will lie within approximately 2sj /n of the population mean etc. (Figure 5). 

Random errors may not follow a gaussian distribution, which is usually assumed for the analysis of data. Once more, statistical tests may be applied to determine whether serious deviation from a gaussian distribution exists and to interpret the data accordingly. 

For small sets of measurements it was found that the relative frequency of occurrence of random errors is not described so well by Gaussian distribution as by another frequency function named t or Student function, f(t, v) (where v = n -1 represents the degree of freedom of the sample) with a known expression [71] ... 

Even if all systematic error could be eliminated, the exact value of a chemical or physical quantity still would not be obtained through repeated measurements, due to the presence of random error (Barford, i985). Random error refers to random differences between the measured value and the exact value the magnitude of the random error is a reflection of the precision of the measuring device used in the analysis. Often, random errors are assumed to follow a Gaussian, or normal, distribution, and the precision of a measuring device is characterized by the sample standard deviation of the distribution of repeated measurements made by the device. [By contrast, systematic errors are not subject to any probability distribution law (Velikanov, 1965).] A brief review of the normal distribution is provided below to provide background for a discussion of the quantification of random error. 

Figure 14-19 Outline of the relation between xl and x2 values measured by two methods subject to random errors with constant standard deviations over the analytical measurement range. A linear relationship between the target values (XI Target.. X2Targeti) Is presumed.The xlj and x2,- values are Gaussian distributed around Xi Target and X2Targeti. respectively, as schematically shown. 021 (Oyx) is demarcated.

Figure N-21 The model assumed in ordinary OLR.The x2 values are Gaussian distributed around the line with constant standard deviation over the analytical measurement range.The x values are assumed to be without random error. 021 is shown.

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