Gaussian distribution analysis

Multiple linear regression is strictly a parametric supervised learning technique. A parametric technique is one which assumes that the variables conform to some distribution (often the Gaussian distribution) the properties of the distribution are assumed in the underlying statistical method. A non-parametric technique does not rely upon the assumption of any particular distribution. A supervised learning method is one which uses information about the dependent variable to derive the model. An unsupervised learning method does not. Thus cluster analysis, principal components analysis and factor analysis are all examples of unsupervised learning techniques. 

The first application of the Gaussian distribution is in medical decision making or diagnosis. We wish to determine whether a patient is at risk because of the high cholesterol content of his blood. We need several pieces of input information an expected or normal blood cholesterol, the standard deviation associated with the normal blood cholesterol count, and the blood cholesterol count of the patient. When we apply our analysis, we shall anive at a diagnosis, either yes or no, the patient is at risk or is not at risk. 

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 degree of data spread around the mean value may be quantified using the concept of standard deviation. O. If the distribution of data points for a certain parameter has a Gaussian or normal distribution, the probabiUty of normally distributed data that is within Fa of the mean value becomes 0.6826 or 68.26%. There is a 68.26% probabiUty of getting a certain parameter within X F a, where X is the mean value. In other words, the standard deviation, O, represents a distance from the mean value, in both positive and negative directions, so that the number of data points between X — a and X -H <7 is 68.26% of the total data points. Detailed descriptions on the statistical analysis using the Gaussian distribution can be found in standard statistics reference books (11). 

In their analysis Richards and Temple assumed that the end-to-end distance of a growing polymer chain follows a Gaussian distribution, the ratio of the probability of a Bis... 

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. 

Each oil-dispersant combination shows a unique threshold or onset of dispersion [589]. A statistic analysis showed that the principal factors involved are the oil composition, dispersant formulation, sea surface turbulence, and dispersant quantity [588]. The composition of the oil is very important. The effectiveness of the dispersant formulation correlates strongly with the amount of the saturate components in the oil. The other components of the oil (i.e., asphaltenes, resins, or polar substances and aromatic fractions) show a negative correlation with the dispersant effectiveness. The viscosity of the oil is determined by the composition of the oil. Therefore viscosity and composition are responsible for the effectiveness of a dispersant. The dispersant composition is significant and interacts with the oil composition. Sea turbulence strongly affects dispersant effectiveness. The effectiveness rises with increasing turbulence to a maximal value. The effectiveness for commercial dispersants is a Gaussian distribution around a certain salinity value. 

In reality, the queue size n and waiting time (w) do not behave as a zero-infinity step function at p = 1. Also at lower utilization factors (p < 1) queues are formed. This queuing is caused by the fact that when analysis times and arrival times are distributed around a mean value, incidently a new sample may arrive before the previous analysis is finished. Moreover, the queue length behaves as a time series which fluctuates about a mean value with a certain standard deviation. For instance, the average lengths of the queues formed in a particular laboratory for spectroscopic analysis by IR, H NMR, MS and C NMR are respectively 12, 39, 14 and 17 samples and the sample queues are Gaussian distributed (see Fig. 42.3). This is caused by the fluctuations in both the arrivals of the samples and the analysis times. 

Fig. 42.3. Time series of the observed queue lengths (n) in a department for structural analysis, with their corresponding histograms fitted with a Gaussian distribution.

The observed Gaussian distribution of the queue lengths in the spectroscopic laboratory (see Fig. 42.3) is not in agreement with the theoretical distribution given by eq. (42.6). This indicates that an analysis station cannot be modelled by the simple M/M/1 model. We will return to this point later. 

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.

In particle size analysis it is important to define three terms. The three important measures of central tendency or averages, the mean, the median, and the mode are depicted in Figure 2.4. The mode, it may be pointed out, is the most common value of the frequency distribution, i.e., it corresponds to the highest point of the frequency curve. The distribution shown in Figure 2.4 (A) is a normal or Gaussian distribution. In this case, the mean, the median and the mode are found to fie in exactly the same position. The distribution shown in Figure 2.4 (B) is bimodal. In this case, the mean diameter is almost exactly halfway between the two distributions as shown. It may be noted that there are no particles which are of this mean size The median diameter lies 1% into the higher of the two distri-... 

Now the variance in free energy difference is described in terms of the variance of work. The analysis above also indicates that the Gaussian distributions /(IF) and g W) are related... 

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. 

Data from phase-modulation fluorometry have been analyzed using an alternative approach to those described above, as expounded by Gratton and co-workers(14 12 13,22) and Lakowicz et al. W> Here, Lorentzian or Gaussian distribution functions with widths and centers determined by least-squares analysis are used to model the unknown distribution function. While this approach may introduce assumptions about the shape of the ultimate distribution function since these trial functions are symmetric, it has the advantage of minimizing the number of parameters involved in the fit. Here, a minimum x2 is sought, where... 

Little is known about the fluorescence of the chla spectral forms. It was recently suggested, on the basis of gaussian curve analysis combined with band calculations, that each of the spectral forms of PSII antenna has a separate emission, with Stokes shifts between 2nm and 3nm [133]. These values are much smaller than those for chla in non-polar solvents (6-8 nm). This is due to the narrow band widths of the spectral forms, as the shift is determined by the absorption band width for thermally relaxed excited states [157]. The fluorescence rate constants are expected to be rather similar for the different forms as their gaussian band widths are similar [71], It is thought that the fluorescence yields are also probably rather similar as the emission of the sj tral forms is closely approximated by a Boltzmann distribution at room temperature for both LHCII and total PSII antenna [71, 133]. 

While radioactive decay is itself a random process, the Gaussian distribution function fails to account for probability relationships describing rates of radioactive decay Instead, appropriate statistical analysis of scintillation counting data relies on the use of the Poisson probability distribution function ... 

The simulation says that the maximum value is. 5188, which is less than the expected value. Remember that for the resistor with the 5% Gaussian distribution, the standard deviation was 1.25%, and the absolute limits on the distribution were 4o = 5%. In the Worst Case analysis, a device with a Gaussian distribution is varied by only 3cr. Had we calculated the maximum value with a 3.75% resistor variation, we would have come up with a maximum gain of 0.51875, which agrees with the PSpice result. To obtain the worst case limits, I prefer to use the uniform distribution. Type CTRL-F4 to close the output file and display the schematic. 

In the Monte Carlo analysis the tolerance distribution has a major effect on the results. It is best to know the distribution of your parts before you trust the results of the Monte Carlo analysis. You can use the PSpice .Distribution statement to define your own probability distributions. If you do not know the distribution of your parts, the Gaussian distribution is a better representation of parts than the uniform distribution. 

The Gaussian distribution is the best to use with the Monte Carlo analysis since it more closely matches the actual distribution of components. 

The formation of a boundary between the dextran solution and the dextran solution containing PVP 360 (concentration 5 kg m 3) yields an apparently normal Gaussian distribution of the material detected by Schlieren optics. The various apparent diffusion coefficients obtained by an analysis of the Schlieren curves, which include diffusion coefficients obtained by the reduced height-area ratio method, the reduced second-moment and the width-at-half-height method, show the same qualitative behavior although quantitative differences do exist. This is seen in Fig. 7 where the... 

Archaebacterial rhodopsins may provide models for signal transduction and ion transport.62 Spin labels were attached to sensory rhodopsin (pSRII) and its transducer (pHTrll) from Natronobacterium pharaonis. Interspin distance determined by line-shape analysis including a Gaussian distribution of distances revealed a 2 2 complex with 2-fold symmetry. Distances for 26 pairs of spin labels defined the orientation of the TM helices of pHtrll relative to the F and G helices of pSRII.63 Light excitation causes a flap-like movement of helix F of NpSRII that induces a rotary motion of a helix in the transmembrane domain of the transducer.62... 

---