Error minimization

The probability of a type 1 error is inversely related to the probability of a type 2 error. Minimizing a type 1 error by decreasing a, for example, increases the likelihood of a type 2 error. The value of a chosen for a particular significance test, therefore, represents a compromise between these two types of error. Most of the examples in this text use a 95% confidence level, or a = 0.05, since this is the most frequently used confidence level for the majority of analytical work. It is not unusual, however, for more stringent (e.g. a = 0.01) or for more lenient (e.g. a = 0.10) confidence levels to be used. 

We want to make the overall error minimal for fixed n. We express the total error in terms of h and n... 

For longer boreholes the data may need to be synchronized by comparing the return temperature not with the current time step but with the time step— n, where n is the travel time. The error minimized is the sum square error of the difference between the calculated and measured borehole heat exchanger return temperature. We have set up the analyses procedure in such a way that it is easy to select discrete data-windows for the calibration. 

The calibration phase focuses on the determination of the planarization length itself. This is a crucial characterization phase since once the planarization length is determined, the effective density, and thus the thickness evolution, can be determined for any layout of interest polished under similar process conditions. The determination of planarization length is an iterative process. First, an initial approximate length is chosen. This is used to determine the effective density as detailed in the previous subsection. The calculated effective density is then used in the model to compute predicted oxide thicknesses, which are then compared to measured thickness data. A sum of square error minimization scheme is used to determine when an acceptably small error is achieved by gradient descent on the choice of planarization length. 

Note that careful evaluation and minimization of uncertainties and errors in CTMs is requested to enable the application of these CTMs to the study of observed changes in 03 as small as < 1.5 %/yr. However, actually 03 concentrations are simulated by the models within 20-50%. Chemical reaction rates are also uncertain, for instance in the 90 s determinations of the rates of CH4 and CH3CC13 reactions with OH suggested that these reactions are about 20% slower than believed. Similarly OH reaction with N02 which is an important sink for NOx in the troposphere is measured to be 10-30% lower than earlier estimates [23]. Thus, the past years a number of studies (mainly based on Monte Carlo simulations) focused on the identification and evaluation of the importance of various chemical reactions on oxidant levels to highlight topics crucial for error minimization. Temperature dependence of reaction rates can also introduce a 20-40% uncertainty in 03 and H20 computations in the upper troposphere. It has been also shown that 03 simulations are particularly sensitive to the photolysis rates of N02 and 03 and to PAN chemistry. 

Another way to characterize the LPC filter is as an autoregressive (AR) spectral envelope model [Kay, 1988], The error minimized by LPC (time-waveform prediction error) forces the filter to model parametrically the upper spectral envelope of the speech waveform [Makhoul, 1975], Since the physical excitation of the vocal tract is not spectrally flat, the filter obtained by whitening the prediction error is not a physical model of the vocal tract. (It would be only if the glottal excitation were an impulse... 

R. W. H. Sargent The method used is given in the Results section of the paper, page 160. The errors between the model and experimental results at 29 points were computed and the sum of squares of the errors minimized by adjusting Dp and P using Powells conjugate gradient method Computer Journal, 1964). 

Remember 8.3 Impedance measurements entail a compromise balance behveen minimizing bias errors, minimizing stochastic errors, and maximizing the information content of the resulting spectrum. The optimal instrument settings and experimental parameters are not universal and must be selected for each system under study. 

Tor a more exten.sive discussion of error minimization, see J. D. Ingle, Jr., and S. R. Crouch, Spectrochemical Analysis, pp. 176-183. Upper Saddle River, NJ Prentice-Hall, 1988. 

To compute a nonlinear map, the distances between all pairs of descriptors are calculated. The initial positions of the compounds on the map are chosen randomly and then modified in an iterative algorithm until all distances are represented as well as possible. The core algorithm of NLM is a partial least-squares error minimization (PLS). The total error of mapping must be smaller than the distances between the molecules and is therefore given on the NLM, e.g., as sum of error squares, E2. 

A comprehensive review is given in S. J. Freeland, T. Wu and N. Keulmann. The case for an error minimizing standard genetic code. Origins of Life, and Evolution of Biospheres, 33 (2003), 457-77. 

The adaption hypothesis assumes that the problem of codon assignment is a typical optimization issue with an error minimization scheme stating that near codons should encode similar amino acids. Thus, upon mutating one of the three bases of the codon, the encoded amino acid should be synonymous to the one encoded by the unmutated codon. This minimizes structural and functional deleterious effects of this mutation on the resulting protein (Di Giulio 2005). 

To eliminate the problems associated with the washing step, Okazaki, Smith, and Moodie proposed a CEC procedure in which salts are not removed between the index cation saturation and extraction steps. Rather, the anion of the salt providing the index cation is analyzed in the final extract. In accordance with electrical neutrality, the CEC is then equal to the total quantity of index cations removed during extraction minus the quantity of index anions removed simultaneously. The main potential source of error from this procedure arises from anion repulsion, if the quantity of index salt remaining after saturation is merely calculated from the weight of solution retained and its initial (or average) concentration. This error minimized if the index solution is lowered to approximately 0.1 M during the final two saturation washes. The error is eliminated if the quantities of index salt are analytically determined instead. 

To test the inverse predictions of i and Spi from C and S , random noise was added to the S, profile predicted by the forward calculation. This simulates a random error in the measurement of S. The random noise was uniformly distributed over the interval [ — (r,5, where ag is 0, 0.01, 0.03, or 0.1%. The perturbed, noisy profiles of 5 used for the inverse analysis are shown in Figure 7d. (No noise was added to C, because ihe effect of such noise has already been tested in Figure 4.) When a direct inversion of Eq. (38) was used to calculate random noise even at very small assumed random measurement errors (not shown). To obtain an acceptably smooth profile of 5p, the least-squares error minimization procedure of Eq. (30) was used with (n, m) = (10,... 

Experimental error is the difference between the true value and the measured value. For example, if the true temperature is 1000 K (1340°F) and the measured value is 950 K (1250°F), then the experimental error would be 50 K (90°F). This section considers the possible sources for experimental errors, minimizing those errors, and estimating the uncertainty in measurements. 

All theses 7 parameters were obtained from squared-error minimizations. The data analysis reveals that the best fits for hydrogen follow a very shallow minimum valleys so that the reported 7 parameters obtained from these fits should be taken with a grain of salt. The value 7j = 2 which was tested here is consistent with - and loosely justified by - the virial ratio for H, V e/E = 2. A reasonable explanation for possible distortions of the 7 value is perhaps linked to the basis used for hydrogen. This point is briefly examined with the help of DFT results (Table 5) and SCF computations using enriched bases. 

It is common that field data quality could be a problem and not all exchanger temperatures are measured online. To resolve these issues, data reconciliation is developed. First, field data are reviewed by engineers and some critical temperatures that are not measured are estimated by engineers. Then the least squares error minimization is applied to make estimation of the rest of the temperatures. It is an interactive procedure and the estimated temperamres are subject to further data reconcUiation. 

Nagy T, Turdnyi T. Reduction of very large reaction mechanism using methods based on simulation error minimization. Combust Hame 2009 156 417-28. 

On the other hand if e, are non-orthogonal, the equation of error minimization (4.63) can be written as... 

For linear functions, single or multiple, it is possible to find analytical solutions of the error minimization partial differential. Therefore, exact mathematical expressions exist for the calculation of slopes and intercepts. It should be noted at this point that a linear function of parameters does not imply a straight fine. A model is linear if the first partial derivative... 

The correction realized at Step 2(b) is determined from the error minimization condition, and is called the perceptron learning rule ... 

FIGURE 12.4 Comonomer concentrations versus time during a 68/32 methyl methacrylate (MMA)/N-methacryloxysuccinimide (MASI) copolymerization reaction in A/,A/-dimethylformamide (DMF) using the error minimization procednre developed in Reference (20). Reprinted (adapted) with permission from Alb AM, Enohnyaket P, Drenski MF, Shnnmngam R, Tew GN, Reed WF. Quantitative contrasts in the copolymerization of acrylate and methacrylate based comonomers. Macromolecules 2006 39 8283-8292. 2006 American Chemical Society. 

The ARX model has the general format as shown in Eqn. (24.3).Using least squares error minimization, the following result is found ... 

High-order models are often a result of models consisting of many differential equations or partial differential equations that have been converted into ordinary differential equations. These types of model are adequate for simulations studies but are not suitable for online use. A popular technique of model reduction that does not make use of error minimization is the model balancing method. The procedure is to find observability and controllability Gramians so as to determine which states have the largest overall contribution to the model. In systems theory and linear algebra, a Gramian matrix is a real-values symmetric matrix that can be used for a test for linear independence of functions. A system is called controllable if all states X can be influenced by the control input vector w, a system is observable if all states can be determined from the measurement vector jp. 

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