Uncertainty maximum interval

It is important that the information and data required for interpretation be complete. Another aspect that must be checked is whether uncertainties, for instance due to estimated data used to fill data gaps, might influence the result. Such uncertainties can be determined by calculation of a minimum-maximum interval covering the potential extremes, then determining its effect on the final result. 

In the case of a single processor, Bolzano s method minimizes the maximum interval of uncertainty. 

The maximum interval of uncertainty is equal to 7, since it is the value obtainable by the worst situation. 

Uncertainly estimates are made for the total CDF by assigning probability distributions to basic events and propagating the distributions through a simplified model. Uncertainties are assumed to be either log-normal or "maximum entropy" distributions. Chi-squared confidence interval tests are used at 50% and 95% of these distributions. The simplified CDF model includes the dominant cutsets from all five contributing classes of accidents, and is within 97% of the CDF calculated with the full Level 1 model. 

Apply a sequential one-dimensional search technique to reduce the interval of uncertainty for the maximum of the function/ = 6.64 + 1.2 - x2 from [0,1] to less than 2 percent of its original size. Show all the iterations. 

The 95% confidence interval amounted to 1.3 log units, which corresponds to a scatter in the ATdoc values by a factor of 20. Burkhard (2000) argues that the uncertainty in Eq. 3.25 originates partly from inter-laboratory variation, and partly from differences in DOC quality. In view of the wide range of DOC sources included in Eq. 3.25, it does not seem likely that the uncertainties in any ATdoc encountered would be more than 1.3 log units. Hence, the worst-case estimate (maximum sorption of dissolved phase residues by DOC) can be described by... 

For determining the estimated amount interval at the lower or minimum end of the regression line three cases arise 1. The lower end can be calculated and is positive. 2. The lower end can be calculated and is negative and/or the interval is excessively long. 3. The lower end cannot be calculated because it is negatively infinite. A similar situation exists at the maximum end of the regression line. In these cases the true uncertainty is properly reflected by the calculated amount uncertainty. The effects found in cases 2 and 3 could be studied to determine their possibility with various types of calibration data. This paper, however, will not delve into that aspect. 

The variables 17, Ua, and are the corresponding uncertainty values for each parameter. They are computed to the 67% confidence interval by taking the standard error of each parameter in the regressions (i.e., ai, U2 and (73, and multiplying by their Student f-score ts (i.e., =ts SEo ), where ts is the Student t-score at the confidence level of interest and SEai is the corresponding standard error for the parameter ai. The period can be chosen based on the maximum value or another statistical parameter. The results of four experiments are given in Figure 9.8. 

Schmidt [16] thoroughly analyzed the problem of uncertainties in measuring the mean lifetime with a small number of detected nuclei, including the presence of a stochastic background. The crucial point was that the measurement was supposed to last until complete decay of the nuclide. The treatment was based on the maximum likelihood approach the 90 percent confidence intervals were tabulated. 

In the shutdown mode, the reactor vessel is fully pressurized or, at different times, in various stages of depressurization. Afterheat from fission product decay is generated at rates of up to about 7 percent of the core power level prior to shutdown, depending on the time interval since shutdown. The core decay heat is removed by the HTS. When the HTS is not available, the heat is removed by the Shutdown Cooling System (SCS). The outer control rods are normally fully Inserted during shutdown, and meet the required shutdown margin, with due allowances for uncertainties, even if the maximvim reactivity worth rod remains fully withdrawn. For cold shutdown, the control rods in the inner reflector are also Inserted and for this case, the maximum reactivity worth control rod is in the inner reflector. The neutron flux level is continuously monitored by the source range detectors. 

Uncertainty intervals represent the maximum from the data point to the fitted curve. 

In the special case of a knovm interval of uncertainty, there is a test that provides a maximum guarantee of reliability. If the algorithm checks the interval of uncertainty [Ia, h] and the best point between (a and ts is selected as the solution, the error on the solution is... 

This is the criterion that ensures maximum reliability in deeming the accuracy of the solution achieved, when the interval of uncertainty is given. 

Given the maximum number N of function evaluations, the goal is to minimize the final interval of uncertainty Lp or its reduction ratio a ... 

Thus far, we have denoted the final interval of uncertainty with Lp even though it would be more correct to denote the maximum final interval of uncertainty by the same symbol. 

The final interval of uncertainty depends on the function. Conversely, the maximum final interval of uncertainty does not change, given the points at which the function is to be evaluated actually, this is the worst final interval achievable by applying the predefined strategy. 

Consider the problem of defining the optimal strategy to minimize the maximum final interval of uncertainty, given the initial interval of uncertainty, and the number of function evaluations. 

This problem can also be investigated in its complementary form given the initial and final ranges of uncertainty, the goal is to find the strategy that minimizes the total amount of function evaluations required to obtain a maximum final interval of uncertainty smaller than the assigned one. 

When only two points are available, their optimal position is easy to find. The two points should be evenly spaced by the center, otherwise the maximum final interval of uncertainty cannot be minimized. Moreover, they should be placed at the minimum acceptable distance d since the closer they are to each other, the smaller the final interval. 

Our problem can now be solved by starting not at the beginning but at the end at the end of the search, after N — 1 function evaluations, the interval of uncertainty is In i and the last function evaluation needs to be positioned. One of the previous points - the identity of which is unknown to us - is already within the interval of uncertainty. To minimize the maximum final interval of uncertainty, Ln, obtained by introducing the last point, t, the last two points should be symmetric with respect to the center of the interval Ln-i and at the minimum distance d (Figure 2.2). 

It minimizes the maximum final interval of uncertainty for any function. 

Unfortunately, some of these advantages may also turn out be disadvantages at times too. The Fibonacci method minimizes the maximum final interval of uncertainty only if the required function evaluations are accomplished. 

It is also possible to modify the Fibonacci method to obtain a series that minimizes the maximum final interval of uncertainty when a point is already positioned in the starting interval. In this case, the final interval is not univocally determined by the number of points, but different widths can be obtained depending on the position of the first point and the function to be minimized. 

Each uncertain parameter, djj, is denoted by a nominal value, dy, and a shift value, djj. Therefore, uncertain parameter dy could be demonstrated via the closed interval [dy—dy, dy + dy]. Contrary to common robust optimization models where it is assumed that all uncertain parameters can deviate from their nominal values, [21] assumed that probability of all uncertain parameters deviating from their nominal values is very small. Therefore, the decision-maker should decide a number F, known as uncertainty budget, which limits maximum number of uncertain parameters deviating from their nominal values. The selection of this number is highly dependent on the level of decision-maker s conservativeness. As the value of r increases, the level of decision-maker s conservativeness increases, and the robust model becomes more similar to minimax model of [17]. Considering /, as the set of uncertain parameters belonging to the row of A, the robust counterpart of problem (26.18)-(26.20) could be written as follows [21] ... 

---