Bias correction

Figure 3. Contour plots of the bias corrected MEM densities in the (110) plane of metallic beryllium (a) uniform prior, (b) non-uniform prior. The plots are on a linear scale with 0.05 el A1 intervals. Truncation at 1.0e/A3. Maximum values in e/A3 are given at the Be position and in the bipyramidal space of the hep structure.

Table 4 shows the projected anomalies of annual and seasonal precipitation and air temperature for the Ebro River, whereas Figs. 7-12 show the spatial variation of these anomalies, computed using ordinary kriging. Anomalies are computed as the difference between projected bias-corrected values for the climate scenarios (January 2071 to December 2100) and the corresponding values observed during the control period (January 1961 to December 1990), and they can be viewed as expected values about which uncertainties of different origin exist. Table 4 shows that both RCMs predict a reduction in the mean annual precipitation, accompanied by an increase in the mean annual temperature with respect to the control period. In particular, the RCAO E model projects a reduction of 21.8% for the mean annual precipitation and an increase of +6.3°C for the mean annual temperature. 

The hydrological simulations for the future time-slice 2071-2100 were performed using the bias-corrected daily time series derived from the two different climate scenarios described in Table 4 as external forcing, and the hydrological parameters obtained during the calibration and verification period described in Sect. 6.1. One operational modification was made to the simulation of subcatchment 057, where a withdrawal corresponding to the Ebro-Besaya water transfer, described in Sect. 6.1, was allowed to remove water outside the catchment. This water transfer was introduced only for the simulations of the future scenarios, while the... 

A relatively simple statistical downscaling technique which may be applied quickly to a large number of models is the use of correction factors based on monthly relationships between observed data collected at a particular weather station and the relevant RCM control data set for the appropriate grid-cell [40], These monthly differences (for temperature) and ratios (for precipitation) between the control and the point observations (i.e. not the gridded interpolated CRU data set) can then be used to correct the daily RCM control and scenario data. This gives bias-corrected scenarios of temperature and precipitation, which can then be used as input to hydrological models for the exploration of various management and policy formulations. 

The bias-correction is necessary to correct both the absolute magnitude and the seasonal cycle to that of the observations. This approach assumes that the same model biases persist in the future climate and thus GCMs more accurately simulate relative change than absolute values. It provides a correction of monthly mean climate only and does not correct biases in higher order statistics including the simulation of extreme events and persistence. 

Figures showing the spatial distribution of the annual mean precipitation and temperature for the Ebro for the bias-corrected control and future periods for two of the most extreme RCMs (RCAO E and HIRHAM H) are given in [80]. To avoid repetition, the following sections discuss the bias-corrected results for the Gallego. As discussed in [38], these results are similar to those derived for the Ebro.

The accuracy of an analysis can be determined by several procedures. One common method is to analyze a known sample, such as a standard solution or a quality control check standard solution that may be available commercially, or a laboratory-prepared standard solution made from a neat compound, and to compare the test results with the true values (values expected theoretically). Such samples must be subjected to all analytical steps, including sample extraction, digestion, or concentration, similar to regular samples. Alternatively, accuracy may be estimated from the recovery of a known standard solution spiked or added into the sample in which a known amount of the same substance that is to be tested is added to an aliquot of the sample, usually as a solution, prior to the analysis. The concentration of the analyte in the spiked solution of the sample is then measured. The percent spike recovery is then calculated. A correction for the bias in the analytical procedure can then be made, based on the percent spike recovery. However, in most routine analysis such bias correction is not required. Percent spike recovery may then be calculated as follows ... 

Slope/bias correction This method, which is really a postprocessing of model outputs, is one of the simplest improvement methods, and can be quite effective in cases where temporal shifts in analyzer response are expected [105]. However, when used, it should be accompanied by a well-defined sampling and measurement protocol, in order to generate a sufficiently large population of time-localized standards that can be used to determine stable estimates of slope and bias correction factors. 

During real-time operation, statistical/-tests can be used to determine whether the slope and bias correction calculated for a particular unknown spectrum are within an expected range [108]. If not, then a warning can be issued indicating that the current sample is not appropriate to apply to the model. 

Cowtan, K. (1999). Error estimation and bias correction in phase-improvement calculations. Acta Crystallogr. D 55, 1555-1567. 

Bias corrections are sometimes applied to MLEs (which often have some bias) or other estimates (as explained in the following section, [mean] bias occurs when the mean of the sampling distribution does not equal the parameter to be estimated). A simple bootstrap approach can be used to correct the bias of any estimate (Efron and Tibshirani 1993). A particularly important situation where it is not conventional to use the true MLE is in estimating the variance of a normal distribution. The conventional formula for the sample variance can be written as = SSR/(n - 1) where SSR denotes the sum of squared residuals (observed values, minus mean value) is an unbiased estimator of the variance, whether the data are from a normal distribution... 

Based on the discussion of criteria for parameter estimation, it is not necessarily important to use estimators that are unbiased in the statistical sense. The emphasis should be on the overall performance of the estimator, considering precision as well as accuracy. If bias is known to be large for practical purposes, bias correction may improve performance (bootstrap bias correction is easy). However, in practice, precision may be a greater concern than bias, particularly with few data, and bias correction may result in lower precision. 

Using the mean of the results for R , calculate the mass bias correction factor, C, using Eqn. B. 1. 

Calculate the observed isotope ratio, R , for the sample, and multiply by the mass bias correction factor to obtain the true isotope ratio, R Hence, calculate C for the sample using Eqn. B.4. 

In any experiment, therefore, all significant systematic errors should be measured and corrected for, and the random errors, including those pertaining to the bias corrections, estimated and combined in the measurement uncertainty. 

If there is an inherent bias in the method, usually estimated and reported in method validation studies, its uncertainty needs to be included. However, if run bias is estimated during the analysis, the method bias will be subsumed, and the local estimate of uncertainty of any bias correction should be used. This might have the effect of lowering the uncertainty because not all the laboratory biases of the participants in the study are now included. 

As reproducibility standard deviation from interlaboratory method validation studies has been suggested as a basis for the estimation of measurement uncertainty if it is known sR can be compared with a GUM estimate. It may be that with good bias correction, the estimate may be less than the reproducibility, which tends to average out all systematic effects including ones not relevant to the present measurement. Another touchstone is the Horwitz relation discussed in section 6.5.4. A rule of thumb is that the reproducibility of a method (and therefore the estimated measurement uncertainty) should fall well within a factor of two of the Horwitz value. 

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