Variance propagation methods

For many mathematical operations, including addition, subtraction, multiplication, division, logarithms, exponentials and power relations, there are exact analytical expressions for explicitly propagating input variance and covariance to model predictions of output variance (Bevington, 1969). In analytical variance propagation methods, the mean, variance and covariance matrix of the input distributions are used to determine the mean and variance of the outcome. The following is an example of the exact analytical variance propagation approach. If w is the product of x times y times z, then the equation for the mean or expected value of w, E(w), is  

Bevington (1969) lists variance propagation solutions like the one above for several mathematical operations. 

In an unmodified Monte Carlo method, simple random sampling is used to select each member of the 777-tuple set. Each of the input parameters for a model is represented by a probability density function that defines both the range of values that the input parameters can have and the probability that the parameters are within any subinterval of that range. In order to carry out a Monte Carlo sampling analysis, each input is represented by a cumulative distribution function (CDF) in which there is a one-to-one correspondence between a probability and values. A random number generator is used to select probability in the range of 0-1. This probability is then used to select a corresponding parameter value. 

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This is an example exposure assessment that illustrates quantitative representations of uncertainty and variability at the higher tiers of an exposure assessment. This case-study is based on human exposures to a persistent, bioaccumulative and lipid-soluble compound through fish consumption. This compound is fictional and referred to here as PBLx, but it has properties that correspond to those of known persistent compounds. Specific goals of this case-study are to illustrate (1) the types of uncertainty and variability that arise in exposure assessments, (2) quantitative uncertainty assessment, (3) how distributions are established to represent variability and uncertainty, (4) differences among alternative variance propagation methods, (5) how to distinguish uncertainty from variability and (6) how to communicate the results of an uncertainty analysis. 

The composition of the case-study includes a conceptual model, the modelling approach, construction of input distributions and variance propagation methods. When evaluating uncertainty, it is important to consider how each of these elements contributes to overall uncertainty. 

In order to illustrate the use of the variance propagation methods described above, we have selected for the case-study a simple three-input exposure model. The three inputs for this model include water concentration, fish BCF and fish consumption rates. The model output is dose expressed in micrograms per day averaged over a one-year exposure period. This model has the form ... 

Table A2.6 Selected statistics of the PBLx intake distribution obtained from different model variance propagation methods. ...

The differences in estimation of these moments for each scenario are graphically illustrated in Figures A2.4, A2.5 and A2.6, where the CDFs obtained from each numerical variance propagation method are compared with the analytical results. The results of the analytical method are assumed to represent the true moments of the model output and, therefore, the true CDF. Mean and standard deviation of ln(x) are used in plotting the analytical CDF. The equations for the transformation from arithmetic moments to the moments of ln(x) are as follows ... 

So far in the case-study, we have focused on variance propagation methods and have not made an effort to distinguish between the relative contributions to overall variance from uncertainty and variability. In the examples above, the cumulative distributions presented in figures all reflect overall variance that includes the combined contributions from both uncertainty and variability. So our last step is to illustrate a two-dimensional analysis in which we distinguish and display separate contributions from uncertainty and variability. We begin this analysis by going back to our inputs and assessing the relative contributions from uncertainty and variability. 

The Projected Variance (PV) method describes robustness as the variance of the response induced by the variance in the independent variable(s) propagated through the response surface. This method was first described by Box [17]. Vuchkov et al. [18] has used this method in case of second... 

Sensitivity analysis methods can be used in combination with methods for variance propagation. For example, Cullen Frey (1999) describe how variance in the sum of random numbers can be apportioned among the inputs to the sum. All of the statistical sensitivity methods mentioned above can be applied to the results of Monte Carlo simulation, in which... 

A parameter such as a rate constant is usually obtained as a consequence of various arithmetic manipulations, and in order to estimate the uncertainly (error) in the parameter we must know how this error is related to the uncertainties in the quantities that contribute to the parameter. For example, Eq. (2-33) for a pseudo-first-order reaction defines k, which can be determined by a semilogarithmic plot according to Eq. (2-6). By a method to be described later in this section the uncertainty in itobs (expressed as its variance associated with cb. Thus, we need to know how the errors in fcobs and cb are propagated into the rate constant k. 

Methods for propagating variance or distributions through models... 

Model variance was propagated using the factorial, DPD, Monte Carlo and Latin hypercube sampling (LHS) methods. Table A2.6 provides a summary comparison of the outputs—the arithmetic mean, arithmetic standard deviation, coefficient of variation (CV), geometric mean (GM), geometric standard deviation (GSD), 5th percentile and 95th percentile outcomes— from each method. 

This relatively simple model illustrates the viability of the straightforward analytical analysis. Most models, unfortunately, involve many more input variables and proportionally more complex formulae to propagate variance. Fortunately, the Latin hypercube sampling and Monte Carlo methods simplify complex model variance analysis. 

The problem in both methods is the error propagation. If an error exists in the measurement, this error will be submitted to the transformation as well. A second problem arises in the variances. Usually the variances of measurement in TLC are constant within the calibration range. The transformation of data will lead to inhomogeneous variances and this is the reason for unreliable regression analysis. 

Standard errors and confidence intervals for functions of model parameters can be found using expectation theory, in the case of a linear function, or using the delta method (which is also sometimes called propagation of errors), in the case of a nonlinear function (Rice, 1988). Begin by assuming that 0 is the estimator for 0 and X is the variance-covariance matrix for 0. For a linear combination of observed model parameters... 

An overview of statistical methods covers mean values, standard deviation, variance, confidence intervals, Student s t distribution, error propagation, parameter estimation, objective functions, and maximum likelihood. 

Statistical methods provide tools for assessing univariate data (replicate measurements of a single parameter) resulting in measurements of i) accuracy, defined as the difference between an experimental value and the true value the latter is generally not known for a real-world analytical sample, so that accuracy must be estimated using a surrogate sample e.g., a blank matrix spiked with a known amount of analytical standard) ii) precision, such as the relative standard deviation (RSD, also known as the coefficient of variance, COV or CV) iii) methods for calculation of propagation of experimental error in calculations. 

In fact, the error term in eqn (2.1), a,-, produces uncertainty in the estimators bg and bj. In order for these estimates to be useful, it is necessary to know how important their uncertainties are. As in any uncertainty (error propagation) calculation, the variance of the data has to be propagated to the estimates. Unfortunately, the LS method does not provide unbiased estimators for the variance of the y-values (cr unless there is no lack of fit between the data and the line. ... 

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