Variance estimates

Fortunately, however, the technique used here does not depend on the magnitude of the variances, but only on their ratios. If estimates of the magnitudes of the variances are wrong but the ratios are correct, the residuals display the random behavior shown in Figure 3. However, the magnitudes of these deviations are then not consistent with the estimated variances. 

Geostatistical techniques, such as variography and kriging, have been recently introduced into the environmental sciences (O Although kriging allows mapping of the pollution plume with qualification of the estimation variance, it falls short of providing a truly risk-qualified estimate of the spatial distribution of pollutants. 

Probabilistic techniques of estimation provide some Insights Into the potential error of estimation. In the case of krlglng, the variable pCic) spread over the site A is first elevated to the status of a random function PC c). An estimator P (2c) is then built to minimize the estimation variance E [P(2c)-P (2c) ], defined as the expected squared error ( ). The krlglng process not only provides the estimated values pCiyc) from which a kriged map can be produced, but also the corresponding minimum estimation variances 0 (39 ) ... 

Secondly, knowledge of the estimation variance E [P(2c)-P (2c)] falls short of providing the confidence Interval attached to the estimate p (3c). Assuming a normal distribution of error In the presence of an Initially heavily skewed distribution of data with strong spatial correlation Is not a viable answer. In the absence of a distribution of error, the estimation or "krlglng variance o (3c) provides but a relative assessment of error the error at location x Is likely to be greater than that at location 2 " if o (2c)>o (2c ). Iso-varlance maps such as that of Figure 1 tend to only mimic data-posltlon maps with bull s-eyes around data locations. 

Figure 1. Isopleths of krlglng estimation variances. (The bull s-eyes reflect the data locations.)...

The estimate p (x) retained need not be at the center of the confidence interval ]qw(2L) (x) Since the confidence intervals are obtained directly, there is no need to calculate the estimation variance, nor to hypothesize any model for the error distribution. 

Assessment of spatial distributions of pollutant concentrations is a very specific problem that requires more than blind mapping of these concentrations. Not only must the criterion of estimation be chosen carefully to allow zooming on the most critical values (the high concentrations), but also the evaluation of the potential error of estimation calls for a much more meaningful characteristic than the traditional estimation variance. Finally, the risks a and p of making wrong decisions on whether to clean or not must be assessed. 

Estimate the sampling variance (S s) by one-way analysis of variance (2 x 6n) using the average value of each well (estimation variance between each portion and each sample). 

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

The relationship between the estimated variances s2 has to consider the number of primary samples, p, subsamples, q, and test samples, q r... 

Cochran WG (1941) The distribution of the largest of a set of estimated variances as a fraction of their total. Ann Eugen [London] 11 47... 

Finally, the contribution to error, that is specific to this laboratory, especially when compared to other laboratories, is not known. On the other hand, there are some very interesting and noteworthy observations to be made from the error study. First, repeated attempts to include polymer type and temperature into the error model failed. This observation implies that temperature control of the device is independent of temperature and that any fluctuation is the same throughout the interval that was used in this study. Secondly, repeated attempts to include sample type into the analysis also failed, implying that the estimated variances were independent not only of temperature but of material. This result suggests very strongly that the variances are due to the magnitude of the measurement, the device that was used, and our own set of laboratory circumstances. 

ANOVA (Analysis of Variance) A statistical procedure that is used (a) to test for significant differences among means of several sets of results or (b) to estimate variances of several influences operating independently. 

The variance of a population is another useful measure of dispersion and reflects the extent of the differences between the data. Denoted by a2, it is equal to the mean squared deviation of the individual values from the population mean. Usually, the symbols V and s2 are used for the variance deduced from sample data. Thus, for a sample of N data drawn from a population with mean /z, the estimated variance is... 

Each of the upper left to lower right diagonal elements of V is an estimated variance of a parameter estimate, si, these elements correspond to the parameters as they appear in the model from left to right. Each of the off-diagonal elements is an estimated covariance between two of the parameter estimates [Dunn and Clark (1987)]. 

Thus, for a single-parameter model such as y,j = p + r,j, the estimated variance-covariance matrix contains no covariance elements the square root of the single variance element corresponds to the standard uncertainty of the single parameter estimate. 

Let sIq be the estimated variance associated with the parameter estimate bo let be the estimated variance associated with b and let slf (or sl ) represent the estimated covariance between and b,. Then... 

A more complete picture is shown graphically in Figure 7.1 where the value of the element for the estimated variance of fe, is shown as a function of the location of the second experiment. The stationary experiment is shown by a dot at j , = +1. As the two experiments are located farther and farther apart, the uncertainty in the slope of the straight line decreases. Note that the curve in Figure 7.1 is symmetrical about the fixed experiment at j , = +1. 

Figure 7.1 Value of the element of for the estimated variance of bi as a function of the...

Figure 8.4 Value of the element of (X X) for the estimated variance of h, as a function of the location of a third experiment, two experiments fixed at x, = -1 and x, = 1.

In Section 6.4, it was shown for replicate experiments at one factor level that the sum of squares of residuals, SS can be partitioned into a sum of squares due to purely experimental uncertainty, SS, and a sum of squares due to lack of fit, SSi f. Each sum of squares divided by its associated degrees of freedom gives an estimated variance. Two of these variances, and were used to calculate a Fisher F-ratio from which the significance of the lack of fit could be estimated. 

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