Variances of estimates

Although the derivation is beyond the scope of this presentation, it can be shown that the estimated variance of predicting a single new value of response at a given point in factor space, is equal to the purely experimental uncertainty variance, plus the variance of estimating the mean response at that point, 5, 0 that is. 

Rotatable designs are most efficient for k=3. Rotatable designs of second order are not orthogonal and they do not minimize the variance of estimates of regression coefficients. They are efficient in solving research problems when trying to find an optimum. 

An extension of the CRB approach can be to minimize not only the minimal uncertainties but also both bias and variance in order to consider the use of biased estimators. Bias and variance result from a trade-off, and so it is possible to reduce the variance of estimates by tolerating an increase in the bias. For this purpose an extension of the CRB has been introduced by Hero et al.,41 which represents the variance of estimates as function of the norm of the bias gradient. This curve shows the achievable trade-off between bias and variance. 

In addition, we would hke the variance of the estimator to be small since this improves the accuracy. Statistical texts devote considerable time on the efficiency (the variance) of estimators. 

Assessing a subjective probability distributiou of au uncotain quautity is a difficult task. An individual formulates his belief by adjustmeut from some convenient starting point. Unfortunately, the adjustmoit is typically not enough, and the resulting probability distribution exhibits bias due to auchor-ing at the starting point. The result is that the variance of estimated probability distributions turns out to be narrower than the variance of actual probability distributions. In tests, anchoring errors in the assessment of subjective probability distributions are common to both expert and non-expert respondents. 

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

The need for good estimates of the variances of the measurements cannot be overemphasized. Unfortunately, these can be accurately estimated only by completely replicated experiments. 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

As discussed above the errors in the trajectory are correlated with the missing rapid motions. In contrast to the friction approach of estimating the variance, which may affect long time phenomena, we identify our errors as the missing ( filtered ) high frequency modes. We therefore attempt to account approximately for the fast motions by choosing the trajectory variance accordingly. 

In spite of the three methods of estimating open question that needs to be investigated further in the future. The above expressions are order of magnitude estimates and not exact formulas. 

So D may be estimated either from the measured delay time, or from the variance of the response pulse. In the former case, comparison of equations... 

Consider next the problem of estimating the error in a variable that cannot be measured directly but must be calculated based on results of other measurements. Suppose the computed value Y is a hnear combination of the measured variables [yj], Y = CL y + Cioyo + Let the random variables yi, yo,. . . have means E yi), E y, . . . and variances G yi), G y, . The variable Y has mean... 

Thus, the variance of the desired quantity Y can be found. This gives an independent estimate of the errors in measuring the quantity Y from the errors in measuring each variable it depends upon. 

Harnhy, N., The Estimation of the Variance of Samples Withdrawn from a Random Mixture of Multi-Sized Particles, Chem. Eng. No. 214 CE270-71 (1967). 

The statistical measures can be calculated using most scientific calculators, but confusion can arise if the calculator offers the choice between dividing the sum of squares by N or by W — 1 . If the object is to simply calculate the variance of a set of data, divide by N . If, on the other hand, a sample set of data is being used to estimate the properties of a supposed population, division of the sum of squares by W — r gives a better estimate of the population variance. The reason is that the sample mean is unlikely to coincide exactly with the (unknown) true population mean and so the sum of squares about the sample mean will be less than the true sum of squares about the population mean. This is compensated for by using the divisor W — 1 . Obviously, this becomes important with smaller samples. 

Hay and Pasquill (5) and Cramer (6, 7) have suggested the use of fluctuation statistics from fixed wind systems to estimate the dispersion taking place within pollutant plumes over finite release times. The equation used for calculating the variance of the bearings (azimuth) from the point of release of the particles, cTp, at a particular downwind location is... 

If the tracer concentration is X measured at each sampling position that has its position at y, on a scale along the arc (either in degrees or in meters), estimates of the mean posiHon of the plume at ground level and the variance of the groundlevel concentration distribution are given by ... 

It can be argued that the main advantage of least-squares analysis is not that it provides the best fit to the data, but rather that it provides estimates of the uncertainties of the parameters. Here we sketch the basis of the method by which variances of the parameters are obtained. This is an abbreviated treatment following Bennett and Franklin.We use the normal equations (2-73) as an example. Equation (2-73a) is solved for <2o-... 

Equation (2-95) gives the variance of y at any Xj. With this equation confidence intervals can be estimated, using Student s t distribution, for the entire range of Xj. In particular, when all Xj = 0, y = Oq. nd we find... 

This weighting procedure for the linearized Arrhenius equation depends upon the validity of Eq. (6-7) for estimating the variance of y = In k. It will be recalled that this equation is an approximation, achieved by truncating a Taylor s series expansion at the linear term. With poor precision in the data this approximation may not be acceptable. A better estimate may be obtained by truncating after the quadratic term the result is... 

Although it sounds reasonable to use the maximum likelihood to define our esfimafe of the displacement, there are two questions that remain. Firstly, what is the variance of the error associated with this estimate This defines N which was used in Eq. 22 fo defermine fhe error in fhe wavefront reconstruction. Secondly, is it possible to do better than the centroid In other words is it optimal ... 

Finally we need to compare the variance of our estimator with the best attainable. It can be shown that The Cramer-Rao lower bound (CRLB) is a lower bound on the variance of an unbiased estimator (Kay, 1993). The quantities estimated can be fixed parameters with unknown values, random variables or a signal and essentially we are finding the best estimate we can possibly make. 

Note that by unbiased we mean that when the noise tends to zero we expect a —> a. The left-hand side can be written simply as Var(d), the variance of the estimate, since the estimator is unbiased. The CRLB for some PDFs of interest is now computed. 

Figure 7 shows that for the maximum likelihood estimator the variance in the slope estimate decreases as the telescope aperture size increases. For the centroid estimator the variance of the slope estimate also decreases with increasing aperture size when the telescope aperture is less than the Fried parameter, ro (Fried, 1966), but saturates when the aperture size is greater than this value. 

The mean and variance of the difference between B colla (from table) and B colu is determined for all 14 diets for each trial combination of dp, dpi and to, and the best values for dp, dpj and w chosen to minimize both the mean and the variance. These values turn out to be dp = +5, dN = +2 and CO = -0.75. Figure All.l shows a plot of the difference between the estimated and calculated collagen values for each diet for this particular DIFF, and it can be seen that, except for one point, the others are correctly estimated to within 1 or 1.5%o. 

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