Variance proportion

The only point to note is how the pattern is contrary to the above example. In the example with halogenated volatile hydrocarbons we find a total redundancy of 3.0% in the X set (variance explained by the Yset) and 2.7% in the Y set (variance explained by the Xset). The extracted variance proportions are 73.3% for the X set and 100% for the Y set. Whereas the last-mentioned proportions are normal, we confirm from the redundancies that we really have independent sets of different halogenated hydrocarbons. 

Fig. 9-3. Plot of the scores of discriminant function df 2 (44.9% variance proportion) vs. scores of discriminant function df 1 (55.1% variance proportion) of the four classes obtained by CA. (The circles correspond to the 5% risk of error of the MANOVA — separation line of discrimination corresponding to highest probability)...

Experiments on Wien-bridge oscillators showed such anomalous fluctuations to exhibit a variance proportional — where /3 is the reaction factor and the corresponding threshold value. The relaxation time... 

Another useful tool is the variance proportion for each x,, which is the proportion of the total variance of its 6, estimate, for a particular principal component. Note that in Table 6.4, the first column is... 

The mean absolute percent error MAPE, which is 9.99 less than 10, indicates that the forecasted values are in a high accuracy. Besides, Thell Inequality Coefficient TIC is 0.057, and Bias Proportion BP 0.059 and Variance Proportion VP 0.005 are relatively small while Covariance Proportion CP 0.935 is relatively large. All these parameters suggest that the result of forecasting is desirable. See the Figure 2. 

The total % variance explained should not exceed the accuracy of the underlying data. If, for example, toxicity data are subject to 20% variability, it is inappropriate to extract more PCs than correspond to 80% (max. 85%) of the total variance (proportion of information). The remainder is, for example, experimental data scatter. 

For a drift Brownian motion, the differences of two consecutive performance measurements are independently, normally distributed with mean and variance proportional to the time lag between two measurements. Utilizing this property, Doksum Normand (1995) discussed the maximum likelihood estimation of the parameters in degradation mean function and variance. Therefore, the solution to model parameter estimation is obtained by maximizing the marginal likelihood function, which in turn is obtained by integrating out the random coefficient from the joint condi-... 

In many types of analytical equipment, the error is a fixed percentage of the maximum instrument reading, which results in a variance proportional to y, i.e. a y,. This gives an uneven variance distribution that calls for using a weight factor, for example 1 /yf, or a logarithm ... 

Note that the relative sampling variance is inversely proportional to the number of particles sampled. Increasing the number of particles in a sample, therefore, improves the sampling variance. 

The procedure for testing the significance of a sample proportion follows that for a sample mean. In this case, however, owing to the nature of the problem the appropriate test statistic is Z. This follows from the fact that the null hypothesis requires the specification of the goal or reference quantity po, and since the distribution is a binomial proportion, the associated variance is [pdl — po)]n under the null hypothesis. The primary requirement is that the sample size n satisfy normal approximation criteria for a binomial proportion, roughly np > 5 and n(l — p) > 5. 

Thus, the variance of the peak is inversely proportional to the number of theoretical plates in the column. Consequently, the greater the value of (n), the more narrow the peak, and the more efficiently has the column constrained peak dispersion. As a result, the number of theoretical plates in a column has been given the term Column Efficiency. From the above equations, a fairly simple procedure for measuring the efficiency of any column can be derived. 

Now, all the curves are describing the same chromatogram thus, by simple proportion, the ratios of the variance of each elution curve to the square of the retention (in the respective units in which the variables are defined) will all be equal. 

Equation (12) indicates that the band variance is directly proportional to the square of the tube radius, very similar to that for a straight tube. At high linear velocities, Tijssen deduced that... 

It is conventional to define (for the case being considered) the weight of a measurement yi to be inversely proportional to the variance of y that is. 

The weight of the ith observation is inversely proportional to the variance of the observation we will use Eq. (2-82) for this quantity, n being the number of observations. 

A die is loaded so diat die probability of any face turning up is directly proportional to die number of dots on die face. Let X denote the outcome of tlirowing die die once. Find die mean and variance of X. 

Four studies suggest that k /kK has a significant temperature dependence (Table 5.5). Although not agreeing on the precise value of ktJkte, all four studies indicate that the proportion of disproportionation increases with increasing temperature. These results are at variance with model studies that suggest that kJkK is independent of temperature. It was also proposed that the preferred termination mechanism is solvent dependent and that disproportionation is favored in more polar media.161... 

One performs so many repeat measurements at each concentration point that standard deviations can be reasonably calculated, e.g., as in validation work the statistical weights w, are then taken to be inversely proportional to the local variance. The proportionality constant k is estimated from the data. 

The variance about the mean, and hence, the confidence limits on the predicted values, is calculated from all previous values. The variance, at any time, is the variance at the most recent time plus the variance at the current time. But these are equal because the best estimate of the current time is the most recent time. Thus, the predicted value of period t+2 will have a confidence interval proportional to twice the variance about the mean and, in general, the confidence interval will increase with the square root of the time into the future. 

For long linear chains the second condition is supported by the Stockmayer bivariate distribution (8,9) which shows the bivariate distribution of chain length and composition is the product of both distributions, and the compositional distribution is given by the normal distribution whose variance is inversely proportional to chain length. 

Hence is the fraction of the total sum of squares (or inertia) c of the data X that is accounted for by v,. The sum of squares (or inertia) of the projections upon a certain axis is also proportional to the variance of these projections, when the mean value (or sum) of these projections is zero. In data analysis we can assign different masses (or weights) to individual points. This is the case in correspondence factor analysis which is explained in Chapter 32, but for the moment we assume that all masses are identical and equal to one. 

The variance diagram obtained for the example discussed before is quite simple. Clusters of pure variables are found at 30 degrees (var = 0.5853) and at 300 degrees (var = 0.4868) (see Fig. 34.36). The distance from the centre of the diagram to each point is proportional to the variance value. Neighbouring points are connected by solid lines. All values were scaled in such a way that the highest variance is full scale. As can be seen from Fig. 34.36, two clusters of pure variables are found. The... 

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