Sums of squares

Sum of squared, weighted residuals °F Number of degrees of freedom... 

BETA cols 11-20 oscillation control parameter default value is set equal to 0.25. To help prevent oscillations (thus slowing convergence) we not only require that the sum of squares, SSQ, decreases... 

FORMAT (34H0DELTA LINEARIZED SUM OF SQUARES WRITE (6,2004) (OELP(L),L=1,LL)... 

The electron density p(r) at a point r can be calculated from the Bom interpretation of the u a Vv tu iiclion as a sum of squares of the spin orbitals at the point r for all occupied molecular orhital-i. For a system of N electrons occupying N/2 real orbitals, we can write ... 

A fourth hierarchical method that is quite popular is Ward s method [Ward 1963]. This method merges those two clusters whose fusion minimises the information toss due to the fusion. Information loss is defined in terms of a function rvhich fdr each cluster i corresponds to the total sum of squared deviations from the mean of the cluster ... 

Vcaic,i is obtained by feeding the appropriate r,- value into the regression equation. Anothe common squared term is the residual sum of squares (RSS), which is the sum of square of the differences between the observed and calculated y values. TSS is equal to the sur of RSS and ESS. The is then given by ... 

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

This method, because it involves minimizing the sum of squares of the deviations xi — p, is called the method of least squares. We have encountered the principle before in our discussion of the most probable velocity of an individual particle (atom or molecule), given a Gaussian distr ibution of particle velocities. It is ver y powerful, and we shall use it in a number of different settings to obtain the best approximation to a data set of scalars (arithmetic mean), the best approximation to a straight line, and the best approximation to parabolic and higher-order data sets of two or more dimensions. 

If the experimental error is random, the method of least squares applies to analysis of the set. Minimize the sum of squares of the deviations by differentiating with respect to m. 

The coefficient matrix and nonhomogeneous vector can be made up simply by taking sums of the experimental results or the sums of squares or products of results, all of which are real numbers readily calculated from the data set. 

The sum of squares of differences between points on the regression line yi at Xi and the arithmetic mean y is called SSR... 

The quadr atic curve fit leads to a number of residuals equal to the number of points in the data set. The sum of squares of residuals gives SSE by Eqs. (3-23) and MSE by Eq. (3-30), except that now the number of degrees of freedom for n points is... 

We wish to cany out a proceduie that is the multivariate analog to the analysis in the section on reliability of fitted parameters. A vector multiplied into its hanspose gives a scalar that is the sum of squares of the elements in that vector. The y vector leads to a vector of residuals... 

The product e e is the sum of squares of residuals from the vector of residuals. The vai iance is... 

The term on the left-hand side is a constant and depends only on the constituent values provided by the reference laboratory and does not depend in any way upon the calibration. The two terms on the right-hand side of the equation show how this constant value is apportioned between the two quantities that are themselves summations, and are referred to as the sum of squares due to regression and the sum of squares due to error. The latter will be the smallest possible value that it can possibly be for the given data. 

There is an obvious similarity between equation 5.15 and the standard deviation introduced in Chapter 4, except that the sum of squares term for Sr is determined relative toy instead of y, and the denominator is - 2 instead of - 1 - 2 indicates that the linear regression analysis has only - 2 degrees of freedom since two parameters, the slope and the intercept, are used to calculate the values ofy . 

To calculate the standard deviation for the analyte s concentration, we must determine the values for y and E(x - x). The former is just the average signal for the standards used to construct the calibration curve. From the data in Table 5.1, we easily calculate that y is 30.385. Calculating E(x - x) looks formidable, but we can simplify the calculation by recognizing that this sum of squares term is simply the numerator in a standard deviation equation thus,... 

Variance was introduced in Chapter 4 as one measure of a data set s spread around its central tendency. In the context of an analysis of variance, it is useful to see that variance is simply a ratio of the sum of squares for the differences between individual values and their mean, to the degrees of freedom. For example, the variance, s, of a data set consisting of n measurements is given as... 

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