Normal equation, 5.24

Solving the normal equations by Cramer s rule leads to the solution set in determinantal fomi... 

The least squares derivation for quadratics is the same as it was for linear equations except that one more term is canied through the derivation and, of course, there are three normal equations rather than two. Random deviations from a quadratic are ... 

The form of the symmetric matrix of coefficients in Eq. 3-20 for the normal equations of the quadratic is very regular, suggesting a simple expansion to higher-degree equations. The coefficient matrix for a cubic fitting equation is a 4 x 4... 

We have already seen the normal equations in matrix form. In the multivariate case, there are as many slope parameters as there are independent variables and there is one intercept. The simplest multivariate problem is that in which there are only two independent variables and the intercept is zero... 

The left side of the normal equations can be seen to be a product including X, its transpose, and m. Matrix multiplication shows that... 

Obtain the normal equations [Eq. set (3-63)] from the minimization conditions... 

We do not know either side of Eq. (6-33), but we do know that E is to be minimized with respect to some minimization parameters. The only arbitrary parameters we have are the a and aa, which enter into the LCAO. Thus our normal equations are... 

Any linearly independent set of simultaneous homogeneous equations we can construct has only the zero vector as its solution set. This is not acceptable, for it means that the wave function vanishes, which is contrai y to hypothesis (the electron has to be somewhere). We are driven to the conclusion that the normal equations (6-38) must be linearly dependent. 

Additional rearrangement gives the so-called normal equations ... 

Noimal equations. The three normal equations are defined by... 

The pressure drop for the exhaust opening, the jet supply opening, and for the hood opening if there are side walls, can be calculated using the normal equations for flow inside ducts and into ducts and openings. 

Because it is of particular interest in the present context, we now obtain the normal equations for linear regression with a single independent vanable. The model function is... 

Because there is only one independent variable, the subscript has been omitted. We now note that Zx/n = x and Zy/n = y, so we find Eqs. (2-75) as the normal equations for unweighted univariate least-squares regression. 

A variable can take any physically admissible value a variate is a variable that must also satisfy a frequency distribution.) These are called unweighted normal equations because each observation is accorded equal weight. 

Carrying through the treatment as before yields Eqs. (2-78) as the normal equations for weighted linear univariate least-squares regression. 

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-... 

Referring to the earlier treatment of linear least-squares regression, we saw that the key step in obtaining the normal equations was to take the partial derivatives of the objective function with respect to each parameter, setting these equal to zero. The general form of this operation is... 

Table 6-1 lists the experimental quantities, k, T, ct, the transformed variables x, y, and the weights w. (It is necessary, in least-squares calculations, to carry many more digits than are justified by the significant figures in the data at the conclusion, rounding may be carried out as appropriate.) The sums required for the solution of the normal equations are... 

Differentiation of the expression for R according to Eq. (2-76) affords the two normal equations ... 

The partial derivative of R with respect to each parameter is then minimized. The normal equations are... 

The two normal equations were obtained in the usual way and solved for the two parameters. The results are... 

Quite recently, Thorn has derived an essentially identical set of normal equations when analyzing the vapor-pressure-temperature relationships (209) he did not deal with its solution. 

---