Squares of residuals

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... 

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

The simplest procedure is merely to assume reasonable values for A and to make plots according to Eq. (2-52). That value of A yielding the best straight line is taken as the correct value. (Notice how essential it is that the reaction be accurately first-order for this method to be reliable.) Williams and Taylor have shown that the standard deviation about the line shows a sharp minimum at the correct A . Holt and Norris describe an efficient search strategy in this procedure, using as their criterion minimization of the weighted sum of squares of residuals. (Least-squares regression is treated later in this section.)... 

Table 11. Sum of the square of residuals (SSR) for A1 and A3 through A5, using the S basis vectors for Al.

Si = -xgm) sum of squares of residuals relative to the appropriate group mean Xmean, (1.31)... 

The k-value can be estimated by minimization of the sum of squares of residuals (the least squares technique) ... 

In this case we minimize a weighted sum of squares of residuals with constant weights, i.e., the user-supplied weighting matrix is kept the same for all experiments, Q,=Q for all i=l,...,N and Equation 3.7 reduces to... 

The sum of squares of residuals has to be minimized according to the general least squares (LS) criterion... 

Fig. 7. Contours of sums of squares of residual rates for isooctene hydrogenation, Eq. (46).

One measure of variation about the mean of a set of data is the variance s ), defined as the sum of squares of residuals, divided by the number of degrees of... 

Although a complete proof of the following is beyond the scope of this presentaiiun, it can be shown that partial differentiation of the sum of squares of residuals with respect to the B matrix gives, in a simple matrix expression, the partial derivative of the sum of squares of residuals with respect to all of the P s. 

If this matrix of partial derivatives is set equal to zero (at which point the sum of squares of residuals with respect to each P will be minimal), the matrix equation... 

This is the general matrix solution for the set of parameter estimates that gives the minimum sum of squares of residuals. Again, the solution is valid for all models that are linear in the parameters. 

Figure 5.3 Squares of the individual residuals and the sum of squares of residuals, as functions of different values of 6g.

Figure 5.3 plots the squares of the individual residuals (rj, and nd the sum of squares of residuals (SS,), for this data set as a function of different values of feo demonstrating that = 4 is the estimate of Pq that does provide the best fit in the least squares sense. 

Equations 5.28 and 5.30 provide a general matrix approach to the calculation of the sum of squares of residuals. This sum of squares, SS divided by its associated number of degrees of freedom, DF, is the sample estimate, s, of the population variance of residuals, CJ. ... 

In general, the sum of residuals (not to be confused with the sum of squares of residuals) will equal zero for models containing a Pq term for models not containing a Po term, the sum of residuals usually will not equal zero. 

The sum of squares of residuals must also be equal to zero. 

Graph 4 ft, = 4 Graph 5 ft, = 5. Why doesn t the minimum occur at the same value of ftp in all graphs Which graph gives the overall minimum sum of squares of residuals ... 

We begin by examining more closely the sum of squares of residuals between the measured response, yi and the predicted response, (y, = 0 for all i of this model), which is given by... 

If (and only if) replicate experiments have been carried out on a system, it is possible to partition the sum of squares of residuals, SS, into two components (see Figure 6.10) one component is the already familiar sum of squares due to purely experimental uncertainty, 55. the other component is associated with variation attributed to the lack of fit of the model to the data and is called the sum of squares due to lack of fit, SS. ... 

Assume the model = 0 + r, is used to describe the nine data points in Section 3.1. Calculate directly the sum of squares of residuals, the sum of squares due to purely experimental uncertainty, and the sum of squares due to lack of fit. How many degrees of freedom are associated with each sum of squares Do and SS add up to give SS l Calculate and What is the value of the Fisher F-ratio for lack of fit (Equation 6.27)7 Is the lack of fit significant at or above the 95% level of confidence ... 

---