Sum of squares for residuals

Comments The cross-validation algorithm is performed identically to PRESS with the exception that rather than sum of squares for residual as the reported results, the cross-validation uses the square root of the mean squares for residuals, using N — I degrees of freedom for each model as ... 

Van der Voet [21] advocates the use of a randomization test (cf. Section 12.3) to choose among different models. Under the hypothesis of equivalent prediction performance of two models, A and B, the errors obtained with these two models come from one and the same distribution. It is then allowed to exchange the observed errors, and c,b, for the ith sample that are associated with the two models. In the randomization test this is actually done in half of the cases. For each object i the two residuals are swapped or not, each with a probability 0.5. Thus, for all objects in the calibration set about half will retain the original residuals, for the other half they are exchanged. One now computes the error sum of squares for each of the two sets of residuals, and from that the ratio F = SSE/JSSE. Repeating the process some 100-2(K) times yields a distribution of such F-ratios, which serves as a reference distribution for the actually observed F-ratio. When for instance the observed ratio lies in the extreme higher tail of the simulated distribution one may... 

In general, the subset of m variables providing the smallest residual sum of squares does not necessarily contain the subset of (m - 1) variables that gives the smallest residual sum of squares for (m - 1) variables. 

Let RSSm denote the residual sum of squares for a model with m variables and an intercept term, Suppose the smallest RSS that can be obtained by adding another variable to the present set is RSS , . The calculated ratio R according to... 

What are the residual sum of squares for the V and V blocks as each successive component is computed (hint start from the centred data matrix and simply sum the squares of each block, repeat for the residuals) What percentage of the overall variance is accounted for by each component ... 

Table IV shows the overall analysis of variance (ANOVA) and lists some miscellaneous statistics. The ANOVA table breaks down the total sum of squares for the response variable into the portion attributable to the model, Equation 3, and the portion the model does not account for, which is attributed to error. The mean square for error is an estimate of the variance of the residuals — differences between observed values of suspensibility and those predicted by the empirical equation. The F-value provides a method for testing how well the model as a whole — after adjusting for the mean — accounts for the variation in suspensibility. A small value for the significance probability, labelled PR> F and 0.0006 in this case, indicates that the correlation is significant. The R2 (correlation coefficient) value of 0.90S5 indicates that Equation 3 accounts for 91% of the experimental variation in suspensibility. The coefficient of variation (C.V.) is a measure of the amount variation in suspensibility. It is equal to the standard deviation of the response variable (STD DEV) expressed as a percentage of the mean of the response response variable (SUSP MEAN). Since the coefficient of variation is unitless, it is often preferred for estimating the goodness of fit.

The variance can be analysed now into between source of raw material (R), between suppliers (S), and the R X S interaction, and the two further components shown below in Table 11.25. The degrees of freedom and sums of squares for the new G term are obtained by pooling them together with the G x S, S x R and G X S X R terms. The Residual is the same as in Table 11.24. 

The analysis of variance proceeds exactly as in the previous section, except that now there is a further item derived exactly in the same way as the first three in Table 12.9, again with (n — 1) degrees of freedom. The Sum of Squares for the Residual, obtained by difference as before, now has (n — 1) (n — 3) degrees of freedom. 

Since GSA requires an estimate of the magnitude of cost function response at the global optimum, an estimate of the residual sum of squares for the best fit was needed. Based on work with similar data and some trial and error, a value of le-4 was found to give good results. 

A plot similar to a scree plot can be made for PARAFAC, by plotting the sum of squares of the individual components. However, in this case, cumulative plots cannot be made directly because the variances of the individual factors are not additive due to the obliqueness of the factors. Furthermore, the sum of squares of the one-component model may not equal the size of the largest component in a two-component model. Hence, the scree plot is not directly useful for PARAFAC models. The cumulative scree plot for PARAFAC models, on the other hand, can be constructed by plotting the explained or residual sum of squares for a one-component model, a two-component model, etc. This will provide similar information to the ordinary two-way cumulative scree plot, with the exception that the factors change for every model, since PARAFAC is not sequentially fit. The basic principle is retained though, as the appropriate number of components to use is chosen as the number of components for which the decrease in the residual variation levels off to a linear trend (see Example 7.3). 

Figure 8.41. Squared Mahalanobis distance plotted against residual sum of squares for the treatments in the peat example.

Figure 1.2 Scatter plots of simulated concentration-time data ( ) and fitted models (solid lines). Data were simulated using a 2-term polyexponential model and were fit with inverse order polynomials up to degree four and to a 2-term exponential equation. The residual sum of squares for each model is shown in Table 2.

Fit separate OLS fits to the two datasets and compute the residual sum of squares for each group and... 

This indicates that the residual sum of squares for the individual is near zero. Check the initial value of the residual error and see if it is small. If so, increase its value. Alternatively, try fitting the logarithms of the concentration with an additive residual error model. 

Each group is successively left out and predicted in the same manner as just described. The predicted residual error sum of squares (PRESS) is computed from all the predictions. The PRESS value is compared with the sums of squares for the dependent variable y (SSY) ... 

Table 4.5 Total and Residual Sums of Squares for the 2 Design of Table 4.2...

Of the total adjusted sum of squares, 28% is accounted for by the first-order terms alone, and 75% by all the terms of the second-order model, first-order, square, and interaction. The residual sum of squares of the first order model includes the sum of squares for the second-order terms. So the difference between the residual sums of squares for the two models is the second-order regression sum of squares... 

The residual and total sum of squares for the testing data are defined, respectively to be... 

We can apply the general strategy outlined above using so-called type I sums of squares. For type I sums of squares, SAS calculates the residual variability for a given factor having adjusted for all other factors previously specified by the model. The order in which the effects are specified is important so that, for example, if we write ... 

This section illustrates application of the PRESS statistic for process model structure selection. Ljung (1987) used data collected from a laboratory-scale Process Trainer to illustrate various identification techniques and examined the sum of squared conventional residuals and Akaike s information theoretic criterion (AIC) for model structure selection. Two sets of input-output data collected from this process are available within MATLAB. We use the entire first set of data M = 1000), called DRYER2 in MATLAB, for this study. Two different model structures are examined here, namely the ARX and FIR model structures, with the objective to find the model within a pcurtic-ular structure that produces the smallest PRESS. 

It can also be calculated by adding the sum of squares for the regression and residual, which results in... 

Besides plotting the sum of squares for each sample, sometimes it can be more useful also to plot, object-wise, the whole vector of residuals, in order to evidence the presence of unmodelled systematic structure (especially when the variables are homogeneous), or to identify blocking effects. Indeed, when there are sources of systematic variation that are not explained by the model, the residuals for that particular sample are not anymore randomly distributed and present a structured shape. As an example, it is possible to consider the case where the UV spectrum of a solution is used to predict the concentration of an analyte in the presence of a possible interferent. Figure 4A shows the residual vector for 20 samples in which only the analyte and the known interferent are present as expected, for each object, the residuals are randomly... 

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