Predicted sum of squares

PRESS is the prediction sum of squares SSTotal is the total sum of squares... 

PRESS, the prediction sum of squares, is the measure for the accuracy of the prediction. It is the sum over all squared differences between cross-validation predicted and true known qualities. 

Finally, a measure of lack of fit using a PCs can be defined using the sum of the squared errors (SSE) from the test set, flSSETEST = Latest 2 (prediction sum of squares). Here, 2 stands for the sum of squared matrix elements. This measure can be related to the overall sum of squares of the data from the test set, SStest = -Xtest 2- The quotient of both measures is between 0 and 1. Subtraction from 1 gives a measure of the quality of fit or explained variance for a fixed number of a PCs ... 

Another criterion that is based on the predictive ability of PCA is the predicted sum of squares (PRESS) statistic. To compute the (cross validated) PRESS value at a certain k, we remove the ith observation from the original data set (for i = 1,. .., n), estimate the center and the k loadings of the reduced data set, and then compute the fitted value of the ith observation following Equation 6.16, now denoted as x, . Finally, we set... 

Since PLS technique is sensitive to outliers and scaling, outliers should be removed and data should be scaled prior to modeling. After data pretreatment, the number of latent variables (PLS dimensions) to be retained in the model is determined. Cumulative prediction sum of squares (CUM-PRESS) versus the number of latent variables or prediction sum of squares (PRESS) versus the number of latent variables plots are used for this purpose. It is usually enough to consider the first few PLS dimensions for monitoring activities, while more PLS dimensions are needed for prediction in order to improve the accuracy of predictions. 

One useful statistic that can be derived from the eXO is called PRESS, an acronym for prediction sum of squares. 

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... 

We can also examine these results numerically. One of the best ways to do this is by examining the Predicted Residual Error Sum-of-Squares or PRESS. To calculate PRESS we compute the errors between the expected and predicted values for all of the samples, square them, and sum them together. 

PRESS for validation data. One of the best ways to determine how many factors to use in a PCR calibration is to generate a calibration for every possible rank (number of factors retained) and use each calibration to predict the concentrations for a set of independently measured, independent validation samples. We calculate the predicted residual error sum-of-squares, or PRESS, for each calibration according to equation [24], and choose the calibration that provides the best results. The number of factors used in that calibration is the optimal rank for that system. 

Fortunately, since we also have concentration values for our samples, We have another way of deciding how many factors to keep. We can create calibrations with different numbers of basis vectors and evaluate which of these calibrations provides the best predictions of the concentrations in independent unknown samples. Recall that we do this by examing the Predicted Residual Error Sum-of Squares (PRESS) for the predicted concentrations of validation samples. 

Just as we did for PCR, we must determine the optimum number of PLS factors (rank) to use for this calibration. Since we have validation samples which were held in reserve, we can examine the Predicted Residual Error Sum of Squares (PRESS) for an independent validation set as a function of the number of PLS factors used for the prediction. Figure 54 contains plots of the PRESS values we get when we use the calibrations generated with training sets A1 and A2 to predict the concentrations in the validation set A3. We plot PRESS as a function of the rank (number of factors) used for the calibration. Using our system of nomenclature, the PRESS values obtained by using the calibrations from A1 to predict A3 are named PLSPRESS13. The PRESS values obtained by using the calibrations from A2 to predict the concentrations in A3... 

The Predicted Residual Error Sum of Squares (PRESS) is simply the sum of the squares of all the errors of all of the samples in a sample set. 

The first equation shows that the data and model predictions are compared at the same values of the (nominally) independent variables. The second equation explicitly shows that the sum-of-squares depends on the parameters in the model. 

The computation result yield acetaldehyde concentration as function of time. The value of kinetics parameters, ki, ka, k3 were adjusted to minimize the sum of square of error between the predicted and measured concentration using Hooke Jeeve method [3]. 

The least squares criterion states that the norm of the error between observed and predicted (dependent) measurements 11 y - yl I must be minimal. Note that the latter condition involves the minimization of a sum of squares, from which the unknown elements of the vector b can be determined, as is explained in Chapter 10. 

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... 

The g-statistic or square of predicted errors (SPE) is the sum of squares of the errors between the data and the estimates, a direct calculation of variability ... 

SSR = sum of squared residuals N = number of observations C° = mean of the observed drug concentration C = predicted drug concentration rii = number of experimental repetitions S2 = variance of the observed concentrations at each data point LL = log likelihood function... 

Another measure for the precision of multivariate calibration is the so-called PRESS-value (predictive residual sum of squares, see Frank and Todeschini [1994]), defined as... 

Although we cannot clearly determine the reaction order from Figure 3.9, we can gain some insight from a residual plot, which depicts the difference between the predicted and experimental values of cA using the rate constants calculated from the regression analysis. Figure 3.10 shows a random distribution of residuals for a second-order reaction, but a nonrandom distribution of residuals for a first-order reaction (consistent overprediction of concentration for the first five datapoints). Consequently, based upon this analysis, it is apparent that the reaction is second-order rather than first-order, and the reaction rate constant is 0.050. Furthermore, the sum of squared residuals is much smaller for second-order kinetics than for first-order kinetics (1.28 X 10-4 versus 5.39 xl0 4). 

Although equation (A) can be integrated analytically (resulting in equations 3.4-9 and 3.4-10), for the sake of illustration we presume that we must integrate equation (A) numerically. For a given kA and n, numerical integration of Equation (A) provides a predicted cA(f) profile which may be compared against the experimental data. Values of kA and n are adjusted until the sum of squared residuals between the experimental and predicted concentrations is minimized. 

If we consider the case where K treatments are being compared such that K = 1,2,K, and we let Xik and Yik represent the predictor and predicted values for each individual i in group k, we can let Xk and Yk be the means. Then, we define the between-group (for treatment) sum of squares and cross products as... 

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