Error predicted values

Time GM Predictive value The error predictive value The final predictive value T statistic... 

Residual error in linear regression, where the filled circle shows the experimental value/, and the open circle shows the predicted value/,. 

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. 

For the purposes of this section, error is simply the difference between the value of the y variable predicted by a regression and the true value (sometimes called the expected value). Naturally, it is impossible to know the true value, so we are forced to settle for using the best available referee value for the y variable. (Note it is possible that the "best available referee values" can have larger errors than the predicted values produced by the calibration.) We will follow the common convention and name the expected value of the variable y and the predicted value of the variable y, pronounced "Y-hat." Then the error is given by p -y. We will also denote the number of samples in a data set by the letter n. 

Pressure drop and heat transfer in a single-phase incompressible flow. According to conventional theory, continuum-based models for channels should apply as long as the Knudsen number is lower than 0.01. For air at atmospheric pressure, Kn is typically lower than 0.01 for channels with hydraulic diameters greater than 7 pm. From descriptions of much research, it is clear that there is a great amount of variation in the results that have been obtained. It was not clear whether the differences between measured and predicted values were due to determined phenomenon or due to errors and uncertainties in the reported data. The reasons why some experimental investigations of micro-channel flow and heat transfer have discrepancies between standard models and measurements will be discussed in the next chapters. 

Test of Model Adequacy. The final step is to test the adequacy of the model. Figure 4 is a plot of the residual errors from the model vs. the observed values. The residuals are the differences between the observed and predicted values. Random scatter about a zero mean is desireable. 

If we believe that D = diag(u), that is, that errors are independent and proportional to the square root of the predicted value, then D" = diag(l/M )/o, where we may further approximate this result by estimating... 

The NO2 molecule offers an example which illustrates this point. The spectrum of N02 molecules rigidly held on MgO at —196° is characterized by gxx = 2.005, gyv = 1.991, and gzz = 2.002 (29). If this molecule were rapidly tumbling, one would expect a value of Qa.v — 1 999. The spectrum of NO2 absorbed in a 13X molecular sieve indicates an isotropic gzv = 2.003 (.80), which is within experimental error of the predicted value for NO2 on MgO. The hyperfine constants confirm that NO2 is rapidly tumbling or undergoing a significant libration about some equilibrium position in the molecular sieve (81). 

The basis for this calculation of the amount of nonlinearity is illustrated in Figure 67-1. In Figure 67-la we see a set of data showing some nonlinearity between the test results and the actual values. If a straight line and a quadratic polynomial are both fit to the data, then the difference between the predicted values from the two curves give a measure of the amount of nonlinearity. Figure 67-la shows data subject to both random error and nonlinearity, and the different ways linear and quadratic polynomials fit the data. 

Near-infrared spectroscopy is quickly becoming a preferred technique for the quantitative identification of an active component within a formulated tablet. In addition, the same spectroscopic measurement can be used to determine water content since the combination band of water displays a fairly large absorption band in the near-IR. In one such study [41] the concentration of ceftazidime pentahydrate and water content in physical mixtures has been determined. Due to the ease of sample preparation, near-IR spectra were collected on 20 samples, and subsequent calibration curves were constructed for active ingredient and water content. An interesting aspect of this study was the determination that the calibration samples must be representative of the production process. When calibration curves were constructed from laboratory samples only, significant prediction errors were noted. When, however, calibration curves were constructed from laboratory and production samples, realistic prediction values were determined ( 5%). 

Standard error of prediction Standard deviation of the predicted value obtained from linear regression. 

From Fig. 2.28 it is obvious that a reasonable fit is easily obtained. A detailed analysis of the results, however, disdoses that there seems to be a systematic deviation in the residuals with high predicted values at the equilibrium conditions (TIME > 1000) and a low prediction between TIME around 50 and 150. These differences can be caused by an inadequacy in the model or by systematic experimental errors. A more appropriate objective function may also be desirable. 

Why did the posterior probability increase so much the second time One reason was that the prior probability was considerably higher in the second calculation than in the first (27% versus 2%), based on the fact that the first test yielded positive results. Another reason was that the specificity of the second test was quite high (98%), which markedly reduced the false-positive error rate and therefore increased the positive predictive value. 

Table 3. Experimental and predicted values of the plait point and the percentage of relative error...

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