Square-error

The problem with plant data becomes more significant when sampling, instrument, and cahbration errors are accounted for. These errors result in a systematic deviation in the measurements from the actual values. Descriptively, the total error (mean square error) in the measurements is... 

The root-mean-square error is the square root of the mean square error. Note that since the root-mean-square error involves the square of the differences, outliers have more influence on this statistic than on the mean absolute error. 

Which measure of scatter is likely to be larger, the mean absolute error or the root-mean-square error ... 

The unknown parameters of the model, such as film thicknesses, optical constants, or constituent material fractions, are varied until a best fit between the measured P and A and the calculated P/ and A/ is found, where m signifies a quantity that is measured. A mathematical function called the mean squared error (MSE) is used as a measure of the goodness of the fit ... 

If the performance index or cost function J takes the form of a summed squared error function, then... 

This argument obviously can be generalized to any number of variables. Equation (2-65) describes the propagation of mean square error, or the propagation of variances and covariances. 

The root-mean-square error (RMS error) is a statistic closely related to MAD for gaussian distributions. It provides a measure of the abso differences between calculated values and experiment as well as distribution of the values with respect to the mean. 

The number of neurons to be used in the input/output layer are based on the number of input/output variables to be considered in the model. However, no algorithms are available for selecting a network structure or the number of hidden nodes. Zurada [16] has discussed several heuristic based techniques for this purpose. One hidden layer is more than sufficient for most problems. The number of neurons in the hidden layer neuron was selected by a trial-and-error procedure by monitoring the sum-of-squared error progression of the validation data set used during training. Details about this proce-... 

A number of modifications to eliminate some less favorable aspects of the Levenberg-Marquardt method were considered by Fletcher. For instance, the arbitrary initial choice of the adjustable parameter A, if poor, can cause an excessive number of evaluations of squared error, before a realistic value is obtained. This is especially noticeable if v, i.e., J R x), is chosen to be small, i.e., v = 2. Another disadvantage of the method is that the reduction of A to v at the start of each iteration may also cause excessive evaluations, especially when V is chosen to be large, i.e., = 10. The... 

Figure 16 Root-mean-squared error progression plot for Fletcher nonlinear optimization and back-propagation algorithms during training.

The Mean Squared Error of Prediction (MSEP) is supposed to refer uniquely to those situations when a calibration is generated with one data set and evaluated for its predictive performance with an independent data set. Unfortunately, there are times when the term MSEP is wrongly applied to the errors in predicting y variables of the same data set which was used to generate the calibration. Thus, when we encounter the term MSEP, it is important to examine the context in order to verify that the term is being used correctly. MSEP is simply PRESS divided by the number of samples. 

It is quite a simple matter to generalize the simple prediction problem just discussed to the situation where we want to obtain the best (in the sense of minimum mean square error) linear estimate of one random variable fa given the value of another random variable fa. The quantity to be minimized is thus... 

For the data the squared correlation coefficient was 0.93 with a root mean square error of 2.2. The graph of predicted versus actual observed MS(1 +4) along with the summary of fit statistics and parameter estimates is shown in Figure 16.7. 

Root mean square error Mean of response Observations (or sum wt)... 

Figure 9. Linearity of response and reproducibility. The error flags indicate the root mean square error for five measurements at each value. The average relative error is about 10%.

Fig. 14. Calcium response of Sf-9 insect cells subjected to different values of e in a stirred bioreactor equipped with a 5.1 cm diameter 6-bladed Rushton impeller (closed circles) or in the capillary flow system (open squares). Error bars for stirred bioreactor are standard deviation for each experiment but for the capillary, data are hard to discern [99]...

---