Apparent error rate

Two methods are used to evaluate the predictive ability for LDA and for all other classification techniques. One method consists of dividing the objects of the whole data set into two subsets, the training and the prediction or evaluation set. The objects of the training set are used to obtain the covariance matrix and the discriminant scores. Then, the objects of the training set are classified, so obtaining the apparent error rate and the classification ability, and the objects of the evaluation set are classified to obtain the actual error rate and the predictive ability. The subdivision into the training and prediction sets can be randomly repeated many times, and with different percentages of the objects in the two sets, to obtain a better estimate of the predictive ability. 

To judge the performance of the discriminant functions and the classification procedure in respect of future samples one can calculate misclassification probabilities or error rates. But these probabilities cannot be calculated in general because they depend on the unknown density functions of the classes. Instead we can usually utilize a measure called apparent error rate. The value of this quantity is easily calculated from the classification or confusion matrix based on the samples of the training set. For example with two classes we can have the following matrix ... 

With the number nwrong of misclassified samples from the training set the apparent error rate, A HR, is ... 

The results of classification techniques examined in this chapter will be assessed by their apparent error rates using all available data for both training and vahdation, in line with most commercial software. 

Of interest here is how well a model will predict some future response in some external or new data. One may look at the average residual error in the data set from which the prediction rule was estimated, called the apparent error rate. However, this estimate of the residual error (apparent error here) will be too optimistic and will underestimate the true prediction error. The problem here is that the training and assessment sample are the same. CV is used to correct this underestimation of the apparent error rate. 

The double bootstrap was a method originally suggested by Efron (15) as a way to improve on the bootstrap bias correction of the apparent error rate of a linear discrimination rule. It is simply a bootstrap iteration (i.e., taking resamples from each bootstrap resample). The double bootstrap has been useful in improving the accuracy of confidence intervals but it substantially increases computation time and most likely increases the incidence of unsuccessfully terminated runs. It has been applied to linear models but not to PM modeling. 

Section II,B, this may be an overestimate of the false positive error rate because many of the apparent consecutive errors correspond to regions of disorder that are ordered in the crystal due to ligand binding or crystal contacts. Also, because disordered regions of length >40 residues are often missed due to false negative predictions of order, the data in... 

Tests at900°C. The long period or equilibrium apparent leak rates of both the mullite and zircon double-walled furnace units are of the order of the limit of error of the author s measuring method, i.e., 1.7 to 3.4 X 10-9 l.-mm. of mercury per second, or 2 to 4 X 10-9 cc. at N.T.P. per second. These values were observed after 15 to 24 hours of pumping and after exposure to a dry gas atmosphere with the zirconium specimen removed. Control experiments were made without the furnace tube present. 

Assessment of the fibril formation process involved measuring the lag preceding the onset of fluorescence increase (the nucleation process) and the apparent maximal rate of fluorescence increase (rate of fibril formation) for 300-600 min, depending on the duration of the lag phase. Mean standard error and the kinetic parameters were calculated using Kaleidagraph Software (Reading, PA, USA). 

Kinetics of the propagation reactions were followed by dllato-metry under high vacuum, at 20"C, at various living ends concentrations, with nearly the same value of [M]q (0.5 mole. 1 1) (10,11,17). Some experiments were performed In toluene Instead of benzene and In some cases, tertiary butylllthlum was used as Initiator Instead of n-butylllthlum. The results were nearly the same within the experimental errors. The values of the apparent propagation rate constant kp Rp/[M]x[C] were determined for each... 

We utilized both the leave-one-out bootstrap as well as the 0.632+ alternative to estimate the error rate of k-NN in the lymphoma dataset. In Table 10.5, LOOB s overestimation of the error rate is quite obvious compared to the previous methods. The 0.632+ estimator correction is also apparent. Both have a significantly lower value for k compared with other methods. A k-value of 1 indicates that the best method of predicting the l5miphoma subtype for a given observation is to look at the subt5q)e of the one closest observation in terms of the genetic expression values. 

Melt rheometers either impose a fixed flow rate and measure the pressure drop across a die, or, as in the melt flow indexer, impose a fixed pressure and measure the flow rate. Equation (B.5) gives the shear stress, but Eq. (B.IO) requires knowledge of n to calculate the shear strain rate. It is conventional to plot shear stress data against the apparent shear rate y, calculated using n = 1 (assuming Newtonian behaviour). If the data is used subsequently to compute the pressure drop in a cylindrical die, there will be no error. However, if a flow curve determined with a cylindrical die is used to predict... 

Equation 4 was discretised by a 5-point central difference formula and thereafter first-order differential equations 1, 2, 4 and 6 were solved by a backward difference method. Apparent reaction rate was solved by summing the average rates of each discretisation piece of equation 4. The reactor model was integrated in a FLOWBAT flowsheet simulator [12], which included a databank of thermodynamic properties as well as VLE calculation procedures and mathematical solvers. The parameter estimation was performed by minimising the sum of squares for errors in the mole fractions of naphthalene, tetralin and the sum of decalins. Octalins were excluded from the estimation because their content was low (<0.15 mol-%). Optimisation was done by the method of Levenberg-Marquard. 

The rate expression from Catalytic Steam Reforming (1984), modified to include the Boudart interpretation of the hydrogen site versus hydrocarbon site difference, is the relationship used in the model. The apparent error in the 2n exponent of the denominator has been corrected, and an exponent of 2 is used. 

Mishaps are assumed by many to be stochastic events, that is, random, haphazard, unpredictable. However, a mishap is more than a random unplanned event with an unpredictable free will. Mishaps are not events without apparent reason they are the result of actuated hazards. Hazards are predictable and controllable and they occur randomly based on their statistical predilection, which is typically controlled by a failure or error rate. Thinking of a mishap as a chance event without justification gives one the sense that mishaps involve an element of destiny and futility. System safety, on the other hand, is built upon the premise that mishaps are not just chance events instead they are seen as deterministic, predictable, and controllable events (in the disguise of hazards). 

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