Leave-one-out method

Table 2 shows results of a number of models for the lateral interactions. They are obtained by a linear regression procedure. We use the leave-one-out method to see how reliable these results are. The idea is to do linear regression with... 

Van der Voet and Doornbos used this method in connection with the classification of French wines, and the result (prediction ability with the leave-one-out method) was as good as that with more sophisticated selection methods, but with a higher number of retained variables. 

The leave-one-out method is among the simplest ones to use because it requires no input parameters. Because it runs N cross-validation procedures, it can take a rather long time if N is large, depending on the processing speed of the computer. The concern about representative validation samples is usually not an issue with this method because it tests... 

Quantitative structure-activity/pharmacokinetic relationships (QSAR/ QSPKR) for a series of synthesized DHPs and pyridines as Pgp (type I (100) II (101)) inhibitors was generated by 3D molecular modelling using SYBYL and KowWin programs. A multivariate statistical technique, partial least square (PLS) regression, was applied to derive a QSAR model for Pgp inhibition and QSPKR models. Cross-validation using the leave-one-out method was performed to evaluate the predictive performance of models. For Pgp reversal, the model obtained by PLS could account for most of the variation in Pgp inhibition (R2 = 0.76) with fair predictive performance (Q2 = 0.62). Nine structurally related 1,4-DHPs drugs were used for QSPKR analysis. The models could explain the majority of the variation in clearance (R2 = 0.90), and cross-validation confirmed the prediction ability (Q2 = 0.69) [ 129]. 

The ability of the derived correlation equation to predict values can be measured with the cross-validated r value, q. Values of greater than 0.5 indicate acceptable ability to predict a q of 0.6 is considered quite respectable. Often q values are less than r values. The simplest cross-validated r value is calculated by excluding each point in turn (leave-one-out method) and using the remaining points to calculate a regression equation. The resulting values are averaged to obtain q. ... 

Fig. 1. Statistical classification strategy (SCS) a schematic road map of how the SCS method is developed for individual databases. GA ORS, genetic algorithm based optimal region selection LDA, linear discriminant analysis LOO, leave-one-out (method of cross-validation) coeff, coefficients.

If the independent sample set method is considered to be too wasteful of data, which may be expensive to obtain, and the use of the training set for validation is considered insufficiently rigorous, then the leave-one-out method can be employed. By this method all samples but one are used to derive the classification rule, and the sample left out is used to test the rule. The process is repeated with each sample in turn being omitted from the training set and used for validation. The major disadvantage of this method is that there are as many rules derived as there are samples in the data set and this can be computationally demanding. In addition, the error rate obtained refers to the average performance of all the classifiers and not to any particular rule which may subsequently be apphed to new, unknown samples. 

The JKK is a direct application of the plug-in principle. To understand the JKK, let us denote the estimator of 9 by 9, where 9 is based on a sample of size n. The JKK estimator kk, of v is defined as follows. Calculate n estimators 9(i), where for each i = 1 to n, % is obtained using the expression defining 9 eliminating the ith observation so that each 9(i) is calculated with a sample of size n - 1. Each observation is removed once from the data and the procedure of interest is carried out on the data. For this reason, the JKK is often also known as the leave-one-out method. If one now defines the mean of the 9a), / = 1,..., n, as... 

Having obtained such a classification rule, it is necessary to test the rule and indicate how good it is. There are several testing methods in common use. Procedures include the use of a set of independent samples or objects not included in the training set, the use of the training set itself, and the leave-one out method. The use of a new, independent set of samples not used in deriving... 

In the case of large data sets, the leave-one-out method can be replaced by leaving out groups of objects. Other criteria for deciding on the number of PCs or factors are introduced as follows. 

However, the instability of tree-based methods implies also here a much higher error in case of cross-validation by the simple leave-one-out method, that is, the cross-validated fraction of misclassified objects for CART is with 2.25%, 10 times higher than the resubstitution error. This error can be only reduced if ensemble methods are included in the model budding step. A bagged CART model revealed a cross-validation error of only 1.0% (Figure 5.38e). The fraction of misclassifica-tions for the cross-validated models increases for QDA, SVM, and A-NN to 5.5%, 5.0%, and 4.75%, respectively. The cross-validated classifications by LDA reveal 58.8% of misclassified objects as expected from the type of data. 

The leave-one-out method. Each data point is successively left out from the sample and used for validation. The performance is calculated for the left-out data point based on the fitted model obtained from the remaining data. The average of all iterations gives an estimate of the overall performance. This method makes maximum use of the data, but is also computationally expensive because the number of iterations is the same as the number of data points. 

An alternative procedure for an investigation of pattern recognition methods is the leave-one-out-method shown in Figure 9. At any given time... 

FIGURE 9. Leave-one-out-method for an investigation of pattern recognition methods, n total number of patterns, j running variable for pattern number (j = 1...n). 

Evaluation of the classifier with a sufficiently large prediction set is absolutely necessary C94, 3273. For small data sets the "leave-one-out-method" was proposed for an objective evaluation C443. The predictive ability is usually higher for that class which is more frequent in the training set C3273. 

Statement about the membership to a certain class (or to several classes). The quality of this answer can be estimated by the leave-one-out method (Chapter 1-4.). Each pattern of the data set is treated as unknown and classified by all other patterns. If the spectral library contains only one spectrum for each compound all classifications in the test are made with "strange spectra". Therefore, this test simulates practical conditions in which unknowns have to be classified by a data set which does not contain the unknowns. 

The actual merit of a classification method can only be judged in connection with practical problems. This merit not only depends on the classification method and problem but also on non-objective criteria like the demands and knowledges of the user. The basis of a first implementation of a classification method however has to consist of objective mathematically defined criteria which characterize the efficiency of a classifier. The efficiency of a classifier is usually estimated by an application to a random sample of known patterns (prediction set) which have not been used for the training. An alternative method is the leave--one-out method (Chapter 1.4 C2921). 

TABLE 13. K-nearest neighbour classification of 13 chemical classes from binary encoded infrared spectra tested with the leave-one-out method. The Taniraoto distance was used because it gave slightly better results than the Hamming distance. P... 

We have seen that there are three strategies for avoiding overtraining the validation set, the leave-one-out, and leave- -out methods. There appears to be no definite consensus on which method is best, although as data sets become smaller, the leave-one-out method is usually preferable, whereas as data sets become larger, the validation set method is usually preferable. For more details on these methods, consult Refs. 22, 29, 216, 217, 223, and 228. 

The prediction performance can be validated by using a cross-validation ( leave-one-out ) method. The values for the first specimen (specimen A) are omitted from the data set and the values for the remaining specimens (B-J) are used to find the regression equation of, e.g., Cj on Ay A2, etc. Then this new equation is used to obtain a predicted value of Cj for the first specimen. This procedure is repeated, leaving each specimen out in turn. Then for each specimen the difference between the actual and predicted value is calculated. The sum of the squares of these differences is called the predicted residual error sum of squares or PRESS for short the closer the value of the PRESS statistic to zero, the better the predictive power of the model. It is particularly useful for comparing the predictive powers of different models. For the model fitted here Minitab gives the value of PRESS as 0.0274584. 

The results have been evaluated using the leave-one-out method (see Example 8.9.1). The first block in the printout shows that using this method of cross-validation the number of components required to model Cj is four. The third block in the table gives the reason for this choice it shows that value of the PRESS is lowest for a four-component model, taking the value 0.0052733. (This, incidentally, is higher than it was for the PCR model.) Note that the predictive value of the model, as measured by the PRESS value, decreases if more components are added. The first column of the last block in the table gives the coefficients in the equation for this model. So the regression equation is... 

Noise in the properties (i.e., variance not related to bioactivity) degrades the forecasting performance of PLS. As the number of latent variables increases, the ratio of noise extracted compared to signal also increases, a condition that increases the risk of fitting the data with noise. Hence a proper number of latent variables must be selected, usually by a process called cross-validation. °> In cross-validation one or more compounds, randomly chosen, are excluded from the input data set and a PLS model is derived from those remaining. This model is used to forecast the potency of the temporarily excluded compounds. For each newly added latent variable, the process of exclusion-prediction is repeated until each compound is predicted once and only once. The leave-one-out method is cross-validation that omits only one compound at a time. 

Here results are presented for three different types namely Calcium Oxalate Monohydrate 80% Calcium Oxalate Dihydrate 20%(17 stones). Calcium Oxalate Monohydrate 90% Calcium Oxalate Dihydrate 10%(16 stones). Calcium Oxalate Monohydrate 60% Calcium Oxalate Dihydrate 30% Carbonate Apatite 10%(18 stones). For each texture class, the GLCM statistical, FFT and quincunx wavelet are estimated out of 112 texture samples with the leave-one-out method [23]. 

The results of the first phase classification with 8 features by using leave-one-out method were tested, and the accuracy... 

It is often necessary to include at least 50 samples in the calibration and prediction sets. Sometimes, measurement of the primary analytical data of so many samples is excessively time consuming. The number of samples can be approximately halved, at the cost of computation time, by using only one calibration set and calculating the root-mean-square error of cross validation (RMSECV), as described in Section 9.9. In general, however, it is preferable to use an independent prediction set to investigate the validity of the calibration but the leave-one-out method significantly reduces the number of samples for which primary analytical data are required. 

Then, the spectroscopic regions devoid of signals were deleted to reduce the number of data points. Factor analysis and general discriminant analysis were appUed to the data set, and a leave-one-out method was used as a cross-validation procedure. 73.1% and 24.9% of total variance were explained by the first and second canonical functions, respectively, with the signals related to glucose, maltose, fmctose, and sucrose representing the correlated variables. The model was able to correctly classify aU of the authentic honeys, and aU of the adulterated honeys were correctly misclassified. Furthermore, the method was so accurate to classify correctly the adulterated honeys in accordance with the syrup addition levels, with a prediction capacity of 90.5%. 

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