ngineering 53 (2006) 149157www.elsevier.com/locate/petrolJournal of Petroleum Science and EFacies identification from well logs: A comparison of discriminantanalysis and nave Bayes classifier

Yumei Li , Richard Anderson-Sprecher

University of Wyoming, Laramie, WY 82071-3332, USA

Received 19 March 2005; received in revised form 1 June 2006; accepted 6 June 2006Abstract

The performance of a nave Bayes classifier is compared with a well-established statistical classification approach, lineardiscriminant analysis, by considering core and log data from marineeolian sediments. The results indicate that both methodsperform adequately, and the Gaussian nave Bayes classifier provides estimates as good as those based on the linear discriminantanalysis for the given data set. Quadratic discriminant analysis, a more conventional Bayesian analysis, and kernel-based densityestimation methods perform unexpectedly poor, probably because of overfitting. We conclude that the normal distribution isappropriate to fit the distribution of log readings in the present data, and the simplifications of nave Bayes provide a robust, simpleapproach for facies identification. 2006 Elsevier B.V. All rights reserved.Keywords: Facies; Well logs; Discriminant analysis; Nave Bayes classifier1. Introduction

Facies identification is important in oil explorationand development because facies often control the var-iation of petrophysical properties. Identification offacies is generally based on core samples and outcropcharacteristics. Because available core and outcrop areusually limited, establishing relationships betweenfacies and more readily available data sources, in par-ticular well logs, is highly desirable.

Some efforts have been made to use statistical me-thods such as discriminant analysis (Sakurai andMelvin,1988; Avseth et al., 2001; Tang et al., 2004) to identify0920-4105/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.petrol.2006.06.001

Corresponding author.E-mail address: liyumei@uwyo.edu (Y. Li).facies from well logs. The past decade has also seenapplications of Artificial Neural Network (ANN) (Dereket al., 1990; Wong et al., 1995; Siripitayananon et al.,2001; Bhatt and Helle, 2002) and fuzzy logic (Cuddy,2000; Saggaf and Nebrija, 2003) in facies classification.Initial successes of ANN for facies prediction haveinspired enthusiasm, leading to claims that it has thepotency to dominate or take over other analytical toolsused in the exploration and production industry(Iloghalu, 2003). However, the reliable use of neuralnetworks requires experience for adjusting parametersand a large amount of training time, especially for largedata sets (Wong et al., 1995; Avseth et al., 2001).

All methods use a training data set consisting ofobserved cases with full information about both pre-dictors (in our application, well-log readings) and groups(in our case, facies). Based on the training data set, one

http://dx.doi.org/10.1016/j.petrol.2006.06.001mailto:liyumei@uwyo.edu

150 Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157creates a rule (called a classifier) by which futureobservations of predictors can be used to infer probablegroup memberships. The ideal classifier would be easyto implement and would give reliable results. Amongstatistical classificationmethods, discriminant analysis isrobust and powerful (Wong et al., 1995; Avseth et al.,2001). Like other statistical methods, discriminant anal-ysis does, however, need a large training data set (>100cases in the training set). The classification rule max-imizes the separation of the pre-defined groups in themulti-dimensional space formed by variables or pre-dictors in the training set. It is widely accepted that thesuccess of discriminant analysis depends on the validityof certain statistical assumptions such as multivariatenormality and homogeneity (Wong et al., 1995).However, experience shows that the technique is fairlyrobust when data size is adequate (at least 20 cases in thesmallest group in the training set) and when there arerelatively few (five or fewer) predictors (Tabachnick andFidell, 1996).

Bayesian classifiers provide another alternative. Con-ceptually, a Bayesian approach to classification isappealing because it allows one to incorporate knowninformation or expert opinions, it explicitly leads to pro-babilities of new cases falling into different classes, and itis easily updated as new information is obtained.Multivariate Bayesian analyses are sometimes problem-atic, however, in that computations may be difficult andmodeling multiple correlations between variables ispotentially both delicate and unwieldy. To take advantageof positive aspects of the Bayesian approach while av-oiding some of the negative aspects, a modified Bayesianmethod, known as nave Bayes, is gaining acceptance.Nave Bayes is easy to implement, and is thus appealing,provided that it gives good results. At first glance theapproach seems dubious because it assumes indepen-dence, and in many settings proper treatment of cor-relations is known to be important for good inference. Fornave Bayes, however, the impact of this simplification isoften surprisingly small and early experience with naveBayes suggests that it may give facies predictions that areat least as accurate as those from neural networks withoutthe burden of lengthy training required by neural networks(Kapur et al., 2000).

Little work has been done on nave Bayes for faciesidentification, probably for three reasons: First, thechoice of prior probability distribution can greatly affectclassification results; although prior probabilities areused in other classification methods, including discrim-inant analysis, the problem of priors is particularly as-sociated with Bayesian methods. Prior informationoriginates from local geological knowledge. In hetero-geneous formations like fluvial deposits, the prior dis-tribution may change from one well to another (Coudertet al., 1994). The heterogeneity of deposits makes thechoice of prior a challenge. Second, probabilities re-quired by a fully Bayesian method are hard to obtain formore than one predictor. The nave Bayes classifierassumes independence among predictors, but well logsare often dependent. It is not clear whether violation ofthe independence assumption will affect the facies class-ification. Third, it is still unknown what distributions areappropriate to fit different log readings and how differentdistributions affect the facies prediction. Kapur et al.(2000) discretized values of predictor variables and useda counting rule to calculate probabilities. They empha-sized the importance of picking appropriate bin sizes: Iftoo few bins are selected, the FOP (facies occurrenceprobability) lacks the ability to discriminate betweenadjacent log readings. If there are toomany bins, the FOPwill not be estimated precisely.

This study evaluates the performance of discriminantanalysis and a normal-based nave Bayes classifier infacies identification from well logs by applying the log-facies correlation derived from the training set in threehold out wells.

2. Methodology

2.1. Nave Bayes classifier

Bayes theorem aims to determine the conditionalprobability of parameter values given the data by com-bining expectations based on previous experience (priorprobabilities) with information from available data. Inthis study, Bayes theorem is used to calculate the pro-bability of the occurrence of a certain facies given thewell-log readings and to assign the facies of the highestposterior probability to that observation depth.

The application of Bayes theorem in facies classifi-cation can be written as follows:

P fjjX x P fjPX xj fjPX x 1

Here P( fj|X=x) is the posterior probability of the jthfacies fj given that a random log reading X is equal to x;P( fj) is the probability of the jth facies obtained fromprevious experience or from our initial belief of thefacies distribution before we have observed any data;and P(X=x| fj) is the conditional probability density fora random log reading x given the occurrence of the jthfacies fj. P(X=x) is the probability density for a random

151Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157log reading x, without conditioning on the facies. Wepredict that a new case X will come from the facies fjthat achieves the highest posterior probability. If thereare n well logs (X1, X2, X3, , Xn), then the aboveformula can be modified as:

P fjjX1 x1;X2 x2; :::;Xn xn P fj

PX1 x1;X2 x2; :::;Xn xnj fjPX1 x1;X2 x2; :::;Xn xn

2

By assuming independence among well logs givencertain types of facies, we get what is called a naveBayes or simple Bayes classifier given by:

PfjjX1 x1;X2 x2; :::;Xn xn

P fj

Yni1

PXi xij fj

Pmj1

P fj Yni1

PXi xij fj3

where m is the number of facies.The above posterior probability is computed for each

facies and the prediction is made for the facies associatedwith the largest posterior probability. This classificationrule requires preliminary knowledge of univariate pro-bability distributions of well logs, which can be extractedfrom training data for each facies. Note that Eq. (3)differs from Eq. (2) in that Eq. (3) treats values of welllogs as though they were independently distributed.

The nave Bayes classifier is simple and computa-tionally efficient. The independence assumption simpli-Fig. 1. Location of seven wells in Teapot dfies the classification task dramatically by allowing theconditional densities to be calculated separately for eachwell log. Although the independence assumption isalmost certainly violated, the classifier has been shown tobe robust to the violation of independence in classificationand to exhibit surprisingly good performance in manydomains that contain clear attribute dependences (Clarkand Niblett, 1989; Langley et al., 1992). A goal of thepresent study is to see whether facies identification is oneof these domains.

2.2. Probability density estimation

Normal probability distributions are often assumedfor data in practical situations. In this study, we assumelog readings x (or, in some cases, natural logarithms oflog readings) given a certain facies f are normallydistributed, with a probability density function givenby:

Pxj f 1ffiffiffiffiffiffiffiffiffiffi2kr2f

q e 1

2r2f

xuf 2 4

where f2 is the variance of log readings given facies f,

and f is the mean of log readings given facies f.Estimates of parameters in the above probability den-sity function can be derived from the training set.Estimation of parameters in Eq. (4) was done usingstandard unbiased univariate estimators, the samplemean for and the sample variance for 2. Thesample mean is both the maximum likelihoodestimator and the least squares estimator. The sampleome, Powder River Basin, Wyoming.

Fig. 2. The matrix plot of GR, NPHI, RHOB and LOGRT shows moderately strong pairwise correlations among NPHI, RHOB and LOGRT.

152 Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157variance is slightly larger than the maximum likeli-hood estimator of variance, and in this situation eitherthe sample variance or the MLE may be used with littleFig. 3. Boxplots of GR, NPHI, RHOB and LOGRT grouped by facies show ththat the most discriminating individual well logs are RHOB and LOGRT.difference in results. Notice that correlations need notbe estimated because they do not enter into the naveBayes approach to classification.at overlap of well-log responses is common among the five facies, and

Table 1Facies description of Upper Tensleep Formation, Wyoming

Facies Description Frequency

Sand dune Fine- to medium-grained sandstone 160High-angle cross-bedding

Interdune Siltstone to very fine-grained sandstone 200Burrowed, crinkly laminations

Sand sheet Dolomitic sandstone 110Horizontal or low angle laminations

Shallowmarine

Dolomite or sandy dolomite 38Massive, fossil (crinoids)

Sabkha Dolomite, vugs (molds after evaporitecrystals) and fractures

85

153Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157An alternative approach, which we also consider, isto use a nonparametric estimate of the density for eachfacies based on kernel density estimation (KDE). InKDE the density function is approximated by the super-position of a set of kernels (Kraaijveld, 1996). As inmost applications, a particularly popular choice, theGaussian or normal kernel was used (Duda and Hart,1973; Specht, 1990). In keeping with nave Bayes, weapplied univariate kernel density estimation to evaluateFig. 4. The nave Bayes posterior probabilities, LDA-predicted facies, and of5=SS, LDA=linear discriminant analysis). For clarity, probabilities of classi5 for linear discriminant analysis (LDA). Probabilities give more detailedProbability curves indicate uncertainty in identification. See also Fig. 5.conditional probability densities given different types offacies. A program written in Matlab estimated the kerneldensity, and the optimal bandwidth for kernel densityestimates (the default bandwidth in Matlab) was cal-culated on the basis of estimated integrated squarederror (Martinez and Martinez, 2002).

2.3. Discriminant analysis

Discriminant analysis may take the form of eitherlinear or quadratic discriminant analysis. Both formsassume each well log and their linear combinations arenormally distributed for each facies, an assumption that isseldom true in practice. Linear discriminant analysisadditionally assumes homogeneity of the variancecovariance structures for the different classes (facies).This assumption is also violated for the given data setaccording to Box's M test. Violation of the homogeneityassumption may lead to overclassification, which meanscases tend to be assigned to facies with higher variancedue to higher posterior probability. Tabachnick and Fidell(1996) recommend quadratic discriminant analysis as analternative to avoid overclassification. However, due tobserved facies columns of well 55 (f1=SD, f2=ID, f3=SM, f4=SB,fication are split into two figures, Fig. 4 for nave Bayes (BAY) and Fig.information than class identification (the highest probability class).

Table 2Classification results of linear discriminant analysis in well 55

Observed Predicted Percent

SD ID SM SB SS correct

SD 8 9 0 0 0 47.1%ID 0 66 0 0 2 97.1%SM 0 4 8 2 2 50%

Overall percent correct: 81.2%.

Fig. 5. The posterior probability, observed facies and BAY-predicted facies columns of well 55 (f1=SD, f2=ID, f3=SM, f4=SB, f5=SS,BAY=nave Bayes classifier). See also Fig. 4. The agreement between nave Bayes and LDA is close. Both methods locate the economicallyimportant stratum f1 but identify a narrower band of f1 than is actually present. F3 is erratically identified by LDA, with similar but slightly superiorperformance by nave Bayes. The dominant facies f2 is identified by both nave Bayes and LDA, although other facies are sometimes labeled as f2 byboth nave Bayes and LDA.

154 Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157overfitting, quadratic discriminant analysis can have poorclassification capability in hold out data sets, especiallywhen a hold out data distribution deviates far from thetraining set distributions.

The steps of a discriminant analysis may be sum-marized as: (1) create discriminant functions from thetraining set; (2) use discriminant functions to calculatediscriminant scores; (3) convert discriminant scores toMahalanobis distances and associated posterior proba-bilities; (4) classify observations to the facies associatedwith highest posterior probability. We performeddiscriminant analysis using the statistical program SPSS.

2.4. Cross-validation

Cross-validation evaluates classification performanceby using two independent samples of data, one to learn therule and another to test it. In this study, seven wells (Fig. 1)were selected on the basis of stratigraphic and geographiccoverage, availability of appropriate well logs, andavailability of core analysis data. Due to limited data andlimited facies types in some wells, instead of leaving out arandomly selected well as the test set, the hold out well waschosen so that there would be enough data for each type offacies in the rest wells (i.e., the training set). Three wells(51, 55, 56) were held out respectively as test sets to studythe consistency of the two classification methods. Multipleanalyses are performed: for each analysis one well is heldout as a test set from the beginning and otherwells are takenas the training set.

3. Results and analyses

The geological data, consisting of 593 core readingsand log signatures, were obtained from seven wells in theUpper Tensleep Formation in Teapot Dome, Powder

Table 3Classification results of nave Bayes classifier in well 55

Observed Predicted Percent

SD ID SM SB SS correct

SD 5 12 0 0 0 29.4%ID 0 60 0 0 8 88.2%SM 0 4 8 4 0 50.0%

Overall percent correct: 72%.

Fig. 7. Comparison of discriminant analysis and the nave Bayesanalysis suggests that both approaches perform consistently in thethree analyzed wells. (LDA = linear discriminant analysis, BAY =nave Bayes classifier).

155Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157River Basin, Wyoming (Fig. 1). In the Powder RiverBasin, the 150-foot-thick Upper Tensleep Formation atdepth 53005800 ft is composed of eolianmarine se-quences, featured by sandstones, dolomitic sandstones,sandy dolomite, and dolomite.

The well-log data consist of gamma-ray (GR),neutron porosity (NPHI), formation density (RHOB),and deep resistivity (LLD). The resistivity data are log-normally distributed, so a natural log transform of thesedata was taken and designated LOGRT. Among the fourwell logs, the variables NPHI, RHOB, and LOGRTshowmoderately strong pairwise correlations with each other(Fig. 2).

Different facies have different responses in well logs,but overlap of well-log responses is very common amongdifferent facies (Fig. 3). The most discriminatingindividual well logs are RHOB and LOGRT. The leastdiscriminating log is GR.

Five facies were identified based on descriptions ofwell cores: sand dune (SD), interdune (ID), shallow ma-rine (SM), sabkha (SB) and sand sheet (SS). The de-cription and frequency of the five facies are presented inTable 1.

Both linear discriminant analysis and the nave Bayesclassifier are applied in three hold out wells with priorsFig. 6. Comparison of kernel density estimation and normal densityestimation of well logs suggests the normal assumption is moreappropriate than is kernel density estimation. (KDE = kernel densityestimation, NOR = normal distribution).being set as percentages of facies in the training data set.For each method, a predicted facies column is producedwith corresponding posterior probability column foreach well. The classification results of the two methodsin one of the three hold out wells are illustrated in Figs. 4and 5. The cross-validation results (Tables 2 and 3)suggest that: (1) Interdune, the most prevalent facies, aremostly correctly classified; (2) Although less than 50%of sand dune, the main hydrocarbon reservoir, is cor-rectly identified, no other facies are misclassified as sanddune. Also, misclassifications of sand dune typicallyoccur physically adjacent to correct classifications of sanddune.

In the current data, the normal-based Bayes classifierachieved a higher success rate than did the KDE-basedBayes classifier (Fig. 6), with increases in classificationrate by up to 20%. Thus the normality assumption isappropriate for probability density estimation of speci-fied well logs when using the nave Bayes classifier.Both linear discriminant analysis and the normal-distribution-based nave Bayes classifier perform con-sistently in three wells with average success rate 74%(Fig. 7).

4. Discussion and conclusions

The performance of the nave Bayes method in faciesidentification from well logs primarily depends on howthe probability densities are estimated and how priors aredistributed. Estimation of probability densities is impor-tant for the calculation of the likelihood and thus forestimation of the posterior distribution of facies. Com-parison of KDE and the normal distribution surprisinglyindicates that a normal distribution gives better results

156 Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157than does KDE from the aspect of prediction. The successof the normal assumption over KDE implies that in-corporating efforts to find the actual distribution does notnecessarily improve prediction. Optimal bandwidthsprobably follow the data too closely, and broader band-width with smoother density estimates could be expectedto perform better. Under the normality assumption, thebandwidth goes to infinity, which leads to an increasedrobustness of the classifier, as the location of the decisionsurface is less affected by noise and outliers in the data(Kraaijveld, 1996). Furthermore, compared with otherdensity estimation methods, fitting log readings to anormal distribution is simple, computationally efficientand reliable for purposes of facies identification.

The choice of priors also plays a role in classification.The cross-validation inwell 55 (Table 3) indicates thatmostof sand dune facies are misclassified as interdune. This isprobably due to interdune's much higher prior probability,the larger variance in well logs for interdune over sanddune, or a combination of these two influences. Classifi-cation based on an alternative prior distribution, whichtakes the average of the prior from the training set and anequal prior (all probabilities=0.2), failed to improveresults. We conclude for the given data that the differencein the variance of well logs among facies plays a moreimportant role than does the prior distribution. Inheterogeneous deposits like fluvial deposits where theprior distribution plays a more important role, theperformance of nave Bayes classifier may be lessconsistent than that in homogeneous marine deposits.

Although linear discriminant analysis requires multi-variate normality and equal variances across groups, pastexperience shows that violation of these assumptions doesnot generally lead to poor prediction, a finding that isjustified by this study. How the degree of violation of thenormality assumption affects the prediction is hard tocharacterize precisely and is still unknown. On the otherhand, violation of the homogeneity assumption is known tolead to overclassification. Our cross-validation (Table 2)demonstrates that some sand dune are misclassified asinterdune, which is probably the result of overclassifica-tion. This explanation is consistent with the observationthat the two most discriminating well logs, RHOB andNPHI, show substantial overlap between interdune andsand dune, and interdune has larger spread than sand dune.Quadratic discriminant analysis, which is a natural remedyto this problem, was also tested, but, with a success rate of67.3%, we judged it to be inferior to linear discriminantanalysis for the present application. The probable difficultywith quadratic discriminant analysis in the current setting isoverfitting of the training set coupled with heterogeneity ofdistributions within facies across physical sites.The nave Bayes classifier assumes independenceamong predictors. Violation of the independenceassumption is substantial but does not adversely affectthe classification in this study. One possible reason isthat although the estimated posteriors are not necessar-ily correct, the group associated with the highestiPxij fj=Pxi is the group associated with the high-est P(X| fj) /P(X). This slightly weaker condition relaxesthe importance of the strict independence assumption.An attempt to replace the four well-log variables withfour corresponding principle components in the naveBayes classifier ends up with 42% success rate in thehold out well 55. This initially surprising result may beexplained by noting that: (1) Estimation of too manyparameters in the variancecovariance matrix for eachfacies may introduce error; (2) The difference in thecorrelation among well logs from one facies to anotherfacies complicates principle component analysis; (3)Although principle components in the training set areindependent of each other, the principle components ofthe test set, which are calculated based on the principlecomponent functions derived from the training set, arenot necessarily independent due to the difference indistribution between the test set and the training set.

In this study, the nave Bayes classifier performs theclassification aswell as does linear discriminant analysis interms of efficiency and consistency. Although we selectednormal likelihoods, nave Bayes requires no assumptionon data distribution, which makes it a more universaltechnique than discriminant analysis.We conclude that thenaveBayes classifier isworthy of consideration in generalfor problems of facies identification.

Acknowledgements

The authors would like to thank Dr. P.G. Yin for pro-viding the data and professional advice, Q.S. Zhang forhis contribution to facies analysis, Huaiyu Yuan for valu-able discussion and insight, and an anonymous reviewer,whose comments substantively improved the paper.

References

Avseth, P., Mukerji, T., Jorstad, A., Mavko, G., Veggeland, T., 2001.Seismic reservoir mapping from 3-D AVO in a North Sea turbiditesystem. Geophysics 66 (4), 11571176.

Bhatt, A., Helle, H., 2002. Determination of facies from well logsusing modular neural networks. Pet. Geosci. 8 (3), 217228.

Clark, P., Niblett, T., 1989. The CN2 induction algorithm. Mach.Learn. 3 (4), 261283.

Coudert, L., Frappa, M., Arias, R., 1994. A statistical method for litho-facies identification. J. Appl. Geophys. 32, 257267.

Cuddy, S., 2000. Litho-facies and permeability prediction from electricallogs using fuzzy logic. SPE Reserv. Evalu. Eng. 3 (4), 319324.

157Y. Li, R. Anderson-Sprecher / Journal of Petroleum Science and Engineering 53 (2006) 149157Derek, H., Johns, R., Pasternack, E., 1990. Comparative study of abackpropagation neural network and statistical pattern recognitiontechniques in identifying sandstone lithofacies. Proceedings 1990Conference on Artificial Intelligence in Petroleum Exploration andProduction. Texas A and M University, College Station, TX, pp.4149.

Duda, R., Hart, P., 1973. Pattern Classification and Scene Analysis.John Wiley and Sons Inc, New York. 482 pp.

Iloghalu, E., 2003. Application of neural networks technique inlithofacies classifications used for 3-D reservoir geological modelingand exploration studies. AAPG Annual Meeting Abstract.

Kapur, L., Lake, L., Sepehrnoori, K., 2000. Probability logs for faciesclassification. In Situ 24 (1), 5758.

Kraaijveld, M., 1996. A Parzen classifier with an improved robustnessagainst deviations between training and test data. Pattern Recogn.Lett. 17 (7), 679689.

Langley, P., Iba, W., Thompson, K., 1992. An analysis of Bayesianclassifiers. Proceedings of the Tenth National Conference onArtificial Intelligence. AAAI Press, San Jose, CA.Martinez,W.,Martinez, A., 2002. Computational StatisticsHandbookwithMATLAB. Chapman and Hall/CRC, Boca Raton, Florida. 616 pp.

Saggaf,M., Nebrija, E., 2003. A fuzzy logic approach for the estimationof facies from wire-line logs. AAPG Bull. 87 (7), 12231240.

Sakurai, S., Melvin, J., 1988. Facies discrimination and permeabilityestimation from well logs for the Endicott field. 29th AnnualAPWLA Symposium. San Antonio, Texas.

Siripitayananon, P., Chen, H., Hart, B., 2001. A new technique forlithofacies prediction: back-propagation neural network. Proceed-ings of the 39th Annual ACM-SE Conference.

Specht, D., 1990. Probabilistic neural networks. Neural Netw. 3, 110118.Tabachnick, B., Fidell, L., 1996. Using Multivariate Statistics.

HarperCollins College Publishers, New York.Tang, H., White, C., Zeng, X., Gani, M., Bhattacharya, J., 2004.

Comparison of multivariate statistical algorithms for wireline logfacies classification. AAPGAnnual Meeting Abstract, vol. 88, p. 13.

Wong, P., Jian, F., Taggart, I., 1995. A critical comparison of neuralnetworks and discriminant analysis in lithofacies, porosity andpermeability predictions. J. Pet. Geol. 18 (2), 191206.

Facies identification from well logs: A comparison of discriminant analysis and nave Bayes cla.....IntroductionMethodologyNave Bayes classifierProbability density estimationDiscriminant analysisCross-validation

Results and analysesDiscussion and conclusionsAcknowledgementsReferences