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Relevance Vector Machines for Earthquake Response Spectra

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This study uses Relevance Vector Machine (RVM) regression to develop a probabilistic model for the average horizontal component of 5%-damped earthquake response spectra. Unlike conventional models, …

This study uses Relevance Vector Machine (RVM) regression to develop a probabilistic model for the average horizontal component of 5%-damped earthquake response spectra. Unlike conventional models, the proposed approach does not require a functional form, and constructs the model based on a set predictive variables and a set of representative ground motion records. The RVM uses Bayesian inference to determine the confidence intervals, instead of estimating them from the mean squared errors on the training set. An example application using three predictive variables (magnitude, distance and fault mechanism) is presented for sites with shear wave velocities ranging from 450 m/s to 900 m/s. The predictions from the proposed model are compared to an existing parametric model. The results demonstrate the validity of the proposed model, and suggest that it can be used as an alternative to the conventional ground motion models. Future studies will investigate the effect of additional predictive variables on the predictive performance of the model.

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  • 1. 2012 American Transactions on Engineering & Applied Sciences 2011 American Transactions on Engineering & Applied Sciences. American Transactions on Engineering & Applied Sciences http://TuEngr.com/ATEAS, http://Get.to/Research Relevance Vector Machines for Earthquake Response Spectra a* b Jale Tezcan , Qiang Chenga Department of Civil and Environmental Engineering, Southern Illinois University Carbondale,Carbondale, IL 62901, USAb Department of Computer Science, Southern Illinois University Carbondale, Carbondale, IL 62901, USAARTICLEINFO A B S T RA C TArticle history: This study uses Relevance Vector Machine (RVM)Received 23 August 2011Received in revised form regression to develop a probabilistic model for the average horizontal23 September 2011 component of 5%-damped earthquake response spectra. UnlikeAccepted 26 September 2011 conventional models, the proposed approach does not require aAvailable online26 September 2011 functional form, and constructs the model based on a set predictiveKeywords: variables and a set of representative ground motion records. TheResponse spectrum RVM uses Bayesian inference to determine the confidence intervals,Ground motion instead of estimating them from the mean squared errors on theSupervised learning training set. An example application using three predictiveBayesian regression variables (magnitude, distance and fault mechanism) is presented forRelevance Vector Machines sites with shear wave velocities ranging from 450 m/s to 900 m/s. The predictions from the proposed model are compared to an existing parametric model. The results demonstrate the validity of the proposed model, and suggest that it can be used as an alternative to the conventional ground motion models. Future studies will investigate the effect of additional predictive variables on the predictive performance of the model. 2012 American Transactions on Engineering & Applied Sciences.*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 25http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 2. 1. Introduction  Reliable prediction of ground motions from future earthquakes is one of the primarychallenges in seismic hazard assessment. Conventional ground motion models are based onparametric regression, which requires a fixed functional form for the predictive model. Because themechanisms governing ground motion processes are not fully understood, identification of themathematical form of the underlying function is a challenge. Once a functional form is selected,the model is fit to the data and the model coefficients minimizing the mean squared errorsbetween the model and the data are determined. This approach, when the selected mathematicalform does not accurately represent the actual input-output relationship, is susceptible tooverfitting. Indeed, using a sufficiently complex model, one can achieve a perfect fit to thetraining data, regardless of the selected mathematical form. However, a perfect fit to thetraining data does not indicate the predictive performance of the model for new data. Kernel regression offers a convenient way to perform regression without a fixed parametricform, or any knowledge of the underlying probability distribution. A special form of kernelregression, called the Support Vector Regression (SVR) (Drucker et al., 1997) is characterized byits compact representation and its high generalization performance. In SVR, the training data isfirst transformed into a high dimensional kernel space, and linear regression is performed on thetransformed data. The resulting model is a linear combination of nonlinear kernel functionsevaluated at a subset of the training input. Combination weights are determined by minimizing apenalized residual function. The SVR has proved successful in many studies since its introductionin 1997. The effectiveness of SVR in ground motion modeling has been recently demonstrated(Tezcan and Cheng, 2011), (Tezcan et al., 2010). A well-known weakness of the SVR is the lackof probabilistic outputs. Although the confidence intervals can be constructed using themean-squared errors, similar to the approach used in conventional ground motion models, theposterior probabilities, which produce the most reliable estimate of prediction intervals, are notgiven. The lack of probabilistic outputs in the SVR formulation has motivated the development ofa new kernel regression model called Relevance Vector Machine (RVM) (Tipping, 2000) whichoperates in a Bayesian framework. To overcome the limitations of parametric regression while obtaining probabilistic 26 Jale Tezcan and Qiang Cheng
  • 3. predictions, this paper proposes a new ground motion model based on the RVM regression.Unlike standard ground motion models, which make point estimates of the optimal value of theweights by minimizing the fitting error, the RVM model treats the model coefficients as randomvariables with independent variances and attempts to find the model that maximizes the likelihoodof the observations. This approach offers two main advantages over the conventional groundmotion models. First, the prediction uncertainty is explicitly determined using Bayesianinference, as opposed to being estimated from the mean squared errors. Second, the complexity ofthe RVM model is controlled by assigning suitable prior distributions over the model coefficients,which reduces the overfit susceptibility of the model. The rest of the paper is organized as follows. In Section 2, the RVM regression algorithm isdescribed. Section 3 is devoted to the construction of ground motion model. Starting with thedescription of the ground motion data and the predictive and target variables, the training resultsare presented, and the prediction procedure for new data is described. Section 4 demonstratescomputational results and compares the RVM predictions to an existing empirical parametricmodel. Section 5 concludes the paper by presenting the main conclusions of this study, anddiscusses the advantages and limitations of the proposed method.2. The RVM Regression Algorithm  Given a set of input vectors , 1: and corresponding real-valued targets , theregression task is to estimate the underlying input-output relationship. Using kernel representation(Smola and Schölkopf, 2004), the regression function can be written as a linear combination of aset of nonlinear kernel functions: , (1)where , 1… are the combination weights and is the bias term.*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 27http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 4. This study uses the radial basis function (RBF) kernel: , , , 0 (2)where is the width parameter controlling the trade-off between model accuracy andcomplexity. In this study, the width parameter has been determined using cross-validation. Assuming independent noise samples from a zero-mean Gaussian distribution,i.e., ~ 0, , the target values can be written as: 1, … , . (3) Recast in matrix from, Equation (3) becomes: Φw , (4)where ,…, , ,…, , and Φ is an 1 basis matrix with 1and , . The likelihood of the entire set, assuming independent observations isgiven by: | , 2 . (5)where ,…, is the vector containing the mean values of the combination weights.To control the complexity of the model, a zero-mean Gaussian prior is used where each weight isassigned a different variance (MacKay, 1992): | 0, 1/ . (6) 28 Jale Tezcan and Qiang Cheng
  • 5. In Eq. (6), ,…, where 1/ is the variance of . The posterior distributionof the weights is obtained as: | , , 2 | | . (7)where the mean vector and covariance matrix are: (8) (9) with … … 0 : . (10) 0 …The marginal likelihood of the dataset can be determined by integrating out the weights (MacKay,1992) as follows: | , 2 | | (11)where and is the identity matrix of size . Ideal Bayesian inferencerequires defining prior distributions over and , followed by marginalization. This process,however, will not result in a closed form solution. Instead, the and values maximizingEq. (11) can be found iteratively as follows (MacKay, 1992):*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 29http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 6. 1 (12) . (13) ∑ 1 Because the nominator in Eq.(12) is a positive number with a maximum value of 1, anvalue tending to infinity implies that the posterior distribution of is infinitely peaked at zero,i.e. 0. As a consequence, the corresponding kernel function can be removed from themodel. The procedure for determining the weights and the noise variance can be summarized asfollows: 1) Select a width parameter of the kernel function and form the basis matrix Φ. 2) Initialize ,…, and . 3) Compute matrix using Eq.(10). 4) Compute the covariance matrix using Eq.(9). 5) Compute the mean vector using Eq.(8). 6) Update and using Eq.(12) and Eq.(13). 7) If ∞, set 0 and remove the corresponding column in Φ. 8) Go back to step 3 until convergence. 9) Set the remaining weights equal to . The training input points corresponding to the remaining nonzero weights are called the“relevance vectors”. After the weights and the noise variance are determined, the predictive meanfor a new input can be found as follows: Φ. (14) T In Eq.(14) Φ 1 x ,r x ,r … x , rN where r , r … , rN are therelevance vectors. 30 Jale Tezcan and Qiang Cheng
  • 7. The total predictive variance can be found by adding the noise variance to the uncertainty dueto the variance of the weights, as follows: ΦT CΦ . (15)3. Construction of the Ground Motion Model    In this section, RVM regression algorithm will be used to construct a ground motion model. InSection 4, the resulting model will be compared to an existing parametric model by Idriss (Idriss,2008), which will be referred to as “I08 model” in this paper. To enable a fair comparison, thedataset and the predictive variables of I08 model have been adopted in this study. The RVMalgorithm is independent of the size of the predictive variable set; additional variables can beintroduced the set of predictive variables can be customized to specific applications.3.1 Ground Motion Data        The ground motion records used in the training have been obtained from the PEER-NGAdatabase (PEER, 2007). Consistent with the I08 model, a total of 942 free-field records have beenselected using the following criteria: • Shear wave velocity at the top 30 m ranging from 450 m/s to 900 m/s, • Magnitude larger than 4.5, • Closest distance between the station and rupture surface (R) less than 200 km. Detailed information regarding these records can be found in the paper by Idriss (Idriss, 2008).3.2 Predictive and Target Variables      The predictive variable set includes moment magnitude (M), natural logarithm of the closestdistance between the station and the rupture surface in kilometers and fault mechanism (F).Idriss finds that with the shear wave velocity ( ) constrained to 450 m/s- 900 m/s range, it has*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 31http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 8. negligible effect on spectral values up to 1 second. Therefore, was not used as a predictivevariable. Following the convention used in I08 model, earthquakes that have been assigned a faultmechanism type 0 and 1 in the PEER database were merged to a single, “strike-slip” group, whilethe rest were considered to be representative of “reverse” events. In the RVM model, strike-slipand reverse earthquakes are assigned 1 and 1, respectively. The input vectorrepresenting ith record has the following form: . (16) A set of eight vibration periods ( 8) ranging from 0.01 second to 4 seconds was used inthe RVM model. The output for the ith record for the vibration period is defined as: for 1 to . (17) In Equation (17), is the natural logarithm of the average horizontal component of 5%-damped pseudo-acceleration response spectrum. The spectral values represent the medianvalue of the geometric mean of the two horizontal components, computed using non-redundantrotations between 0 and 90 degrees (Boore, 2006).3.3 Training of the RVM Regression Model  As a pre-processing step, and values were linearly scaled to [-1 1] to achieveuniformity between the ranges of the predictive variables. There is no need to scale the faultmechanism identifier ( as it was already defined to take either -1 or 1. Because kernel functionsuse Euclidean distances between pairs of input vectors, such scaling will help prevent numericalproblems due to large variations between the ranges of the values that variables can take. In theground motion data used in this study, the ranges of the predictive variables are4.53 7.68 , and 0.32 199.27 . Therefore, input scaling takes thefollowing form: 32 Jale Tezcan and Qiang Cheng
  • 9. 2 12.21 2 4.16 , , . (18) 3.15 6.44 The optimal value of the kernel width parameter for each vibration period wasdetermined using 10-fold cross validation (Webb, 2002). In 10-fold cross validation, the trainingdata is randomly partitioned into 10 subsets of equal size; and the model is trained using 9 subsets,and the remaining subset is used to compute the validation error. This process is repeated 10 times,each time with a different validation subset, and the average validation error for a particular iscomputed. By computing the average validation error over a range of possible values, theoptimal with the smallest average validation error is determined. The resulting values foreach period are listed in Table 1, along with the standard deviation of noise ( ), the mean value ofthe constant term and the number of relevance vectors. The relevance vectors and thecombination weights are listed in Table 2. After the RVM models, one for each vibration period, were trained, standardized residualswere computed. Figure 1 shows the distribution of the standardized residuals, corresponding toT=1 second, with respect to , and . The residual distribution patterns for other periodswere similar, not indicating any systematic bias.Table 1: Kernel width parameter , logarithmic standard deviation of noise ( ), mean value of the bias term and the number of relevance vectors ( ), for each period. T (sec) 0.01 0.23 0.633 -3.069 7 0.05 0.32 0.666 -0.664 7 0.10 0.13 0.718 0.002 7 0.20 0.15 0.661 -15.042 6 0.50 0.25 0.695 -8.359 7 1.00 0.36 0.748 -4.670 5 2.00 0.28 0.869 -6.0548 5 4.00 0.26 0.983 -7.794 5*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 33http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 10. Figure 1: Standardized residuals for T=1 second. Table 2: Mean values of the combination weights and the relevance vectors T=0.01 s. T=0.05 s.i Wi ri i Wi ri1 13.258 [-0.1937 0.2676 -1] 1 -6.177 [0.7905 -0.4227 1]2 15.393 [0.5238 -0.2268 1] 2 6.355 [-0.3841 -0.1783 -1]3 0.4861 [ 0.8921 0.9414 -1] 3 28.555 [0.5238 0.5856 1]4 -5.073 [0.9619 -1.0000 1] 4 -7.930 [-0.5111 0.7896 -1]5 -4.275 [0.9619 -0.6751 1] 5 -0.402 [0.7460 -0.4021 -1]6 -14.173 [-0.2889 0.7862 -1] 6 -12.622 [0.9619 0.9545 1]7 -8.086 [ 0.0603 0.9789 1] 7 -16.194 [0.0603 0.9789 1] T=0.1 s. T=0.2 s.i Wi ri i Wi ri1 64.423 [0.4159 -0.1499 1] 1 29.569 [-0.8921 -0.0837 -1]2 -6.991 [ 0.9619 0.9545 1] 2 2.293 [0.7905 -0.4227 1]3 -36.297 [0.9619 -1.0000 1] 3 35.440 [0.8921 0.6543 -1]4 15.875 [1.0000 0.4559 -1] 4 5.7412 [0.9619 -1.0000 1]5 -5.599 [-0.3143 0.0809 1] 5 3.5036 [-0.8222 0.1385 1]6 -17.361 [ 0.6508 0.9961 -1] 6 -48.496 [0.0603 0.4955 -1]7 -25.799 [-0.1302 0.9056 1] 34 Jale Tezcan and Qiang Cheng
  • 11. Table 2 (continued). T=0.5 s. T=1.0 s. i Wi ri i Wi ri 1 6.4551 [0.7905 -0.4227 1] 1 1.9699 [0.7905 -0.4227 1] 2 12.825 [-0.2317 -0.2931 -1] 2 4.8873 [0.0540 -0.2785 -1] 3 0.0283 [-0.7714 0.1214 1] 3 -4.1425 [-0.7524 0.7892 1] 4 -0.806 [ 0.8921 -0.0318 -1] 4 -3.9593 [-0.7651 0.8672 -1] 5 8.4335 [0.8921 0.9414 -1] 5 3.7352 [-0.1302 -0.0121 1] 6 -0.089 [ 0.9619 0.9545 1] 7 -12.9 [ 0.0603 0.5786 -1] T=2.0 s. T=4.0 s. i Wi ri i Wi ri 1 7.3574 [-0.2317 -0.2931 -1] 1 0.4747 [0.7460 -0.4021 -1] 2 4.5548 [-0.0730 0.4691 1] 2 11.936 [0.7460 0.5118 -1] 3 3.0086 [ 0.9619 -1.0000 1] 3 6.8109 [0.3714 -0.0296 1] 4 -6.4695 [-1.0000 0.5142 -1] 4 -5.6050 [-0.7524 0.7892 1] 5 -5.3630 [-0.7524 0.7892 1] 5 -10.180 [0.3778 1.0000 -1]3.4 Prediction Phase    After training, the spectral values for a new input vector , , can be determinedas follows: 1. Scale the input to the range [-1 1] using Eq. (18); T 2. Construct the basis vector Φ 1 , , … , using the relevance vectors from Table 2 and the kernel width parameter from Table 1; 3. Determine the median value of using Eq.(14); 4. Obtain the standard deviation of the noise from Table 1. Total uncertainty, if needed, can be determined using Eq.(15).4. Computational Results  The RVM model was tested using different magnitude, distance and fault mechanisms, and theresults were compared to the I08 model. Figure 2 shows the median spectral acceleration at T=1*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 35http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 12. second, along with the 16th and 84th percentile values bounds for strike-slip faults, forM=5 (left) and M=7 (right). The circles in the figure show the spectral values from earthquakeswith the same fault mechanism and within ±0.25 magnitude units. Figure 3 shows the sameinformation for reverse faults. For periods about 1 second and longer, it was observed that themedian estimates from the RVM model were generally lower than those from the I08 model. Atvery short distances, within ~20 km of the source, RVM estimates were higher for M=7, for bothstrike-slip and reverse faulting earthquakes. Figure 2: Median ±σ bounds for spectral acceleration at T=1 second, strike-slip faults. Figure 3: Median ±σ bounds for spectral acceleration at T=1 second, reverse faults. 36 Jale Tezcan and Qiang Cheng
  • 13. Figure 4 presents the results for vibration period T=0.2 second, for strike-slip earthquakes.The results for the reverse faulting earthquakes were similar. For shorter vibration periods, andM=7, RVM estimates were lower than those from the I08 model. For M=5, however, RVMpredictions equaled or exceed the I08 predictions. Regarding the variation about the median (noisevariance), the predictions from the two models were in general agreement for all vibration periods. Figure 4: Median ±σ bounds for spectral acceleration at T=0.2 second, strike-slip faults.5. Conclusion  This paper proposes an RVM-based model for the average horizontal component ofearthquake response spectra. Given a set of predictive variable set, and a set of ground motionrecords, the RVM model predicts the most likely spectral values in addition to its variability. Anexample application has been presented where the predictions from the RVM model have beencompared to an existing, parametric ground motion model. The results demonstrate the validity ofthe proposed model, and suggest that it can be used as an alternative to the conventional groundmotion models. The RVM model offers the following advantages over its conventional counterparts: (1) Thereis no need to select a fixed functional form. By determining the optimal variances associated with*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 37http://TUENGR.COM/ATEAS/V01/25-39.pdf
  • 14. the weights, the RVM automatically detects the most plausible model; (2) The resulting RVMmodel has a simple mathematical structure (weighted average of exponential basis functions), andis based on a small number of samples that carry the most relevant information. Samples that arenot well supported by the evidence (as measured by the increase in the marginal likelihood) areautomatically pruned. (3) Because the model complexity is controlled during the training stage, theRVM has lower risk of over-fitting. One limitation of the proposed approach is that the resulting model may be difficult tointerpret. Because the RVM is not a physical model, it does not allow any user-defined, physicalconstraints, not allowing extension of the model to scenarios not represented in the training dataset. However, in our opinion, this does not constitute a shortcoming, considering that the reliabilitysuch practice is questionable in any regression model. Another potential limitation is that the RVMrequires a user-defined kernel width parameter, which does not have a very clear intuitive meaning,especially when working with high dimensional input vectors. However, the optimal value of thekernel width parameter can be determined using cross-validation, as has been done in this study.Future studies will investigate the effect of using additional predictive variables on theperformance of the model.6. Acknowledgements  This material is based in part upon work supported by the National Science Foundation underGrant Number CMMI-1100735.7. References Boore, D.M., J. Watson-Lamprey, and N.A. Abrahamson. (2006). Orientation-independent measures of ground motion. Bulletin of the Seismological Society of America, 96(4A), 1502-1511.Bozorgnia, Y. and K. W. Campbell. (2004). The vertical-to-horizontal response spectral ratio and tentative procedures for developing simplified V/H and vertical design spectra. Journal of Earthquake Engineering, 8(2), 175-207.Campbell, K. W. and Y. Bozorgnia. (2003). Updated Near-Source Ground-Motion (Attenuation) Relations for the Horizontal and Vertical Components of Peak Ground Acceleration and Acceleration Response Spectra. Bulletin of the Seismological Society of America, 93(1), 314-331. 38 Jale Tezcan and Qiang Cheng
  • 15. Drucker, H., C. J. C. Burges, L. Kaufman, A. Smola and V. Vapnik. (1997). Support vector regression machines, Advances in Neural Information Processing Systems 9, MIT Press.Idriss, I. M. (2008). An NGA empirical model for estimating the horizontal spectral values generated by shallow crustal earthquakes. Earthquake spectra, 24(1), 217-242.MacKay, D. J. C. (1992). Bayesian interpolation. Neural computation, 4(3), 415-447.MacKay, D. J. C. (1992). The evidence framework applied to classification networks. Neural Computation, 4(5), 720-736.PEER. (2007). PEER-NGA Database. http://peer.berkeley.edu/nga/index.html.Smola, A. J. and B. Schölkopf. (2004). A tutorial on support vector regression. Statistics and Computing, 14(3), 199-222.Tezcan, J. and Q. Cheng. (2011). A Nonparametric Characterization of Vertical Ground Motion Effects. Earthquake Engineering and Structural Dynamics (in print).Tezcan, J., Q. Cheng and L. Hill. (2010). Response Spectrum Estimation using Support Vector Machines, 5th International Conference on Recent Advances in Geotechnical Earthquake Engineering and Soil Dynamics, San Diego, CA.Tipping, M. (2000). The relevance vector machine. Advances in Neural Information Processing Systems MIT Press.Webb, A. (2002). Statistical pattern recognition, New York, John Wiley and Sons. Dr.Jale Tezcan is an Associate Professor in the Department of Civil and Environmental Engineering at Southern Illinois University Carbondale. She earned her Ph.D. from Rice University, Houston, TX in 2005. Dr.Tezcan’s research interests include earthquake engineering, material characterization, and numerical methods. Dr.Qiang Cheng is an Assistant Professor in the Department of Computer Science at Southern Illinois University Carbondale. He earned his Ph.D. from the University of Illinois at Urbana Champaign, IL in 2002. Dr.Cheng’s research interests include pattern recognition, machine learning and signal processing.Peer Review: This article has been internationally peer-reviewed and accepted for publication according to the guidelines given at the journal’s website.*Corresponding author ( J. Tezcan). Tel/Fax: +001-618-4536125. E-mail address:jale@siu.edu. 2012. American Transactions on Engineering & Applied Sciences.Volume 1 No.1 ISSN 2229-1652 eISSN 2229-1660. Online Available at 39http://TUENGR.COM/ATEAS/V01/25-39.pdf