Filling up gaps in wave data with genetic programming

Transcription

1 Filling up gaps in wave data with genetic programming Ketaki Ustoorikar, M.C. Deo Department of Civil Engineering, Indian Institute of Technology, Mumbai , India Abstract A given time series of significant wave heights invariably contains smaller or larger gaps or missing values due to a variety of reasons ranging from instrument failures to loss of recorders following human interference. In-filling of missing information is widely reported and well documented for variables like rainfall and river flow, but not for the wave height observations made by rider buoys. This paper attempts to tackle this problem through one of the latest soft computing tools, namely, genetic programming (GP). The missing information in hourly significant wave height observations at one of the data buoy stations maintained by the US National Data Buoy Center is filled up by developing GP models through spatial correlations. The gap lengths of different orders are artificially created and filled up by appropriate GP programs. The results are also compared with those derived using artificial neural networks (ANN). In general, it is found that the in-filling done by GP rivals that by ANN and many times becomes more satisfactory, especially when the gap lengths are smaller. Although the accuracy involved reduces as the amount of gap increases, the missing values for a long duration of a month or so can be filled up with a maximum average error up to 0. m in the high seas. Keywords: Data gaps; Neural networks; Genetic programming; Wave heights. Introduction The time series of ocean wave heights has applications in many studies related to coastal, offshore and ocean engineering. Analysis of such series can yield a variety of information

2 78 ARTICLE IN PRESS for design and operational use, like the long-term wave height corresponding to a certain return period and exceedance probabilities of a given wave height, respectively. An analysis of the time series usually dictates that the sequential observations contained in it are equally spaced, made over long periods and reported in an uninterrupted manner. However despite precautions, gaps in the collected records cannot be avoided. This is due to many reasons like, failure of collection and transmission equipments, noise and synchronization problems between the buoy and the receivers, hardware- as well as software-related failures, aging of equipments, accidental or weather-induced snapping of mooring lines, severe weather rendering the system in-operational, thefts and the cloud cover problem in satellite image-based records. The amount of gaps in a given record changes from one buoy location to another, but in general appears to range anywhere from less than % a year to more than 40% as shown in Table, that gives an example of the percentage missing values in the wave rider buoy measurements made by the US National Data Buoy Center (NDBC) [] at selected locations in the Gulf of Mexico. Arena et al. [] based on the Italian national wave measurement programme involving 4 buoys mention the occurrence of 5% gaps over a -year period, associated with a maximum repair time of 4 days. In the Sea Wave Monitoring Network (SWAN) of 0 buoys around Italy Puca et al. [3] report loss of data ranging from less than 5 5%. While analysing data collected under the Indian data buoy programme in a separate exercise the authors have noted that for three locations along the west coast, namely, DS, SW and SW4 the loss of data was around 9%, 39% and 9% and over periods of 4.0,.5 and 5.5 year, respectively, during The loss of information can be valuable and hence needs to be retrieved. The presence of gaps obviously affects the quality of information obtained through analysis of such gappy time series and also the performance of the application made, like say real-time wave forecasting and derivation of wave height duration curves. The presence of missing information may introduce a bias in the results so obtained. The techniques of substituting missing values in a given time series of a random variable have been well studied and routinely employed in case of variables like river discharge and runoff [4], but the same cannot be said for the variable of ocean waves. This might probably be due to relatively smaller sample sizes involved in many hydrological studies, like analysis of annual peak river flows or that of monthly rainfalls, where a single missing value may introduce very large bias in the results. The methods employed in such applications include a random choice within the observed range, linear and non-linear interpolation [4], autoregressive schemes [5], chaos theory [6] and artificial neural networks [7]. The problem of gappy data in general oceanography has been addressed by Table Percentage of missing values (NDBC locations) Station Year , FPSN , ,

3 79 investigators like Thompson [8] who suggested that a random sampling of data points might be an optimally efficient approach and Sturges [9] who used a Monte Carlo technique to make up gaps at random in a known time series of monthly mean sea-level. Emery and Thomson [0] gave an account of such attempts in a wider domain of oceanography. As regards the time history of wave heights (rather than that of other variables in the works referred to earlier) is concerned there are relatively sparse studies directly addressing the issue. Stefanokos and Athanassoulis [] made use of a residual wave height series with the same probability distribution as the original one created after removing the trend and periodicity from the observed series. Use of the soft computing tools like artificial neural network (ANN) s for the in-filling of wave data is very recent. Puca et al. [3] filled up gaps at one location by developing the spatial correlation with two nearby sites, while Balas et al. [] resorted to temporal correlations probably due to smaller gaps ( 4 h or so) in their series and also smaller period of observation (4 months). In general small gaps a few in number appeared to have been filled up by simple interpolation, medium gaps by stochastic model fitting and large gaps by spatial correlation, although the distinction made between the small, large, medium is not very clear [,3]. A review of past works indicates that when it comes to in-filling of missing values in the wave height series there is a scope to carry out a systematic analysis based on various sizes of gap lengths and using a large database and the present useful information accordingly to future investigators. Further, since recent past researchers dealing with uncertainties in data are finding new soft computing approaches more attractive compared to traditional schemes and it is necessary to try out such newer techniques to retrieve the missing information. The present work is directed towards this. It involves application of one of the latest and so far untried soft tool of genetic programming (GP) for filling up the missing information at a given location based on the same being collected at nearby stations. The GP can iteratively generate new values till such values reach a certain level of acceptance as per the selected criterion and thus looks attractive in the current problem of retrieval of missing values. In the present work suitability of this new approach is assessed for different lengths of the gap and its outcome is compared with that of an ANN. Unlike past, a large amount of wave rider as well as satellite wave data are now becoming increasingly available for multiple locations and over long durations, at many parts in the world and this study would therefore be useful while analysing such database.. The database used The wave rider buoy observations pertaining to four stations in the Gulf of Mexico maintained by the US National Data buoy Center were downloaded form the web site []. These stations were: FPSN7, 4,00, 4,008 and 4,004 (Fig. ) located in the Gulf of Mexico-off the US SE coast. The measurements were made by frying pan shoals and moored type of buoys. The station 4,00 is in deep water (depth ¼ 3786 m) while stations 4,004, FPSN7 and 4,008 are in shallower depths of 33.5, 3.5 and 8.0 m, respectively. The sample size used was of four years for each of these locations and it ranged from January 00 to December 004. The hourly values of the significant wave heights (H s ) were involved. As mentioned in the earlier section the amount of gaps present in the available time series is shown in Table from which it is apparent that the station FPSN7 had the largest volume of missing information ranging from 9.5% to 4.8% per

4 80 ARTICLE IN PRESS Fig.. Location plan (Gulf of Mexico; off US SE Coast). year while the one identified as 4,008 had the smallest number of gaps with their maximum value of 3.5% per year. It was decided to fill up the gaps at the central location of 4,004 (where the volume of the missing information is relatively moderate) from the measurements made at the three adjoining stations, using GP... Genetic programming GP is modelled out of the process of evolution occurring in nature, where the species survive following the principle of survival of the fittest. Unlike the more widely known genetic algorithms (GAs), its solution is a computer programme or an equation as against a set of numbers in the GA. Koza [4] explains various concepts related to GP. For readers of this paper not familiar with GP an information on basic GP operations like reproduction, mutation, and cross-over is given in Appendix A. In GP a random population of individuals (equations or computer programs) is created, the fitness of individuals is evaluated and then the parents are selected out of these individuals. The parents are then made to yield offspring s by following the process of reproduction, mutation and cross-over. The creation of offspring s continues (in an iterative manner) till a specified number of offspring s in a generation are produced and further till another specified number of generations is created. The resulting offspring at the end of all this process (an equation or a computer programme) is the solution of the problem. The GP thus transforms one population of individuals into another one in an iterative manner by following the natural genetic operations like reproduction, mutation and

5 8 cross-overs. The step-by-step procedure involved in this connexion is further explained below:. Create initial random population of individuals (equations or programs) of a certain size by randomly picking up the same from a set of terminals (consisting of input variables and constants) and functions (involving operators like, multiplication, addition, subtraction, division, square root, log, etc.). (Refer to Appendix A.). Evaluate the fitness of each individual in a population through some criterion like the root-mean-square error. 3. Select individuals or parents (normally probabilistically typically through a tournament involving comparing two parents at a time and thereafter short listing the winner for further competition). 4. Generate new offspring s (individuals) from these parents by (a), (b) and (c) below (see Figs. 0 referred to later). (a) Reproduction: Copy the best programme as it is as per the fitness criterion and include it in the new population. Increase individuals by. (b) Cross-over: Select individuals as per the fitness. Perform cross-over. Insert the two individuals into the new population. Increase individuals by. (c) Mutation: Select one individual as per the fitness. Perform mutation. Insert the mutant into the new population. Increase individuals by. 5. If the number of individuals (offspring s) equals a maximum (selected) number, increase the number of generations by and go to step 6; otherwise increase the individuals by repeating steps If the number of generations is equal to a certain maximum value, terminate the programme; otherwise repeat steps 5. There is hardly any application of GP in ocean engineering so far, although the same in civil engineering related to water flows started around 5 years ago. The tool of GP has been used for a variety of purposes like pattern recognition, classification and regression. Unlike the other soft computing tools like artificial neural networks, the GP applications are restricted to relatively fewer areas and include rainfall-runoff modelling [5,6], estimation of settling velocities of faecal pellet [7], modelling of risks in water supply [7,8], evaluation of ocean component concentration from sunlight reflectance or luminance values [9], modelling of waste water treatment plants [0], ground water level changes due to storm water infiltration, [] and estimation of river discharge from rainfall and soil and air properties []. Applications of GP to solve problems in coastal engineering are conspicuous by its absence and the current work is probably one of the first studies in this regard. 3. Implementing GP The present problem of establishing spatial correlations can be handled by GP either through the equation mode or by the programme mode. Considering more flexibility in data mining offered by the latter approach the individuals consisting of computer programs only were used. The software Discipulus [3] was used to generate the GP programs. TurboC in the C environment was employed to run the evolved programs and to implement them by applying to a new data set (applied data set). The statistical

6 8 ARTICLE IN PRESS measures of correlation coefficient, root-mean-square error and mean absolute error, have been used in this study to compare the GP estimations with actual observations and these were evaluated by using Matlab, which also facilitated generation of scatter plots between the target output and the one obtained through GP. 4. Results and discussion The objective of the study was to fill up gaps in the time series of H s values at the location: 4,004 given the measurements of H s at the three surrounding locations (Fig. ). The calibration as well as the testing of the GP model was done for those cases only where observations at all the four locations were simultaneously available. The calibration or training was thus done with the help of 60% of total data (input output pairs) of 48 months belonging to the initial observed sequence. Once the final GP programme was obtained after such training it was tested with the help of the remaining or following 40% pairs of input output. The number of training pairs was thus,750 while the same of testing was Appendix B gives the final GP programme obtained at the end of training. The choice of control parameters used in this process depended on the case being analysed. The typical value of the population size was 500, of number of generations 5 and that of the number of tournaments was 90,00,000. The mutation and the cross-over frequency also varied for different testing exercises and it ranged from 0% to 80%. The fitness criterion was the mean squared error between actual observations and corresponding predictions. To begin with the testing exercises, all observations (simultaneous with all the source stations) at the target station: 4004 were treated as missing and then evaluated from the GP programme. Fig.. H s values evaluated by GP during testing exercise vis-a-vis actual observations. (Period of observations: February 003 December 004.)

7 83 A comparison of the GP output so resulted with the corresponding actual observations during such testing or validation is shown in Fig. in the form of a scatter plot. The quantitative comparison is made in terms of 3 alternative error statistics namely, correlation coefficient (R), root-mean-square error (RMSE) and mean average error (MAE). (For a good prediction R should be high (maximum value being.0) and MAE, RMSE should be as low as possible). The accompanying error statistics (referring to Fig. ) were R ¼ 0.9, RMSE ¼ 0.3 m; MAE ¼ 0.6 m. It may thus be said that the GP has performed very well over most of the levels of the wave heights, except for the H s values above 4.0 m or so for which there is under-estimation. A possible reason behind this observation is the availability of very few training data for higher waves. Attempts were made to overcome this problem by retraining with higher values again or by applying empirical corrections. However they met with limited success and the conclusions reached could not be generalised. This problem needs to be addressed separately in future. It was also noticed that the GP was able to capture the non-linear relationships in a given univariate wave height series well. This can be seen in example Figs. 3 and 4. Figs. 3a and 4a show the GP evaluation of missing values vis-a-vis their actual observations while Figs. 3b and 4b indicate their evaluations (shown by dotted lines) made by GP. These figures illustrate that the GP was able to capture the non-linear variations in the time series of H s well. In order to see how GP carries out the in-filling task for various lengths of missing information the actual observations of H s over periods of month as well as 5 days, 5 days, and day were treated as missing and thereafter evaluated based on the trained GP programme. The locations of these last four missing patches selected purely randomly. The results can be seen in Figs. 5 8 in which the GP evaluations are compared with the actual observations for a typical month of January, 004 selected as an example and also in Tables 5 that give a quantitative performance of GP in terms of the error statistics, of R, RMSE and MAE fro example months of January, March and August, 004. A close match between the GP outcome and the actual observations can be noted through these figures and tables, both qualitatively and quantitatively. It may be seen from these tables that the GP is able to capture the underlying non-linear relationship between the observations at the source stations with the measurements at the target station reasonably well, as reflected in high values of R and low values of RMSE and MAE. The figures and tables also indicate that longer intervals of gap filling are associated with lesser success, probably as expected. Although the performance of the method of GP in the task of gap in-filling can best be judged by comparing the GP estimations with corresponding actual observations through overall error statistics an effort was also made to see how the gap filling affects the basic statistics of observed series. The statistical parameters of mean, standard deviation, skewness and kurtosis were considered. Table 6 shows these parameters for the example month of January 004 for various gap in-filling cases. The second column in this table shows the statistics of the observed time series as it is, while columns 3 6 indicate the same when gaps of month, 5 days, 5 days and day were artificially and randomly created and filled up using the GP. It may be seen that in general the mean and standard deviation values are excellently restored for all cases of gap in-filling. When the gap lengths did not exceed or 5 days all the four statistics were faithfully reproduced. For the case of longer gap in-fillings 5 days and month the skewness seems to be fairly restored but this was not the case with kurtosis, owing probably to the fact that the higher moments are generally unstable in nature. In applications where higher order moments are important a caution would be necessary to observe.

8 84 ARTICLE IN PRESS GP Derived Hs (m) Observed data Hs (m) Actual Predicted Hs in m Time in days Fig. 3. Evaluation of missing values by GP. (November 003) (Note: The dotted line indicates evaluation by GP). In order to see how the results of GP modelling performs with respect to the more familiar soft tool of artificial neural network (ANN) a common feed forward type of the network was developed to repeat the same tasks done by the GP. Such ANN had therefore 3 input nodes belonging to data at the three source stations and one output node pertaining to the H s value evaluated at the target station of 4,004 (Fig. ). The network (Fig. 9) was trained using a variety of schemes to ensure that adequate training is imparted. This included Levenberg Marquredt, Resilent propagation as well as variants of the

9 GP Derived Hs (m) Observed data Hs (m) Actual Predicted Hs in m Time in days Fig. 4. Evaluation of missing values by GP. (June 003) (Note: The dotted line indicates evaluation by GP). conjugate gradient. The best learning however resulted from the conjugate gradient scheme of Bowden type. For details of these methods see Demuth et al. [4]. The number of hidden nodes was determined using the trial and error approach aimed at obtaining the best possible testing results and this was 4. The hidden layer and the output layer had the transfer functions of Log-sigmoid and Purelin type, respectively, determined again by trials

10 86 ARTICLE IN PRESS GP Derived Hs (m) Observed data Hs (m) Actual Predicted.5 Hs in (m) Time in days Fig. 5. Comparison of GP-derived and actual H s values for -month duration in January 004. till the best outcome was noticed. The number of epochs was 000 while the training goal was The outcome of this exercise is also shown in Tables 5. Table indicates the error statistics for observations made over a month while Tables 3 5 show the same when the measurement period considered was 5 days, 5 days and day, respectively. It can be noticed that the GP produced results that were marginally more satisfactory than ANN, especially when the number of gaps become smaller with the observation period changing

11 GP Derived Hs (m) Observed data Hs (m) Actual Predicted Hs in m Time in days Fig. 6. Comparison of GP-derived and actual H s values for 5 days duration in January 004. from month to day. The difference between the GP and ANN was more pronounced in terms of MAE and RMSE, which unlike R give an overall error perspective. While R is very sensitive to the highest deviations, RMSE is more suitable for iterative methods of evaluation, like the current one and further in engineering applications the MAE giving average absolute deviations is a well-understood parameter.

12 88 ARTICLE IN PRESS Fig. 7. Comparison of GP-derived and actual H s values for 5 days duration in January 004 (6 January January 004). Fig. 8. Comparison of GP-derived and actual H s values for -day duration in January 004 (4 January 004). It may be seen that when we move towards the smaller amount of gaps from the larger one (from month to over a day) the accuracy in general improves. The month of August was noticed as the worst to estimate the missing values (due to stormy sea

13 89 Table Error statistics for observations over a month: testing data Month and year Method R MAE (m) RMSE (m) January 004 GP ANN March 004 GP ANN August 004 GP ANN Table 3 Error statistics for observations over 5 days: testing data Month and year Method R MAE (m) RMSE (m) January 004 GP ANN March 004 GP ANN August 004 GP ANN Table 4 Error statistics for observations over 5 days: testing data Month and year Method R MAE (m) RMSE (m) January 004 GP ANN March 004 GP ANN August 004 GP ANN Table 5 Error statistics for observations over day: testing data Month and Year Method R MAE (m) RMSE (m) January 004 GP ANN March 004 GP ANN August 004 GP ANN

14 90 ARTICLE IN PRESS Table 6 An example of statistics of different time series Analysis for the month of January 004 Observed series for month Series with gaps assumed and filled up for month Series with gaps assumed and filled up for 5 days Series with gaps assumed and filled up for 5 days Mean Std. dev Skewness Kurtosis Series with gaps assumed and filled up for day Feed forward Bias Hs Hs Hs 4 Output Layer Hs 3 Input Layer Connections weights Hidden Layer Fig. 9. The feed forward network. conditions); however even in August it was possible to evaluate the missing information with an average error of 0. m only. The new approach of GP adopted in this study thus seems to perform well in filling up the gaps in a given time series of H s values. 5. Conclusions GP was found to be an effective tool to retrieve the missing information in a given wave height time history by establishing spatial correlation with neighbouring locations. The technique of GP was able to learn the non-linear trends in the underlying time series satisfactorily. The results obtained by adoption of the GP were marginally better than the best-trained feed-forward type of ANN, especially when a small interval (a few days or so) of gaps was intended to be filled up.

15 9 Although in general smaller gaps were more accurately retrieved than the larger ones, the missing information even within a period of month was filled up with an average error of only 0. m in the high seas. The success of the present spatial correlation study may inspire similar applications of GP to carry out other works like establishing temporal correlations or evaluating a cause effect relationship in maritime engineering. Acknowledgement The authors gratefully acknowledge the financial support given to the above project by the Department of Science and Technology, Government of India. Appendix A A.. Examples of genetic operations Let H s, H s, and H s3 represent three input variables of wave heights at surrounding locations. Consider a simple programme ½ð H s ðþh s3 Þ = Þ=3H s Š in the form of a tree structure as in Fig. 0. A.. Generating population A population of random trees representing the programs is initially constructed and genetic operations are performed on these trees to generate individuals with the help of two distinct sets; the terminal set T and the function set F. For Fig. 0: f ; þ; p ;=gf and fh s3 ; 3; H s ; H s gtg () In order to generate a random tree one has to pick randomly from T[F, until all branches end up in terminals. Cross-over: Two random nodes are selected from inside such programme (parents) and thereafter the resultant sub-trees are swapped, generating two new programs as in Fig.. Mutation: A sub-tree is replaced by another one randomly (Fig. ). Reproduction: This means an exact duplication of the programme if it is found to be acceptable by the fitness criteria. / 3 Hs Hs Hs3 Fig. 0. Programme ½ð H s ðþh s3 Þ = Þ=3H s Š in the form of a tree structure.

16 9 ARTICLE IN PRESS / / x x Hs Hs 5 Hs Hs Hs 3 Hs Hs 3 CROSSOVER Parent Parent / / Hs 3 x Hs Hs Hs Hs 3 Child x Hs 5 Hs Child Fig.. The crossover. x Hs / MUTATION / x Hs 5 Hs Hs Hs 3 Hs Hs 3 Fig.. The mutation. Appendix B. The optimum GP programme (Note: This programme can be run in the Discipulus software [3] (interactive evaluator mode), although it can also be compiled in C environment)

17 93 float DiscipulusCFunction(float H s ; H s ; H s3 ) { double f[]; float v[3]; double tmp ¼ 0; v[0] ¼ H s ; v[] ¼ H s ; v[] ¼ H s3 ; f[] ¼ 0; f[0] ¼ v[0]; L: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L: tmp ¼ f[]; f[] ¼ f[0]; f[0] ¼ tmp; L3: f[0] ¼ v[]; L4: f[0]* ¼ 0.5; L5: f[0]* ¼ v[]; L6: f[0] ¼ v[]; L7: f[0] ¼ f[]; L8: f[0]* ¼ 0.5; L9: f[0] ¼ f[]; L0: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L: f[0] ¼ fabs(f[0]); L: tmp ¼ f[]; f[] ¼ f[0]; f[0] ¼ tmp; L3: f[0] ¼ v[]; L4: f[0]* ¼ 0.5; L5: f[0] ¼ f[]; L6: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L7: f[0]* ¼ 0.5; L8: f[0] ¼ f[]; L9: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L0: f[0] ¼ fabs(f[0]); L: tmp ¼ f[]; f[] ¼ f[0]; f[0] ¼ tmp; L: f[0]* ¼ f[0]; L3: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L4: f[0]* ¼ 0.5; L5: f[0]* ¼ f[0]; L6: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L7: f[0]* ¼ 0.5; L8: f[0]* ¼ 0.5; L9: f[0]* ¼ 0.5; L3: f[0]* ¼ 0.5; L3: f[0]* ¼ 0.5; L3: f[0] ¼ f[0]; L33: f[0]* ¼ 0.5; L34: f[0]* ¼ 0.5; L35: f[0]* ¼ 0.5; L36: f[0] ¼ f[];

18 94 ARTICLE IN PRESS L37: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L38: f[0]* ¼ v[]; L39: f[0]* ¼ 0.5; L40: f[0]* ¼ 0.5; L4: f[0]* ¼ 0.5; L4: f[0]* ¼ 0.5; L43: f[0]/ ¼ v[0]; L44: f[0] ¼ f[]; L45: tmp ¼ f[0]; f[0] ¼ f[0]; f[0] ¼ tmp; L46: f[0]* ¼ 0.5; L47: f[0]* ¼ 0.5; return f[0]; } References [] [] Arena G, Briganti G, Corsini S, Franco L. The italian wave measurement buoy network: years management experience. In: Proceedings of the fourth international symposium waves 00, San Francisco, CA, 00. p [3] Puca S, Tirozzi B, Arena G, Corsini S, Inghilesi R. A neural network approach to the problem of recovering lost data in a network of marine buoys. In: Proceedings of International Society of Offshore and Polar Engineers, ISOPE-00, vol. III, Norway, 00. p [4] Mutreja KN. Applied hydrology. New York: Tata McGraw-Hill Publications; 987. [5] Bennis S, Berrada F, Kang N. Improving single variable and multi-variable techniques for estimating missing hydrological data. J Hydrol 997;9: [6] Elshorbagy A, Simonovic SP, Panu US. Estimation of missing streamflow data using principles of chaos theory. J Hydrol 00;55:3 33. [7] Khalil M, Panu US, Lennox WC. Groups and neural networks based streamflow data in-filling procedures. J Hydrol 00;4: [8] Thompson R. Spectral estimation from irregularly spaced data. IEEE Trans Geosci Electron 97;GE- 9:07 9. [9] Sturges W. On interpolating gappy records for time series analysis. J Geophys Res 983;88: [0] Emery WJ, Thomson RS. Data analysis methods in physical oceanography. Amsterdam: Elsevier Publication; 004. [] Stefanakos ChN, Athanassoulis GA. A unified methodology for the analysis, completion and simulation of nonstationary time series with missing values, with application to wave data. Appl Ocean Res 00;3(4):07 0. [] Balas CN, Koc L, Balas L. Predictions of missing wave data by recurrent neuronets. J Waterway Port Coastal Ocean Eng ASCE 004;30(5): [3] Makarynskyy O, Makarynska D. Wave prediction and data supplementation using artificial neural networks. J Coastal Res 007;3(4): [4] Koza JR. Genetic programming: on the programming of computers by means of natural selection. A Bradford book. Cambridge: MIT Press; 99. [5] Whigham PA, Crapper PF. Modeling rainfall-runoff using genetic programming. In: Mathematical and computer modeling, vol. 33, Canberra, Australia, 00. p [6] Muttil N, Liong SY. Improving runoff forecasting by input variable selection in genetic programming. In: Proceedings of world water congress 00, ASCE, Downloaded 04 August 005 from / ascelibrary.org/s. [7] Babovic V, Kanizares R, Jenson HR, Klinting A. Neural networks as routine for error updating of numerical models. J Hydraulic Eng ASCE 00;7(3):8 93. [8] Babovic V, Drecourt J-P, Keijzer M, Hansen PF. A data mining approach to modeling of water supply assets. Urban Water 00;4:40 4.

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