Group-based estimation of missing hydrological data: II. Application to streamflows

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1 Hydrological Sclences~Journal-des Sciences Hydrologiques, 45(6) December Group-based estimation of missing hydrological data: II. Application to streamflows AMIN A. ELSHORBAGY Civil and Geological Engineering Department, University of Manitoba, Winnipeg, Manitoba R3T5V6, Canada umelshor(5>,cc.umanitoba.ca U. S. PANU Civil Engineering Department, Lakehead University, Thunder Bay, Ontario P7B 5EI, Canada timed.panu(q;lakeheadu.ca S. P. SIMONOVIC Department of Civil and Environmental Engineering, University of Western Ontario, London, Ontario N6A 5B9, Canada Abstract The group approach that treats hydrological data as groups rather than as single-valued observations was proposed in a companion paper. Various models representing four techniques are briefly presented and applied to single series and biseries cases, respectively, in this paper. The techniques represented by these models are regression, time series analysis, partitioning modelling, and artificial neural networks. The utility of the models for estimating missing streamflow data using the group approach is investigated. It turns out that the group approach is valid for estimating missing values, and possibly other applications, when data are significantly auto-correlated. L'estimation de groupe pour l'estimation des données hydrologiques manquantes: II. Application aux débits Résumé L'approche de groupe, traitant les données hydrologiques comme des suites de groupes plutôt que comme des suites d'observations singulières a été présentée dans un papier préliminaire. Divers modèles utilisant quatre techniques différentes sont brièvement présentés et ont été appliqués aux cas de séries simples ou doubles. Ces techniques sont la régression, l'analyse de séries chronologiques, la segmentation et les réseaux de neurones artificiels. L'utilité de ces modèles pour estimer les données de débit manquantes en utilisant l'approche de groupe a été examinée. Il apparaît que l'approche de groupe est appropriée pour estimer les valeurs manquantes, et probablement pour d'autres applications, quand les données sont significativement autocorrélées. INTRODUCTION The approach introduced in Elshorbagy et al. (2000a), hereafter referred to as Part I, is utilized for estimating the missing data. The aim of this paper is to demonstrate the application of the techniques presented in Part I, using the group approach. The application consists of two cases: (a) single series case, where only one time series is available for the analysis and (b) bi-series case, where two correlated time series are available, the reference one with complete record and the target one that has missing observations. Open for discussion until I June 2001

2 868 Amin A. Elshorbagy et al. APPLICATION OF THE GROUP APPROACH TO STREAMFLOW RECORDS Different models representing four techniques are applied to monthly streamflows of three rivers. These rivers are the English River at Umferville, Ontario, Canada, and the Little River and Reed Creek, Virginia, USA. Table 1 summarizes information such as drainage area, length of records used for calibration and verification of the different models, and lag-1 autocorrelation (AC) coefficient for each river. Prior to segmenting the time series records into groups, the data are pre-analysed as single-valued observations. The absence of outliers and other indications of major interventions in the natural flows need to be verified. In spite of the importance of such pre-analysis, it is a step that is usually overlooked by analysts (Salas et al., 1980). For the case of the English River, flow record at another station, Sioux Lookout, is used to verify the flows at Umferville. The Little River and Reed Creek are used together for the bi-series analysis. Therefore, they can be used to verify each other. The two rivers are from the same basin with cross-correlation coefficient equal to Original monthly data and mass curve are plotted for each river. Also, for each pair of comparative records (Little River vs Reed Creek and English River at Umferville vs Sioux Lookout), double mass curves and original flows are plotted. These graphs help detect the existence of extreme values, outliers, or any significant slope change in the curves. The need for deletion or correction of observations is not detected in the preliminary investigations on the data sets. Table 1 Basic information on the three rivers under consideration. River English River Little River Reed Creek Drainage area (km") Calibration Verification AC Segmentation The English River is a challenging example for any segmenting or clustering technique because the annual and monthly values of the mean flows and the variances are larger than normally expected. The mean value of the whole series is calculated to be 56.9 m 3 s" 1 and is taken as a reference point from which distances to all months are calculated. Distances are allowed to be of either positive or negative value. For simplicity of illustration, the average values of the distances are plotted (Fig. 1). By visual inspection of Fig. 1, one can draw vertical lines in two places, first between April and May and second between July and August. At the two partitions there are jumps in the calculated distances, which are big enough to justify the chopping. So, tentatively there are three pattern classes (PCs) as follows: PCI (January, February, March, April); PC2 (May, June, July); and PC3 (August, September, October, November, December). Each pattern class consists of one segment type with N realizations; where Nis the number of years. For PCI each segment can be considered as a point in the four-dimensional Euclidean space. Likewise, PC2 and PC3 are considered to be of three and five dimensions, respectively, in the Euclidean space.

3 Group-based estimation of missing hydrological data: II. Application to stream/lows 869 -~ 60 O 0 S on j 5 ^o Month ( ) Fig. 1 Monthly distance from the series mean flow. One of the useful ways of representing the data to help segment them and also to judge the resulting segments is by plotting them on a box-plot graph. The box-plot for the monthly flows of the English River (Fig. 2) shows the interquartile, the median, the range, the outliers (shown as asterisks) for every month. It should be noted that outlier is defined here as the value more than 1.5 box-lengths from the 25th or 75th percentiles. The monthly box-plots provide further support for data partitioning into three classes. It can be noted from Fig. 2 that the first four months form a class, the following three or four months are close to each other, forming another class, and the last four months can be considered as forming a third class. Any claims that one can reduce the variability of the third class by splitting it into more classes can be refuted by a simple fact: the variability is not concentrated in one month, but rather exists in four out of the five months constituting the class. The correlation matrix of the monthly flow data of the English River is depicted in Table 2. Moving horizontally from each month can give an idea about the trend of the correlation among months. Assuming a threshold correlation coefficient value of 0.5, below which the two variables (months) could be considered insignificantly correlated, we can cluster the data into classes. A vertical boundary can be drawn when the value before is more than 0.5 and the value after is less than 0.5. It is clear from Table 2 that a vertical boundary between April and May can be drawn. One can argue that the 400 f "i i i i i i i r JANFEBMARAPRMAYJUNJULAUGSEPOCTNOVDEC Fig. 2 Box-plots of the monthly flows of the English River at Umferville.

4 870 Amin A. Elshorbagy et al. Table 2 Correlation matrix of the monthly flows of English River at Umferville. Month Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. Jan. Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec boundary can be drawn between March and April, but this can be refuted by presenting the difficulty of associating April with May because of their weak correlation. So April is better classified with March. The more difficult situation arises in the vicinity of August and September. Drawing the boundary after September means taking out from October, November, and December, a month that is significantly correlated with all of them. The line could be drawn after either July or August. In both cases there is no clear cut boundary, and the trade-off should be balanced and judged by the analyst. In this paper, the boundary is considered after August to make the case easier by having three seasons of equal lengths. Two important observations can be derived from the correlation matrix. The first is that within each class lag-1 and lag-2 autocorrelation, for example, could exist while jumping from one class to the other might not be statistically justified (e.g. lag-1 correlation between April and May < 0.5). Secondly, in the case of applying the traditional ACF and PACF to the data, an AR(1) or AR(2) time series model might be recommended. It happens due to the process of averaging the lag-& autocorrelation coefficients over the entire data set. Therefore, care must be exercised in the case of applying traditional time series analysis, which ignores the concept of groups, on such a data set. It is believed that these two observations lend a strong support to the grouping approach proposed in this paper. The same procedure of segmentation is repeated with the Little River and Reed Creek except the correlation matrix (because of the weakness of the autocorrelations for both of the two rivers). Two pattern classes PCI and PC2, each of six months duration, are identified for the two rivers. Table 3 summarizes the percentage of the correct and mis-classified rivers segments using the Euclidean distance norm. A segment belonging to PCI of the English River has low probability of being mis-classified while a segment belonging to PC3 has a higher chance of being mis-classified due to the high variance of PC3. As a concluding remark for the process of data clustering or grouping, one can say that grouping is not only a necessary step for our further analysis, but it also helps provide a deep insight into the data at hand. Searching the data for a structure of natural groupings is an important explanatory technique. Grouping can provide an informal means for assessing dimensionality, identifying outliers, and suggesting interesting hypotheses concerning relationships (Johnson & Wichern, 1988).

5 Group-based estimation of missing hydrological data: II. Application to stream/lows 871 Table 3 Percentage of classification error using Euclidean distances for the data segments. River Class Euclidean distance (d E ) PCI PC2 PC3 English River Little River Reed Creek PCI PC2 PC3 PCI PC2 PCI PC2 93% 7% 41% 69% 15% 84% 2% 0% 68% 22% 31% 85% 16% 98% 7% 25% 37% N/A N/A N/A N/A Identifying inter-group relationships towards model selection Having the data set segmented as consecutive groups, one should proceed with studying the inter-group structure. The traditional or single-valued approach is based on understanding the time series as a sequence of individual observations. Therefore, identification of the structure linking the individual observations is required to model the time series in the single-valued approach (whether in regression or time series analysis). In the proposed approach, the model entities are groups of observations rather than individuals, and the links sought are supposed to represent the relationships among vectors rather than single-valued observations. Characteristics such as trend, seasonality, independence, autocorrelation and cross-correlation coefficients, and normality distribution should be checked in relation to data groups. Autocorrelation and cross-correlation Autocorrelation and cross-correlation coefficients among groups are ideally represented in a form of (k*k) matrix where k is (a) (b) 2 1 t c ' 5 o o 8? o JD a «,-_-CL Lag number t Q. -1 J CT> T - CO Lag number 1 coefficient - Confidence limits Fig. 3 (a) Autocorrelation function of data groups, and (b) partial autocorrelation function of data groups (English River).

6 872 Amin A. Elshorbagy et al. c 1i o 1 c 0.5 -I _u_o_^ n_ 81 S a T- CO IO N- CO Lag number Fig. 4 Autocorrelation function of data groups (Little River). 3 coefficient - Confidence limits 1 T- 13 c 0.5 ô.2 t -S 0 o c o < -1 CO 1X5 N- CO Q-U u g p-o-o-o-tj-n-~- - -a-n-o-n^^-p- Lag number 3 coefficient - Confidence limits Fig. 5 Autocorrelation function of data groups (Reed Creek). the number of elements in each segment. For English River, the autocorrelation coefficient of any lag is (4*4) matrix while auto- and cross-correlation coefficients are represented by (6*6) matrices in the cases of Little River and Reed Creek. Such matrices are difficult to interpret and cannot be represented on two-dimensional plots as autocorrelation function (ACF) and partial autocorrelation function (PACF). For the purpose of this paper, it is considered that a mean value of each segment can satisfactorily represent that segment in studying the correlation structure. The ACF and PACF of the English River data groups are respectively plotted as Fig. 3(a) and 3(b). The slow decay of the ACF and the cutoff in the PACF after lag-1 demonstrate that a lag-1 autoregressive group, ARG(l), model can be successful in modelling the English River data as groups. The independence of the Little River and Reed Creek data represented by the ACFs in Figs 4 and 5, as well as the significant group cross-correlation (Fig. 6), refutes the utility of time series analysis in this case (bi-series case). It should be noted that the lag-1 autocorrelation may look significant according to Fig. 4, but there is no gradual decay that indicates existence of serial dependence. The regression technique lends itself to studying the Little River and Reed Creek as a bi-series case. Two other techniques, ANNs and MPM, are used along with regression and time series analysis for comparison purposes. Information about autocorrelation or independence is not required for the application of ANNs and MPM techniques. Deterministic components Trends, shifts, and cyclicity in the mean and standard deviation can be removed before modelling the data groups. Perceiving the data as groups does not affect the definition of these deterministic components. Removing these components can be done in the same way as in the traditional single-valued approach. Details of such procedures are provided by Salas (1992).

7 Group-based estimation of missing hydrological data: II. Application to stream/lows 873 c o 1 - * ^ «*- e 0.5 a <u t: '3 0 8* -0.5» «o o o ~TTgp~ m-m-^juim U.,» ffoj '- r g~,r ~ t -1 -J i-oom(na)noco<ocricnu-)00->- CM * - i- <- ' ' T - T - T - C M Lag number! I i coefficient Confidence limits Fig. 6 Cross-correlation function of data groups (Little River and Reed Creek). Multivariate normality It is generally recommended that data follow normal distribution given in equation (1), while using time series analysis and MPD technique: 1 -l/2(x-(jc-'(x- i) f(x) = (1) (27t)* /2 C I where k is the dimension of the group, which is equal to number of elements in each segment, and C is the covariance matrix of the pattern class. In regression technique, the residuals have to be normally distributed while the variables themselves need not follow normal distribution (Makhuvha et al, 1997). To check the multivariate normality, the following relationship should hold for a ^-dimensional normal distribution: 'x-^c-'(x-^<xhp) (2) where x*( a ) ' s me upper 100th percentile of a chi-square distribution with k degrees of freedom (Johnson & Wichern, 1988). For the three rivers involved in this study, it is found that log-transformation is sufficient to bring the data close to multivariate normal distribution. Other methods of transforming the data to multivariate normal may lead to better transformation and improvement of final results, but only logtransformation is applied in this paper. Application to the single series case (English River) In the following, three techniques for estimating missing data, namely, time series modelling, multivariate partitioning modelling, and artificial neural networks are presented. Time series model, ARG(p) Analysing the ACF and PACF of the data groups reveals that ARG(l) is worth testing. In general AR(1) and AR(2) are the most commonly used models of the AR(p) type in hydrology. Needless to say, perceiving the data as groups can further weaken the autocorrelation links among data constituents (groups rather than single-valued observations) due to the local autocorrelation within each group. ARG(l) or even independent groups type of modelling might be the most dominant structure in modelling monthly streamflows using the group approach. The model represented by equations (8) and (9) in the companion paper is used with the calibration data to estimate the model parameters.

8 874 Amin A. Elshorbagy et al. The parameter matrices A\ and B are estimated as follows (Salas et al, 1980): À, = M, M: 1 BB' =M 0 -A I M; (3) (4) Pk >P*. PÏ" M k =E[z,Zl k ] = (5) P* > P* _ where T denotes the transpose of the matrix, p\ = corrfz/'-*, Z,^] is the lag-& crosscorrelation coefficient between Z, and Z t -k f r ; ^ i As a model verification step, the random vector of the English River is checked. It is uncorrelated in time and space, that is, Elzfz^ ] = 0 for i it j. Periodic auto regressive groups PARG(l) model To check the influence of the periodicity of the autocorrelation among the three classes of the English River data, a periodic model given in equation (6) is developed: Z ViI =/É lit Z Vjt.,+i? t e v, t (6) where three (4*4) periodic coefficient matrices represented by A\, z and three (4*4) matrices represented byb T are estimated. Model verification procedures similar to those mentioned earlier are conducted. Multivariate partitioning modelling (MPM) technique The form of MPM given in equation (10) of Part I (Elshorbagy et al., 2000a) is rewritten to handle the case of three classes to be applied to the English River. The mean values of classes 1, 2, and 3 are given below. Mean 1 = p (l) + C n C;l (x3 - y <3) ) Mean 2 = u/ 2) + C 21 C n ' (A"1 - u_ ( Mean 3 = p. (3) + C 32 C 2 " 2 ' (x2 - y (2) ) (7) (8) (9) x = ~x\~ X2 X3 andc = c c c ""ii *-n '-13 c c c c c c ^32 33 For simplicity, each of the three above equations is considered to represent the mean of the conditional distribution of a specific class, given the values of the preceding class and independently of the third class. These equations are used to estimate the missing segments of the English River data at Umferville.

9 Group-based estimation of missing hydrological data: II. Application to stream/lows 875 Artificial neural network (ANN) technique Multilayer feed forward ANN with back propagation learning algorithm is applied to the English River data. After many trials, the network configuration that estimates the missing data in the best way is ANN (7-6-4) with seven input nodes, six hidden nodes, and four output nodes. The reason for having three extra input nodes is the use of the monthly mean, annual mean, and monthly variance as inputs. The trials showed that these additional inputs help accelerate the convergence process of the ANNs. One can easily notice that the three additional inputs represent three components that are recognized during the preanalysis of time series analysis. These components are the trend, monthly average value, and monthly variance. A lag-1 transition model is assumed such that any segment of four observations is estimated based on the previous segment. Therefore, data are arranged in a way that makes a (180*7) matrix of the input data and a (180*4) matrix of the output data of the ANN model. The 180 rows represent 60 years of data with three segments each. The estimated values of the English River corresponding to the missing data using the four models are given in Fig. 7 along with the observed values for comparison. It should be noted that the estimated values are calculated as segments of four consecutive values. Each segment is estimated in one step at a time (i.e. the estimated value within the segment is not used for estimating the successive one). ^r~ Cfl m im fc U) 5 o U , to >> o. r-- >* o_ co >. a. o> ^o.o >» a. «- >-. O. inffljjcotllsibibltoosjjolffldlolfflo co co ca co ce co Actual - - ARG(1) -* RûRG(1) ANSI -* IVPM Month (198&-1991) Fig. 7 Estimated and observed values of the English River. Application to the bi-series case In the following, three techniques, namely, multivariate multiple regression, multivariate partitioning modelling, and artificial neural networks are presented. Multivariate multiple regression (MMR) technique The model equation and the parameter estimation technique are given in the companion paper (Elshorbagy et al., 2000a): F is a (102*6) data matrix of the "target river" (Little River), and Z is a (102*6) matrix of the "reference river" (Reed Creek). The 102 rows represent 51 years of data with two segments each. The model is verified by checking the independence of the errors and their distribution. The errors are found to be independent and their distribution is close to normal.

10 876 Amin A. Elshorbagy et al. MPM technique Equation (10) in the companion paper is ideal for the bi-series case where X2 is the segment known from the "reference river" (Reed Creek) and X\ is the segment to be estimated for the "target river" (Little River). The data of both rivers are arranged in a way that collates the data from each of the two concurrent classes of both rivers in one matrix. The (102*12) data matrix can be used to calculate the main covariance matrix, C. The constituent components of this matrix are C\\, Cn, C21, and C22. ANN technique Data in 102 rows are arranged in such a way that there are six inputs to the network and six outputs. The six inputs are the number of elements in each class of the Reed Creek while the six outputs are the elements in each class of the Little River. It should be noted that the 102 refers to the number of data rows used for training the ANN model. ANN (6-8-6) is found to provide the best estimate of the missing segments for the Little River. The estimated values of the Little River missing data using the three models, for the bi-series case, are shown in Fig. 8 along with the observed values. It should be noted that estimated values are calculated as segments of six consecutive values (one segment at a time) in one step rather than one by one single-valued outputs. 50 -T _ o 00 I c,a^-1( tf! O T - T - N C N C O n ^ * l O U ( D ( D N N «) C O I I l O ) cooooooooooocooooocacooooooooocooococo c c c c c c c c Month ( ) Fig. 8 Estimated and observed values of the Little River. -Actual ~MMR ANN --MPM Estimation of partially missing segment Sometimes, one might face the case of having only part of a segment missing. Due to adoption of the group approach in this paper, each segment is considered as a single object. Thus, a segment is estimated as an inseparable set of consecutive values. When missing values constitute only a part of a segment, a solution should be found in a way that integrates explicitly the information available from the known part of the segment. It is assumed here that the first value of the missing segment is known and the rest are to be estimated. In other words, only three and five observations respectively are missing out of the segments in the English and Little rivers. Similarly, other varieties of partial segments are possible, but, for demonstration purposes, only the first value of the segment is assumed to be known. First, keeping the modelling procedures

11 Group-based estimation of missing hydrological data: II Application to stream/lows 877 unchanged, the whole segment (including the first known observation) is assumed missing and estimated in one step as done before. Second, the percent error of the first value is calculated as follows. Pe del ~~ d k d., (10) where d e \ is the estimated value of the first observation of the segment, and dt\ is the known value of the first observation. Third, the rest of the estimated values within each segment are modified according to the (p e ) of the first known value using the following equation: ^,,,=(i-(/0)*<, where d mei is the modified value of any estimated value in the missing segment, and d ei is the estimated value of any observation after the first one in the missing segment. Figures 9 and 10 show the modified estimated values of the English River and Little River assuming that the first observation of the missing segment is known. (11) Actual - - Mod.MPM I o o ' - ^ N N n n ^ ' j i i i i n i o i o s N t o c o i i i i D ODODCOCOOOCOODOOCOOOCOOOOOODOOOOOOCOODOO -* Mod.ANN Months ( ) Fig. 9 Modified estimates of the missing segments (English River). ou r ^ 40 tfl E 30 in 5 o u M ri * \Rk f ^\JL \$ \j uhjt& jf yy*w AAiïL 0, 0 0 ' - r - ( \ N l ) n t ' t l 0 1 I ) l D ( O N S ( D «) ( J ) r a 000D0O0O ODCOCOCOCOCOCOCO(K)0O0O Actual Mod.MPM -» Mod.ANN Months ( ) Fig. 10 Modified estimates of the missing segments (Little River).

12 878 Amin A. Elshorbagy et al. DISCUSSION To evaluate the relative performance of different models, the error statistics should be used to quantify that performance. The mean squared error (MSE) and the mean relative error (MRE) are two of the most commonly used statistics for evaluation of model performance in water resources. Table 4 summarizes the results of performance of various models for estimating the missing values in the single series case (English River) based on MSE and MRE values. The last two columns of the table show the model performance when the first observation of the missing segment is known. The results reflect the superiority of the ANN model in estimating the missing values when the whole segment is missing. On the other hand, the ANN model becomes inferior to the rest of the models when the first observation of the missing segment becomes known. The results of the ANN model do not improve with the acquisition of additional information (first observation in the segment). This is because the modification process conducted using equations (10) and (11) is based on the group concept, and the whole segment is assumed to have consistent signs (+ve or -ve) of the relative errors for all elements within the segment. It seems that the ANN model, during its search to minimize the squared error, tends to violate the group-structure of the data. Perhaps it treats each element within the segment differently. Therefore, the apparent superiority of the ANN model when the whole segment is missing should not be generalized to include all cases of missing data. Similar to Table 4, Table 5 summarizes the results for the bi-series case (the Little River). The ANN model tends to provide better estimates for the missing values when the whole segment is missing. The last two columns of Table 5 indicate that none of the adopted techniques is able to benefit from the availability of the first element of the missing segment. It is noted that the differences between the MSE values of different models in Table 5 are not big; however, the lower the MSE value, the better the model performance. A modified measure that can show the significance of such a difference is proposed by Elshorbagy et al. (2000b). In this paper, the MSE and MRE are considered sufficient as model performance measures. Keeping in mind that the three techniques follow different routes to approach the problem solution, the utility of the Table 4 Summary of model performance for the English River case. Models ARG(l) PARG(l) MPM ANN Whole missing segment: MSE MRE Partial MSE missing segment: MRE Table 5 Summary of model performance for the Little River case. Models Whole missing segment: MSE MRE Partial MSE missing segment: MRE MMR MPM ANN

13 Group-based estimation of missing hydrological data: II Application to streamfiows 879 grouping becomes questionable in this bi-series case. One explanation, supported by the relative success of the group approach in the English River case and by the improper results in the Little River case, is the role of the autocorrelation. The idea of tying each group of elements in one segment to behave as a single object seems to require significant autocorrelation among the single-valued elements. Therefore, the significant autocorrelation of the English River, unlike that of the Little River, plays a major role in supporting the group approach. Another point of interest in light of the modification process described by equations (10) and (11) is the significant improvement in the accuracy of estimating the values of the partially-missing segments of the English River which diverts attention toward the utility of the group approach for forecasting purposes. A complete segment of four elements (monthly values) can be forecast as a first step. Then, when the first value becomes available, a significant improvement in the following three monthly values can be achieved without remodelling or re-estimation of the model parameters. This result not only saves the time and effort of re-estimating model parameters, but also provides satisfactory estimates for multiple steps ahead (three steps in this example). CONCLUSION This paper demonstrates that monthly streamflow data can be clustered into a definite number of classes according to the group approach. The flow values and their variances are reasonable characteristics that can be used for data segmentation. The results of the application of different techniques for estimation of missing segments indicate that the autocorrelation among elements of each segment plays a major role in the success of the group approach. In both, single series and bi-series cases, the artificial neural network model shows superiority over other models, applied in this paper, in estimating missing segments using the group approach. However, care must be exercised when ANN models are applied to cases of partially-missing segments. ANNs tend to violate the group structure of the data to minimize the overall errors of the estimates. Other techniques represented by group time series models (ARG(l) and PARG(l)), developed in this study, and MPM are more consistent in handling data groups. Their accuracy of estimating missing data improves when a part of the missing segment becomes available. The discussion given earlier on the utility of the group approach for forecasting applications indicates that the group approach may have potential for other applications in water resources. The success of the group approach is contingent on the segmenting process and the selection of the models that can handle groups. Segmentation procedures and models used in this paper are not claimed to be exhaustive. Improvement in these two aspects of the group approach reflects on the final results of the application. Finally, the group approach is not proposed as a replacement for the traditional single-valued approach but rather as a sound alternative when the hydrological data show clustering characteristics and as a strong candidate when consecutive observations are missing. Acknowledgements This research has been supported by the Natural Sciences and Engineering Research Council of Canada.

14 880 Amiii A. Ehhorbagy et al. REFERENCES Elshorbagy, A., Panu, U. S. & Simonovic, S. P. (2000a) Group-based estimation of missing hydrological data: I. Approach and general methodology. Hydrol. Sci. J. 45(6), (this issue). Elshorbagy. A., Simonovic, S. P. & Panu, U. S. (2000b) Performance evaluation of artificial neural networks for runoff prediction. J. Hydrol. Engng ASCE 5(4), Johnson, R. A. & Wichern, D. W. (1988) Applied Multivariate Statistical Analysis. Prentice Hall, New Jersey, USA. Makhuvha, T., Pegram, G., Sparks, R. & Zucchini, W. (1997) Patching rainfall data using regression methods. 2. Comparisons of accuracy, bias and efficiency. J. Hydrol. 198, Salas, J. D Delleur, J. W.. Yeyjevich, V. & Lane, W. L. (1980) Applied Modelling of Hydrologie Time Series. Water Resources Publications, Littleton, Colorado, USA. Salas, J. D. (1992) Analysis and modeling of hydrologie time series. In: Handbook of Hydrology (ed. by D. R. Maidment), McGraw-Hill, New York, USA. Received 1 November 1999; accepted 26 June 2000