Using probabilistic finite automata to simulate hourly series of global radiation

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1 Solar Energy 74 (2003) Using probabilistic finite automata to simulate ourly series of global radiation a L. Mora-Lopez *, M. Sidrac-de-Cardona a, b Dpto. Lenguajes y C. Computacion, E.T.S.I. Informatica, Universidad de Malaga, Campus de Teatinos, Malaga, Spain b Dpto. Fısica Aplicada II, E.T.S.I. Informatica, Universidad de Malaga, Campus de Teatinos, Malaga, Spain Abstract A model to generate syntetic series of ourly exposure of global radiation is proposed. Tis model as been constructed using a macine learning approac. It is based on te use of a subclass of probabilistic finite automata wic can be used for variable-order Markov processes. Tis model allows us to represent te different relationsips and te representative information observed in te ourly series of global radiation; te variable-order Markov process can be used as a natural way to represent different types of days, and to take into account te variable memory of cloudiness. A metod to generate new series of ourly global radiation, wic incorporates te randomness observed in recorded series, is also proposed. As input data tis metod only uses te mean montly value of te daily solar global radiation. We examine if te recorded and simulated series are similar. It can be concluded tat bot series ave te same statistical properties Elsevier Ltd. All rigts reserved. 1. Introduction next values of te series from teir predecessors. Te approac is as follows: first, te model must be identified; Different approaces ave been followed to caracterize to do tis, te recorded series are statistically analyzed in te ourly series of solar global radiation. Taking into order to select te best model for te series. Ten te account te nature of tese series, we propose te use of a parameters of te model must be estimated. After tis, a new model for teir caracterization and simulation. Tis new series of values can be generated using te estimated new model is easy to use once it as been constructed and model. For example, tis approac as been followed in it allows us to represent te relationsips observed in te Brinkwort (1977), Bendt et al. (1981), Aguiar et al. ourly series of global radiation. Moreover, it can be (1988), Aguiar and Collares-Pereira (1992), and Moraembedded in engineering software by including te esti- Lopez and Sidrac-de-Cardona (1997). mated probabilistic finite automata and te algoritm One of te problems wit most of tese metods is tat presented in Section 5. Before explaining te model, we te probability distribution functions of te generated briefly review te existing models, paying special attention series are normal wen stocastic models are used. Tis to teir simplicity, requirements and limitations. problem can be solved for daily series using first-order Several studies ave been carried out to obtain models Markov models (see Aguiar et al., 1988). For ourly series, wic allow us to simulate te ourly series of solar global to circumvent tis problem, a differenced series and radiation. Traditionally, te analysis of time series as ARMA models can be used (e.g. Mora-Lopez and Sidracbeen carried out using stocastic process teory. One of de-cardona, 1997); owever, in tis case te simulation of te most detailed analyses of statistical metods for time a new series uses a complex iterative process: te use of series researc was performed by Box and Jenkins (1970). te differences operator makes it difficult to generate new Te goal of data analysis by time series is to find models series of global radiation because it is necessary to wic are able to reproduce te statistical caracteristics of eliminate te negative values wic appear in te series. te series. Moreover, tese models allow us to predict te Te aim of tis work was to study te use of a matematical model called probabilistic finite automata *Corresponding autor. Tel.: ; fax: (PFA) as a means of representing te relationsips ob served in ourly global solar radiation series. PFAs are address: llanos@lcc.uma.es (L. Mora-Lopez). matematical models developed witin te fields of Artifi X/ 03/ $ see front matter 2003 Elsevier Ltd. All rigts reserved. doi: / S X(03)00149-X

2 236 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) cial Intelligence and Macine Learning. Te macine te year and latitude of te location were tey were learning models are very useful for studying systems in recorded (Nort latitude). Tis table also includes annual wic te goal concept presents probabilistic beavior. average values of daily global solar radiation, G dy, for eac Te prediction of climatic variables is an example of tis location. type of concept. Recently, some autors ave used differ- Te weater caracteristics of te locations are very ent types of neural networks and finite automata to model different. Tere is a location wit a moderate Atlantic te values of global solar radiation on orizontal surfaces climate (Oviedo). Tere are interior locations wic ave a (e.g. Moandes et al., 1998; Kemmoku et al., 1999; continental climate, suc as Madrid, Tortosa, etc. Finally, Moandes et al., 2000; Sfetsos and Coonick, 2000; Mora- te coastal locations ave a Mediterranean climate, wit Lopez et al., 2002). Wen neural network models ave softer temperatures bot in winter and in summer (Malaga, been used, only mean values of daily or ourly global Mallorca, etc.). radiation ave been analyzed. In te paper of Sfetsos and Coonick (2000) te developed models can be used to predict te ourly solar radiation time series, but tese models are obtained using only data from summer monts 3. Probabilistic finite automata (63 days). In all cases, te obtained models are black boxes, and no significant information can be obtained. We propose using a matematical model called prob- Tere are several programs tat allow us to generate abilistic finite automata (PFA). One of te first applications daily and ourly series of global radiation based on of tis model was proposed by Rissanen (1983) for publised models (see, for instance, Scarmer and Greif, universal data compression. Various oter practical tasks 2000), and different international projects provide metodas ave been approaced using tis matematical model, suc ologies for syntetic data generation for instance, te te analysis of biological sequences, for DNA and European JOULE III Climed Project. proteins (Krog et al., 1993), and te analysis of natural After tis brief review of te models used to caracter- language, for andwriting and speec (Nadas, 1984; ize ourly solar radiation series, we present te outline of Rabiner, 1994; Ron et al., 1998). te paper. First, we describe te matematical model Different classes of automata ave been developed. For used Probabilistic Finite Automata. We ten explain ow instance, acyclic probabilistic finite automata ave been to use tis model for te analysis and prediction of solar used for modeling distributions on sort sequences (Ron et radiation series. Ten we propose a generalization of te al., 1998). Probabilistic suffix automata, based on variable- model, based on te use of general cumulative probability order Markov models, ave been used to construct a model distribution functions and mean values of te daily clearautomata of te Englis language (Ron et al., 1994). All tese ness index. Finally, we test te model using data from allow us to take into account te temporal several Spanis locations. relationsips in a series. We propose te use of tis matematical model, probabilistic finite automata, to represent a univariate time 2. Data set series. Formally, a PFA is a 5-tuple (Q,S,t,g,q 0), were Te data of ourly exposure series of global radiation, S is a finite alpabet; tat is, a set of discrete symbols G (t)j used in tis work were recorded over several years corresponding to te different continuous values of te at nine Spanis meteorological stations. Te number of analyzed parameter. Te different symbols of S will be available observations is reported in Table 1, specifying represented by x i. In one series, te values observed can be x5x 3...x 3. To represent te different observable Table 1 Data set series for a period t to t we will use te symbols 1 m y y...y. Terefore, in te series x x...x, te 1 2 m symbol y corresponds to te value x, te symbol y to Location Years Monts Lat. Gdy x 3, etc. (8N) 22 (MJ m ) Q is a finite collection of states. Eac state corresponds to a subsequence of te discretized time series. Te Badajoz Castellon maximum size of a state number of symbols is Madrid bounded by a value N fixed in advance. Tis value is Malaga related to te number of previous values wic will be Murcia considered to determine te next value in te series and Oviedo depends on te memory of te series. Mallorca t : Q 3 S Q is te transition function. Sevilla g : Q 3 S [0,1] is te next symbol probability func- Tortosa tion.

3 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) q 0 [ Q is te initial state. Te function g satisfies te following requirement: For every q [ Q and for every x i[ S, ox [ S g(q,x i) 5 1. i Moreover, te following conditions are required: te transition function t can be undefined only on states q [ Q and symbols x [ S for wic g(q,x) 5 0; te function t can be extended to be defined on Q 3 S * in te following recursive manner: t(q,y 1,y 2,...,y t) 5 t(t(q,y 1,y 2,...,y t21),y t), (1) Fig. 2. Simplified probabilistic finite automata. were y i [ S. Te maximum size of a state number of symbols is te same probability vector as state 1. Tat is, wen te bounded by a value N fixed in advance. Tis value is symbol 1 appears, it is not necessary to know te precedrelated to te number of previous values wic will be ing value to determine te probabilities of te next symbol, considered to determine te next value and depends on te since, in bot cases (0 or 1), te probability vector of te memory of te series. next symbol is (0.5,0.5). Terefore, te PFA of Fig. 1 can Grapically, eac state is represented by a node and te be converted into te PFA sown in Fig. 2. Tis class of edges going out of eac state are labelled by symbols PFA is used to represent variable-order Markov models. drawn from te alpabet. Moreover, eac state as an Tese simplified automata are te automata proposed in associated probability vector wic includes te probabili- tis paper. Tey capture te same information wit fewer ty of te next symbol for eac of te symbols of te states tan te original automata. Moreover, tey allow us alpabet. Fig. 1 sows a simple PFA as an example. to take into account, for eac state, a different number of In tis PFA, te alpabet is composed of te symbols 0 previous values in te series. and 1. Te states of te system are described in eac node Let us define some concepts tat we will use to build te of te automata: initial (i), 0, 1, 00, 01, 10 and 11. For PFAs for ourly global radiation series. Let S 5 instance, te state labeled 01 corresponds to te following x 1,x 2,...,xnj be te set of discrete values of te analyzed sequence of values in te series: 1 as te last value and 0 variable and let S * denote te set of all possible sequences as te previous value. Te associated vectors at eac state wic can be obtained wit tese values. For any integer (node) are te probabilities wic eac symbol of te N N, S denotes te set of all possible sequences of lengt N alpabet as to appear in te next moment, after te #N and S is te set of all possible sequences wit lengt symbol sequence tat label te node as appeared. For less tan or equal to N. For any subsequence Y, represented instance, te node labeled 10 as te associated vector by y 1... y m, were y i[ S, te following notation will be (0.25,0.75); tis means tat if te current state is 10, ten used: te next symbol can be 0, wit a probability of 0.25, and 1, te longest final subsequence of Y, different from Y, wit a probability of Te continuous and discontinu- will be final(y) 5 y... y ; 2 m ous arrows represent te transition function between states te set of all final subsequences of Y will be last(y) 5 (discontinuous for 0, continuous for 1). For instance, if te y i...ym u 1 # i # mj. current state is 10, and te next symbol is 0, ten te following state will be labeled 00; but if te next symbol is 1, ten te following state will be labeled 01. In te PFA sown in Fig. 1, te states 01 and 11 ave In te next section we explain ow to build a PFA for ourly global radiation series. 4. Building PFA for ourly global radiation series Te parameter used to build te PFA is te ourly clearness index, defined as K 5 G /G, (2),0 Fig. 1. Example of probabilistic finite automata: (- - -) transition wit 0; ( ) transitions wit 1. were G is te ourly global radiation and G,0 is te extraterrestrial ourly global radiation. Te ourly clearness index series ave been constructed in an artificial way because data from different days ave been linked togeter: te last observation of eac day is followed by te first observation of te following day. Tis assumption as already been used in previous papers

4 238 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) (see, for instance, Mora-Lopez and Sidrac-de-Cardona, series, compute te frequency of Y. If 4a and 4b are 1997), and te results obtained confirm to us te validity of true, ten go to 5, else go to 6: tis ypotesis. On te oter and, te number of ours 4a. Te frequency of tis sequence is greater tan te considered for eac series (mont) is constant and equal tresold frequency. for all locations considered. Te number of ours consid- 4b. For some x p [ S, te probability of occurrence of ered for eac mont is 10 for January, February, te subsequence Yxp is not equal to te probability November and December, 12 for Marc, April, September of te subsequence final(y)x p, tat is and October and 14 for May, June, July and August. To discretize te continuous values of te clearness P(xp u Y) ± P(xp u final(y)) (5) index we ave used eigt different discrete values. Te (not equal: wen te ratio between te probabilities symbolic discrete values used to construct te PFA are is significantly greater tan one; for instance, S 5 0,1,...,7j. (3) greater tan 1.2). 5. Do Tese values form te alpabet of te PFA. 5a. Add to te PFA a node, labeled Y, and compute its Te relationsip between te values of te clearness corresponding probabilities vector. index and te symbols of te alpabet is as follows: 5b. For eac amplified sequence, Yx p: if te probability of tis amplified sequence is greater tan te 0, 0 # K, 0.35, tresold probability, ten include it in PSS. K Y 5 ]]] 1 1, 0.35 # K, 0.65, (4) 6. Remove te analyzed subsequence, Y, from PSS If tere are no elements of order o in PSS, add 1 to te 57, K $ 0.65, To generate new series we need an initial state. Te initial state we ave used is te discrete value corre- sponding to te mean value of te clearness index for eac series. Let qt be te current state. Te next symbol, y, is Q 5 S * 5 0,1,2,...,00,01,02,...,77,000,001,..., generated as follows: first, a random number r [ [0,1] is generated. Ten, we coose te only component of te 777,...,77777j. probabilities vector for te current state, q t wic In tis set, te state can correspond to te following satisfies sequence of values for te clearness index: 0.62, 0.58, j j , 0.53, y 5 yj uog(q t,y i) $ r and yj21 uog(q t,y i), r. (6) From all possible subsequences observed in te series, i51 i51 only tose wit a sufficient probability will be used to Te process continues until te lengt of te required build te PFA. Tis tresold of probability must be sequence is reaced. defined wen te PFA is built. Tus, we can generate different series wit te same Te montly series of te ourly clearness index ave initial day. Once we generate a syntetic series, we test if been grouped using te montly mean value of te ourly it is possible to accept te null ypotesis tat tis series clearness index. Te ranges for eac group are te same as and te recorded one ave te same mean and variance, tose defined for te discretization of tis parameter. For wit significance level If tis is te case, tis every interval, one PFA as been built. syntetic series is selected as a proxy for te recorded one; Te following algoritm is used to construct eac PFA: oterwise, we generate anoter syntetic series, and te 1. Compute te series of discrete values. process continues until we find a syntetic series for wic 2. Initialize te PFA wit a node, wit label null sequence. te null ypotesis is not rejected in all cases, less tan 3. Te set PSS Possible Subsequence Set is initialized 10 syntetic series ad to be generated. were A is te integer value of A. We ave not used uniform intervals to discretize te series because, in te lower and upper intervals, te frequency of values in te series is less tan in te oter intervals. Using tese expressions and te ourly clearness index series, te discrete series Yj are obtained. For instance, if te maximum order of PFA is 5, te set of all possible states will be value of o. Ifo # N and tere are elements of lengt o in PSS, ten go to 4, else Stop. 5. Generating new series of ourly global radiation wit all sequences of order 1. Eac element in tis set Ten, for eac selected syntetic series, we compare its corresponds to a sequence of discrete values. Take cumulative probability distribution function (cpdf) wit te o 5 1 as te initial value of te order tat is, te size cpdf of te recorded series. Tis comparison as been of subsequences to consider. made using te Kolmogorov Smirnov two-sample test- 4. If tere are elements of order o in PSS, pick any of statistic, wic focuses on te absolute value of te tese elements, Y. Using all discrete sequences in te maximum difference between te two empirical distribu-

5 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) tion functions (a detailed description of te Kolmogorov in Fig. 3. To analyze te relationsip between tese two Smirnov statistic can be found, for instance, in Roatgi, parameters, we ave computed te correlation coefficient 1976, Section 13.3). However, to perform te statistical between tem. Tis correlation coefficient proves to be test of te null ypotesis te two series ave te same 0.992, wic indicates a very strong relationsip between cpdf, te standard critical values of te Kolmogorov tem. Terefore, we can conclude tat te mean montly Smirnov two-sample tests cannot be used wit time series. daily clearness index wic is usually available can be Instead, we ave computed for eac statistic a bootstrap used in te proposed model instead of te mean montly P-value, following te block resampling metod wic is ourly clearness index. usually employed wit time series data (a detailed descrip- On te oter and, in te previous section we compared tion of tis procedure can be found, for instance, in te empirical distribution functions of te recorded series Davidson and Hinkley, 1997, Section 8.2.3). A large and te selected syntetic series to test te ypotesis tat bootstrap P-value means tat te two series wic are teir cpdfs are te same. However, for most locations te being compared ave a very similar cpdf specifically, if cpdfs are not available, and ence tis comparison cannot te bootstrap P-value is greater tan 0.05, te null ypot- be performed. Several autors ave sown tat te cpdf of esis tat bot series ave te same cpdf is not rejected te daily clearness index is related to its montly mean wit te usual significance level Tis procedure can value (see, e.g., Bendt et al., 1981; Hollands and Huget, be used to compare eac recorded series wit te syntetic 1983; Saunier et al., 1987). Terefore, for locations were series selected for it. te cpdf is not available, we can examine if te generated syntetic series is similar to te unobserved real series by comparing te cpdfs of te generated syntetic series and 6. Generalization of te simulation model te standard curves of cpdf described in te aforementioned articles. Te model wic as been proposed to generate a new To sum up, te general model to simulate new series of series of te ourly clearness index uses, as input data, te ourly global radiation wic we propose only uses, as mean montly value of te daily clearness index and te input data, te mean montly value of te daily solar cumulative probability distribution function of te recorded global radiation. Te procedure is as follows: first, using mont. For most meteorological stations, tese values are tis value, te mean montly value of te clearness index not available and only te mean montly values of te is calculated. Second, using tis oter value, a series of te daily global radiation are usually recorded. Tus, one of clearness index is generated, and ten it is tested if te te aims of tis paper is to caracterize te observed mean value of tis generated series is te same as te input relationsip between te recorded data and te parameters data used. Tird, to determine if te generated series can used for te proposed model. be accepted as a true series, te cpdf of tis generated On te one and, te relation between te mean montly series is compared wit te standard cpdf proposed in te daily clearness index and te mean montly ourly clear- literature for tat mean value tis comparison is perness index as been analyzed. Te former as been formed by computing a bootstrap P-value for te Kolobtained from te mean montly value of daily radiation, mogorov Smirnov two-sample statistic. and te latter from te data series. Bot values are sown In te following section, we compare te results ob- Fig. 3. Mean ourly clearness index vs. mean daily clearness index.

6 240 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) tained wen te input data are obtained from te recorded data for eac mont wit te results obtained wen te input data are te mean values of global radiation and te estimated mean cpdf. Table 2 Results obtained for eac interval of te clearness index, wit N 5 4 and tresold 2. Te tird column reports te number of monts in wic it is accepted tat te cpdf of te generated series and te cpdf of te real series are te same Interval Number Number of Percentage of monts simulated series 7. Results wit cpdf similar to real one To select te best values of te parameters to build te [0 0.35) PFAs, we ave cecked te results obtained wit different [ ) values of tese parameters. Te values we ave used are: [ ) Order of te PFA (N): from 2 to 14. N is te maximum [ ) lengt of te subsequences or number of symbols wic [ ) label eac state. Tis value can be called te mem- [ ) ory of te series. [ ) Tresold minimum number of appearances of a [ ) sequence from 1 to 5. For eac mont, syntetic series of te ourly clearness global clearness index and te standard cpdf proposed in index ave been generated, using, as input data: te literature for te corresponding mean value, using te Te mean montly value of te ourly clearness index aforementioned bootstrap procedure. Te results are reand te cumulative probability distribution function, ported in Table 3. It can be observed tat, for almost all bot obtained from recorded data. monts, te generated syntetic series are statistically Te corresponding PFA for te interval to wic te equal to te corresponding standard series. Tis means tat clearness index belongs by using te mean value of te daily clearness index only, In order to compare te simulated series wit te real a new ourly series of te clearness index can be generseries, several statistical tests ave been used. First, te ated, and, terefore, a series of global radiation. For ypoteses tat bot series ave te same mean and example, Figs. 4 and 5 sow te recorded and generated variance ave been tested. Second, te cpdfs of te series for te data from Murcia (January 1977). Fig. 6 recorded and simulated series ave been compared using sows te cpdf of bot series and te standard cpdf for te te Kolmogorov Smirnov two-sample test statistic wit a interval of tese series. bootstrap P-value, as described above. We ave also computed te ourly series of global For most of te intervals of te mean clearness index, if irradiation from te ourly clearness index series. Using te order used for te PFA is 2, te results are similar to te same metodology as before, we ave also tested tose wen te order is 4; owever, for intervals 5 and 6, weter te statistical caracteristics (mean, variance and using order 4, te PFA captures te relationsip observed cpdf) of te recorded and simulated series are te same. In in te series better tan wen using order 2. Tus, te almost all tests, te null ypotesis of equality is not selected order (maximum) for te PFA is 4. Te selected rejected, using 0.05 as significance level. For instance, minimum number of occurrences required to use a Figs sow te recorded and simulated values of sequence to build a PFA is 2. Wit te built PFAs, new sequences of te ourly Table 3 clearness index ave been generated. Te original and Results obtained using te mean values of te clearness index and generated series ave been compared again using te estimated cpdfs. Te tird column reports te number of monts statistical tests described above first it is tested weter in wic it is accepted tat te cpdf of te generated series and te te mean and variance are te same, and wen tese standard cpdf are te same ypoteses are accepted (wit significance level 0.05) te Interval Number Number of simulated cpdfs are compared wit te Kolmogorov Smirnov twoto of monts series wit cpdf similar sample statistic wit a bootstrap P-value. Te results proposed standard cpdf obtained for eac interval of te clearness index are sown [0 0.35) in Table 2. It can be observed tat, in 97.8% of te [ ) monts, it is accepted tat te recorded series as te same [ ) cpdf as te generated series using te proposed model [ ) only for interval 7 is tis percentage less tan 96%. [ ) On te oter and, to validate te generalization of te [ ) proposed model, we ave also performed a comparison [ ) [ ) between te cpdf of te generated series of te ourly

7 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) Fig. 4. K values calculated from recorded G data. Location: Murcia, January 1977 (K ). m Fig. 5. K values simulated from te PFA of te interval ourly global radiation for two Spanis locations (Malaga, January 1977 and Murcia, Marc 1977). Finally, we ave also compared te daily series of global irradiation obtained from recorded and simulated data of ourly exposure wit te same statistical metodology. Again, in tis case te null ypotesis is never rejected using 0.05 as significance level. 8. Conclusions In tis paper, a new model to simulate ourly global radiation series is proposed. Tis model is based on te use of Probabilistic Finite Automata and as been developed witin te macine learning field. We ave verified tat tis model allows us to keep all te relevant information Fig. 6. Cpdfs: standard, recorded data and simulated data. Interval Murcia, January 1977.

8 242 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) Fig. 7. Recorded values of G. Malaga, January Mean montly value of K Fig. 8. G values simulated from te PFA of interval [ ). contained in te univariate time series in an easy way. Moreover, wit tis matematical model, te different relationsips observed between different subsequences can be left out; eac subsequence only uses te memory lengt tat it requires. Using tis matematical model a set of PFAs were built, Fig. 9. Recorded values of G. Murcia, Marc Mean montly value of K

9 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) Fig. 10. G values simulated from te PFA of interval [ ). one for eac interval of te analyzed parameter te Box, G.E.P., Jenkins, G.M., Time Series Analysis Forecast- ourly clearness index. A metod to generate new series of ing and Control. Prentice-Hall, Engelwood Cliffs, NJ. ourly global radiation as been proposed. Tis metod Davidson, A.C., Hinkley, D.V., Bootstrap Metods and Teir Application. Cambridge University Press. only uses te montly mean value of te daily solar global Hollands, K.G.T., Huget, R.G., A probability density radiation. Using tis value, te constructed PFAs, and te function for te clearness index, wit applications. Solar Energy proposed standard cpdf, new series of ourly global 30, radiation, similar statistically to te real series, can be Kemmoku, Y., Orita, S., Nakagawa, S., Sakakibara, T., generated. Daily insolation forecasting using a multi-stage neural network. Te model as been cecked using several tests. Te Solar Energy 66 (3), obtained results demonstrate tat te generated and retat Krog, A., Mian, S.I., Haussler, D., A idden Markov model corded series are equal: tey ave te same mean, variance finds genes in E. coli DNA. Tecnical report UCSC-CRL- and cpdf. We can terefore conclude tat Probabilistic 93-16, University of California at Santa-Cruz. Finite Automata can be used to caracterize and predict Moandes, M., Reman, S., Halawani, T.O., Estimation of global solar radiation using artificial neural networks. Renewnew series of ourly global solar radiation series. able Energy 14 (1 4), Te data used to estimate te collection of PFAs, i.e. te Moandes, M., Balgonaim, M., Kassas, M., Reman, S., data used in te generation of new series of global Halawani, T.O., Use of radial basis functions for estimatradiation, correspond only to locations in Spain. In order to ing montly mean daily solar radiation. Solar Energy 68 (2), obtain a more general PFA it would be desirable to 161. estimate te PFA using data for oter climates. Generaliza- Mora-Lopez, L., Sidrac-de-Cardona, M., Caracterization tion of te PFA is easy to do: it is only necessary to and simulation of ourly exposure series of global radiation. recalculate tese PFA using te proposed algoritm. Solar Energy 60 (5), Readers interested in te collection of estimated PFAs and Mora-Lopez, L., Morales-Bueno, R., Sidrac-de-Cardona, M., te program to generate new series of ourly global Triguero, F., Probabilistic Finite Automata and random- ness in nature: a new approac in te modelling and prediction radiation can request tem from te autors by . of climatic parameters. In: Proceeding of te International Environmental Modelling and Software Society Congress, Lugano, Suiza, June. References Nadas, A., Estimation of probabilities in te language model of te IBM speec recognition system. IEEE Trans. ASSP 32 (4), Aguiar, R.J., Collares-Pereira, M., Conde, J.P., Simple Rabiner, L.R., A tutorial on idden Markov models and procedure for generating sequences of daily radiation values selected applications in speec recognition. In: Proceedings of using a library of Markov Transition Matrix. Solar Energy 40, te Sevent Annual Worksop on Computational Learning Teory. Aguiar, R., Collares-Pereira, M., Tag: a time-dependent, Rissanen, J., A universal data compression system. IEEE autoregressive, gaussian model for generating syntetic ourly Trans. Inf. Teory 29 (5), radiation. Solar Energy 49 (3), Roatgi, V.K., An Introduction to Probability Teory and Bendt, P., Collares-Pereira, M., Rabl, A., Te frequency Matematical Statistics. Wiley, New York. distribution of daily insolation values. Solar Energy 27, 1 5. Ron, D., Singer, Y., Tisby, N., Learning probabilistic Brinkwort, B.J., Autocorrelation and stocastic modelling automata wit variable memory lengt. In: Proceedings of te of insolation series. Solar Energy 19, Sevent Annual Worksop on Computational Learning Teory.

10 244 L. Mora-Lopez, M. Sidrac-de-Cardona / Solar Energy 74 (2003) Ron, D., Singer, Y., Tisby, N., On te learnability and Scarmer, K., Greif, J., Database and Exploitation Software. usage of acyclic probabilistic finite Automata. J. Comput. Te European Solar Radiation Atlas, Vol. 2, pp System Sci. 56, Sfetsos, A., Coonick, A.H., Univariate and multivariate Saunier, G.Y., Reddy, T.A., Kuman, S.A., A montly forecasting of ourly solar radiation wit artificial intelligence probability distribution function of daily global irradiation tecniques. Solar Energy 68 (2), values appropriate for bot tropical and temperate locations. Solar Energy 38,