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Fuzzy Inf. Eng. (2009)2:191-204 DOI 10.1007/s12543-009-0015-z ORIGINAL ARTICLE Application of System NCF Method to Ice Flood Prediction of the Yellow River Yu Guo Received: 20 May 2008 / Revised: 10 February 2009/ Accepted: 22 February 2009/ © Springer and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract Combined forecasts is a well-established procedure for improving fore- casting accuracy which takes advantage of the availability of both multiple informa- tion and computing resources for data-intensive forecasting. Therefore, based on the combination of engineering fuzzy set theory and artiﬁcial neural network theory as well as genetic algorithms and combined forecast theory, the system Non-linear Com- bined Forecast (NCF) method is established for accuracy enhancement of prediction, especially of ice ﬂood prediction. The NCF values from single forecast model for Inner Mongolia Reach of the Yellow River are given. The case shows that the method has clear physical meanings and precise consequences. Compared with any single model, the system NCF method is more rational, eﬀective and accurate. Keywords Intelligent forecast · System NCF · Ice ﬂood of the Yellow River 1. Introduction Since the inﬂuential work of Bates and Granger [1] several schemes for combining forecasts of diﬀerent models have been constructed. An important motive to com- bine forecasts from diﬀerent models is the fundamental assumption that one cannot identify the true process exactly, but diﬀerent models may play a complementary role in the approximation of the data generating process [11]. We follow this idea and consider the combination of several models (Fuzzy sets, neural network and genetic algorithm (GA)) for analyzing data which show, possibly, nonlinear characteristics. Fuzzy systems are an alternative to classical notions of set membership and logic that have their origins in ancient Greek philosophy. It was not until relatively recently that the notion of a multi-valued logic took hold. That was in the mid-sixties when Zadeh [15] published his seminal work fuzzy sets which described the mathematics of fuzzy sets theory. A large literature has evolved in the forty years since the semi- nal work on fuzzy sets, such as fuzzy recognition, fuzzy optimization, fuzzy decision Yu Guo () Pearl River Hydraulic Research Institute, Ministry of Water Resources, Guangzhou 510611, P. R. China e-mail: gymail@sohu.com 192 Yu Guo (2009) making and fuzzy forecasts etc., all those schemes and methods of fuzzy sets have been established to express people’s knowledge and experience [4]. The technique of artiﬁcial neural networks (ANNs) has been proposed as an eﬃcient tool for modeling and forecasting in recent years, mainly because of ANNs’ wide range of applicability and their capability to treat complicated and non-linear problems. They can identify and learn correlated patterns between input data sets and corresponding target values. After training, ANNs can be used to predict the output of the new independent in- put data. Thus, ANNs are ideally suited for the modeling of ecological data, which are known to have very complex and often non-linear relationships [9]. GA could provide versatile problem solving mechanism for search, adaptation, and learning in a variety of application domains, especially for those problems in which heuris- tic methods lead to unsatisfactory results. They are random search and optimization techniques guided by the principles of evolution and natural genetics. They are ef- ﬁcient, adaptive, and robust search processes, producing near-optimal solutions and having a large amount of implicit parallelism [10]. Therefore, the adoption of GAs for ice ﬂood prediction, which needs optimization of computation requirements and robust, fast, and close approximate solution, appears to be appropriate and natural. In the presented study, based on fuzzy optimization BP neural network (FONN)[2], we develop a system NCF method that incorporating FONN with engineering fuzzy sets, GA and combined forecast, and apply it in ice ﬂood prediction of Inner Mongolia Reach of the Yellow River [6]. 2. NCF principle Basic model of system combined forecast Assume that f , f ,··· , f are forecast values obtained by m diﬀerent forecast mod- 1 2 m els, w (i = 1, 2,··· , m) is weight to forecast value of ith model and it satisﬁes nor- malized condition: w = 1, then the combined forecast value is i=1 f = w f. (1) i i i=1 The equation (1) just is basic model of combined forecast, obviously it is a linear model, in modern science ﬁelds from system theory as synergetics and dissipation structure we know that nature and society are all possess nonlinear characteristics, so in this paper, according to FONN theory that we establish a system NCF method to improve precision of forecasts. System NCF method under FONN The combined forecast in this paper is developed from intelligent forecast theory [3] and its basic principles are as follows. 1) Constructing topological structure of FONN for combined forecast. Here we take a 3-layered FONN system as an example to show relationship be- tween input and output. In the neural network system, there are m nodal points in the input layer, here m can select models number of concerned combined forecast; l Fuzzy Inf. Eng. (2009) 1:191-204 193 i h Input Output Input layer Hidden layer Output layer Fig.1: BP Neural network structure nodal points in the hidden layer, which nodes number can be determined by samples distributing number; and q output nodes in the output layer (see Fig. 1). 2) Constructing an adjusted model of linked-weights by FONN for combined fore- cast. Firstly, we normalize observed values of forecast objects and forecast results of forecast models, thus we get corresponding normalized matrixes. Normalized equa- tion for forecast results of forecast models is it r = , f x , (2) it it max max where f refers to forecast value in forecast period t regarding to model ith; x is it max observed values’ maximum of forecast object at each forecast period t ;if f , itx max then let r = 1. Accordingly, normalized equation for observed values of forecast it objects is α = , (3) max where x is an observed value of forecast object at forecast period t . Assume that there are n forecast periods, as to forecast period t, its input is r , here i = 1, 2,··· , m it and t = 1, 2,··· , n . For nodal point i in the input layer, it transmits messages directly to nodal point in the hidden layer, therefore, its output equals to its input, i.e. u = r . (4) it it For nodal point k in the hidden layer, its input is as follows: I = w r . (5) kt ik it i=1 We adopt fuzzy optimization model [3] as stimulated function of FONN, here rule parameter of model optimizationα = 2 and distance parameter p = 1, then the output of nodal point k in hidden layer is 1 1 u = = , (6) kt ⎡ ⎤ 2 2 −1 −1 ⎢ m ⎥ 1+ I − 1 ⎢ ⎥ ⎢ ⎥ kt ⎢ ⎥ 1+ ⎢ w r − 1⎥ ik it ⎣ ⎦ i=1 194 Yu Guo (2009) where w is the linked-weight between the nodal point i of the input layer and the ik nodal point k of the hidden layer. The input of nodal point p in output layer is I = w u , (7) pt kp kt k=1 where w is the linked-weight between the nodal point k of the hidden layer and the kp nodal point p of the output layer. The output of nodal point p in output layer is 1 1 u = = . (8) pt ⎡ ⎤ 2 2 −1 −1 ⎢ m ⎥ ⎢ ⎥ 1+ I − 1 ⎢ ⎥ pt ⎢ ⎥ 1+ ⎢ w u − 1⎥ kp kt ⎣ ⎦ i=1 The actual output u of networks is just the response to the input r of FONN pt it system. Assume that expected output at period t isα , and the diﬀerence between the actual output and expected output is expressed by square error E 1 2 E = u −α . (9) t pt t In equation (9), the linked-weights w and w are adjusted to make E be min- ik kp t imum, less than or equal to the permissible error. At this time, FONN system is desirable and can be used for combined forecast. The weight-adjustment formulae for w and w are set out below. ik kp First, the gradient descent method is applied to obtain the adjusting values of linked-weight w and w : ik kp ∂E Δ w = −η , (10) t kp ∂w kp ∂E Δ w = −η . (11) t ik ∂w ik Hereη is learning eﬃciency. Through equation (10), the following can be derived ∂I ∂E pt Δ w = −η . (12) t kp ∂I ∂w pt kp From equation (7) we get ∂I pt = u . (13) kt ∂w pt Let −∂E ∂E ∂u pt t t δ = = − . (14) pt ∂I ∂u ∂I pt pt pt According equation (9) we have ∂E = u −α . (15) pt t ∂u pt Fuzzy Inf. Eng. (2009) 1:191-204 195 By using equation (8) we obtain ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1− w u ⎢ kp kt ⎥ ⎢ ⎥ ∂u ⎢ ⎥ pt ⎢ k=1 ⎥ ⎢ ⎥ ⎢ ⎥ = 2u . (16) ⎢ ⎥ pt ⎢ 3 ⎥ ⎢ ⎥ ∂I l pt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ w u kp kt k=1 Substituting equations (15) and (16) into (14) what we get ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 1− w u ⎥ ⎢ kp kt ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ k=1 ⎥ ⎢ ⎥ ⎢ ⎥ δ = 2u α − u . (17) pt ⎢ ⎥ t pt pt ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ w u kp kt k=1 Then the adjusting value of linked-weight w between the nodal point k of the kp hidden layer and the nodal point p of the output layer is as follows ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1− w u ⎢ kp kt ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ k=1 ⎥ ⎢ ⎥ ⎢ ⎥ δ w = 2 u α − u . (18) t kp kt ⎢ ⎥ t pt pt ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ w u kp kt k=1 Similarly, through equations (11) and (5), the following can be derived: ∂E ∂I ∂R pt t t Δ w = −η = −η r . (19) t ik it ∂I ∂w ∂I pt ik kt Let ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1− w r ⎢ ik it ⎥ ⎢ ⎥ ∂I ⎢ ⎥ ∂E ∂E ∂E pt i=1 t t t ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ δ = − = − = 2u . (20) kt ⎢ ⎥ kt ⎢ 3 ⎥ ⎢ ⎥ ∂I ∂u ∂u ∂u m kt kt kt kt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ w r ik it i=1 So ∂E ∂E ∂I pt t t = = −δ w . (21) pt kp ∂u ∂I ∂u kt pt kt Then ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1− w r ⎢ ik ik ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i=1 ⎥ ⎢ ⎥ δ = 2δ w u ⎢ ⎥ . (22) kt pt kp ⎢ ⎥ kt ⎢ 3 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ w r ⎦ ik ik i=1 Therefore, the adjusting value of linked-weight w between the nodal point i of ik the input layer and the nodal point k of the hidden layer is as follows: ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 1− w r ⎢ ik ik ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ i=1 ⎥ ⎢ ⎥ ⎢ ⎥ Δ w = 2ηr w u δ . (23) t ik it kp ⎢ ⎥ pt kt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ w r ik ik i=1 196 Yu Guo (2009) 3) Training of FONN for combined forecast Taking forecast results obtained by i− th(i = 1, 2,··· , m) forecast model at period t(t = 1, 2,··· , n) as network training samples, and according to follow method that we can obtain network training results: Step 1: Randomly presenting initial weights as initial training values in FONN of combined forecast, and according equations (23) and (18) to calculating the adjusted values of linked-weight Δ w and linked-weight Δ w that between input layer and t it t kp hidden layer, hidden layer and output layer , respectively. Step 2: According to adjusted weight equations (24), (25) to calculate adjusted weights: w (l+ 1) = w (l)+Δ w (l+ 1)+αΔ w (l), (24) ik ik t ik t ik w (l+ 1) = w (l)+Δ w (l+ 1)+αΔ w (l). (25) kp kp t kp t kp where l is the iterations number, α is momentum parameter and 0<α< 1. Step 3: Calculating actual output of samples and average error E(l) of samples set. Step 4: Assuming ΔE(l)isdiﬀerence of output between before and after itera- tion results, i.e., ΔE(l) = E(l) − E(l − 1), if ΔE(l) less than stated judging index of convergence velocity, then turn to accelerated GA [10]. Step 5: Based on assumed parameters of GA (colony scope, optimal individual number, accelerated number, weight change size etc.) and according to GA in Jin et al [10] that the iterative calculation can be run. After iteration, the program switches to FONN to train data again. Step 6: Reiteration step1-step 5 until global error less than stated calculating error precisionε . Step 7: Outputting trained linked-weights of network for forecast. 4) Forecast ij Step 1: Applying normalizing equation r = to normalize forecast values of ij max j−th forecast object by each forecast model (here f is forecast values of j−th forecast ij object under i− th model) and inputting them into FONN which has been trained. Step 2: Then FONN make response to input and output forecast value u of j− th j pt training object at period t. Step 3: Substituting maximal observed values of forecast samples into equation (26) to get forecast values: f = u x . (26) j t j pt max 5) Results check of combined forecast For checking forecast precision of jth sample that the mean absolute error (MAE) E can be took as evaluating index, and its calculating equation is j ma E = | x − f |. (27) j ma j t j t j=1 3. Case Study Ice ﬂood is a rise in water when ice is blocked in the river course. In general, ice ﬂood occurs suddenly, water rises violently and remains a long period. It is diﬃcult Fuzzy Inf. Eng. (2009) 1:191-204 197 to master its law of operation. Ice in a river course constitutes a hazard to shipping, traﬃc, hydropower, water supply and sewerage works. Furthermore, it may lead to ice ﬂood and cause huge losses [13]. Ice ﬂood in the Inner Mongolia reach of the Yellow River is quite common, mainly resulting from its special geographical location, hydro-meteorological conditions and river course characteristics. The Inner Mongolia reach lies in the top north of the Yellow River and ﬂows from south to north. The temperature of the upper river is higher than that of the lower river, so it freezes up from lower river to upper river, while it breaks up from upper river to lower river. That will cause ice to block the river course. The Inner Mongolia reach, being far from the ocean, is often controlled by Mongolia high-pressure. It always presents a continental climate, i.e., annual rainfall is scarce, summers are extremely hot and of short duration; winters are bitterly cold and long, and the annual ice period is about four or ﬁve months. Thirdly, there are many river bends and shoals in the Inner Mongolia reach. Ice ﬂoods often occur in these places during the break-up period. During every winter and the following spring, ice ﬂoods occur frequently in the Inner Mongolia reach. In recent years ice damage was severe because of changes in air temperature, human activities and scour-and-ﬁll of riverbed. Therefore, it is of particular important for ice prevention to forecast freeze-up date and break-up data accurately [7][12]. The freeze-up dates and break-up dates are calculated from reference points: 1 November (forwards) and 1 May (backwards) respectively. That is, a freeze-up date of 45 days means that the river froze up 45 days after 1 November, i.e., on 15 December; and a break-up date of 42 days means that the river ice broke up 42 days before 1 May, i.e., on 19 March. Ice ﬂood prediction of the Yellow River includes freeze-up date, break-up date etc. Based on cause analysis that we select thermodynamic factors, kinetic factors and river characteristics, which determining ice ﬂood of the Yellow River, and take them as feature factors of singular forecast model. After applying diﬀerent forecast models to obtain corresponding forecast results, we employ system NCF model presented in this paper and get combined forecast results which synthesizing data of multi-forecast model. Here we take 1968-2002 years ice ﬂood data of Bayangaole Station at the Inner Mongolia reach of the Yellow River as example and use system NCF model presented in this paper to forecast freeze-up date of the Bayangaole Station. Firstly, according to feature factors we apply intelligent forecast model [2] to operate forecast. In this paper, forecast factor x refers to accumulated air temperature day by day in the same year from date of air temperature stably turning to negative to freeze-up date; x to average water level during freeze-up date of the same year; x to average discharge during freeze-up date of the same year; freeze-up date y to days number from cal- culating reference point, viz., on 1st November, to freeze-up date of the same year. Corresponding statistic data are listed in Table 1. (i) Forecast with intelligent forecast model According to feature factors and normalized equation, we normalize the observed values data of former 28 years and apply it to construct training samples of forecast, and take latter 6 years data as check samples. Topologic structure of network is: 3- layered network structure, input layer has 3 nodal points (corresponding to 3 feature 198 Yu Guo (2009) factors), hidden layer has 5 modal points, and output layer has one nodal point. Parameters of GA are set as: error precision of network training is 0.0002, net- work tardiness judgment index between before and after iteration is 0.0001, learning eﬃciency is 0.8, momentum parameter is 0.8, weight changing size is 0.5 times of itself, colony scope is 30, optimal individual number is 5, and accelerated number is So we apply intelligent forecast method to train intelligent network and obtain linked-weights matrix of nodal points between input layer and hidden layer as ⎛ ⎞ ⎜ 0.2225 1.0986 0.3669 0.8021 0.1687⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ W = 0.6689 0.5625 0.3689 0.5697 0.0891 . ik ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.4669 0.2012 0.5369 0.9489 0.3002 Linked-weights vector between hidden layer and output layer as W = 0.7689 0.4914 0.4645 0.6089 0.7236 . kp Then we take latter 6 years data to run forecast check: normalize forecast values of feature factors that sequence from 29 to 34 in Table 1 and obtain relative membership degree (RMD) matrix of 6 years feature factors as ⎛ ⎞ ⎜ 0.380 0.404 0.573 0.567 0.718 0.425⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R = . 0.995 0.998 0.998 0.999 0.999 0.998 x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.496 0.538 0.562 0.565 0.555 0.535 Inputting above matrix into trained intelligent network and the output are normal- ized vector of feature values corresponding to forecast object f r = . 0.7432 0.7284 0.9186 0.8840 0.9600 0.7934 f t Substituting above vector into equation (26) and forecast check results are obtained as Table 2. From Table 2 we can see that, the maximum of absolute error is 3.11 days, the forecast precision can satisfy demand. (ii) Combined forecast with system NCF model Zhang [14] respectively employed fuzzy pattern recognition (FPR) model and FPR neural network model and obtained two groups forecast values of freeze-up date at Bayangaole Station; and in this paper at above section, we apply intelligent forecast model and also get corresponding forecast values. Now we use these three diﬀerent groups forecast values by three diﬀerent models and system NCF model to operate combined forecast. Forecast values of three models are listed in Table 3. Then we use normalized equation and normalize forecast values data of former 28 years ice ﬂood under three models and apply it to construct training samples of fore- cast, and take latter six-year data as check samples. Topologic structure of network is: 3-layered network structure, input layer has 3 nodal points (corresponding to 3 forecast models), hidden layer has 5 modal points (corresponding to ﬁve grades in [14]), and output layer has one nodal point. Fuzzy Inf. Eng. (2009) 1:191-204 199 Table 1: The feature values of forecast factors and observed freeze-up date at Bayangaole Station Accumulated air Discharge Freeze-up o 3 Sequence Year temperature( C) Water level(m) (m /s) date(day) 1 1968-1969 110.7 1049.45 239 46 2 1969-1970 91.1 1050.29 667 28 3 1970-1971 68.6 1048.89 411 23 4 1971-1972 55.1 1050.59 461 32 5 1972-1973 65.2 1051.80 506 27 6 1973-1974 146.2 1052.41 447 50 7 1974-1975 115.7 1051.89 595 39 8 1975-1976 156.6 1051.62 443 41 9 1976-1977 98.2 1051.66 302 23 10 1977-1978 114.4 1051.52 480 54 11 1978-1979 210.1 1052.48 500 54 12 1979-1980 191.7 1051.63 390 54 13 1980-1981 76.8 1052.32 280 45 14 1981-1982 110.5 1051.52 820 35 15 1982-1983 130.4 1052.52 270 42 16 1983-1984 194.7 1051.35 698 54 17 1984-1985 164.3 1051.06 577 46 18 1985-1986 122.1 1051.54 506 37 19 1986-1987 141.1 1051.83 462 38 20 1987-1988 90.3 1052.77 504 33 21 1988-1989 217.3 1054.97 850 59 22 1989-1990 317.9 1052.93 440 74 23 1990-1991 181.3 1052.90 742 55 24 1991-1992 183.8 1051.96 495 57 25 1992-1993 187.0 1053.18 500 55 26 1993-1994 193.2 1054.00 740 35 27 1994-1995 202.6 1053.78 520 58 28 1995-1996 221.6 1053.46 530 55 29 1996-1997 120.7 1051.12 422 40 30 1997-1998 128.5 1052.70 457 41 31 1998-1999 182.1 1053.00 478 51 32 1999-2000 180.3 1050.46 480 54 33 2000-2001 228.5 1053.24 472 54 34 2001-2002 143.6 1053.30 455 45 Table 2: Comparison of the observed values and forecast values for freeze-up date at Bayangaole Station (unit: day) Sequence 29 30 31 32 33 34 Observed values 40 41 51 54 54 45 Forecast values 43.11 42.25 53.28 51.27 55.68 46.02 Absolute error 3.11 1.25 2.28 2.73 1.68 1.02 200 Yu Guo (2009) Parameters of GA are set as: error precision of network training is 0.0002, net- work tardiness judgment index between before and after iteration is 0.0001, learning eﬃciency is 0.8, momentum parameter is 0.8, weight changing size is 0.5 times of itself, colony scope is 30, optimal individual number is 5, and accelerated number is So we apply system NCF method to train intelligent network and obtain linked- weights matrix of nodal points between input layer and hidden layer as: ⎛ ⎞ ⎜ 0.2225 0.6986 0.3669 0.0021 0.9687⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ W = ⎜ ⎟ . 0.3689 0.3625 0.0689 0.8697 0.6891 ik ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.2669 0.0012 0.5389 0.2489 0.0002 Linked-weights vector between hidden layer and output layer as W = 0.2337 0.5431 0.4368 0.3001 0.6774 . kp Therefore we take latter six-year data to run forecast check: normalize forecast values of three models that sequence from 29 to 34 in Table 3 and obtain RMD matrix of 6 years by three models as ⎛ ⎞ ⎜ ⎟ 0.75 0.77 0.94 0.93 0.97 0.77 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ R = 0.57 0.57 0.71 0.71 0.75 0.61 . x ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ 0.72 0.70 0.88 0.85 0.92 0.76 Inputting above matrix into trained intelligent network and the output are normal- ized vector of feature values corresponding to forecast object f : r = . 0.5724 0.5438 0.7043 0.7093 0.7516 0.5965 f j Substituting above vector into equation (26) and forecast check results are obtained as Table 4. From Table 4 we can see that, the maximum of absolute error is 2.36 days, the combined forecast precision can also satisfy demand. Here we employ equation (27) to analysis errors of forecast results under three forecast models and system NCF model and list them into Table 5. From Table 5 we can see that, MAEs of system NCF model and FPR neural net- work are almost equal and all have high precision. Comparison with singular forecast model, precisions of system NCF model and FPR neural network are equal and higher than other two forecast models. Analogously, according to data in [14] we employ system NCF model and ob- tain forecast results of freeze-up dates of Sanhuhekou station, Toudaoguai station and break-up dates of Bayangaole station, Sanhuhekou station, Toudaoguai station at In- ner Mongolia reach of Yellow River (see Table 6). The course is that: ﬁrst use annual observed freeze-up dates and break-up dates of ﬁve stations (in [8]) and intelligent forecast method (presented by[5]Ďto obtain forecast values of freeze-up dates and break-up dates of ﬁve stations; then according to FPR and FPR neural network model we obtain another two series forecast values of freeze-up dates and break-up dates of Fuzzy Inf. Eng. (2009) 1:191-204 201 Table 3: The forecast values and observed values of freeze-up date under three diﬀerent models at Bayangaole Station (unit:day) FPR neural Intelligent Sequence Year FPR network forecast Actual date 1 1968-1969 38.774 47.32 49.32 46 2 1969-1970 32.19 29.15 26.13 28 3 1970-1971 31.15 22.85 21.76 23 4 1971-1972 32.37 30.93 29.55 32 5 1972-1973 31.45 26.55 25.56 27 6 1973-1974 43.66 52.30 51.01 50 7 1974-1975 40.81 38.76 41.23 39 8 1975-1976 44.73 42.39 43.14 41 9 1976-1977 33.69 24.88 29.89 23 10 1977-1978 40.44 55.81 56.12 54 11 1978-1979 56.44 52.61 57.11 54 12 1979-1980 55.15 53.10 54.67 54 13 1980-1981 31.03 47.23 38.24 45 14 1981-1982 38.88 34.01 34.87 35 15 1982-1983 43.24 41.37 40.36 42 16 1983-1984 55.99 56.03 57.24 54 17 1984-1985 46.83 45.60 46.89 46 18 1985-1986 42.61 38.22 39.25 37 19 1986-1987 43.57 40.20 39.54 38 20 1987-1988 31.27 32.25 29.74 33 21 1988-1989 55.56 58.92 57.55 59 22 1989-1990 49.77 73.57 60.22 74 23 1990-1991 52.96 54.60 53.79 55 24 1991-1992 53.76 58.02 55.51 57 25 1992-1993 54.65 56.36 55.90 55 26 1993-1994 55.66 33.10 42.80 35 27 1994-1995 56.58 59.42 57.87 58 28 1995-1996 55.74 56.38 60.12 55 29 1996-1997 42.32 42.10 43.11 40 30 1997-1998 43.40 41.85 42.25 41 31 1998-1999 53.18 52.37 53.28 51 32 1999-2000 52.58 52.28 51.27 54 33 2000-2001 55.12 55.32 55.68 54 34 2001-2002 43.59 44.92 46.02 45 Table 4: Comparison of the observed values and combined forecast values for 6 time freeze-up dates at Bayangaole Station (unit:day) Sequence 29 30 31 32 33 34 Observed values 40 41 51 54 54 45 Forecast values 42.36 40.24 52.12 52.49 55.62 44.14 Absolute error 2.36 -0.76 1.12 -1.51 1.62 -0.86 202 Yu Guo (2009) Table 5: Comparison of errors for the diﬀerent models Model FPR FPR neural network Intelligent forecast System NCF model MAE 4.88 1.1 2.71 1.17 Table 6: Comparison of forecast values and observed values for 5 freeze-up dates and break-up dates (unit: day) Forecast items Intelligent Observed Stations (Sequence) FPR FPRNN forecast System NCF values Sanhuhekou Freeze-up (29) 33 33 32 32 30 Sanhuhekou Freeze-up (30) 34 34 35 34 33 Sanhuhekou Freeze-up (31) 36 36 36 37 39 Sanhuhekou Freeze-up (32) 42 44 45 42 43 Sanhuhekou Freeze-up (33) 36 36 36 35 33 Sanhuhekou Freeze-up (34) 36 37 36 37 38 Toudaoguai Freeze-up (32) 33 34 33 34 35 Toudaoguai Freeze-up (33) 54 53 53 54 55 Toudaoguai Freeze-up (34) 43 41 42 42 43 Bayangaole Break-up (29) 53 53 53 53 53 Bayangaole Break-up (30) 49 50 51 51 52 Bayangaole Break-up (31) 52 53 53 53 54 Bayangaole Break-up(32) 50 49 48 48 47 Bayangaole Break-up (33) 52 52 52 52 52 Sanhuhekou Break-up (29) 39 41 41 41 42 Sanhuhekou Break-up (30) 38 39 38 39 40 Sanhuhekou Break-up (31) 41 43 43 44 44 Sanhuhekou Break-up (32) 41 41 42 42 40 Sanhuhekou Break-up (33) 41 41 41 41 43 Toudaoguai Break-up (32) 39 38 38 38 38 Toudaoguai Break-up (33) 39 40 39 40 41 ﬁve stations; thus we have three diﬀerent series forecast values by three model, ﬁ- nally we apply system NCF model and obtain combined forecast values of freeze-up dates and break-up dates of ﬁve stations (see 7th column of Table 6). And we employ equation (27) to analysis errors of forecast results under three forecast models and system NCF model for these ﬁve stations and list them into Table 7. From Table 7 we know one side is that, forecast results’ precision of system NCF model is very high, maximum of MAE for total time series of ﬁve freeze-up dates and break-up dates is 1.56, if taking above MAE of Bayangaole Station into account that the maximum value is 1.17, it’s obvious that the precision of combined forecast is perfect; other side is that, comparison with results of FPR, FPR neural network and intelligent forecast model, except that MAE of FPR neural network for freeze-up date of Sanhuhekou Station is equal with system NCF model, that the precision of system NCF model is higher than any singular forecast model. Fuzzy Inf. Eng. (2009) 1:191-204 203 Table 7: Comparison of forecast results errors under the diﬀerent models FPR neural Intelligent System NCF Model FPR network forecast model Sanhuhekou (freeze-up) 5.82 1.29 2.21 1.29 Toudaoguai (freeze-up) 6.01 1.35 1.62 1.14 Bayangaole (break-up) 3.83 1.76 1.98 1.56 Sanhuhekou (break-up) 3.20 1.39 2.02 1.25 Toudaoguai (break-up) 3.12 1.27 2.14 1.07 4. Conclusions The combined forecast method to system NCF that combined engineering fuzzy sets, artiﬁcial neural network and GA, can get better forecast results for integrating multi- forecast models information. When training network, fuzzy optimization model [5] is employed as stimulated functions of network nodes, and it makes an artiﬁcial neural network posses in physical sense; at the same time it adopts FONN and GA to train network weights by turns, so at a certain extent the method can solve convergence slow and global optimum issues of neural network. The forecast results of ice ﬂood in Inner Mongolia reach of Yellow River show that forecast precision of combined forecast method of system NCF optimal is higher than that one of any single forecast model. References 1. Bates JM, Granger CWJ (1969) The combination of forecasts. Operational Research Quarterly 20:451-468 2. Chen SY (1998a) Multiobjective decision making theory and application of neural network with fuzzy optimum selection. The Journal of Fuzzy Mathematics 6(4):957-967 3. Chen SY (1998b) Theory and application engineering fuzzy sets. Beijing: National defense industry press (in Chinese) 4. Chen SY, Nie XT (1999) Forecasting model of neural network with fuzzy pattern recognition and evolutionary simples method. The Journal of Fuzzy Mathematics 7(4):913-923 5. Chen SY, Guo Y, Wang, D G (2006) Intelligent forecasting mode and approach of mid and long term intelligent hydrological forecasting. Engineering Science 8(7):30-35 6. 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Fuzzy Information and Engineering – Taylor & Francis
Published: Jun 1, 2009
Keywords: Intelligent forecast; System NCF; Ice flood of the Yellow River
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