An Accurate Heave Signal Prediction Using Artificial Neural Network

Transcription

1 Internatonal Journal of Multdsclnary and Current Research Research Artcle ISSN: Avalale at: htt://jmcr.com Mohammed El-Dasty 1,2 1 Hydrograhc Surveyng Deartment, Faculty of Martme Studes, Kng Adulazz Unversty 2 Engneerng Deartment of Pulc Wors, Faculty of Engneerng, Mansoura Unversty Acceted 05 Oct 2014, Avalale onlne 15 Oct 2014, Vol.2 (Set/Oct 2014 ssue) Astract An accurate heave modelng s reured for several alcatons, ncludng hydrograhc surveyng. Ths aer rooses an adatve heave sgnal modelng, whch uses a neural networ-ased modellng. A recurrent neural networ and three-layer feed forward neural networ traned usng the Levenerg-Maruardt learnng algorthm s used for ths urose. Comutatonal results wth fve dfferent datasets of real tme heave are rovded to valdate the effectveness of the artfcal neural networ-ased model. It s shown that the new neural networ-ased model gve a relale heave model wth excellent erformance. Also, a comarson s made etween the develoed artfcal neural networ models and the autoregressve models. It s shown that the new artfcal neural networ-ased model gve the est erformance results (.e., the mean suare error MSE). Keywords: Heave Predcton, Neural Networ, Autoregressve, Recurrent Networ 1. Introducton Accurate heave modelng s reured for several alcatons, ncludng hydrograhc surveyng. The hyscal models and Kalman Flter estmaton of the heave rocess can e found n [1]. Freuency resonse methods have een used n the ast to dentfy have models. The model s ased on the freuency content of the heave record. The freuency content s used as the ass to formulate a state sace reresentaton of the heave model [1]. A lot of studes have een reorted to solve ths rolem, wth the Kalman flter eng the most common [1]. The desgn of the Kalman flter s ased on the assumton of the comlete structural nowledge of the model whch descres the heave dynamcs and t s ased the Gaussan assumton of the state sace nose statstcs [2]. It s well nown that there s a degradaton of the KF-ased estmaton ualty when the actual nose statstcs s not Gaussan [3]. In order to overcome these lmtatons, another aroach was followed ased on autoregressve movng average model [4]. Ths aer rooses an adatve heave sgnal modelng, whch uses a neural networ-ased modellng. A recurrent neural networ (RNN) and three-layer feed forward neural networ (FFNN) traned usng the Levenerg-Maruardt algorthm s used for ths urose. Comutatonal results wth fve dfferent datasets of real tme heave are rovded to valdate the effectveness of the ANN-ased model. It s shown that the new artfcal neural networased model gve a relale heave model wth excellent erformance. Also, a comarson s made etween the develoed artfcal neural networ models and the autoregressve models. It s shown that the new artfcal neural networ-ased model gve the est erformance results (.e., the mean suare error MSE). 2. Autoregressve Models (AR) Autoregressve rocess s lnear model. The ref lterature of autoregressve rocess can e found n [4] and [5]. Autoregressve (AR) rocess s tme seres consstng lnear comnaton of mmedate ast values of the tme seres. Fgure 1 shows an examle of a fully connected AR strucure, whch s referred to as -1 (_AR, 1_out). AR() rocess of order can e summarzed n the followng lnear euaton (see Fgure 1): y ˆ( n) a( ). n )) (1) DOI: htt://dx.do.org/ /ijmcr/2/5/2014/ Int. J. of Multdsclnary and Current research, Set/Oct Where, yn ˆ( ) s the estmated outut, n ) s the mmedate oserved heave value and a () s autoregressve (AR) model arameter. AR() model arameters are estmated usng least suares method [5]. To consder the non-statonary art (.e., resduals) n the tme seres, the lnear autregressve movng average (ARMA) model. ARMA rocess s tme seres consstng of random error comonent and lnear comnaton of mmedate heave values and mmedate ast resduals. Fgure 1 shows an examle of a fully connected AR strucure, whch s referred to as (+)-1 (_AR + _MA,

2 Mohammed El-Dasty 1_out). The ARMA(, ) rocess can e summarzed n the lnear euaton (see Fgure2): n) 1 a( ). n ) j1 c( j). e( n j) ( n) Where, yn ˆ( ) s the estmated outut, n ) s the mmedate ast value of the records, a () s autoregressve (AR) model arameter, e( n j) yˆ ( n j) n j) s mmedate ast resdual, and c( j ) s movng average (MA) model arameter. ARMA(, ) model arameters are estmated usng least suares method [6]. (2) redcton model (See [6]; [7]). Artfcal neural networs can e develoed n many dfferent forms, deendng on networ structure and the learnng algorthms. Fgure 3 reresents a loc dagram of a smle model of a neuron showng the weghts of the varous lns. w 1 w2 y w 3 w y = Inut neuron y = Outut neuron a = AR arameters 1 x 1 x 1 x a 1 a 2 a Fg. 3: Smle neural networ structure The aove structure contans nut layer and outut layer and can e reresented n the comact form as: Fg. 1: Autoregressve (AR) structure [(_AR)-(1_out)] a 1 a +- 1 n-2) e (n) = (y (n) - y (n)) n-) y = Inut neuron y = Outut neuron a = AR arameters a j = MA arameters -j = Delay oerator yˆ f ( 1 w x ) Where, x reresents one varale n the nut layer, f j s the actvaton functon, yˆ reresents the outut varale, s the as varale, and w reresents one of the neural networ arameter. Artfcal neural networs are a flexle nonlnear form n whch the actvaton functons f can e chosen artrary. Fgure 4 show examle of two common actvaton functons (.e. Sgmod and hyerolc tangent functon). (3) n-) e(n-1) e(n-) Fg. 2: Autoregressve Average Movng (ARMA) structure [(_AR+_MA)-(1_out)] 3. Artfcal Neural Networs (ANN) Models 3.1 Bascs of Artfcal Neural Networs In ths secton we refly descre the fundamentals of artfcal neural networs (for more detals see [5]). Artfcal neural networ can e smly nterreted as a hghly nonlnear autoregressve model whch s constructed y comnng nonlnear actvaton functon through a multlayer structure. Neural networ have een successfully aled n many dfferent felds as a a. Sgmod functon. Hyerolc tangent Fg. 4: Actvaton functons - sgmod and hyerolc tangent functon. In ths aer we nvestgated the use of the feedforward neural networs and recurrent neural networ n heave redcton model. It should e noted that for the redcton model, the nut layer varale n ) reresents the mmedate ast value of the tme seres records. 990 Int. J. of Multdsclnary and Current research, Set/Oct

3 Mohammed El-Dasty 3.2 Feed forward and Recurrent Neural Networ Heave Predcton Models h y h h yˆ( n) f wn. f w. n ) wj. e( n j) n 1 1 j1 (5) A feedforward networs are characterzed y one nut layer, one outut layer and one or more hdden layers contanng a numer of networ actvated varales [5]. Fgure 5 shows an examle of a fully connected threelayer feedforward networ n AR strucure, whch s referred to as whch s referred to as [(_AR)-(s_hdden)- (1_out)]. The nut varales are smultaneously actvated y a nonlnear actvaton functon f and s rovde hdden varales n a hdden layer. Then, the resultng hdden varales are actvated y another actvaton functon f and roduce the networ outut J. j h Hdden Layer Inut Layer Outut Layer w 1 0 h 1 Fg. 5: Three-Layer Feedforward Neural Networ wth the Structure [(_AR)-(s_hdden)-(1_out)]. The aove structure can e reresented n a comact form as: h y h yˆ( n) f wn. f w. n ) n 1 1 (4) Alternatvely, a recurrent neural networ s a hghly nonlnear dynamc structure and smlar to ARMA model n the structure [9]. From a dynamc ont of vew, t s mortant to nclude the lagged deendent varales as feedac loos. The lagged varales are the resdual errors etween the desred oututs and the estmated oututs are consdered as nut varales. Hence, a recurrent neural networ has een roosed n ths aer. A recurrent neural networs are smlar to the feedforward networs, wth excet that the former have at least one feedac loo). Fgure 6 shows an examle of a fully connected three-layer recurrent networ n ARMA structure, whch s referred to as [_AR + _MA)- (s_hdden)-(1_out)]. The followng structure can e reresented n a comact form as: w c s s y = Outut neuron = Transfer functon w s = ANN arameters h s n-) y = Inut neuron h = Hdden neuron w s h Hdden Layer Inut Layer Outut Layer - + h 1 n-) Fg. 6 Recurrent Neural Networ structure [(_AR + _MA)-(s_hdden)-(1_out)] Tranng the feed forward neural networ and recurrent neural networ s accomlshed through teratve adjustments of the free arameters,.e., the weghts and as, of the networ tll we otan the otmal values. There exst varous learnng algorthms, whch are fundamental to the desgn of neural networs. The Levenerg-Maruardt learnng algorthm s the most wdely used for feedforward neural networs and recurrent neural networ, whch s used n ths research (see [9] for more detals). 4. Results and dscusson e (n) = (y (n) - y (n)) w c s e(n-1) w s (+) = Transfer functon -j = Delay oerator Fve dfferent data seres of real tme heave, namely MBheave-1, MBheave-2, MBheave-3, MBheave-4, and MBheave-4, were used to valdate the neural networ model. The structure of the neural networ was ult usng the Matla [8]. Several tests were conducted to otmze the structure of the neural networ for each of the fve data seres. As mentoned efore, we have nvestgated two neural networ-ased models. Frst, we used the feedforward neural networ structure. In ths aroach, mmedate ast values of the heave records are used as nut to the networ, whle future values of the tdal records are used as the desred outut. Then, we followed another neural networ structure, whch s recurrent neural networ. In ths aroach, the mmedate ast values of the heave records and the mmedate ast of value of the resduals are used as nut to the networ, whle future values of the heave records are used as the desred outut. Each of the data seres was dvded nto three datasets: tranng, testng and valdaton datasets. The frst 10% heave records were assgned to the testng dataset, whle the last 10% heave records were assgned to the valdaton dataset. The tranng dataset was 991 Int. J. of Multdsclnary and Current research, Set/Oct 2014 h s e(n-) -1 -r y = Inut neuron h = Hdden neuron y = Outut neuron w j = ANN arameters

4 Magntude (numer) Mohammed El-Dasty Tale 1: Summary results Data Model AR ARMA FFNN RNN MB-1 MSE-tran 1.13E E E E-06 MSE-vald 1.88E E E E-06 MB-2 MSE-tran 7.53E E E E-06 MSE-vald 5.89E E E E-06 MB-3 MSE-tran 1.34E E E E-06 MSE-vald 1.01E E E E-06 MB-4 MSE-tran 1.55E E E E-06 MSE-vald 1.14E E E E-06 MB -5 MSE-tran 1.52E E E E-06 MSE-vald 1.99E E E E-06 selected to reresent the mddle orton of the data seres, whch vared from one MBheave to another. The tranng was stoed ased on testng the generalzaton erformance the neural networ usng the testng dataset. After tranng and testng the networ, we generalzed the model to redct ahead the last 10% values of the data seres and comared the results wth the oserved heave records. Ths was done n a seuental manner to emulate the real-tme condton. It was concluded that the recurrent neural networ n ARMA structure [(6_AR+ 6_MA)-(12_hdden)-(1_out)] gves the est results,.e., has the lowest MSE (MSE) error (See Tale 1). To further valdate our model, we comared t wth the autoregressve aroach. Autoregressve (AR) structure and autoregressve Movng Average (ARMA) structure were ult usng the Matla [8]. Each of the data seres was dvded nto two datasets: tranng, and valdaton datasets. The frst 90 % heave records were assgned to the tranng dataset, whle the last 10% heave records were assgned to the valdaton dataset. Several tests were conducted to otmze the structure of the AR and ARMA for each of the fve data seres. It was concluded that the ARMA wth the structure [6_AR + 6_MA)-(1_out)] gves the est results,.e., has the lowest MSE (MSE) error Hstogram of models resduals AR ARMA FFNN RNN Resduals (m) Fg. 7: Hstogram for all models resduals The overall concluson from Tale 1 and Fgure 7 was that the recurrent neural networ wth the structure [6_AR + 6_MA)-(12_hdden)-(1_out)] gves the est results,.e., has the lowest MSE (MSE) error. Fgures 8 shows that the redcted heave versus the desred values, and the redcton errors (.e., resduals) for MBheave-1. Smlar results were otaned for the other four data seres. It can e seen that the maxmum redcton error s aroxmately 1 cm, wth the ul of the resduals fall wthn the +/-0.5 cm range (see Fgure 7). Ths shows that the develoed neural networ model has the caalty to recsely redct the heave values. Fg. 8: Predcted heave versus the desred (.e., actual) values, and the redcton error usng Artfcal Neural Networ (RNN) Concluson A seuental heave redcton model usng Artfcal Neural Networs was develoed n ths aer. The recurrent neural networ structure gave the est erformance results (.e., the mnmum RMS), and therefore was used n redctng the heave values. Heave data seres, wth varyng lengths, from fve dfferent real tme heave data seres were used to verfy the model. It s shown that the maxmum redcton error s aroxmately 1 cm, wth the ul of the resduals fall wthn the +/-0.5 cm range. A erformance comarson was made etween the develoed neural networ model and the autoregressve method for heave redcton. It s shown that erformance of recurrent neural networ s etter than autoregressve method.e., has the lowest MSE (MSE) error. Future develoment wll nclude a comlete atttude and heave soluton, whch taes advantage of GNSS heghts 992 Int. J. of Multdsclnary and Current research, Set/Oct 2014

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