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Application of ensemble neural networks to prediction of towboat shaft power

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Abstract

In this paper towboat shaft power was predicted using various artificial neural networks. This work is a step toward reducing errors in the prediction of towboat power as well as providing better understanding of powering characteristics by the crew of the towboat. An ensemble neural network (ENN) and the single neural network (ANN) with two hidden layers are proposed to predict towboat shaft power. These two models were compared on the basis of their calculated root mean squared errors, mean absolute errors and relative errors. The database used for training and testing of the proposed ANN and ENN has been collected from the full-scale speed-power trials. Trials are conducted on selected towboats and convoys of barges. The goal of the paper is to show that ENN can be applied on towboat shaft power prediction and can improve the accuracy of the results over the single ANN. Computational results from this numerical example show that ENN definitely outperforms single ANN with two hidden layers. The contribution of this paper is a proposal to use an AIC-based ENN method for predicting towboat shaft powers. The paper is the first one that addresses AIC-based ENN method to predict towboat shaft powers.

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Notes

  1. A bias neuron is invented to adjust only weights during the training process and always emits +1 value.

  2. In the case of Neural network model sample size corresponds to number of trained data (N tr).

Abbreviations

AIC:

Akaike information criterion

AICc :

Corrected AIC value

B :

Pushed convoy beam in meters (m)

\(\beta\) :

Constant

∆AICc :

Modified delta_AIC value

DIV:

Diversity of the component network

η 1 :

Relative error calculated for training data set

η 2 :

Relative error calculated for testing data set

f 1 :

Relative frequencies of the relative errors calculated for training data set

f 2 :

Relative frequencies of the relative errors calculated for testing data set

F n :

Froude number

K :

Total number of estimated parameters of the ANN model

L :

Pushed convoy length in meters (m)

Λ :

Weight of the component network

N :

Sample size or number of all measured data

N:

The set of all natural numbers

\(n_s^1\) :

Number of times the relative errors appear in subinterval s calculated for training data set

\(n_s^2\) :

Number of times the relative errors appear in subinterval s calculated for testing data set

N h :

Number of hidden neurons

N i :

Number of input neurons

N tr :

Number of training data

O :

Set of all shaft powers obtained through measurements (real outputs)

Ô :

Set of all predicted shaft powers

Ô 1 :

Set of all towboat shaft powers predicted with training data set by ENN

Ô 2 :

Set of all towboat shaft powers predicted with testing data set by ENN

Ô 1(r) :

Set of all towboat shaft powers predicted with training data set by r-th component network

\(\hat{O}_{j}^{1\left( r \right)}\) :

J-th towboat shaft power predicted with training data set by r-th component network

R :

Number of component networks

RMSE:

Root mean squared error

S :

Number of subintervals

\(\hat{\sigma }^{2}\) :

Mean squared error (MSE)

SHP:

Towboat shaft power (kW)

T :

Pushed convoy mean drought in meters (m)

V :

Pushed convoy volume of displacement in cubic meters (m3) (volume of the underwater shape of the pushed convoy which is equal to the displacement of the pushed convoy on the rivers)

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Acknowledgments

This work has been supported by Serbian Ministry of Science and Technological Development, Grant No. TR36002.

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Authors and Affiliations

  1. University of Belgrade-Faculty of Transport and Traffic Engineering, Belgrade, Serbia

    Aleksandar Radonjic & Katarina Vukadinovic

Authors
  1. Aleksandar Radonjic
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  2. Katarina Vukadinovic
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Correspondence to Aleksandar Radonjic.

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Radonjic, A., Vukadinovic, K. Application of ensemble neural networks to prediction of towboat shaft power. J Mar Sci Technol 20, 64–80 (2015). https://doi.org/10.1007/s00773-014-0273-2

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