Diagnostics Model for Estimating Mean Sea Level Change using Hybrid Model of Exponential Smoothing and Neural Network
Dadan Kusnandar1*, Muhlasah Novitasari Mara 2, Naomi Nessyana Debataraja3
123Department of Mathematics Tanjungpura University
Abstract. A diagnostics model was proposed to estimate the mean sea level change by hybridizing exponential smoothing and neural network. The model integrated the linear characteristics of the exponential smoothing model and the nonlinear pattern of the neural network. Mean sea level data were obtained from the measurements of Jason-2 satellite altimeter mission from 2008 – 2014. The results showed that the diagnostics model obtained by hybridization was not significantly different from the neural network model in term of the MAE and MSE.
Keywords: Accurary, time series, satellite altimetry.
INTRODUCTION
Three related issues that concern many people are the increase of greenhouse gases, global warming and the rising sea level. Many studies have been conducted by many researchers to investigate the phenomenon. Studies on sea level rise in Indonesia mainly conducted in Java, such as in the coastal areas of Indramayu [1], Surabaya [2] and in Tuban coastal areas [3]. This study investigated the mean sea level rise in West Kalimantan. The Province of West Kalimantan covers more than 1,500 km long of coastal areas. The data were obtained from the measurement of global sea level produced by the Jason-2 satellite altimeter. The nearest area to the Province of West Kalimantan is the measurement of sea level in the South China Sea. The data consisted of time series data for about six years measurements.
Investigation of time series data often reveals the pattern of both linear and nonlinear models. In model prediction, both models are generally combined to improve accuracy and precision. An application of linear and nonlinear model integration was carried out by [4] that hybridize the ARIMA and the neural network models for time series forecasting. The data were three types of time series data obtained from different areas and different statistical characteristics.
The mean square error (MSE) and the mean absolute deviation (MAD) resulting from the hybrid model were smaller than those of the ARIMA and the neural network models [4]. Other application of linear- nonlinear integration was used in prediction of financial data time series. A hybrid model of exponential smoothing and neural network was
applied to predict the daily fluctuation of two major international currency exchange rates, i.e., Euros/US dollar and Japanese Yen/US dollar. It was found that the hybrid technology performs better than the exponential smoothing and the neural network [5].
A theoretical study on hybridizing exponential smoothing and neural network for prediction in time series data was carried out by [6]. A hybrid model of exponential smoothing and neural network is a model that combines a linear and nonlinear model. As a nonlinear model, neural network advantage the conventional statistical methods as it does not require prior assumption of model form. Hence, it is generally appropriate to be applied to big data set that continuously updated. On the other hand, exponential smoothing is a linear model. It is a simple model and give similar result to the of ARIMA model [7].
In this paper a hybrid model of exponential smoothing and neural network is applied to predict the global mean sea level in South China Sea region. The prediction results were compared to that of exponential smoothing and neural network model by means of the MSE and the mean absolute error (MAE).
METHOD OF ANALYSIS
In this section the diagnostic model for estimating the mean sea level change using the hybrid model of the exponential smoothing and neural network is discussed.
Exponential Smoothing Model
Exponential smoothing is a method used for forecasting time series data. Forecasting for future observations was carried out by calculating a weighted average of previous period’s forecast and the actual value of that period. In this paper the method of simple exponential smoothing was used for prediction. Let be time series data that can be described by the following model
(1)
where [ ] and is random error.
The estimate of made in time t is given by the smoothing equation
(2)
where and is the estimate of at . The point forecast made in time for is
̂ ̂ (3) From equation (2) we have
(4) Substituting equation (4) to equation (2) recursively, we obtain
̂
(5) This method is referred as exponential smoothing. As we can see that of coefficients of the observations, i.e., , , decrease exponentially with time. The advantage of the exponential smoothing method is that it is capable of fitting the linear patterns of the time series well and easy to use [5].
Neural Network Model
The neural network model is an information processing system that resembles the neural network in living organism. A neural network model is characterized by its architecture, learning algorithm and activation function. In this paper the back- propagation neural network was used to analyze the data. The back-propagation neural network consisted of three layers, i.e., the input layer, hidden layer and output layer (Figure 1). For time series forecasting, the relationship between the output and the inputs
( ) has the following mathematical representation [5].
∑ ∑ (6)
where is a bias on the jth unit and
is referred to the connection weights between layers of the model, is the transfer function of the hidden layer, p is the number of input nodes and q is the number of hidden nodes.
FIGURE 1. A model of Neural Network
Hybrid Model of Exponential Smoothing and Neural Network
Time series data often contain both linear and non linear pattern. Exponential smoothing is an appropriate technique for forecasting in linear domains, whereas the neural network has achieved success in nonlinear conditions. To take full advantage of the individual strengths of the exponential smoothing and the back- propagation neural network model, it is necessary to integrate the two models [5].
FIGURES 2. The Hybrid Model
The architecture of hybridizing exponential smoothing and neural network is presented in Figure 2.
At initial stage, the prediction is carried out separately using exponential smoothing technique and the neural network. The exponential smoothing model should generate prediction result, while the neural network also generates prediction result. The two prediction results are then synergized via hybridization process [5].
̂ ̂ ̂ (7) where ̂ is the prediction result obtain from the exponential smoothing model, ̂ is the prediction result obtained from the neural network model and is the weight parameter. The weight parameter can be obtained by solving the linear programming resulting minimizing the sum of absolute error between the estimated and actual values [5].
Comparison among the prediction models, i.e., the exponential smoothing, the neural network and the hybrid model was carried out by means of the MSE and the MAE, as follows.
∑ ( ̂ ) (8) ∑ | ̂ | (9)
RESULTS AND DISCUSSIONS
Diagnostic models of data time series of mean sea level are predicted by using the exponential smoothing, the neural network and hybrid model of exponential smoothing and neural network. The data used in this paper were time series data resulting from measurement of global mean sea level obtained from Jason-2 satellite altimetry during the period of 2008 – 2014 [8]. Jason-2 satellite was launched on June 20, 2008 following the Jason-1 satellite (launched December 7, 2001) and the Topex/Poseidon satellite (launched August 10, 1992, decommissioned 2006).
The satellites have a mission to provide oceanographic data time series. The time series data from Jason-2 satellite consisted of 203 measurements of global mean sea level in 10 days interval.
In this paper exponential smoothing modeling via R programme kemudian nilai prediksi ̂ dihitung dengan menggunakan zaitun programme. The mean squared error (MSE) and mean absolute deviation (MAD) are selected to be the forecasting accuracy measures. Plot Data Jason 2 terdapat beberapa missing data di beberapa titik observasi.
Untuk mengatasi hal tersebut dilakukan screening data seperti yang telah dilakukan pada [9].
Figures 3. Jason-2 Data
Dengan menggunakan program R, model Exponential Smoothing untuk data time series Jason 2 ditentukan pada persamaan (..)
̂ ̂
Kemudian nilai pada persamaan (10) dimasukkan ke dalam zaitun programme untuk memperoleh nilai prediksi. Data asli Jason-2 (black) dan hasil prediksi dengan menggunakan exponential smoothing (green) diplotkan pada Figures 4. Data dan hasil prediksi kemudian dihitung dan mendapatkan nilai MSE sebesar 5.998352 dan MAE sebesar 1.980393.
FIGURES 4. Jason-2 Data and Exponential smoothing prediction of Jason
Setelah itu dilakukan prediksi dengan menggunakan model neural network. Pada penelitian ini digunakan arsitektur back-propagation neural network dengan 48 unit input ). Pada penelitian ini digunakan 12 unit hidden layer dan menghasilkan satu nilai output. Fungsi aktifasi yang digunakan adalah fungsi bipolar sigmoid. Data asli Jason-2 (black) dan hasil prediksi dengan menggunakan neural network (red) diplotkan pada Figures 5. Data dan hasil prediksi kemudian dihitung dan mendapatkan nilai MSE sebesar 0.0993902 dan MAE sebesar 0.113289
.
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050100150200
time
MSL
Data
Exponential Smoothing
FIGURES 5. Jason-2 Data and Neural Network prediction of Jason-2
Data hasil prediksi Exponential Smoothing ( ̂ dan hasil prediksi Neural Network ̂ kemudian di hybridizing menggunakan persamaan (11). Pada penelitian ini, dibuat algoritma pemrograman dengan R untuk memperoleh estimasi . Dari algoritma pemrograman tersebut didapat estimasi nilai untuk hasil pengukuran satelit altimetri Jason 2 adalah 0.01. Alasan pemilihan nilai ini karena menghasilkan MSE dan MAE terkecil.
Model dapat dituliskan model Hybridizing Exponential Smoothing dan Neural Network sebagai berikut
̂ = 0.01 ̂ + 0.99 ̂
Data asli Jason-2 (black) dan hasil prediksi dengan menggunakan hybrid model (purple) diplotkan pada Figures 6 (a). Data dan hasil prediksi kemudian dihitung dan mendapatkan nilai MSE sebesar 0.09975692 dan MAE sebesar 0.1188349.
FIGURES 6. (a) Jason-2 Data and Neural Network prediction of Jason-2 (b) Perbandingan Jason-2 Data, exponential smoothing model, neural network and hybrid
model
In this paper, we compare the linear exponential smoothing model, the nonlinear neural network model and hybrid exponential smoothing and neural network.
Pperbandingan plot untuk ketiga model dapat dilihat pada Figures 6(b) dan nilai MSE dan MAE untuk tiga model tersebut disajikan pada Tabel 1.
TABLE 2. The Experiment Results of Sea Level Rise
ES NN Hybridizing
MAE 1,980393 0,11 0,1188349
MSE 5,998352 0,099 0,09975692
Dari nilai MAE dan MSE exponential smoothing nilai MAE dan MSE yang cukup besar diantara ketiga model lainnya namun pada Figures 6 dapat dilihat trend hybridizing model lebih baik, mengikuti data asli daripada model exponential smoothing atau model neural network. Dari penelitian yang telah dilakukan dapat disimpulkan bahwa, hasil prediksi tinggi muka air laut dari data satelit altimetri Jason 2 menggunakan Hybridizing Exponential Smoothing dan Neural Network tidak lebih baik daripada hasil prediksi Neural Network. MAE serta MSE hasil prediksi dengan Hybridizing Exponential Smoothing dan Neural Network adalah 0.1188349 serta 0.09975692.
Sedangkan MAE serta MSE hasil prediksi dengan Neural Network adalah 0.113289 serta 0.09939024.
ACKNOWLEDGMENTS
Data Provided by B. Beckley, supported by NASA’s Measures program
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