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PEMODELAN ELMAN NETWORKS PADA DATA TIME SERIES NONLINEAR MELALUI PROSEDUR BOTTOM-UP MENGGUNAKAN INFERENSI R 2 inc oleh APRIL WICAKSONO M0106027 SKRIPSI ditulis dan diajukan untuk memenuhi sebagian persyaratan memperoleh gelar Sarjana Sains Matematika JURUSAN MATEMATIKA FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM UNIVERSITAS SEBELAS MARET SURAKARTA 2011 i

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PEMODELAN ELMAN NETWORKS PADA DATA TIME SERIES

NONLINEAR MELALUI PROSEDUR BOTTOM-UP

MENGGUNAKAN INFERENSI R2inc

oleh

APRIL WICAKSONO

M0106027

SKRIPSI

ditulis dan diajukan untuk memenuhi sebagian persyaratan memperoleh gelar

Sarjana Sains Matematika

JURUSAN MATEMATIKA

FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM

UNIVERSITAS SEBELAS MARET

SURAKARTA

2011

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ABSTRAK

April Wicaksono, 2011. PEMODELAN ELMAN NETWORKS PADADATA TIME SERIES NONLINEAR MELALUI PROSEDUR BOTTOM UPMENGGUNAKAN INFERENSI R2

inc. Fakultas Matematika dan Ilmu Penge-tahuan Alam, Universitas Sebelas Maret.

Neural networks merupakan model nonlinear yang fleksibel untuk memo-delkan data dalam berbagai bidang aplikasi. Elman networks adalah salah satujenis neural networks yang dapat digunakan untuk memodelkan data time se-ries. Proses dalam Elman networks dilakukan dengan cara menransfer sinyal in-put berbobot melalui beberapa neuron yang saling terhubung. Berdasarkan haltersebut perlu ditentukan terlebih dahulu nilai bobot dan banyak neuron yangdigunakan. Tujuan penelitian ini yaitu menentukan prosedur pemodelan Elmannetworks untuk data time series, menentukan nilai bobot dan menentukan banyakneuron yang optimal.

Penelitian ini mengkaji ulang beberapa jurnal yang telah dipublikasikan.Dalam pembahasan, nilai bobot pada Elman networks ditentukan dengan memi-nimumkan fungsi eror. Proses ini diselesaikan menggunakan metode optimalisasigradient descent melalui algoritma backpropagation. Untuk banyaknya neuronditentukan melalui prosedur bottom up. Prosedur bottom up merupakan prosedurpenentuan banyak neuron yang dimulai dari arsitektur yang paling sederhanamenuju arsitektur yang lengkap. Prosedur ini dilakukan dengan cara menam-bahkan neuron pada setiap arsitektur yang sederhana dan dihentikan ketikapenambahan neuron tidak lagi memiliki kontribusi yang signifikan. Kontribusitersebut dihitung menggunakan uji F terhadap nilai R2

inc.Berdasarkan hasil pembahasan, prosedur pemodelan Elman networks un-

tuk data time series meliputi perancangan data, uji nonlinearitas, penentuannilai bobot, penentuan banyak neuron dan verifikasi terhadap sejumlah data uji.Nilai bobot optimal dipilih dari hasil akhir pelatihan menggunakan algoritmabackpropagation yang menghasilkan eror minimum, sedangkan banyaknya neuronoptimal ditentukan dari hasil akhir prosedur bottom up. Dalam hal ini, neuronyang dipilih adalah neuron yang memiliki kontribusi yang signifikan berdasarkanuji F dalam prosedur bottom up.

Kata kunci : time series, nonlinear, Elman networks, backpropagation, gradientdescent, bottom up.

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ABSTRACT

April Wicaksono, 2011. MODELING OF ELMAN NETWORKS ON NON-LINEAR TIME SERIES WITH BOTTOM UP PROCEDURE BY USING R2

inc

INFERENCE. Faculty of Mathematics and Natural Sciences, Sebelas MaretUniversity.

Neural networks are a nonlinear model whose flexible to modeling the datain various fields of application. Elman networks is one kind of neural networksthat used to modeling nonlinear time series. The proses in Elman networks iscarried by transferring weighted input signal through many neurons that areconnected, so we need to determine the value of weight and number of neurons.The purposes of this research are to determine procedure of Elman networksmodeling for time series data, determine the value of weights and determine theoptimal number of neurons.

This research reviewed many journals that have been published. In the dis-cussion, the value of weights in Elman networks are determined by minimizing thefunction of error . This process is solved by gradient descent optimization meth-ods through backpropagation algorithm. The number of neurons are determinedby bottom-up procedure. Bottom-up procedure is a procedure to determine thenumber of neurons that starts from the simplest architecture to the full archi-tecture. This procedure is done by adding neurons in simple architecture and isstopped when the addition of neurons no longer has a contribution that signifi-cant. That contributions is calculated by using the F test against the value ofR2

inc.The result shows that the procedure of Elman network modeling for time

series data includes data design, nonlinearity test, determination the value ofweights , determination the number neurons, and verification with some testingdata. The optimal value of weights are selected from the final result of trainingby using backpropagation algorithm that produces the minimum error, while theoptimal number of neurons are determined from the final result of the bottom-up procedure. In this case, the selected neurons are the neurons that having asignificant contribution based on the F test in a bottom up procedure.

Key words : time series, nonlinear, Elman networks, backpropagation, gradientdescent, bottom up.

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DAFTAR PUSTAKA

[1] Cryer, J. D., Time Series Analysis, Duxbury Press, Boston, 1986.

[2] Elman, J. L., Finding Structure in Time, Cognitive Science, University ofCalifornia, San Diego 14 (1990), 179–211.

[3] Faraway, J. and C. Chatfieldt, Time Series Forecasting with Neural Network,Applied Statistics 47 (1998), 231–250.

[4] Fausett, L., Fundamentals of Neural Networks, Prentice Hall, EnglewoodCliffs, (1994).

[5] Gruning, A., Elman Backpropagation as Reinforcement for Simple Re-current Networks, Cognitive Neuroscience Sector, S.I.S.S.A. (2007),(http://citeseerx.ist.psu.edu/).

[6] Kaashoek, J. F. and H. K. Van Dijk, Neural Network Pruning Applied toReal Exchange Rate Analysis, Jurnal of Forecasting 21 (2002), 559–577.

[7] Kuncoro, A. H. dan R. Dalimi, Aplikasi Jaringan Syaraf Tiruan untuk Pera-malan Beban Tenaga Listrik Jangka Panjang pada Sistem Kelistrikan di In-donesia, Jurnal Teknologi Universitas Indonesia (2005), no. 3, ISSN 0215-1685, 211–217.

[8] McCullogh, W. S. and Pitts, W., A Logical Calculus of The Ideas Immanentin Nervous Activity, Bulletin of Mathematical Biophysics 5 (1943), 115–133.

[9] Mittelhammer, R.C., Mathematical Statistics for Economics and Business,Springer. New York, 1996.

[10] Sembiring, R. K., Analisis Regresi, Institut Teknologi Bandung, (1995).

[11] Soedjianto, F., R. Adipranata, dan R.Gunawan, Pengenalan Tulisan TanganBerdasarkan Arah Gerakan Tangan Menggunakan Metode Dominant Point,Proseeding Seminar Nasional Soft Computing, Intelligent Systems and In-formation Technology (2005).

[12] Subanar, dan Suhartono, Uji Linearitas Tipe Lagrange Multiplier denganEkspansi Taylor untuk Deteksi Hubungan Nonlinear pada Data Time Series,Journal of The Indonesian Mathematical Society (MIHMI) 12 (2006), 17–32.

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[13] Suhartono, Feedfoward Neural Network untuk Pemodelan Runtun Waktu,Ph.D. thesis, Universitas Gadjah Mada Yogyakarta, (2007).

[14] Suhartono, Subanar, and S. Guritno, Model Selection in Neural Networksby using Inference of R2 Incremental, PCA and SIC Criterion for TimeSeries Forecasting, JOURNAL OF QUANTITATIVE METHODS: JournalDevoted to The Mathematical and Statistical Application in Various Fields2 (2006), 41–57.

[15] Warsito, B., Penentuan Unit Hidden Optimal pada Model Neural Networkdengan Analisis Kontribusi Incremental Sel, Jurnal Matematika UniversitasDiponegoro Semarang 8 (2005), no. 2, 42–46.

[16] Yan Fang and T. J. Sejnowski, Faster Learning for Dynamic Recurrent Back-propagation, Neural Computation 2 (1990), 270–273.

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