A Comparison of Parameters Estimation Based on Forecasting Accuracy in Singular Spectrum Analysis (SSA)

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1 A Comparison of Parameters Estimation Based on Forecasting Accuracy in Singular Spectrum Analysis (SSA) Meri Andani Rosmalawati, Iman Bimantara M 2, Gumgum Darmawan, Toni Toharudin Statistics Department, University of Padjadjaran - Indonesia 2 PT Kharisma Pemasaran Bersama Nusantara Indonesia andaniros@gmail.com, imanbimantaramumin@yahoo.com, gumstat@yahoo.com, toni.toharudin@unpad.ac.id Abstract. Singular Spectrum Analysis (SSA) has been used as a powerful technique on Time Series Analysis. In this paper, we use SSA to forecast Crude Palm Oil (CPO) price in Rotterdam s Physical Market data and MDEX (Malaysia Derivatives Exchange) data by means of parameter approximation, both reccurent and vector. Based on comparison of parameters estimation of those methods (recurrent and vector), we can conclude that the recurrent method has better level of accuracy. Keywords : Crude Palm Oil (CPO) price, Recurrent, Singular Spectrum Analysis, Vector.. Introduction Singular Spectrum Analysis (SSA) is a technique non parametric time series analysis that combines elements of classical time series, multivariate statistics, multivariate geometry, dynamic system and signal processing (Hassani an Zhilglzavsky, 2009). Broomhead and Raja (986a, 986b) contribute to the birth of SSA, they show that Singular Value Decomposition (SVD) can reduce noise. SSA consists of two terms, that is singular and spectrum. Singular derived from spectral decomposition or eigen decomposition of trajectory matrix. Spectrum is defined for the sum of a set eigenvalue after spectral decomposition. The goal of SSA is to make a decomposition from initial data into the sum of a small number of independent components and interpreted as a trend pattern, periodic component that identified as a signal and component besides noise. There are two methods SSA forecasting, recurrent (R-forecasting) and vector (Vforecasting). Recurrent method is the basic method and often used because relatively easy, the LRF can be described in polynomial form (Golyandina et al., 200). Recurrent forecasting performs recurrent continuation directly (with the help of LRF), while vector forecasting deals with L-continuation. In the case of approximate continuation, the two forecasting algorithms usually give different results.

2 Here, using Crude Palm Oil (CPO) price in Rotterdam s Physical Market data and MDEX (Malaysia Derivatives Exchange) in 203 to compare accuracy of Forecasting Result. 2. SSA Modelling Basic algorithm of SSA is to partition initial data into the new data that consists of trend, seasonal, periodic component and noise (Hassani, 2007). Algorithm of SSA is consists of two stage. The first stage is decomposition and the second stage is reconstruction. 2. Decomposition In the decomposition stage consists of embedding and Singular Value Decomposition (SVD). The parameter which have an important role ini this stage is window length L. a. Embedding Suppose the time series data ( ) has a length of N and there are no missing data. First stage in basic algorithm of SSA is embedding where Y transformed into a trajectory matrix size. In this stage it is necessary to determine the parameter of window length L with the provisions. Embedding can be regarded as a mapping that transfers time series unidimentional Y into a multidimensional with lag vector ( ) for where. From the lag vector then formed the trajectory matrix size. This trajectory matrix is a Hankel matrix in which all of anti-diagonal elements have the same value X i, j Y y y2 yl Y2 y2 y3 yl Y y y y N K K N b. Singular Value Decomposition (SVD) The second stage is made Singular Value Decomposition (SVD) from trajectory matrix X. Let are eigenvalue from S where, with the order of decreasing and are eigenvector for each eigenvalue. Let trajectory matrix is. If we denote (i=,...,d) then the SVD from The matrix X is SVD formed from eigen vector, singular value and principal component from S. Three elements forming the SVD is called eigentriple. 2

3 2.2 Reconstruction In the reconstruction stage consists of grouping and diagonal averaging. The parameter which have an important role ini this stage is grouping effect (r). a. Grouping In this session the trajectory matrix dimension is composed into sub-groups according to trend pattern, periodic and noise. The grouping step related to the separation of element matrix into several sub-group and summing the matrix in each group. Matrix will be partitioned into m disjoint subsets. Let is the resultant matrix with index according to group I can be defined as. Then adjusted to group. Thus, can be expanded into : b. Diagonal Averaging In the basic algorithm on SSA, diagonal averaging step is the transformation result of grouping of matriks into a new series of length N. The purpose of this stage is to obtain the singular value from components that have been separated to be used in forecasting. The results of diagonal averaging will change the trajectory matrix dimension into new matrix that have the same dimension. To look for diagonal averaging of matrix can be used the following equation. { where ( ) and ( ). The above equation when applied to the resultant matrix will be formed series ( ) ( ( ) ( ) ) Therefore original series will be decomposed into a sum of m- series: ( ) 3. Forecasting Method in SSA : R-forecating and V-forecasting There are two method of forecasting in SSA, that is recurrent method (R-forecasting) and vector method (V-forecasting). Recurrent method is a basic method often used because it is relative easily (Golyandina. Et al., 200). Vector method is a modification of recurrent 3

4 method. In SSA forecasting, model can describe with the help of Linear Recurrent Formula (LRF). In polynomial form as follows. The difference between the two methods of forecasting is forecasting with recurrent method perform continuation directly (with the help of LRF). The method of forecasting with vector related to L-continuation. This leads to the approximate continuation usually give different results (Golyandina, et. al, 200) 3. Algorithm of Forecasting with Recurrent Method in SSA Estimate LRF coefficient that is can be used eigenvector obtained from SVD step. Let ( ) and ( ) and is the last component from eigenvector U or it can be written as. Then LRF coefficient can be calculated by the following formula. ( ) where In this recurrent method, time series used is series of reconstruction results obtained from the diagonal averaging result, then determined M point for forecast. So that will form a series of forecasting results, that is ( ) based formulation below: { where is result of SSA forecasting with recurrent method. 3.2 Algorithm of Forecasting with Vector Method in SSA a. After SVD step in in the initial algorithm, and trajectory matrix has formed a Hankel matrix H, the next step is calculate the matrix. Matrix is a matrix of linear operators from ortogonal projection ( ) ( ) Where [ ] b. Calculate the linear operator for vector method as follow. ( ) ( ) Where is a vector containing L- of end component from vector Y and is a vector containing L- of initial component from vector Y. c. Calculate value 4

5 { ( ) d. Calculate diagonal averaging from matrix [ ]. Diagonal averaging obtained has a series value e. M new value results of forecasting using vector method is 4. Comparison Choice of the proper parameter is important thing in SSA. In the embedding step, window length L is a main parameter to determine the dimension of the trajectory matrix. Let Y be the initial time series with no missing values of length N. The range of window length L is. Another parameter is the grouping effect (r) in the reconstruction stage. Grouping as the main parameter plays a significant role to determine the trend, smooth, halfyearly, quarterly, monthly series and white noises for the graphical inference (Myung, 2009). Its determination performed by looking the several eigentriples at one-dimension and twodimension as the result of SVD stage. In this paper, L=70 and r = 5 are selected. In this case will forecast for 0 future point. The table below present a comparison between two methods of forecasting in SSA. Table. A Comparison of R-Forecasting and V-Forecasting Rotterdam s Physical Residual MDEX Market Measure Recurrent Vector Recurrent Vector MSE 0,9 0,52 0,327 0,386 MAPE,664 2,643 9,499 2,676 5

6 Data Data International Symposium on Forecasting-Economic Forecasting, June 29 July 2, Time Series Plot of R-Forecasting; V-Forecasting Variable C4 R-Forecasting R-Forecasting 2 V-Forecasting V-Forecasting Index Figure. Time Series Plot of R-Forecasting and V-Forecasting From Rotterdam s Physical Market Data Time Series Plot of R-Forecasting2; V-Forecasting2 Variable C4 R-Forecasting2 R-Forecasting2 2 V-Forecasting2 V-Forecasting Index Figure 2. Time Series Plot of R-Forecasting and V-Forecasting From MDEX Data 6

7 5. Conclusion Based on table, comparison of parameters estimation of those methods (recurrent and vector), we can conclude that in this case, the recurrent method has better level of accuracy. References. Abraham, B.,Ledolter, J Statistical Methods for Forecasting. John Wiley & Sons, Inc. USA. 2. Hassani, Hossein.: Singular Spectrum Analysis: Methodology and comparison. Journal of data science 5 : Cardif University and Central Bank of the Islamic Republic of Iran Golyandina, N., Nekrutkin, V., Zhigljavsky, A.: Analysis of time series structure: SSA and related techniques. Chapman & Hall/crc, Myung, No Kang.: Singular Spectrum Analysis. Tesis. University of California Pepelyshev, Andrey.:On the choice of a linear reccurent formula for the SSA forecast. St.Petersburg State University Mathematical Department. Cardiff, Wales, UK

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