Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Holt’s exponential smoothing model for interval-valued time series This work is part of a paper submitted to International Journal of Forecasting André Luis Santiago Maia and Francisco de A. T. de Carvalho Universidade Federal de Pernambuco Centro de Informática Av. Prof. Luiz Freire, s/n - Cidade Universitária CEP: 50740-540 - Recife - PE - Brasil {alsm3,fatc}@cin.ufpe.br
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Contents Introduction 1 Interval-valued time series 2 Holt’s method for interval-valued time series 3 Application in stock market 4 Conclusions 5
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Symbolic Data Analysis Interval-valued data has been also considered in the field of Symbolic Data Analysis (SDA) (Bock and Diday (2000)). This field, related to multivariate analysis, pattern recognition and artificial intelligence, aims to extend classical exploratory data analysis and statistical methods to symbolic data. These new variables make it possible to take into account the variability and/or uncertainty present in the data. In the field of SDA, interval-valued data appear when the observed values of the variables are intervals of the set of real numbers I R .
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Nowadays, different approches have been introduced to analyse interval-valued data. Patiño-Escarcina et al. (2004) propose a one layer perceptron for classification tasks, where inputs, weights and biases are represented by intervals. Roque et al. (2007) propose and analyse a new model of multilayer perceptron based on interval arithmetic that facilitates handling input and output interval data. Others authors have shown success with interval-valued data, but on interval analysis approach. We manage interval-valued time series in the context of SDA, without use of operations and functions of interval arithmetic. This is a main feature that differs our paper from those cited above.
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions In the field of SDA for interval-valued data, Ichino et al. (1996) have introduced a symbolic classifier as a region oriented approach. Cazes et al. (1997) and Lauro and Palumbo (2000) introduced principal component analysis methods. Billard and Diday (2003) have introduced central tendency and dispersion measures. Chavent et al. (2006) and De Carvalho (2007) provides a number of clustering methods. Linear regression models have been also considered by Billard and Diday (2000) and Lima–Neto and De Carvalho (2008). Maia et al. (2008) propose approaches to interval-valued time series forecasting.
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Contents Introduction 1 Interval-valued time series 2 Holt’s method for interval-valued time series 3 Application in stock market 4 Conclusions 5
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Interval-valued time series (ITS) When interval data is collected in an ordered sequence against time, we say that we have a interval-valued time series . The interval is described by a two-dimensional vector with elements in I R represented by upper bound , X U t , and by lower bound , X L t . [ X L 1 ; X U 1 ] , [ X L 2 ; X U 2 ] , . . . , [ X L n ; X U n ] Specifically, an observed interval at time t is noted I t and it is represented as � � X U I t = t X L t Tools for interval-valued time series data analysis are also very much required.
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Interval-valued time series (rigth) obtained from a set of classical time series (top left) and from time series of higher frequency (bottom left). 30 30 20 20 Temperature Temperature 10 10 0 0 −10 −10 Jan 74 Jul 74 Jan 75 Jul 75 Jan 76 Jul 76 Dec 73 May 74 Oct 74 Mar 75 Aug 75 Jan 76 Jun 76 Time Time 120 120 100 100 Stock price Stock price 80 80 60 60 40 40 Dec 1, 06 Jan 1, 07 Feb 1, 07 Mar 1, 07 Apr 1, 07 May 1, 07 Dec 06 Jan 07 Feb 07 Mar 07 Apr 07 Time Time
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Motivation Given an ITS, how to solve the problem of forecast in the context of SDA?
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Contents Introduction 1 Interval-valued time series 2 Holt’s method for interval-valued time series 3 Application in stock market 4 Conclusions 5
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Standard Holt’s method In classical data, the standard Holt’s method is given by � α y t + ( 1 − α )( � L t − 1 + � L t = T t − 1 ) , � β ( � L t − � L t − 1 ) + ( 1 − β ) � T t = T t − 1 . � L t is the smoothed level of the series, computed after y t is observed � T t is the smoothed trend at the end of period t y t = � L t + � � T t 0 < α, β < 1 are the smoothing parameters start values: � L 2 = y 2 and � T 2 = y 2 − y 1
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Holt’s method for ITS The interval Holt’s exponential smoothing method (Holt I ) follows the representation L I L I T I A I t + ( I − A )( � � t − 1 + � = t − 1 ) , t T I L I L I T I � B ( � t − � t − 1 ) + ( I − B ) � = t − 1 . t A and B denote the (2 × 2) smoothing parameters matrices, � � � � α 11 α 12 β 11 β 12 A = and B = α 21 α 22 β 21 β 22 and I is an (2 × 2) identity matrix.
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Review Standard Holt � α y t + ( 1 − α )( � L t − 1 + � L t = T t − 1 ) , � β ( � L t − � L t − 1 ) + ( 1 − β ) � T t = T t − 1 . ⇓ Interval Holt L I L I T I A I t + ( I − A )( � � t − 1 + � = t − 1 ) , t T I L I L I T I t − 1 ) + ( I − B ) � � B ( � t − � = t − 1 . t
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Expanding the expressions, the Holt I method is given by: � � � � � � � � � t − 1 − � X U L U T U α 11 α 12 1 − α 11 − α 12 L I � t t − 1 = + t X L � t − 1 − � α 21 α 22 − α 21 1 − α 22 L L T L t t − 1 � � t + ( 1 − α 11 )( � t − 1 − � t − � t − 1 + � α 11 X U L U T U t − 1 ) + α 12 ( X L L L T L t − 1 ) = t + ( 1 − α 22 )( � t − 1 − � t − � t − 1 + � α 22 X L L L T L t − 1 ) + α 21 ( X U L U T U t − 1 ) and � � � � � � � � � � t − � L U L U T U β 11 β 12 1 − β 11 − β 12 T I � t − 1 t − 1 = + t � t − � 1 − β 22 � β 21 β 22 L L L L − β 21 T L t − 1 t − 1 � � β 11 ( � t − � t − 1 ) + ( 1 − β 11 ) � t − 1 + β 12 ( � t − � t − 1 − � L U L U T U L L L L T L t − 1 ) = β 22 ( � L L t − � L L t − 1 ) + ( 1 − β 22 ) � T L t − 1 + β 21 ( � L U t − � L U t − 1 − � T U t − 1 ) The main advantage of the model presented here is the consideration that the trajectory of the upper boundary of the series can be affected by realizations of the lower boundary of this series and vice-versa.
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions A and B with elements constrained to the range ( 0 , 1 ) can be estimated by minimizing n � ( I t − � I t ) ⊤ ( I t − � I t ) R ( A , B ) = t = 3 � � ⊤ � � n � t − � t − 1 − � t − � t − 1 − � X U L U T U X U L U T U t − 1 t − 1 = t − � t − 1 − � t − � t − 1 − � X L L L T L X L L L T L t − 1 t − 1 t = 3 n n � � t − 1 ) 2 + ( X U t − � L U t − 1 − � T U ( X L t − � L L t − 1 − � T L t − 1 ) 2 . = t = 3 t = 3 L I T I 2 = I 2 and � 2 = I 2 − I 1 Start vectors: �
Introduction Interval-valued time series Holt’s method for interval-valued time series Application in stock market Conclusions Optimum smoothing parameters matrices for the Holt I The estimation as a constrained non-linear programming problem: α ij ,β ij R ( A , B ) , min subject to 0 ≤ α ij , β ij ≤ 1 The solution obtained by the limited memory BFGS method for bound constrained optimization (L-BFGS-B); Byrd et al. (1995) This method allows box constraints (each parameter can be given a lower and upper boundary) The L-BFGS-B algorithm is implemented in R software package; R Development Core Team (2008)
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