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Pseudo-Additive (Mixed) Fourier Series Model of Time Series



Emmanuel John Ekpenyong
 
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ABSTRACT

This study improves on the Additive Fourier Series and traditional model of discrete periodic time series. It seeks to formulate a mixed (multiplicative-additive) Fourier Series model which decomposes a time series into multiplicative trend, seasonal components and additive error component together with additive trend. It is discovered that for time series with strongly marked and obviously fluctuating seasonal effects a multiplicative-additive (mixed) Fourier series model is suitable. The relevance of the new model is shown by analyzing the rainfall data of Uyo metropolis with the use of the model. The resulting model gives Yt = 210.1 (1-0.984 cos ωt) which fits well to the original data and can be used in forecasting future values of the rainfall data.

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  How to cite this article:

Emmanuel John Ekpenyong , 2008. Pseudo-Additive (Mixed) Fourier Series Model of Time Series. Asian Journal of Mathematics & Statistics, 1: 63-68.

DOI: 10.3923/ajms.2008.63.68

URL: https://scialert.net/abstract/?doi=ajms.2008.63.68

INTRODUCTION

Most periodic time series have been assumed to be modeled by the use of additive Fourier Series model or any other seasonal models, but not much have been done on the modeling of a special case of such time series with a strongly marked and obviously fluctuating seasonal effects. The researcher discovered that for such time series a pseudo-additive (mixed) Fourier series model is very suitable. The development is as a result of the work of Hernmann et al. (2006), who demonstrated that a series with sharply pronounced seasonal fluctuations and trend-cycles movement, which is extremely weather-dependent could be modeled suitably using a multiplicative-additive (Mixed) model given by Yt = D (Ct + St + It). The further said that this can be implemented in X-12-ARIMA. In a similar development, Findley et al. (1998) earlier suggested a Pseudo-additive decomposition with a relative working day factor (D) as Yt = CtDt (St+ It-1), which made it possible to estimate relative calendar factors based on a logarithmic REGARIMA model. Traditionally, time series are decomposed into four basic components: the trend, seasonal, cyclical and irregular components. Nkpodot and Usoro (2005) stated three traditional models of time series as completely additive, completely multiplicative and mixed models. They outlined the models mathematically as:

Yt = Ct + St + It-completely additive model
Yt = Ct St It-completely multiplicative model
Yt = CtSt + It-Mixed model.

Where:

Yt =   Observation at time t
Ct =   Current mean or trend-cycle effect
St =   Seasonal effect
It =   Residual or random error.

Also in time series analysis, it is assumed that an unadjusted series (Y) may be decomposed into four unobservable components. The first of those is the trend cycle component (C), which includes not just the long-term trend but also cyclical fluctuations. Then comes the calendar component (D), derived from the effects of working-day variations, for example. There is additionally the seasonal component (S), which includes annual fluctuations that recur to almost the same degree in the same period (Anonymous, 2001).

According to Priestly (1981), one of the most important advantages of Fourier Series Analysis is its simple way of modeling a series with seasonality or cyclicalness. He further said that Fourier series method can be used to model seasonal effects using several seasonal peaks per year.

Chatfield (1975), stated that a seasonal component of a time series can be decomposed into underlying sine and cosine functions of different frequencies known as Fourier series. One way of doing this is to cast the issue as a linear multiple regression problem, where the dependent variable is the detrended time series and the independent variable are the sine and cosine functions of all possible (discrete) frequencies (Priestly, 1981).

Roerink et al. (2000) used Fourier Series Analysis or Harmanic Analysis of Time Series to screen and remove cloud affected observations and temporarily interpolate the remaining observations to reconstruct gapless images at a prescribed time.

METHODS OF ANALYSIS

The analysis of this study is done through the help of a statistical package called MINITAB. The model is given by:

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
(1)

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
(2)

The estimated model is given by:

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
(3)

Where:

Yt =   The observation at time t
b0 =   The trend parameter estimate
a0 =   The constant used to set the level of the series
ai =   The parameter estimates.
et =   The error term
ω =   2π f/n

Where:

f =   The fourier frequency
k =   The highest harmonic

It is noteworthy here that the highest harmonic, k in Fourier Series Analysis model is the number of observations per season divided by two (2) for an even number of observation and n-1/2 for an odd number of observations.

Hence; The model given above can be represented traditionally as

Yt = Tt (St + 1) It = TtStIt + TtIt

T = a0 + b0t is used in estimating the trend, while the expressions containing the sine and cosine terms gives the estimated model for seasonality. The method used in estimating the parameters of the model is the method of least squares in Multiple Regression Analysis. Firstly the trend is removed from the series by multiplicative procedure:

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
(5)

Then DT (the detrended series) is then estimated by method of ordinary least squares

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
(6)

A statistical test of significance of the general model and that of the parameter estimates are done using a computer package known as MINITAB.

It is noted here that if the trend parameter b0 is not significant (i.e., bo = o), then the estimated model becomes.

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
(7)

DATA ANALYSIS

From the trend analysis,

b0 = 0
ao = 210.1
Therefore T = a0 = 210.1

Also, since
k = 6, I = 1,2,3,...,6 I = 1 and ω = π/6

Therefore,

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series

is modeled using in Table 1 and 2:

From Table 1 , the estimated model with the significant parameter estimates is; DT = -0.984 cos ω.

Table 1: Results for testing the significance of the detrended series model
Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series

Table 2: Result for testing the significance of general detrended series model
Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series

Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series
Fig. 1: Plot of actual and estimated values

RESULTS AND DISCUSSION

From the above analysis, the general estimated model is obtained as Yt = 210.1 (-0.984 cos ωt+1) or Yt = 210.1-206.728cosùt. The plot of the actual and fitted values of the rainfall data is given in Fig. 1. The plot indicates a good fit of the model.

It is a known fact that there are many models used in modeling a periodic time series. In this study, developments are made in the use of Fourier Series model to fit a time series. It is discovered that a pseudo-additive Fourier series model so developed is quite suitable for the modeling and forecasting of a time series with strongly marked and obviously fluctuating seasonal effects. This is shown in Appendix 1 and 2, where the estimated values fit well to the actual values.

Appendix 1: Actual values (Yt)
Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series

Appendix 2: Fitted values (Y)
Image for - Pseudo-Additive (Mixed) Fourier Series Model of Time Series

REFERENCES

1:  Anonymous, 2001. X-12-ARIMA-Reference Manual; Version 0.2.8. US Census Bureau. Time Series Staff, Statistical Research Division, Washington DC.

2:  Chatfield, C., 1975. Analysis of Time Series: Theory and Practice. Chapman and Hall, London.

3:  Findley, D.F., B.C. Mansell, W.R. Bell, M.C. Otto and B.C. Chen, 1998. New capabilities and methods of the x-12-arima seasonal-adjustment program. J. Bus. Econ. Stat., 16: 127-152.
CrossRef  |  Direct Link  |  

4:  Hernmann, H., T. Liebig and K. Todter, 2006. A new time series model for seasonally adjusting economic data with trend-cycle movement and irregular sharply pronounced seasonal fluctuations. A Paper Presented at the Eurostat Conference on Seasonality, Seasonal Adjustment and Their Implications for Short-Term Analysis and Forecasting.

5:  Nkpodot, U.T.C. and A.E. Usoro, 2005. Applied Statistics for Business and Sciences. Neakso International Company, Ikot Ekpene.

6:  Priestley, M.B., 1981. Spectral Analysis and Time Series. 1st Edn., Academic Press, London, ISBN-10: 0125649010, pp: 653
Direct Link  |  

7:  Roerink, 2000. Harmonic analysis of time series. Int. J. Remote Sensing, 2: 1911-1917.

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