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 pseudoadditive
(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 trendcycles
movement, which is extremely weatherdependent could be modeled suitably
using a multiplicativeadditive (Mixed) model given by Y_{t }=
D (C_{t} + S_{t} + I_{t}). The further said that
this can be implemented in X12ARIMA. In a similar development, Findley
et al. (1998) earlier suggested a Pseudoadditive decomposition
with a relative working day factor (D) as Y_{t} = C_{t}D_{t}
(S_{t}+ I_{t}^{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:
• 
Y_{t} = C_{t} +_{ }S_{t }+ I_{t}completely
additive model 
• 
Y_{t} = C_{t} S_{t }I_{t}completely
multiplicative model 
• 
Y_{t} = C_{t}S_{t }+ I_{t}Mixed
model. 
Where:
Y_{t } 
= Observation at time t 
C_{t } 
= Current mean or trendcycle effect 
S_{t } 
= Seasonal effect 
I_{t } 
= 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
longterm trend but also cyclical fluctuations. Then comes the calendar
component (D), derived from the effects of workingday 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:
The estimated model is given by:
Where:
Y_{t } 
= The observation at time t 
b_{0 } 
= The trend parameter estimate 
a_{0 } 
= The constant used to set the level of the series 
a_{i } 
= The parameter estimates. 
e_{t } 
= 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 n1/2 for an
odd number of observations.
Hence; The model given above can be represented traditionally
as
Y_{t} = T_{t }(S_{t}
+ 1) I_{t} = T_{t}S_{t}I_{t }+ T_{t}I_{t}

T = a_{0} + b_{0}t 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:
Then DT (the detrended series) is then estimated by method
of ordinary least squares
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 b_{0}
is not significant (i.e., b_{o} = o), then the estimated model
becomes.
DATA ANALYSIS
From the trend analysis,
b_{0} 
= 0 
a_{o} 
= 210.1 
Therefore T 
= a_{0 }= 210.1 
Also, since
k = 6, I = 1,2,3,...,6 I = 1 and ω = π/6
Therefore,
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 

Table 2: 
Result for testing the significance of general detrended
series model 


Fig. 1: 
Plot of actual and estimated values 
RESULTS AND DISCUSSION
From the above analysis, the general estimated model
is obtained as Y_{t} = 210.1 (0.984 cos ωt+1) or Y_{t}
= 210.1206.728cosùt. 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 pseudoadditive 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 (Y_{t}) 

Appendix 2: 
Fitted values (Y) 
