Improved Circular Model on Forecasting Arrivals from Western European Countries to Sri Lanka

The Circular Model (CM) is a newly joined member to the family of Univariate Statistical Models. Development of the CM was based on; Newton’s Law of Circular Motion, Fourier Transformation and Multiple Regression Analysis. Most important property of the CM is that; the model could be applied for either stationary or non-stationary series. Further, the model is capable in capturing both seasonal and cyclical patterns of a time series. However, the applicability of CM is restricted to trend free series. As such, it was intended to improve the CM, by using the differencing technique. The Improved Circular Model; named as, "Sama Circular Model, is tested on tourist arrival data from Western European countries to Sri Lanka. Monthly arrival data for the period of April 2008 to December 2016 were used for the analysis. Time Series plots and Auto Correlation Functions were used for pattern recognition. The Auto Correlation Functions (ACF) of residuals and Ljung-Box Q statistics (LBQ) were used to test the independence of residuals. The Anderson Darling test was used to test the normality of residuals. Forecasting ability of the models was assessed by Mean Square Error (MSE) and Mean Absolute Deviation (MAD). Forecasting ability of Sama Circular Model (SCM) was compared with the Decomposition techniques and Seasonal Auto Regressive Moving Average (SARIMA). It is concluded that the SCM is capable in forecasting arrivals from Western European countries and the SCM is superior to the other tested models. It is recommended to test the SCM for different fields; Agriculture, Meteorology, Economics, Financial markets and many more.


Introduction
Modeling wave patterns was an immense interest over the centuries, as wave like patterns is common in; Natural Sciences, Medical Sciences, Economic Sciences and many more .In general, waves are viewed in time domain or frequency domain (Philippe, 2008). Time domain analyses waves with respect to time, whilst the frequency domain analyses a signal with respect to frequency. The time domain analysis is known as "Time Series Analysis"; the frequency domain analysis is known as the "Spectral Analysis or Fourier Analysis". The Decomposition Technique; Auto Regressive Integrated moving Average (ARIMA); Seasonal Auto Regressive Integrated moving Average (SARIMA) are well known time series techniques used for analyzing wave patterns.
The Circular Model (CM) analyzes a wave with respect to its spectrum.

Research Problem
Sri Lanka has been an attractive tourism destination for centuries. The tourism industry in Sri Lanka shows a rapid growth from year 2009; drawn the attention of researchers. Konarasinghe (2016) has emphasized the importance of forecasting arrivals to Sri Lanka from various regions of the world, and attempted to forecast arrivals from Western European region. Konarasinghe (2017)

has tested; Decomposition techniques and Seasonal Auto
Regressive Moving Average Models (SARIMA), and found them successful for the purpose. Yet, the author was unable to differentiate the Seasonal Variations and Cyclical Variations of the series.
The Cyclical variations are long term wave like patterns, while the seasonal variations are short term wave like patterns. Seasonal patterns are observed within a year, but the cyclical patterns are observed in longer period; at least more than a year. In general, Decomposition techniques are used to capture the cyclical patterns. In Decomposition models; a time series is described as a function of four components; Trend (T), Cyclical influence (C), Seasonal influence (S) and the random error (e). In order to capture the cyclical pattern, Decomposition technique follows several steps; firstly, fit the trend model and then obtain the de-trend series; secondly, find the seasonal indices for de-trended data and de-seasonalize them; finally model the de-seasonalized series by trigonometric functions. However, this method is time consuming and cumbersome. In contrast, the Circular Model (CM) is easy to use and less time consuming. Yet, the applicability of the CM is limited to trend free series (Konarasinghe, 2016).

Objectives of the Study
Objectives of the study were twofold; primary and secondary.
Primary Objective: Improve the Circular Model, in order to forecast the series with trend Electronic copy available at: https://ssrn.com/abstract=3411481

Materials and Methods
The study is based on the Circular Model of Konarasinghe, Abeynayake & Gunaratne (2016). The development of the CM was based on the theory of Uniform Circular motion and the Multiple Regression Analysis.
A particle P, which is moving in a horizontal circle of centre O and radius a is given in Figure 1. The ω is the angular speed of the particle; In circular motion, the time taken for one complete circle is known as the period of oscillation. In other words, the period of oscillation is equal to the time between two peaks or troughs of a wave (Stephen, 1998). If a time series follows a wave with f peaks in N observations, its period of oscillation (T) can be given as; Share returns of individual companies of Sri Lankan share market follows wave like patterns. Hence the return at time t (R t ) was modeled as; Electronic copy available at: https: The model (3) was named as the "Circular Model". Amplitudes, a k and b k are found by regressing R t on t k sin and t k cos for k= 1 to 6. Model assumptions of CM are; the series R t is trend free; trigonometric series, t k sin and t k cos are independent; residuals are Normally distributed and independent.

Improved Circular Model
For a random variable Y t , the CM can be written as; The CM cannot be applied, if Y t has a trend. This study suggests the method of differencing to mitigate the limitation of the CM. In usual notation, differencing series of Y t are as follows; First differenced series: The model (9); improved Circular Model, is named as "Sama Circular Model (SCM)".

Population and Sample of the Study
It is a known fact that the Western European countries contribute highly into the Sri Lankan tourism market. As such; monthly arrival data for three leading counties; UK, German and France were collected for the

Results and Findings
At first, box plots of monthly arrival series were obtained and check whether outliers exist. If so, they were adjusted by taking moving average of previous three months. Time series plots were used for pattern recognition of the series. Figure 2 shows wave like patterns with increasing trend for all countries.
Log transformed arrivals were used in the analysis; the differencing technique was used to obtain the trend free series. For example; Figure 3 shows the arrivals from UK (Y t ), while Figure 4 shows the first difference of the same series (X t ); Hence estimates of arrival from UK can be obtained by the model; Same procedure was repeated with arrival series of; German, France and Netherlands; summary of analysis is given in Table 1; Electronic copy available at: https://ssrn.com/abstract=3411481 The model (11) comprises only one periodic function, sinωt. The period of oscillation of sinωt is 6 months, hence the arrivals from UK follows only seasonal pattern. The fitted model for German comprises two periodic functions; sinωt and cosωt , with period of oscillation 5 months and 6 months respectively. Therefore arrivals from German follow only seasonal patterns, but not cyclical patterns. The fitted model for France comprises three periodic functions, but period of oscillation of each function is less than 12 months. Hence, arrivals from France also follow only seasonal patterns.
The Decomposition Models and SARIMA model also tested on same data series. Results reveal that the Decomposition Multiplicative Model serve the purpose with satisfactorily low measurements of errors, but the SARIMA models did not fit for any of the three series. Hence the forecasting ability of Decomposition Multiplicative Models and SCM were compared and found that, measurements of errors are small in SCM for all the three data series. Time series plots of Actual Vs Fits showed that the patterns of SCM forecast are more close to the actual values. Hence the SCM is superior to Decomposition method in forecasting arrivals from Western European countries to Sri Lanka.

Conclusions
The Circular Model (CM) is a univariate time series technique, which can be used to model the wave like patterns. Literature reveals that the CM can be applied only for trend free series. Hence the present study was focused on improving the CM and testing the applicability of improved model in real life data sets. The CM was improved by adopting the differencing technique; the improved model is named as Sama Circular Model (SCM).
The SCM was tested on tourism arrivals to Sri Lanka from Western European countries, and found that the SCM fits into tested data sets. Then the forecasting ability of the SCM was compared with SARIMA and Decomposition models and found that the SCM is superior to the other two methods in forecasting arrivals from Western European countries to Sri Lanka. The study concludes that the SCM mitigates the restriction of CM; can be applied for a series with trend. It is recommended to test the SCM for more real life data sets in different fields of research.