Comparison of Two Time Series Decomposition Methods : Least Squares and Buys-Ballot Methods

This paper discusses comparison of two time series decomposition methods: The Least Squares Estimation (LSE) and Buys-Ballot Estimation (BBE) methods. As noted by Iwueze and Nwogu (2014), there exists a research gap for the choice of appropriate model for decomposition and detection of presence of seasonal effect in a series model. Estimates of trend parameters and seasonal indices are all that are needed to fill the research gap. However, these estimates are obtainable through the Least Squares Estimation (LSE) and Buys-Ballot Estimation (BBE) methods. Hence, there is need to compare estimates of the two methods and recommend. The comparison of the two methods is done using the Accuracy Measures (Mean Error (ME)), Mean Square Error (MSE), the Mean Absolute Error (MAE), and the Mean Absolute Percentage Error (MAPE). The results from simulated series show that for the additive model; the summary statistics (ME, MSE and MAE) for the two estimation methods and for all the selected trending curves are equal in all the simulations both in magnitude and direction. For the multiplicative model, results show that when a series is dominated by trend, the estimates of the parameters by both methods become less precise and differ more widely from each other. However, if conditions for successful transformation (using the logarithmic transform in linearizing the multiplicative model to additive model) are met, both of them give similar results.


Introduction
The two major goals of time series analysis are 1) identification of the nature of the phenomenon represented by the sequence of observations and 2) forecasting (predict-ing future values of the time series variable).Identification of the pattern and choice of model in time series data is critical to facilitate forecasting.Thus, these two goals of time series analysis require that the pattern of observed time series data is identified and described.Two patterns that may be present are trend and seasonality.
The trend represents a general systematic linear or (most often) nonlinear component that changes over time and does not repeat or at least does not repeat within the time range captured by data.As long as the trend is monotonous (consistently increasing or decreasing) the identification of trend component is not very difficult.Trend analysis (methods for estimating the trend parameters) can be done by three important methods; 1) smoothing [1] [2] [3] [4] [5] 2) fitting a mathematical function [6] [7] and 3) differencing to make the series stationary in the ARIMA methodology [1] [8] [9].
Tests for trend are given in Kendall and Ord [7].Correlation analysis can also be used to assess trend.If a time series contains a trend, then the values of the autocorrelations will not come to zero except for very large values of the lag [6].
Many time series exhibit a variation which repeats itself in systematic intervals over time and this behaviour is known as seasonal dependency (seasonality).By seasonality, we mean periodic fluctuations.It is formally defined as correlational dependency of order k between each ith element of the series and the (i − k)th element [10] and measured by autocorrelation (a correlation between t X and t k X − ); k is usually called the lag.Seasonality can be visually identified in the series as a pattern that repeats every k elements.The following graphical techniques can be used to detect seasonality: 1) A run sequence plot [11], 2) A seasonal sub-series plot [12], 3) Multiple box plots [11] and 4) the autocorrelation plot [1].Both the seasonal sub-series plot and the box plot assume that the seasonal periods are known.In most cases, the seasonal periods are easy to find (4 for quarterly data, 12 for monthly data etc).If there is significant seasonality, the autocorrelation plot should show spikes at multiples of lags equal to the period, the seasonal lag (for quarterly data, we would expect to see significant spikes at lag 4, 8, 12, 16, and so on).Davey and Flores [13] proposed a method which adds statistical tests of seasonal indexes for the multiplicative model that helps identify seasonality with greater confidence.Tests for seasonality are also given in Kendall and Ord [7].
In time series analysis, it is assumed that the data consist of a systematic pattern and random noise (error).The systematic pattern consists of the: trend (denoted as t T ), seasonal (denoted as t S ) and cyclical (denoted as t C ) components.The random noise is the error or irregular component (denoted as t I or t e ), where t stands for the particular point in time.These four classes of time series components may or may not coexist in real-life data.
In addition to identifying the patterns (the components), the main two goals of a time series analysis are better achieved if and only if the correct model is used.The specific functional relationship between these components can assume different forms.
However, the possibilities are that they combine in an additive (additive seasonality) or a multiplicative (multiplicative seasonality) fashion, but can also take other forms such as pseudo-additive/mixed (combining the elements of both the additive and multiplica- .It follows that we can study the additive model (1.1) and apply the results obtain to the multiplicative model after linearization.The pseudo-additive model is used when the original time series contains very small or zero values.For this reason, this work will discuss only the additive and multiplicative models.
As far as the traditional method of decomposition is concerned (to be referred to as the Least Squares Estimation (LSE) Method), the first step will usually be to estimate and eliminate t M for each time period from the actual data either by subtraction for Equation (1.4) or division for Equation (1.5).The de-trended series is obtained as t t X M − for Equation (1.4) or t t X M for Equation (1.5).In the second step, the seasonal effect is obtained by estimating the average of the de-trended series at each season.The de-trended, de-seasonalized series is obtained as t t t X M S − − for Equation (1.4) or ( ) X M S for Equation (1.5).This gives the residual or irregular component.Having fitted a model to a time series, one often wants to see if the residuals are purely random.For detailed discussion of residual analysis, see [1] [14].
This LSE as describe above is known to be associated with computational difficulties and does not give an insight for choice of models for time series decomposition and detection of presence of seasonal effect in a series.
However, by arranging a series of length n into m rows  , Iwueze et al. [15] showed that the row and column averages and variances can be used to 1) determine the need for and appropriate transformation 2) choose the appropriate model for decomposition, 3) detect the presence of seasonal effect in the series and 4) obtain estimates of the trend parameters and seasonal indices of the entire series.In particular, for a seasonal series, the rows represent the periods/years while the columns are the seasons.This two-dimensional arrangement of a series is referred to as Buys-Ballot table, see [16] [17].

Using the periodic means ( )
. , Iwueze and Nwogu [20] have shown that, for seasonal time series data and for all trending curves, the row, column and overall averages and variances of the Buys-Ballot table are 1) functions of the trend parameters and 2) different for the additive and multiplicative models.Therefore, the row, column and overall averages and variances can be used for i) choice of appropriate model, ii) detection of presence of seasonal effect in a series in addition to estimation of trend parameters and seasonal indices.Since estimates of the trend parameters and seasonal indices can also be derived from the Least Squares method, there is the need to compare the two decomposition methods.
Therefore, the ultimate objective of this study is to compare the estimates of trend

Methodology
The comparison of Least Squares Estimates and Buys-Ballot Estimates in this study is done using measures often referred to in the literature as Forecasting Accuracy Measures.These include the Mean Error (ME), Mean Square Error (MSE), the Mean Absolute Error (MAE), and the Mean Absolute Percentage Error (MAPE).
Given the actual values of the parameters , 1, 2, i i L θ =  these accuracy measures may be defined in terms of the deviations of the parameter estimates from the corresponding actual values used in the simulations.Thus, if ˆi θ is the estimate of the para- In terms of i e , the accuracy measures adapted from the forecasting Accuracy Measures are ( ) In these definitions, the comparison of parameter estimates is done directly using the actual and estimated values of the parameters.
The summary of the estimates of trend parameters and seasonal indices obtained by Iwueze and Nwogu [20] are given in Table 1 for Linear, Table 2 for Quadratic and Table 3 Exponential trend-cycle component for the additive and multiplicative models.
We note the following for Tables 1-3: 1) a′ , b′ and c′ are estimates derived from the regression equations of row av- erages on row number.
2) Additive and Multiplicative models give different estimates. 3) Estimates of parameters of Linear trend-cycle components and seasonal indices.

Parameter Model
Additive model Multiplicative model

Simulation Examples
This Section presents some simulations examples used to compare the estimates of the parameters of the selected trend curves and seasonal indices using Buys-Ballot method with those based on the conventional Least Squares Estimation method.For the linear trend-cycle, the simulated examples consist of 106 series of 120 observations each simulated from ( )    For the additive model the results are the same for the selected values of the slope parameter ( ) 0.02, 0.2 and 2.0 b = .Therefore, the summary statistics for b = 2 only is shown in Table 5.As Table 5 shows, the summary statistics (ME, MSE and MAE) for the two estimation methods (LSE and BBE) are equal both in magnitude and direction all the simulations.This indicates that the two methods are equally effective in estimating the trend parameters and seasonal indices when the model for decomposition is additive.
For the multiplicative model, Table 6 shows that for b = 0.02, the values of the sum-

Summary, Recommendation and Conclusion
This study has compared the estimates of trend parameters and seasonal indices from the Buys Ballot method with the results from the Least Squares Estimation (LSE) method for the linear, quadratic and exponential trending curves.The rationale for this study is to assess the performance of the Buys Ballot Estimation (BBE) method in relation to the Least Squares Estimation (LSE) method.
The comparison of Least Squares Estimates and Buys-Ballot Estimates in this study is done using Accuracy Measures (the Mean Error (ME), Mean Square Error (MSE), the Mean Absolute Error (MAE), and the Mean Absolute Percentage Error (MAPE)).
These Accuracy Measures are defined, for each estimation procedure, in terms of the deviations of the parameter estimates (using simulated series) from the corresponding actual values used in the simulations.
The results of the analyses show that, for the additive model the summary statistics = , Iwueze and Nwogu[17] constructed two derived variables, the Chain Base Estimation (CBE) and the Fixed Base Estimation (FBE)-derived variables, on bases of which they derived the estimates of linear trend-cycle parameters.Iwueze and Ohakwe[18] derived the estimates of quadratic trend-cycle parameters from the corresponding CBE and the FBE-derived variables, while Iwueze and Nwogu[19] derived the estimates of the exponential trend-cycle parameters from the corresponding CBE and the FBE-derived variables.
parameters and seasonal indices from the Buys Ballot method with the results from the conventional least squares method.The specific objectives are to: 1) Review the Buys Ballot method for estimation of trend parameters and seasonal indices for some selected trending curves, 2) obtain estimates of the trend parameters and seasonal indices using simulated examples, 3) compare the estimates of the trend parameters and seasonal indices from the Buys Ballot method with estimates from the traditional method (LSE) using simulated examples.Based on the results, recommendations are made.
multiplicative model.These values of a, b and c were arbitrarily chosen for simulation purposes.The values of , 1, 2, ,12 j S j =  for the simulated additive and the multiplicative models are given in mary statistics (ME, RMSE, MSE, MAE and MAPE) are equal in almost all the simulations as in the additive model.However, for values of b = 0.2 and 2.0, Table 7 and Table 8 show that the values of the summary statistics (ME, RMSE, MSE, MAE and MAPE) are unequal in most all the simulations and the difference increased as the value of b increased from 0.2 to 2.0.In other words, when a series is dominated by trend, not only do the estimates of the parameters by both methods become less precise; estimates of the parameters from the two methods differ more widely from each other.It could be recalled that logarithm transformation can be used to transform the multiplicative model to become the additive model.In order to preserve the linearity of the trend, Iwueze and Akpanta [21] have shown that the trend parameters (a and b) satisfy the condition; 0.01 0.06 b a − ≤ ≤ .Therefore, this lack of agreement between estimates from the two methods and the actual values when b = 0.2 and 2.0 may be attributable to the violation of the condition for successful transformation (linearization in this case).For b = 0.2 and a = 1, 0.2 b a = and for b = 2.0 and a = 1, 2.0 b a = .For the Quadratic trend-cycle component, the empirical examples also consist of 106 series of 120 observations each simulated from

(
ME, MSE and MAE) are equal both in magnitude and direction all the simulations for the two estimation methods (LSE and BBE) and all the selected trending curves.This indicates that the two estimation methods are equally effective in estimating the trend parameters and seasonal indices when the model for decomposition is additive.For the multiplicative model (shown for linear trend only), results of the analyses show that when the slope b = 0.02, the values of the summary statistics (ME, RMSE, 633 −0.257 −0.627 −0.263 −0.728 −0.162 −0.710 −0.180 −1.026 0.136 −1.002 0503 0.283 −1.523 0.303 −1.029 −0.191 −1.010 −0.210 −0.886 −0.334 −0.901 −0078 −0.072 −0.117 −0.033 −0.075 −0.075 −0.109 −0.041 −0.523 0.373 −0.501 0455 0.365 −0.413 0.323 −0.021 −0.069 −0.009 −0.081 −0.554 0.464 −0.600 0.510 8 [6]lical variation refers to the long term oscillation or swings about the trend and only long period sets of data will show cyclical fluctuation of any appreciable magnitude.If short period of time are involved (which is true of all examples of this study), the cyclical component is superimposed into the trend[6]and we obtain a trend-cycle component denoted by M t .In this case Equations (1.1), (1.2) and (1.3) may be written

Table 2 .
Estimates of parameters of quadratic trend-cycle component and seasonal indices.

Table 4 .
The Buys-Ballot estimates of trend parameters and seasonal indices are computed using the expressions in

Table 1 .
The estimates of the error means and standard devia-tions are computed from the residuals from the fitted models.The deviations of these estimates from the parameters used in the simulations are given in Appendix A for the Additive model and Appendix B for the Multiplicative model, while the summary statistics for 20 simulations only are given in Table5for the Additive model and Tables6-8for the Multiplicative model.

Table 9 .
Table 4for the addi- tive model only.However, for want of space, the summary statistics for 20 simulations only are given in Table9.As Table9shows, the summary statistics (ME, MSE and MAE) for the two estimation methods (LSE and BBE) are equal in almost all the simulations.This indicates that the two methods are equally effective in estimating the trend parameters and seasonal indices when the model for decomposition is additive.The summary statistics for 20 simulations given in Table10shows that the summary statistics (ME, MSE and MAE) for the two estimation methods (LSE and BBE) are equal in almost all the simulations.This indicates that the two methods are equally effective in estimating the trend parameters and seasonal indices when the model for decomposition is additive.Summary statistics for additive quadratic model (a = 1, b = 2 and c = 3).