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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.

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 (predicting 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 [

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 [

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

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 multiplicative models).

Additive model (when trend, seasonal and cyclical components are additively combined):

Multiplicative model (when trend, seasonal and cyclical components are multiplicatively combined):

Pseudo-Additive/Mixed Model: (combining the elements of both the additive and multiplicative models):

Cyclical 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 [_{t}. In this case Equations (1.1), (1.2) and (1.3) may be written as:

and

where

that the sum of the seasonal component over a complete period is zero

while for the multiplicative model (1.2), the sum of the seasonal component over a

complete period is

The multiplicative model (1.5) can be linearized to become the additive model (1.4).

where

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

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

Using the periodic means

Iwueze and Nwogu [

Therefore, the ultimate objective of this study is to compare the estimates of trend 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.

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

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 [

We note the following for Tables 1-3:

1)

2) Additive and Multiplicative models give different estimates.

3)

Parameter | Model | |
---|---|---|

Additive model | Multiplicative model | |

Source: Iwueze and Nwogu [

Parameter | Model | |
---|---|---|

Additive model | Multiplicative model | |

Source: Iwueze and Nwogu [

Parameter | Model | |
---|---|---|

Additive model | Multiplicative model | |

Source: Iwueze and Nwogu [

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

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

Additive | Multiplicative | |

1 | −0.89 | 0.91 |

2 | −1.22 | 0.88 |

3 | 0.10 | 1.00 |

4 | −0.15 | 0.98 |

5 | −0.09 | 0.98 |

6 | 1.16 | 1.12 |

7 | 2.34 | 1.26 |

8 | 1.95 | 1.20 |

9 | 0.64 | 1.05 |

10 | −0.73 | 0.92 |

11 | −2.14 | 0.80 |

12 | −0.97 | 0.90 |

Series | ME | MAE | MSE | |||
---|---|---|---|---|---|---|

LSE | BBE | LSE | BBE | LSE | BBE | |

1 | −0.009 | −0.009 | 0.221 | 0.219 | 0.074 | 0.076 |

2 | 0.000 | 0.001 | 0.156 | 0.160 | 0.038 | 0.041 |

3 | 0.003 | 0.002 | 0.226 | 0.228 | 0.080 | 0.085 |

4 | −0.003 | −0.002 | 0.213 | 0.221 | 0.074 | 0.078 |

5 | 0.002 | 0.004 | 0.157 | 0.159 | 0.062 | 0.065 |

6 | 0.016 | 0.017 | 0.200 | 0.199 | 0.071 | 0.069 |

7 | −0.003 | −0.002 | 0.207 | 0.214 | 0.073 | 0.078 |

8 | 0.026 | 0.027 | 0.192 | 0.191 | 0.072 | 0.069 |

9 | 0.000 | 0.003 | 0.180 | 0.173 | 0.076 | 0.075 |

10 | 0.024 | 0.025 | 0.252 | 0.256 | 0.097 | 0.101 |

11 | 0.004 | 0.004 | 0.167 | 0.170 | 0.048 | 0.049 |

12 | 0.000 | −0.002 | 0.224 | 0.235 | 0.131 | 0.128 |

13 | 0.015 | 0.015 | 0.220 | 0.226 | 0.077 | 0.078 |

14 | −0.011 | −0.010 | 0.230 | 0.232 | 0.093 | 0.094 |

15 | 0.019 | 0.020 | 0.215 | 0.226 | 0.083 | 0.092 |

16 | 0.010 | 0.011 | 0.167 | 0.172 | 0.057 | 0.060 |

17 | 0.003 | 0.004 | 0.143 | 0.152 | 0.041 | 0.044 |

18 | 0.015 | 0.015 | 0.274 | 0.275 | 0.146 | 0.148 |

19 | 0.002 | 0.003 | 0.203 | 0.203 | 0.070 | 0.067 |

20 | 0.006 | 0.006 | 0.222 | 0.228 | 0.082 | 0.085 |

S/N | ME | MAE | MAPE | MSE | RSME | |||||
---|---|---|---|---|---|---|---|---|---|---|

LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | |

1 | −0.001 | −0.001 | 0.005 | 0.012 | 0.840 | 2.834 | 0.000 | 0.000 | 0.006 | 0.016 |

2 | 0.000 | 0.000 | 0.003 | 0.013 | 0.470 | 3.271 | 0.000 | 0.000 | 0.004 | 0.018 |

3 | 0.000 | 0.000 | 0.005 | 0.014 | 0.799 | 3.368 | 0.000 | 0.000 | 0.006 | 0.018 |

4 | 0.000 | −0.001 | 0.004 | 0.015 | 0.798 | 4.924 | 0.000 | 0.001 | 0.006 | 0.023 |

5 | 0.000 | 0.000 | 0.003 | 0.011 | 0.578 | 2.873 | 0.000 | 0.000 | 0.005 | 0.014 |

6 | 0.001 | 0.000 | 0.004 | 0.014 | 0.766 | 4.834 | 0.000 | 0.000 | 0.006 | 0.021 |

7 | 0.000 | −0.001 | 0.004 | 0.013 | 0.773 | 4.223 | 0.000 | 0.000 | 0.006 | 0.017 |

8 | 0.001 | 0.000 | 0.004 | 0.012 | 0.883 | 3.054 | 0.000 | 0.000 | 0.006 | 0.016 |

9 | 0.000 | 0.000 | 0.004 | 0.010 | 0.713 | 2.069 | 0.000 | 0.000 | 0.005 | 0.014 |

10 | 0.001 | 0.000 | 0.006 | 0.013 | 1.055 | 4.062 | 0.000 | 0.000 | 0.007 | 0.017 |

11 | 0.000 | −0.001 | 0.004 | 0.018 | 0.554 | 6.065 | 0.000 | 0.001 | 0.005 | 0.023 |

12 | 0.000 | −0.001 | 0.005 | 0.012 | 1.065 | 3.591 | 0.000 | 0.000 | 0.008 | 0.019 |

13 | 0.000 | 0.000 | 0.004 | 0.014 | 0.837 | 3.310 | 0.000 | 0.000 | 0.006 | 0.017 |

14 | 0.000 | −0.001 | 0.005 | 0.010 | 0.989 | 2.517 | 0.000 | 0.000 | 0.006 | 0.015 |

15 | 0.001 | 0.000 | 0.004 | 0.009 | 0.893 | 2.073 | 0.000 | 0.000 | 0.006 | 0.013 |

16 | 0.000 | 0.000 | 0.003 | 0.014 | 0.577 | 3.555 | 0.000 | 0.000 | 0.005 | 0.018 |

17 | 0.000 | −0.001 | 0.003 | 0.015 | 0.442 | 4.974 | 0.000 | 0.001 | 0.004 | 0.023 |

18 | 0.000 | 0.000 | 0.006 | 0.008 | 1.215 | 1.468 | 0.000 | 0.000 | 0.008 | 0.012 |

19 | 0.000 | 0.000 | 0.004 | 0.013 | 0.743 | 3.077 | 0.000 | 0.000 | 0.006 | 0.018 |

20 | 0.000 | −0.001 | 0.005 | 0.018 | 0.809 | 4.686 | 0.000 | 0.001 | 0.000 | 0.024 |

Series | ME | MAE | MSE | MAPE | RMSE | |||||
---|---|---|---|---|---|---|---|---|---|---|

LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | |

1 | 0.998 | 0.998 | 0.998 | 0.998 | 0.996 | 0.996 | 13.107 | 13.097 | 0.998 | 0.998 |

2 | 1.001 | 1.002 | 1.001 | 1.002 | 1.002 | 1.004 | 13.302 | 13.315 | 1.001 | 1.002 |

3 | 1.000 | 1.001 | 1.000 | 1.001 | 1.001 | 1.003 | 13.292 | 13.304 | 1.001 | 1.002 |

4 | 1.000 | 1.000 | 1.000 | 1.000 | 1.001 | 1.000 | 13.238 | 13.218 | 1.000 | 1.000 |

5 | 1.001 | 1.001 | 1.001 | 1.001 | 1.002 | 1.003 | 13.278 | 13.281 | 1.001 | 1.001 |

6 | 1.002 | 1.003 | 1.002 | 1.003 | 1.005 | 1.007 | 13.420 | 13.454 | 1.002 | 1.004 |

7 | 0.999 | 0.998 | 0.999 | 0.998 | 0.998 | 0.996 | 13.171 | 13.143 | 0.999 | 0.998 |

8 | 1.000 | 1.000 | 1.000 | 1.000 | 1.001 | 1.000 | 13.342 | 13.338 | 1.000 | 1.000 |

9 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 13.234 | 13.232 | 1.000 | 1.000 |

10 | 1.002 | 1.002 | 1.002 | 1.002 | 1.005 | 1.005 | 13.423 | 13.440 | 1.002 | 1.002 |

11 | 1.001 | 1.001 | 1.001 | 1.001 | 1.003 | 1.004 | 13.333 | 13.364 | 1.001 | 1.002 |

12 | 0.998 | 0.998 | 0.998 | 0.998 | 0.997 | 0.997 | 13.169 | 13.166 | 0.998 | 0.998 |

13 | 0.999 | 1.000 | 0.999 | 1.000 | 0.999 | 1.000 | 13.280 | 13.280 | 1.000 | 1.000 |

14 | 0.999 | 0.996 | 0.999 | 0.996 | 0.999 | 1.000 | 13.166 | 13.183 | 0.999 | 1.000 |

15 | 1.001 | 1.001 | 1.001 | 1.001 | 1.003 | 1.003 | 13.374 | 13.379 | 1.001 | 1.001 |

16 | 0.999 | 0.999 | 0.999 | 0.999 | 0.998 | 0.998 | 13.225 | 13.226 | 0.999 | 0.999 |

17 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 13.251 | 13.257 | 1.000 | 1.000 |

18 | 1.001 | 1.001 | 1.001 | 1.001 | 1.003 | 1.002 | 13.346 | 13.340 | 1.001 | 1.001 |

19 | 0.998 | 0.998 | 0.998 | 0.998 | 0.997 | 0.996 | 13.170 | 13.148 | 0.999 | 0.998 |

20 | 1.000 | 1.001 | 1.000 | 1.001 | 1.001 | 1.002 | 13.296 | 13.304 | 1.000 | 1.001 |

Series | ME | MAE | MSE | MAPE | RMSE | |||||
---|---|---|---|---|---|---|---|---|---|---|

LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | |

1 | 0.995 | 0.995 | 0.995 | 0.995 | 0.991 | 0.991 | 1.948 | 1.942 | 0.996 | 0.996 |

2 | 1.002 | 1.002 | 1.002 | 1.002 | 1.004 | 1.005 | 2.062 | 2.065 | 1.002 | 1.003 |

3 | 1.001 | 1.002 | 1.001 | 1.002 | 1.002 | 1.004 | 2.056 | 2.057 | 1.001 | 1.002 |

4 | 1.000 | 0.999 | 1.000 | 0.999 | 1.000 | 0.999 | 2.025 | 2.021 | 1.000 | 1.000 |

5 | 1.001 | 1.002 | 1.001 | 1.002 | 1.003 | 1.004 | 2.051 | 2.051 | 1.001 | 1.002 |

6 | 1.005 | 1.006 | 1.005 | 1.006 | 1.012 | 1.014 | 2.147 | 2.159 | 1.006 | 1.007 |

7 | 0.997 | 0.996 | 0.997 | 0.996 | 0.995 | 0.993 | 1.984 | 1.978 | 0.998 | 0.997 |

8 | 1.001 | 1.001 | 1.001 | 1.001 | 1.003 | 1.003 | 2.093 | 2.095 | 1.002 | 1.001 |

9 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 2.022 | 2.020 | 1.000 | 1.000 |

10 | 1.005 | 1.006 | 1.005 | 1.006 | 1.012 | 1.013 | 2.155 | 2.168 | 1.006 | 1.006 |

11 | 1.003 | 1.003 | 1.003 | 1.003 | 1.006 | 1.006 | 2.083 | 2.095 | 1.003 | 1.003 |

12 | 0.996 | 0.996 | 0.996 | 0.996 | 0.993 | 0.993 | 1.979 | 1.977 | 0.997 | 0.996 |

13 | 0.999 | 1.000 | 0.999 | 1.000 | 0.999 | 1.000 | 2.047 | 2.044 | 0.999 | 1.000 |

14 | 0.998 | 0.998 | 0.998 | 0.998 | 0.996 | 0.997 | 1.984 | 1.985 | 0.998 | 0.999 |

15 | 1.003 | 1.003 | 1.003 | 1.003 | 1.007 | 1.007 | 2.114 | 2.119 | 1.003 | 1.004 |

16 | 0.998 | 0.998 | 0.998 | 0.998 | 0.996 | 0.997 | 2.015 | 2.014 | 0.998 | 0.998 |

17 | 0.999 | 1.000 | 0.999 | 1.000 | 0.999 | 1.000 | 2.031 | 2.032 | 1.000 | 1.000 |

18 | 1.003 | 1.003 | 1.003 | 1.003 | 1.006 | 1.006 | 2.093 | 2.097 | 1.003 | 1.003 |

19 | 0.997 | 0.996 | 0.997 | 0.996 | 0.994 | 0.993 | 1.981 | 1.975 | 0.997 | 0.996 |

20 | 1.001 | 1.001 | 1.001 | 1.001 | 1.002 | 1.003 | 2.056 | 2.058 | 1.001 | 1.001 |

For the additive model the results are the same for the selected values of the slope parameter

For the multiplicative model,

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 [

For the Quadratic trend-cycle component, the empirical examples also consist of 106 series of 120 observations each simulated from

For the Exponential trend-cycle component, the empirical examples also consist of 106 series of 120 observations each simulated from

S/No | ME | MAE | MAPE | MSE | RMSE | |||||
---|---|---|---|---|---|---|---|---|---|---|

LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | |

1 | 0.00 | 0.00 | 0.00 | 0.00 | −0.15 | −0.11 | 0.88 | 0.99 | 0.94 | 1.00 |

2 | 0.00 | 0.00 | 0.00 | 0.00 | −0.06 | −0.02 | 0.93 | 0.99 | 0.96 | 1.00 |

3 | 0.00 | 0.00 | 0.00 | 0.00 | −0.07 | −0.09 | 0.88 | 0.99 | 0.94 | 1.00 |

4 | 0.00 | 0.00 | 0.00 | 0.00 | 0.12 | 0.07 | 0.87 | 0.94 | 0.93 | 0.97 |

5 | 0.00 | 0.00 | 0.00 | 0.00 | 0.07 | 0.04 | 0.90 | 0.97 | 0.95 | 0.98 |

6 | 0.00 | 0.00 | 0.00 | 0.00 | −0.34 | −0.16 | 0.99 | 0.96 | 0.99 | 0.98 |

7 | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | −0.02 | 0.89 | 0.98 | 0.94 | 0.99 |

8 | 0.00 | 0.00 | 0.00 | 0.00 | 0.06 | 0.07 | 0.86 | 0.89 | 0.93 | 0.94 |

9 | 0.00 | 0.00 | 0.00 | 0.00 | 0.19 | 0.15 | 0.89 | 0.93 | 0.94 | 0.97 |

10 | 0.00 | 0.00 | 0.00 | 0.00 | −0.01 | 0.03 | 0.83 | 0.90 | 0.91 | 0.95 |

11 | 0.00 | 0.00 | 0.00 | 0.00 | 0.13 | 0.10 | 0.91 | 0.94 | 0.95 | 0.97 |

12 | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.04 | 0.82 | 0.89 | 0.90 | 0.94 |

13 | 0.00 | 0.00 | 0.00 | 0.00 | −0.03 | −0.03 | 0.88 | 0.95 | 0.94 | 0.97 |

14 | 0.00 | 0.00 | 0.00 | 0.00 | 0.10 | 0.15 | 0.85 | 0.91 | 0.92 | 0.96 |

15 | 0.00 | 0.00 | 0.00 | 0.00 | 0.15 | 0.20 | 0.86 | 0.92 | 0.93 | 0.96 |

16 | 0.00 | 0.00 | 0.00 | 0.00 | 0.13 | 0.14 | 0.90 | 0.96 | 0.95 | 0.98 |

17 | 0.00 | 0.00 | 0.00 | 0.00 | 0.04 | 0.05 | 0.94 | 0.96 | 0.97 | 0.98 |

18 | 0.00 | 0.00 | 0.00 | 0.00 | −0.14 | −0.08 | 0.78 | 0.86 | 0.88 | 0.93 |

19 | 0.00 | 0.00 | 0.00 | 0.00 | −0.22 | −0.15 | 0.89 | 1.02 | 0.95 | 1.01 |

20 | 0.00 | 0.00 | 0.00 | 0.00 | −0.03 | −0.04 | 0.88 | 0.95 | 0.94 | 0.97 |

S/No | ME | MAE | MAPE | MSE | RMSE | |||||
---|---|---|---|---|---|---|---|---|---|---|

LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | LSE | BBE | |

1 | 0.03 | 0.03 | 0.03 | 0.03 | 0.09 | 0.09 | 0.89 | 0.89 | 0.94 | 0.94 |

2 | 0.07 | 0.07 | 0.07 | 0.07 | 0.05 | 0.05 | 0.96 | 0.96 | 0.98 | 0.98 |

3 | 0.05 | 0.05 | 0.05 | 0.05 | 0.09 | 0.09 | 0.89 | 0.89 | 0.94 | 0.94 |

4 | 0.10 | 0.10 | 0.10 | 0.10 | 0.06 | 0.06 | 0.95 | 0.95 | 0.98 | 0.98 |

5 | 0.06 | 0.06 | 0.06 | 0.06 | 0.03 | 0.03 | 0.92 | 0.92 | 0.96 | 0.96 |

6 | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 | 0.06 | 0.88 | 0.88 | 0.94 | 0.94 |

7 | 0.05 | 0.05 | 0.05 | 0.05 | 0.09 | 0.09 | 0.90 | 0.90 | 0.95 | 0.95 |

8 | −0.03 | −0.03 | −0.03 | −0.03 | 0.02 | 0.02 | 1.00 | 1.00 | 1.00 | 1.00 |

9 | 0.07 | 0.07 | 0.07 | 0.07 | 0.02 | 0.02 | 0.90 | 0.90 | 0.95 | 0.95 |

10 | 0.03 | 0.03 | 0.03 | 0.03 | 0.00 | 0.00 | 0.83 | 0.83 | 0.91 | 0.91 |

11 | 0.06 | 0.06 | 0.06 | 0.06 | 0.10 | 0.10 | 0.93 | 0.93 | 0.96 | 0.96 |

12 | 0.03 | 0.03 | 0.03 | 0.03 | 0.20 | 0.20 | 0.83 | 0.83 | 0.91 | 0.91 |

13 | −0.02 | −0.02 | −0.02 | −0.02 | 0.05 | 0.06 | 0.98 | 0.98 | 0.99 | 0.99 |

14 | 0.08 | 0.96 | 0.08 | 0.96 | 0.04 | 0.04 | 0.91 | 0.91 | 0.96 | 0.96 |

15 | 0.03 | 0.03 | 0.03 | 0.03 | 0.08 | 0.08 | 0.87 | 0.87 | 0.93 | 0.93 |

16 | −0.02 | −0.02 | −0.02 | −0.02 | 0.01 | 0.01 | 1.03 | 1.03 | 1.01 | 1.01 |

17 | 0.04 | 0.04 | 0.04 | 0.04 | 0.07 | 0.07 | 0.94 | 0.95 | 0.97 | 0.97 |

18 | −0.63 | 0.06 | −0.63 | 0.06 | −2.88 | 0.07 | 2.77 | 0.79 | 1.66 | 0.89 |

19 | 0.02 | 0.02 | 0.02 | 0.02 | 0.07 | 0.07 | 0.96 | 0.96 | 0.98 | 0.98 |

20 | 0.06 | 0.14 | 0.06 | 0.14 | 0.15 | 0.47 | 0.88 | 0.90 | 0.94 | 0.95 |

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 (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, MSE, MAE and MAPE) are equal in all the simulations as in the additive model. However, as the value of b increased from 0.02 to 2.0 the results show that the values of the summary statistics (ME, RMSE, MSE, MAE and MAPE) are unequal in all the simulations and the difference increases with an increase in b. In other words, when a series is dominated by the trend, the estimates of the parameters by both methods become less precise and differ more widely from each other. This has been attributed to the violation of the condition for successful transformation (linearization in this case). It could be recalled that logarithm transformation of the multiplicative model to the additive model can preserve the linearity of a linear trend only if the trend parameters (a and b) satisfy the condition;

Therefore, when the model is additive, the estimates of trend parameters and seasonal indices are the same for both estimation procedures. However, because of the insight it gives into choice of model and detection of presence of seasonal effect, the BBE is recommended.

Iwueze, I.S., Nwogu, E.C., Nlebedim, V.U. and Imoh, J.C. (2016) Comparison of Two Time Series Decomposition Methods: Least Squares and Buys-Ballot Methods. Open Journal of Statistics, 6, 1123-1137. http://dx.doi.org/10.4236/ojs.2016.66091

Deviations of the Buys-Ballot and Least Squares estimates of the linear trend parameters and seasonal indices from the Parameter values (for Additive model with a = 1and b= 2.0).

Deviations of the Buys-Ballot and Least Squares estimates of the linear trend parameters and seasonal indices from the Parameter values (Multiplicative Model and b = 0.02).

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