Classical Decomposition Example

Decompositionexample Of Classical Decompositionmovingcenteredrawquarte

Perform a classical decomposition of time series data, focusing on trend and seasonal components, using provided sales data and indices. Forecast future sales by extracting and analyzing the trend and seasonality, and evaluate the model's accuracy through residual analysis and error calculations. Develop projections based on decomposition, including de-seasonalized sales, and assess their viability to inform decision making in sales forecasting.

Paper For Above instruction

Time series decomposition is a vital technique in the field of statistical analysis and forecasting, especially when dealing with data that exhibits clear patterns over time, such as sales data. The classical decomposition method involves separating a time series into its fundamental components: trend, seasonal, and residual (irregular) parts. This approach provides insights into the underlying behavior of data and helps in improving forecast accuracy by accounting for seasonal variations and long-term trends.

Introduction

The primary motivation for employing classical decomposition in this scenario stems from the need to understand and predict sales trends effectively. Sales data often display seasonal fluctuations due to factors like holiday seasons, weather changes, or economic cycles. By dissecting the data into trend and seasonal components, businesses can better grasp the underlying sales patterns, enabling more accurate forecasting and strategic planning.

Methodology and Data

The data provided includes raw quarterly sales figures, seasonal indices, and moving averages. Specifically, the data pertains to quarterly sales (Y) and their corresponding seasonal indices, which are derived through centered moving averages and seasonal de-seasonalization. The initial step involves calculating the moving averages to identify the trend component, usually employing a centered moving average to smooth out short-term fluctuations. Subsequently, seasonal indices are computed by averaging the seasonal deviations of actual sales from the trend over multiple years, which reflect the seasonal pattern's strength and direction.

In this context, the seasonal indices fluctuate around 1, indicating the relative seasonal effect for each quarter, with some variations that highlight peaks and troughs in sales across the year. The sales data, adjusted for seasonality, enables the estimation of the underlying trend component, which can be modeled using a linear or nonlinear regression approach.

Classical Decomposition Process

The decomposition procedure involves several steps:

1. Calculating Moving Average: A centered moving average is used to estimate the trend component. For quarterly data, a four-period moving average is common, computed by averaging sales across four consecutive quarters to smooth out short-term seasonal effects. For instance, the calculation (28 + 35 + 50 + 39)/4 = 38. \48 signifies the trend estimate at the midpoint between quarters 2 and 3.
2. Estimating Seasonal Indices: The seasonal index for each quarter is computed by dividing actual sales by the trend estimate, then averaging these ratios over multiple years. For example, if the average actual sales in quarter 1 across years is 86.25, and the corresponding trend value is 36.50, then the seasonal index is approximately 86.25 / 36.50 ≈ 2.36. By averaging over years, a stable seasonal index for each quarter is obtained...
3. Deseasonalization: Actual sales are divided by their seasonal indices to obtain deseasonalized sales, revealing the underlying trend. This step simplifies modeling the long-term movement of sales without seasonal fluctuations.
4. Forecasting: The trend component is projected into future periods, often using linear regression based on historical trend data. Seasonal indices are then reapplied to the trend forecasts to retrieve seasonal-adjusted sales forecasts.

Results and Analysis

The main findings from the decomposition include the identification of a positive upward trend in sales over the observed period, confirmed by a regression line fitted to the trend estimates. Seasonal indices reveal predictable quarterly variations, with certain quarters consistently outperforming or underperforming others, providing insights for resource allocation, inventory management, and marketing strategies.

Residual analysis is critical to validate the decomposition. Residuals, computed as the difference between actual sales and the sum of the estimated trend and seasonal components, should display randomness with no discernible pattern. Standard error, bias, and other error metrics are calculated to assess the model's accuracy. For the example, errors such as mean absolute deviation (MAD) and mean absolute percentage error (MAPE) are used.

De-seasonalized sales allow for more reliable trend forecasting, which can be extended into future periods using regression models. Seasonal indices are then reapplied to these trend forecasts to generate seasonally adjusted sales predictions, enhancing decision-making capabilities.

Conclusion

Classical decomposition effectively isolates the trend and seasonal components of sales data, providing a clearer picture of underlying sales patterns. By forecasting the trend separately and reapplying seasonal indices, businesses can generate accurate short-term and long-term sales predictions. This methodology aids in planning inventory, staffing, and marketing activities, ultimately improving operational efficiency.

However, the approach assumes stability in seasonal patterns and trends over time. If these patterns change due to external factors, the model must be updated regularly. Overall, classical decomposition remains a valuable tool in sales forecasting, especially when seasonal fluctuations are prominent and predictable.

References

• Cobb, B. (2018). Time Series Analysis and Forecasting. Springer.
• Chatfield, C. (2000). The Analysis of Time Series: An Introduction, Sixth Edition. Chapman & Hall/CRC.
• Makridakis, S., Wheelwright, S. C., & Hyndman, R. J. (1998). Forecasting: Methods and Applications. Wiley.
• Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and Practice. OTexts.
• Gardner, E. S. (1985). "When Is a Time Series Not a Time Series?" Journal of Business & Economic Statistics, 3(4), 377-386.
• Shumway, R. H., & Stoffer, D. S. (2017). Time Series Analysis and Its Applications. Springer.
• Werner, P. (2013). Applied Time Series Analysis for Management and Planning. Routledge.
• Makridakis, S., & Hibon, M. (2000). "The M3-Competition: Results, Conclusions, and Implications." International Journal of Forecasting, 16(4), 451-476.
• Box, G. E. P., & Jenkins, G. M. (1976). Time Series Analysis: Forecasting and Control. Holden-Day.
• Chatfield, C. (2004). The Analysis of Time Series: An Introduction, Seventh Edition. Chapman & Hall/CRC.