BFO 7010 Module 5 Written Assignment: Classical Decomposition of a Seasonal Series Example

Taking the Winter Out: Decomposing Five Years of Propane Deliveries

Student Name

MBA in Data Analytics, Thomas Edison State University

BFO-7010: Business Forecasting

Mentor Name

Month Day, Year

Taking the Winter Out: Decomposing Five Years of Propane Deliveries

Method

Classical decomposition treats each month's deliveries as the sum of three parts: a slowly moving trend-cycle, a seasonal effect that repeats every twelve months and a remainder. Because the period is even, the trend-cycle is estimated with a 2x12 moving average, which weights the twelve central months fully and the two ends by half so that every calendar month counts equally (Hyndman & Athanasopoulos, 2018). Subtracting that average from the data leaves the seasonal effect plus noise. Averaging those detrended values for each calendar month and then centering the twelve averages so they sum to zero gives the seasonal indices. What is left after removing trend and season is the remainder. The calculation used the 60 training months, January 2019 to December 2023, with 2024 held back as before.

Part A: Seasonal Indices

A typical January delivers about 246,000 gallons more than the trend level, and a typical August about 166,000 fewer. April is the hinge month, close to trend. The swing from August to January is roughly 413,000 gallons, larger than the average trend level of about 370,000, which is why every method that ignored the season failed so badly in Module 2.

The multiplicative indices express the same pattern as ratios: January runs at 166 percent of trend and August at 55 percent. The choice between the two forms depends on whether the seasonal swing grows as the business grows. Over these five years the trend rose only modestly, so the two forms give nearly the same seasonally adjusted series. If Pinelands added customers quickly, the winter peak would grow with them and the multiplicative form would fit better.

Part B: Trend and Remainder

The 2x12 average climbs from about 349,000 gallons in mid-2019 to about 398,000 in mid-2024, a slow rise consistent with the 1,100 to 1,200 gallons a month found in the regressions of Module 4. The trend is not a straight line, however: it bulges in 2022, the coldest winter of the six years, with 4,945 heating degree days against about 4,700 in the others. A moving average cannot tell a cold year from business growth, so part of what it labels trend-cycle is weather.

The remainder has a standard deviation of about 22,900 gallons. Its largest value, -52,900 gallons, falls in December 2021, a month with 752 heating degree days against a normal of 860. The decomposition records that warm month as noise because it has no weather information, but a planner would recognize it as a cause. The strength-of-seasonality measure, one minus the variance of the remainder over the variance of season plus remainder, is about 0.98, so the seasonal component dominates; the corresponding trend strength is only about 0.50.

Part C: Forecasting From the Components

The seasonally adjusted series, the data minus the seasonal indices, ended 2023 at 392,400 gallons. Two forecasts were built by projecting that series and adding the seasonal indices back: a naive projection holding it flat and a drift projection extending its average monthly rise of 560 gallons.

The drift version cuts the benchmark's RMSE by about 30 percent, because it starts from a level that averages out one year's weather rather than copying last year whole. It still falls short of the weather regression, which knows why a month is high or low. The decomposition forecast is a useful middle step: cheap, transparent and easy to explain to the dispatch manager as "this year's trend plus the usual winter."

Part D: Limits of Classical Decomposition

Classical decomposition has known weaknesses. The moving average loses six months at each end, so the most recent trend estimate, the one a forecaster needs most, is missing. It assumes the seasonal pattern is identical every year, and it lets single unusual months pull the indices. Statistical agencies have long refined the moving-average approach, and the X-12-ARIMA program, for example, extends the series with ARIMA forecasts so the ends are not lost (Findley et al., 1998). Cleveland et al. (1990) developed STL, a decomposition using locally weighted regression that lets the seasonal pattern change slowly and resists outliers; Hyndman and Athanasopoulos (2018) recommend it over the classical method for most work. For Pinelands, an STL fit would allow a winter peak that slowly flattens as customers install heat pumps, a change the classical indices would miss.

Conclusion

Decomposition shows that almost all of Pinelands' monthly variation is seasonal, that the trend is a slow climb partly confused with weather and that the remainder holds identifiable causes such as warm Decembers. Seasonal adjustment alone improves on the benchmark, but the gap to the weather regression says the next models should either include weather or model the remaining structure in the errors.

References

Cleveland, R. B., Cleveland, W. S., McRae, J. E., & Terpenning, I. (1990). STL: A seasonal-trend decomposition procedure based on loess. Journal of Official Statistics, 6(1), 3-73.

Findley, D. F., Monsell, B. C., Bell, W. R., Otto, M. C., & Chen, B.-C. (1998). New capabilities and methods of the X-12-ARIMA seasonal-adjustment program. Journal of Business & Economic Statistics, 16(2), 127-152. https://doi.org/10.1080/07350015.1998.10524743

Hyndman, R. J., & Athanasopoulos, G. (2018). Forecasting: Principles and practice (2nd ed.). OTexts.