On the Strength of Trend and Seasonality

The strength of the trend and the strength of seasonality are two crucial indicators that help in understanding the significance of

trend and seasonal components in a time series. Let us explore these in more detail.

1) Strength of Trend (Ft)

Definition:

The strength of the trend indicates the significance of the long-term trend in the time series data. It is the ratio of

the variance of the trend to the total variance of the series.

Formula:

Ft=max⁡(0,1−var(remainder)var(trend+remainder))Ft=max(0,1−var(trend+remainder)var(remainder)​)

Interpretation:

Ft ≈ 1: The trend dominates the data. The time series has a strong long-term tendency.

Ft ≈ 0: The trend is nearly absent. There is no significant long-term tendency in the data.

2) Strength of Seasonality (Fs)

Definition:

The strength of seasonality indicates the significance of regular fluctuations in the time series data that repeat at

equal intervals.

Formula:

Fs=max⁡(0,1−var(remainder)var(seasonal+remainder))Fs=max(0,1−var(seasonal+remainder)var(remainder)​)

Interpretation:

Fs ≈ 1: Seasonality dominates the data. The time series has significant regular fluctuations.

Fs ≈ 0: Seasonality is nearly absent. There are no significant regular fluctuations in the data.

3) Acceptable Ratios for Interpretation

High Ft and Low Fs (Ft ≈ 1, Fs ≈ 0):

The data have a pronounced trend, but seasonal fluctuations are negligible. This might be typical for time series with

a clear long-term tendency and minor regular fluctuations.

Low Ft and High Fs (Ft ≈ 0, Fs ≈ 1):

The data have strong seasonal fluctuations, but the long-term trend is insignificant. This could be typical for data

subject to regular changes, such as monthly sales of seasonal products.

High Ft and High Fs (Ft ≈ 1, Fs ≈ 1):

The data have both a pronounced trend and significant seasonal fluctuations. This might be typical for time series

subject to both a long-term trend and regular seasonal changes.

Low Ft and Low Fs (Ft ≈ 0, Fs ≈ 0):

The data do not have significant trend or seasonality. The time series likely consists mainly of random fluctuations

without obvious structural components.

Examples of Acceptable Ratios:

Ft = 0.9, Fs = 0.5:

Strong trend and moderate seasonality. The long-term tendency dominates, but there are noticeable seasonal fluctuations.

Ft = 0.2, Fs = 0.8:

Weak trend and strong seasonality. Regular seasonal changes are significant, and the long-term tendency is weak.

Ft = 0.6, Fs = 0.6:

Moderate trend and seasonality. The data have both a long-term trend and regular seasonal fluctuations, but neither component dominates.

Conclusion:

Analyzing the strength of the trend and the strength of seasonality helps in understanding the structure of the time series better

and making more informed decisions in modeling and forecasting. These indicators allow highlighting important components and

considering them when building a model, which in turn improves the accuracy of forecasts.