futurera | open data, time series, forecasting

Equation:

Y^t = at² + bt + c

, where Y^t is the value of the time series at time t, and a, b, and c are estimated from the data using least-squares estimation.

An example serie is in the Examples Data as General curve.

Instead the original data, we could smooth it with a moving average and use it for the slopes and trend calculations. See some summary charts at the end of the page.

From the table from introduction:
The slope of a parabola changes uniformly with time, and if the slope is plotted against time a straight line at an angle to the horizontale is obtained, which is the slope characteristic.

We can imagine that there are two straight lines in the slopes chart. From 2007 to 2011 and the other from 2011 to 2019. It is difficult to be sure that there is a trend with too little data . If we are confident that there is a parabolic trend - say from 2010 to 2021 -, then we can calculate it with the application.

There some differences between the equation and the table data due to rounding effects.

One that the application suggest is shown in next graph.

This is calculated with Trend, then the Parabolic option in the application.
Parameters: start year (2010) and end year (2021).

Equation where Y is the parabolic trend line.

Y^t = -0.24t² + 7.98t + 18.67

The parabolic curve type is useful for data that changes direction over time, either increasing then decreasing or vice versa. All other curve types assume that the values continuously increase or decrease over time.

Demand of serie is in the Examples Data as General curve.

Trend of parabola calculated with a moving average of Demand.

What if we use the moving average of Demand. Will there be different results?

From Demand we make series with moving averages length 3. It is a repeat from the earlier steps: slope, etc. We calculate a trend; we will compare a trend that was ccaluted directly from Demand.

Equation where Y is the parabolic trend line with moving averages.

Y^t = -0.27t² + 8.18t + 18.55

SUMMARY