HelpTrend Extrapolation


By the end of this lesson, you should be able to:

1. Discuss the importance of critically selecting appropriate assumptions.

2. Classify trends.

3. Identify and describe the use of trend extrapolation as a technology assessment technique.

Trend Analysis & Assumptions

Trend extrapolation is one aspect of the larger field of trend (or trendline) analysis. It attempts to extend known data points to regions beyond the timeframe of known datapoints, almost always in an attempt to predict future values with some degree of probability. However, the assumptions made are critical.

For example, let's look at "Biomass as feedstock for a Bioenergy and Bioproducts Industry: The Technical Feasibility of a Billion-Ton Annual Supply," which was published in April 2005 under the auspices of the US Department of Agriculture and the US Department of Energy (required visit):

Please look at the first page of the Executive Summary, where the purpose is stated:

"The purpose of this report is to determine whether the land resources of the United States are capable of producing a sustainable supply of biomass sufficient to displace 30 percent or more of the country's present petroleum consumption.... The short answer to the question of whether that much biomass feedstock can be produced is yes."

That is worth repeating, "The short answer ... is yes." However, now look at the assumptions under which this answer was made.

"Important assumptions that were made include the
  • yields of corn, wheat, and other small grains were increased by 50 percent;
  • the residue-to-grain ratio for soybeans was increased to 2:1;
  • harvest technology was capable of recovering 75 percent of annual crop residues (when removal is sustainable);
  • all cropland was managed with no-till methods;
  • 55 million acres of cropland, idle cropland, and cropland pasture were dedicated to the production of perennial bioenergy crops;
  • all manure in excess of that which can applied on-farm for soil improvement under anticipated EPA restrictions was used for biofuel; and
  • all other available residues were utilized." (Page 2 of Executive Summary)


Are these assumptions reasonable? Are they likely? What would the probability be that all of these assumptions would be shown, over time, to be valid? Maybe a better approach would be for the authors to say, "In order to achieve Level X, we would have to do Activities Y, with a probability of Y leading to X determined to be ____ %. If we do less than Activities Y, such as Activities Y2, the results would not be at Level X, but would likely be at the lesser Level X2, with the same probability."

Trend Extrapolation

There are many techniques used to project past data into the future. These tend to be powerful forecasting techniques that are sometimes subject to unforeseen events.

Some Common Types of Trends

Trends are often shown graphically (as line graphs) with the level of a dependent variable on the y-axis and the time period on the x-axis. There are different "levels" of trends:

  • constant
  • linear
  • exponential
  • damped
The following graphs each contain 100 points of fictitious data connected with a blue line, and the trend superimposed with a black line. They are based on Figure 3-1 on Page 112 of:
Levin, R., Rubin, D., & Stinson, J. (1986). Chapter 3: Forecasting. in Quantitative approaches to management. NY: McGraw-Hill.  (1222 K pdf).


Constant trends are those where there is no net increase or decrease.

However, there may be seasonality, or a periodic fluctuation (as there would be in a graph of the temperature over a 1-year period, with daytime temperatures higher than nighttime):

The above graph shows the same data with periodic additions and subtractions. The blow graph shows this data, but the periods are based on a multiple of the data in the x-axis. Often, it is instead a multiplicative factor of the y-axis that is used.

Linear trends show a steady, straight-line increase or decrease. So the trendline may go up or down, and the angle may be steep or shallow.

Exponential trends are those where the data rises or falls not at a steady rate, but at an increasing rate. The x-value (plotted horizontally) is an exponent of the trendline formula to derive the y-value.

Damped trends are those that approach a horizontal asymptote:

Polynomial trends are those best modeled by a polynomial equation. They may be second-order (quadratic) equations of the form y = ax2 + bx + c, resulting in a parabolic shape:

Polynomial trendlines may also be third order (y = ax3 + bx2 + c) or higher:



Creating graphs of trends is easy. You can do it manually, but there are some powerful forecasting tools designed just for this purpose. But even without purchasing those, you might want to try using a spreadsheet, such as Microsoft Excel. In Excel, if you create a vertical list of data, then highlight the data, you can click Insert, Chart to add a line chart. But then, in Excel 2003 you can highlight the chart and click Chart, Add Trendline (or in Excel 2007, you can select the data line on the chart, then right-click and select "Add Trendline.") You not only can map a best-fitting trendline to the existing data, you can also mape future or previous periods.

Excel 2003 Add Trendline dialog box

Excel 2007 Format Trendline dialog box

Excel 2010 is not much changed from 2007 in this feature. Notice that Excel lets you try to fit different types of trendlines to data. Regression is a statistical tool that is used to fit a straight line to data, but your data may be best described by some other relationship. Trendline analysis (beyond the scope of this lesson) provides a tool to determine the equation that best fits your data.

Tutorial on Forecasting

Regardless of the trend type that best fits your data, you can extend trendlines into future periods using Excel. This is known as "trend extrapolation" (as opposed to "interpolation" which is the approximation of a point that is between to known points.) Some of these trend extrapolation techniques are explained in a short tutorial I wrote on forecasting techniques, available at the following page (Required Visit):
It briefly covers:

  • Naive Forecasts
  • Naive Trends
  • Moving Averages
  • Weighted Moving Averages
  • Simple Exponential Smoothing
  • Double Exponential Smoothing
A separate version of this tutorial exists as a Microsoft Excel spreadsheet. You can download this, as well:

Excel 2003 Spreadsheet:

Excel 2007 Spreadsheet:

(The tutorial was based on Chapter 3 from the text by Levin, et al., cited above.)

Forecasting with Trendlines using Microsoft Excel

There is a separate lesson that discusses how to use Microsoft Excel's graphing tools to add trendlines to a graph of data, extending those trendlines to future periods (Required Visit).

Forecasting Exercise / Example

There is another spreadsheet that contains an example of using Microsoft Excel's forecasting functions to predict future values. Using data on the number of compact fluorescent lamps (CFLs) shipped, and on the recycling rate of CFLs and their mercury content, this spreadsheet guides you through the process of predicting the cumulative environmental discharge of toxic mercury from CFLs that are not recycled (Required Visit):.

Excel 2003 Spreadsheet:

Excel 2007 Spreadsheet:

There is a neat JavaScript Learning Object on "Forecasting by Smoothing Techniques from Professor Hossein Arsham at:

Try it out.

Nearly every typical technology assessment project makes use of some type of forecasting technique, and while trend extrapolation may not always be the technique chosen, it offers powerful tools.

Examples of Trend Analysis
in Technology Assessment
(Which has a historic trend analysis in Appendix 2.1)

"Trend Extrapolation"
All information is subject to change without notification.
© Jim Flowers
Department of Technology, Ball State University