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 BillionTon 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):
www1.eere.energy.gov/bioenergy/pdfs/
final_billionton_vision_report2.pdf
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
following:
 yields of corn, wheat, and other
small grains were increased by 50 percent;
 the residuetograin 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
notill 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 onfarm for soil improvement under anticipated
EPA restrictions was used for biofuel; and
 all other available residues
were utilized." (Page 2 of Executive Summary)
Source:
www1.eere.energy.gov/bioenergy/pdfs/
final_billionton_vision_report2.pdf 
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 Y_{2}, the results would not be at Level X, but would
likely be at the lesser Level X_{2,} 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 yaxis and the
time period on the xaxis. 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 31 on Page 112 of:
Levin,
R., Rubin, D., & Stinson, J. (1986). Chapter 3: Forecasting. in Quantitative
approaches to management. NY: McGrawHill. (1222 K pdf).
www.bsu.edu/libraries/protected/ereserves/FlowersJ/ITEDU510/Levin1986Ch3Forecasting.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 1year
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 xaxis. Often, it
is instead a multiplicative factor of the yaxis that is used.
Linear trends show a steady,
straightline 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 xvalue
(plotted horizontally) is an exponent of the trendline formula to derive the
yvalue.
Damped trends are those that approach a horizontal
asymptote:
Polynomial trends are those best
modeled by a polynomial equation. They may be secondorder (quadratic)
equations of the form y = ax^{2} + bx + c,
resulting in a parabolic shape:
Polynomial trendlines may also be third order (y
= ax^{3} + bx^{2} + c) or higher:
Tools
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 rightclick and select "Add Trendline.") You not only can map a
bestfitting 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):
jcflowers1.iweb.bsu.edu/rlo/forecasting.htm 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:
http://jcflowers1.iweb.bsu.edu/rlo/PracticeWithForecasting.xls
Excel 2007 Spreadsheet:
http://jcflowers1.iweb.bsu.edu/rlo/PracticeWithForceasting.xlsx
(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).
http://jcflowers1.iweb.bsu.edu/rlo/trendlines.htm
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:
http://jcflowers1.iweb.bsu.edu/rlo/ForecastingCFLMercury.xls
Excel 2007 Spreadsheet:
http://jcflowers1.iweb.bsu.edu/rlo/ForceastingCFLMercury.xlsx
There is a neat JavaScript Learning Object on
"Forecasting by Smoothing Techniques from Professor Hossein Arsham at:
http://home.ubalt.edu/ntsbarsh/Businessstat/otherapplets/ForecaSmo.htm
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.
