Data Analysis:
The Residual Effect
CO2 Concentration
Year
Teaching Contemporary Mathematics Conference
January 27-28, 2006
Tamar Castelloe
NC School of Science and Mathematics
Atmospheric CO2 Concentration
Scientists at the Mauna Loa Observatory in Hawaii have been monitoring carbon dioxide concentration in the earth’s atmosphere since 1958. The observatory is operated by the National Oceanic and Atmospheric Administration's Climate Monitoring and Diagnostics Laboratory, and it stands approximately 11,141 feet above sea level. (Associated Press) “The Mauna Loa site is considered one of the most favorable locations for measuring undisturbed air because possible local influences of vegetation or human activities on atmospheric CO2 concentrations are minimal, and any influences from volcanic vents may be excluded from the records.” (US Department of Energy) Researchers have measured an approximate 20% increase in the average annual concentration since measurements were first taken 48 years ago, and the concentration continues to rise.
In this problem, we will model the CO2 level in parts per million by volume of dry air (ppmv) in the atmosphere over time from 1958 to 1989. This is a wonderful real-world problem for Precalculus students, as it involves data analysis, modeling, and trigonometry. It can be done as a class or in small groups as a project completed outside of class.
We begin by examining the scatter plot of the over 350 measurements provided in the data set, which is available at http://cdiac.esd.ornl.gov/ftp/maunaloa-co2/maunaloa.co2. The data reports the CO2 concentration in the atmosphere for each month as well as the average of the monthly values for each year.
Figure 1: CO2 concentration plotted against years since 1900
(Teague)
Two functions seem to describe this data. One is an increasing, monotonic function, and the other is a sinusoidal one that seems to “wrap around” it. We consider that our model may be a combination of functions of the form , where looks to be either a polynomial or exponential function, and a sinusoidal one.
We will model the increasing, monotonic function by examining the annual average data.
CO2 Concentration
Year
Figure 2: Annual average CO2 concentration plotted against years since 1900
We identify a horizontal asymptote at approximately 300, so we consider the exponential function . (If this problem is being done as a classroom activity, you may consider having students in groups of 3 to 4 use slightly different values for the horizontal asymptote and have them compare their models.) In order to linearize our data, we use the reexpression .
Figure 3: Linearized annual data plotted against years since 1900
We observe that this data is reasonably linear and fit a least squares regression line to it, which leads us to the following:
Figure 4 shows this model superimposed on the original annual data. (Students can then examine a residual plot to analyze the fit of this model as an exercise.) We find the fit to be a relatively good one and decide that the first function in our expression is .
CO2
Concentration
Year
Figure 4: Our model superimposed on the annual average scatter plot
Superimposing this model on the monthly data, we observe that it fits the “middle” of the data quite well.
CO2 Concentration
Year
Figure 5: Our model with monthly CO2 data
We are now interested in modeling the second part of our function. According to our “guess,” our model looks like with . To obtain, we subtract to get . If we consider the actual monthly data to be represented by, we see that represents the residuals associated with the model and the actual data. Consequently, our goal is to model the residuals associated with our exponential model in order to construct our ultimate model for the monthly data.
We examine the residuals associated with our entire data set, but for purposes of brevity in this handout, we view a portion of the data to illustrate the construction of our model for the residuals. We will consider the measurements of CO2 concentration from 1960 to 1963.
Year / Jan / Feb / Mar / Apr / May / June / July / Aug / Sept / Oct / Nov / Dec / Annual1960 / 316.5 / 317.1 / 317.8 / 319.2 / 320.1 / 319.7 / 318.3 / 316 / 314.2 / 314.1 / 315.1 / 316.2 / 316.9
1961 / 317.9 / 317.8 / 318.5 / 319.5 / 320.6 / 319.9 / 318.7 / 317 / 315.2 / 315.5 / 316.2 / 317.2 / 317.7
1962 / 318.1 / 318.7 / 319.8 / 320.7 / 321.3 / 320.9 / 319.8 / 317.6 / 316.5 / 315.6 / 316.9 / 317.9 / 318.5
1963 / 318.8 / 319.3 / 320.1 / 321.5 / 322.4 / 321.6 / 319.9 / 317.9 / 316.4 / 316.2 / 317.1 / 318.5 / 319
Table 1: Monthly CO2 concentration 1960-1963
CO2
Concentration
Year Year
Figure 6: superimposed on graph of data 1960-1963 Figure 7: Residual plot associated with this data
As we had “guessed” earlier, the residual plot for the data set implies that our model will be sinusoidal of the form . The data appears to “cycle” every 1 year, and the range values indicate that the amplitude of our function is approximately 3 units. The plot also displays a small horizontal shift of approximately 0.04 units (or about half a month), but no significant vertical shift. This gives us our model for the residuals . (A nice activity to consider here would be for students to work on different models for the residuals in their groups and investigate their respective fits.) We look at our model for the residuals superimposed on the portion of the graph we viewed earlier.
Figure 8: Sinusoidal model superimposed
on residual plot 1960-1963
Combining our two functions and , we obtain the model , which appears to be quite an accurate fit!
Figure 9: Model superimposed on data 1960-1963 Figure 10: Model superimposed on data 1976-1979
CO2
Concentration
Year
Figure 11: Our modelsuperimposed on the entire CO2 data set
Associated Press. "CO2 buildup accelerating in atmosphere." USA Today 22 Mar 2004. 22 Jan 2006 http://www.usatoday.com/weather/news/2004-03-21-co2-buildup_x.htm
Teague, Daniel. "Data Analysis and Mathematical Modeling." Metropolitan Mathematics Club of Chicago. Chicago, IL. 9 May 2003.
US Department of Energy, Office of Science. "Mauna Loa CO2 Measurements are the Longest Continuous Record in the World." 17 September 2001. 22 Jan 2006 <http://www.er.doe.gov/Science_News/feature_articles_2001/september/Maun_%20Loa/CDIAC-Mauna%20Loa.htm>.
5
Castelloe TCM 2006